title: Flux

text: A (and its infinitesimal) is the vector area, combination of the magnitude of the area through which the property passes, "A", and a unit vector normal to the area, formula_7. The relation is formula_12.


title: Flux

text: Finally, we can integrate again over the time duration "t" to "t", getting the total amount of the property flowing through the surface in that time ("t" − "t"):


title: Flux

text: Eight of the most common forms of flux from the transport phenomena literature are defined as follows:


title: Flux

text: These fluxes are vectors at each point in space, and have a definite magnitude and direction. Also, one can take the divergence of any of these fluxes to determine the accumulation rate of the quantity in a control volume around a given point in space. For incompressible flow, the divergence of the volume flux is zero.


title: Flux

text: As mentioned above, chemical molar flux of a component A in an isothermal, isobaric system is defined in Fick's law of diffusion as:


title: Flux

text: where the nabla symbol ∇ denotes the gradient operator, "D" is the diffusion coefficient (m·s) of component A diffusing through component B, "c" is the concentration (mol/m) of component A.


title: Flux

text: For dilute gases, kinetic molecular theory relates the diffusion coefficient "D" to the particle density "n" = "N"/"V", the molecular mass "m", the collision cross section formula_15, and the absolute temperature "T" by


title: Flux

text: where the second factor is the mean free path and the square root (with the Boltzmann constant "k") is the mean velocity of the particles.


title: Flux

text: In turbulent flows, the transport by eddy motion can be expressed as a grossly increased diffusion coefficient.


title: Flux

text: In quantum mechanics, particles of mass "m" in the quantum state "ψ"(r, "t") have a probability density defined as


title: Flux

text: Then the number of particles passing perpendicularly through unit area of a cross-section per unit time is the probability flux;


title: Flux

text: This is sometimes referred to as the probability current or current density, or probability flux density.


title: Flux

text: where F is a vector field, and d"A" is the vector area of the surface "A", directed as the surface normal. For the second, n is the outward pointed unit normal vector to the surface.


title: Flux

text: The surface has to be orientable, i.e. two sides can be distinguished: the surface does not fold back onto itself. Also, the surface has to be actually oriented, i.e. we use a convention as to flowing which way is counted positive; flowing backward is then counted negative.


title: Flux

text: Conversely, one can consider the flux the more fundamental quantity and call the vector field the flux density.


title: Flux

text: Often a vector field is drawn by curves (field lines) following the "flow"; the magnitude of the vector field is then the line density, and the flux through a surface is the number of lines. Lines originate from areas of positive divergence (sources) and end at areas of negative divergence (sinks).


title: Flux

text: See also the image at right: the number of red arrows passing through a unit area is the flux density, the curve encircling the red arrows denotes the boundary of the surface, and the orientation of the arrows with respect to the surface denotes the sign of the inner product of the vector field with the surface normals.


title: Flux

text: If the surface encloses a 3D region, usually the surface is oriented such that the influx is counted positive; the opposite is the outflux.


title: Flux

text: The divergence theorem states that the net outflux through a closed surface, in other words the net outflux from a 3D region, is found by adding the local net outflow from each point in the region (which is expressed by the divergence).


title: Flux

text: If the surface is not closed, it has an oriented curve as boundary. Stokes' theorem states that the flux of the curl of a vector field is the line integral of the vector field over this boundary. This path integral is also called circulation, especially in fluid dynamics. Thus the curl is the circulation density.


title: Flux

text: We can apply the flux and these theorems to many disciplines in which we see currents, forces, etc., applied through areas.


title: Flux

text: An electric "charge," such as a single proton in space, has a magnitude defined in coulombs. Such a charge has an electric field surrounding it. In pictorial form, the electric field from a positive point charge can be visualized as a dot radiating electric field lines (sometimes also called "lines of force"). Conceptually, electric flux can be thought of as "the number of field lines" passing through a given area. Mathematically, electric flux is the integral of the normal component of the electric field over a given area. Hence, units of electric flux are, in the MKS system, newtons per coulomb times meters squared, or N m/C. (Electric flux density is the electric flux per unit area, and is a measure of strength of the normal component of the electric field averaged over the area of integration. Its units are N/C, the same as the electric field in MKS units.)


title: Flux

text: This quantity arises in Gauss's law – which states that the flux of the electric field E out of a closed surface is proportional to the electric charge "Q" enclosed in the surface (independent of how that charge is distributed), the integral form is:


title: Flux

text: If one considers the flux of the electric field vector, E, for a tube near a point charge in the field of the charge but not containing it with sides formed by lines tangent to the field, the flux for the sides is zero and there is an equal and opposite flux at both ends of the tube. This is a consequence of Gauss's Law applied to an inverse square field. The flux for any cross-sectional surface of the tube will be the same. The total flux for any surface surrounding a charge "q" is "q"/"ε".


title: Flux

text: In free space the electric displacement is given by the constitutive relation D = "ε" E, so for any bounding surface the D-field flux equals the charge "Q" within it. Here the expression "flux of" indicates a mathematical operation and, as can be seen, the result is not necessarily a "flow", since nothing actually flows along electric field lines.


title: Flux

text: The magnetic flux density (magnetic field) having the unit Wb/m (Tesla) is denoted by B, and magnetic flux is defined analogously:


title: Flux

text: with the same notation above. The quantity arises in Faraday's law of induction, where the magnetic flux is time-dependent either because the boundary is time-dependent or magnetic field is time-dependent. In integral form:


title: Retina

text: The retina (from "net") is the innermost, light-sensitive layer of tissue of the eye of most vertebrates and some molluscs. The optics of the eye create a focused two-dimensional image of the visual world on the retina, which then processes that image within the retina and sends nerve impulses along the optic nerve to the visual cortex to create visual perception. The retina serves a function which is in many ways analogous to that of the film or image sensor in a camera.


title: Retina

text: The neural retina consists of several layers of neurons interconnected by synapses and is supported by an outer layer of pigmented epithelial cells. The primary light-sensing cells in the retina are the photoreceptor cells, which are of two types: rods and cones. Rods function mainly in dim light and provide monochromatic vision. Cones function in well-lit conditions and are responsible for the perception of colour through the use of a range of opsins, as well as high-acuity vision used for tasks such as reading. A third type of light-sensing cell, the photosensitive ganglion cell, is important for entrainment of circadian rhythms and reflexive responses such as the pupillary light reflex.


title: Retina

text: Light striking the retina initiates a cascade of chemical and electrical events that ultimately trigger nerve impulses that are sent to various visual centres of the brain through the fibres of the optic nerve. Neural signals from the rods and cones undergo processing by other neurons, whose output takes the form of action potentials in retinal ganglion cells whose axons form the optic nerve. Several important features of visual perception can be traced to the retinal encoding and processing of light.


title: Retina

text: In vertebrate embryonic development, the retina and the optic nerve originate as outgrowths of the developing brain, specifically the embryonic diencephalon; thus, the retina is considered part of the central nervous system (CNS) and is actually brain tissue. It is the only part of the CNS that can be visualized noninvasively. Much like the rest of the brain is isolated from the vasular system via the blood–brain barrier, the retina is similarly protected by the blood–retinal barrier.


title: Retina

text: The vertebrate retina is "inverted" in the sense that the light-sensing cells are in the back of the retina, so that light has to pass through layers of neurons and capillaries before it reaches the photosensitive sections of the rods and cones. The ganglion cells, whose axons form the optic nerve, are at the front of the retina; therefore, the optic nerve must cross through the retina en route to the brain. No photoreceptors are in this region, giving rise to the blind spot. In contrast, in the cephalopod retina, the photoreceptors are in front, with processing neurons and capillaries behind them. Because of this, cephalopods do not have a blind spot.


title: Retina

text: Although the overlying neural tissue is partly transparent, and the accompanying glial cells have been shown to act as fibre-optic channels to transport photons directly to the photoreceptors, light scattering does occur. Some vertebrates, including humans, have an area of the central retina adapted for high-acuity vision. This area, termed the fovea centralis, is avascular (does not have blood vessels), and has minimal neural tissue in front of the photoreceptors, thereby minimizing light scattering.


title: Retina

text: The cephalopods have a non-inverted retina, which is comparable in resolving power to the eyes of many vertebrates. Squid eyes do not have an analog of the vertebrate retinal pigment epithelium (RPE). Although their photoreceptors contain a protein, retinochrome, that recycles retinal and replicates one of the functions of the vertebrate RPE, cephalopod photoreceptors are likely not maintained as well as in vertebrates, and that as a result, the useful lifetime of photoreceptors in invertebrates is much shorter than in vertebrates. Having easily replaced stalk eyes (some lobsters) or retinae (some spiders, such as" Deinopis") rarely occurs.


title: Retina

text: The cephalopod retina does not originate as an outgrowth of the brain, as the vertebrate one does. This difference suggests that vertebrate and cephalopod eyes are not homologous, but have evolved separately. From an evolutionary perspective, a more complex structure such as the inverted retina can generally come about as a consequence of two alternate processes - an advantageous "good" compromise between competing functional limitations, or as a historical maladaptive relic of the convoluted path of organ evolution and transformation. Vision is an important adaptation in higher vertebrates.


title: Retina

text: A third view of the "inverted" vertebrate eye is that it combines two benefits - the maintenance of the photoreceptors mentioned above, and the reduction in light intensity necessary to avoid blinding the photoreceptors, which are based on the extremely sensitive eyes of the ancestors of modern hagfish (fish that live in very deep, dark water).


title: Retina

text: These layers can be grouped into four main processing stages - photoreception; transmission to bipolar cells; transmission to ganglion cells, which also contain photoreceptors, the photosensitive ganglion cells; and transmission along the optic nerve. At each synaptic stage, horizontal and amacrine cells also are laterally connected.


title: Retina

text: The optic nerve is a central tract of many axons of ganglion cells connecting primarily to the lateral geniculate body, a visual relay station in the diencephalon (the rear of the forebrain). It also projects to the superior colliculus, the suprachiasmatic nucleus, and the nucleus of the optic tract. It passes through the other layers, creating the optic disc in primates.


title: Retina

text: Additional structures, not directly associated with vision, are found as outgrowths of the retina in some vertebrate groups. In birds, the pecten is a vascular structure of complex shape that projects from the retina into the vitreous humour; it supplies oxygen and nutrients to the eye, and may also aid in vision. Reptiles have a similar, but much simpler, structure.


title: Retina

text: In adult humans, the entire retina is about 72% of a sphere about 22 mm in diameter. The entire retina contains about 7 million cones and 75 to 150 million rods. The optic disc, a part of the retina sometimes called "the blind spot" because it lacks photoreceptors, is located at the optic papilla, where the optic-nerve fibres leave the eye. It appears as an oval white area of 3 mm. Temporal (in the direction of the temples) to this disc is the macula, at whose centre is the fovea, a pit that is responsible for sharp central vision, but is actually less sensitive to light because of its lack of rods. Human and non-human primates possess one fovea, as opposed to certain bird species, such as hawks, that are bifoviate, and dogs and cats, that possess no fovea, but a central band known as the visual streak. Around the fovea extends the central retina for about 6 mm and then the peripheral retina. The farthest edge of the retina is defined by the ora serrata. The distance from one ora to the other (or macula), the most sensitive area along the horizontal meridian, is about 32 mm.


title: Retina

text: In section, the retina is no more than 0.5 mm thick. It has three layers of nerve cells and two of synapses, including the unique ribbon synapse. The optic nerve carries the ganglion-cell axons to the brain, and the blood vessels that supply the retina. The ganglion cells lie innermost in the eye while the photoreceptive cells lie beyond. Because of this counter-intuitive arrangement, light must first pass through and around the ganglion cells and through the thickness of the retina, (including its capillary vessels, not shown) before reaching the rods and cones. Light is absorbed by the retinal pigment epithelium or the choroid (both of which are opaque).


title: Retina

text: The white blood cells in the capillaries in front of the photoreceptors can be perceived as tiny bright moving dots when looking into blue light. This is known as the blue field entoptic phenomenon (or Scheerer's phenomenon).


title: Retina

text: Between the ganglion-cell layer and the rods and cones are two layers of neuropils, where synaptic contacts are made. The neuropil layers are the outer plexiform layer and the inner plexiform layer. In the outer neuropil layer, the rods and cones connect to the vertically running bipolar cells, and the horizontally oriented horizontal cells connect to ganglion cells.


title: Retina

text: The central retina predominantly contains cones, while the peripheral retina predominantly contains rods. In total, the retina has about seven million cones and a hundred million rods. At the centre of the macula is the foveal pit where the cones are narrow and long, and arranged in a hexagonal mosaic, the most dense, in contradistinction to the much fatter cones located more peripherally in the retina. At the foveal pit, the other retinal layers are displaced, before building up along the foveal slope until the rim of the fovea, or parafovea, is reached, which is the thickest portion of the retina. The macula has a yellow pigmentation, from screening pigments, and is known as the macula lutea. The area directly surrounding the fovea has the highest density of rods converging on single bipolar cells. Since its cones have a much lesser convergence of signals, the fovea allows for the sharpest vision the eye can attain.


title: Retina

text: Though the rod and cones are a mosaic of sorts, transmission from receptors, to bipolars, to ganglion cells is not direct. Since about 150 million receptors and only 1 million optic nerve fibres exist, convergence and thus mixing of signals must occur. Moreover, the horizontal action of the horizontal and amacrine cells can allow one area of the retina to control another (e.g. one stimulus inhibiting another). This inhibition is key to lessening the sum of messages sent to the higher regions of the brain. In some lower vertebrates (e.g. the pigeon), control of messages is "centrifugal" – that is, one layer can control another, or higher regions of the brain can drive the retinal nerve cells, but in primates, this does not occur.


title: Retina

text: Using optical coherence tomography (OCT), 18 layers can be identified in the retina. The layers and anatomical correlation are:


title: Retina

text: Retinal development begins with the establishment of the eye fields mediated by the SHH and SIX3 proteins, with subsequent development of the optic vesicles regulated by the PAX6 and LHX2 proteins. The role of Pax6 in eye development was elegantly demonstrated by Walter Gehring and colleagues, who showed that ectopic expression of Pax6 can lead to eye formation on Drosophila antennae, wings, and legs. The optic vesicle gives rise to three structures: the neural retina, the retinal pigmented epithelium, and the optic stalk. The neural retina contains the retinal progenitor cells (RPCs) that give rise to the seven cell types of the retina. Differentiation begins with the retinal ganglion cells and concludes with production of the Muller glia. Although each cell type differentiates from the RPCs in a sequential order, there is considerable overlap in the timing of when individual cell types differentiate. The cues that determine a RPC daughter cell fate are coded by multiple transcription factor families including the bHLH and homeodomain factors.


title: Retina

text: In addition to guiding cell fate determination, cues exist in the retina to determine the dorsal-ventral (D-V) and nasal-temporal (N-T) axes. The D-V axis is established by a ventral to dorsal gradient of VAX2, whereas the N-T axis is coordinated by expression of the forkhead transcription factors FOXD1 and FOXG1. Additional gradients are formed within the retina. This spatial distribution may aid in proper targeting of RGC axons that function to establish the retinotopic map.


title: Retina

text: The retina is stratified into distinct layers, each containing specific cell types or cellular compartments that have metabolisms with different nutritional requirements. To satisfy these requirements, the ophthalmic artery bifurcates and supplies the retina via two distinct vascular networks: the choroidal network, which supplies the choroid and the outer retina, and the retinal network, which supplies the retina's inner layer.


title: Retina

text: At first glance, one may think that the vertebrate retina is "wired wrongly" or "badly designed"; but in fact, the retina could not function if it were not inverted. The photoreceptor layer must be embedded in the retinal pigment epithelium (RPE), which performs at least seven vital functions, one of the most obvious being to supply oxygen and other necessary nutrients needed for the photoreceptors to function. These nutrients include glucose, fatty acids, and retinal. The mammalian photoreceptor amplification process uses large quantities energy for vision in photopic conditions (requiring less under scotopic conditions) and, thus, requires the large supply nutrients supplied by the blood vessels in the choroid, which lies beyond the RPE. The choroid supplies about 75% of these nutrients to the retina and the retinal vasculature only 25%.


title: Retina

text: When light strikes 11-cis-retinal (in the disks in the rods and cones), 11-cis-retinal changes to all-trans-retinal which then triggers changes in the opsins. Now, the outer segments do not regenerate the retinal back into the cis- form once it has been changed by light. Instead the retinal is pumped out to the surrounding RPE where it is regenerated and transported back into the outer segments of the photoreceptors. This recycling function of the RPE protects the photoreceptors against photo-oxidative damage and allows the photoreceptor cells to have decades-long useful lives.


title: Retina

text: The bird retina is devoid of blood vessels, perhaps to give unobscured passage of light for forming images, thus giving better resolution. It is, therefore, a considered view that the bird retina depends for nutrition and oxygen supply on a specialized organ, called the "pecten" or pecten oculi, located on the blind spot or optic disk. This organ is extremely rich in blood vessels and is thought to supply nutrition and oxygen to the bird retina by diffusion through the vitreous body. The pecten is highly rich in alkaline phosphatase activity and polarized cells in its bridge portion – both befitting its secretory role. Pecten cells are packed with dark melanin granules, which have been theorized to keep this organ warm with the absorption of stray light falling on the pecten. This is considered to enhance metabolic rate of the pecten, thereby exporting more nutritive molecules to meet the stringent energy requirements of the retina during long periods of exposure to light.


title: Retina

text: The bifurcations and other physical characteristics of the inner retinal vascular network are known to vary among individuals, and these individual variances have been used for biometric identification and for early detection of the onset of disease. The mapping of vascular bifurcations is one of the basic steps in biometric identification. Results of such analyses of retinal blood vessel structure can be evaluated against the ground truth data of vascular bifurcations of retinal fundus images that are obtained from the DRIVE dataset. In addition, the classes of vessels of the DRIVE dataset have also been identified, and an automated method for accurate extraction of these bifurcations is also available. Changes in retinal blood circulation are seen with aging and exposure to air pollution, and may indicate cardiovascular diseases such as hypertension and atherosclerosis. Determining the equivalent width of arterioles and venules near the optic disc is also a widely used technique to identify cardiovascular risks.


title: Retina

text: The retina translates an optical image into neural impulses starting with the patterned excitation of the colour-sensitive pigments of its rods and cones, the retina's photoreceptor cells. The excitation is processed by the neural system and various parts of the brain working in parallel to form a representation of the external environment in the brain.


title: Retina

text: The cones respond to bright light and mediate high-resolution colour vision during daylight illumination (also called photopic vision). The rod responses are saturated at daylight levels and don't contribute to pattern vision. However, rods do respond to dim light and mediate lower-resolution, monochromatic vision under very low levels of illumination (called scotopic vision). The illumination in most office settings falls between these two levels and is called mesopic vision. At mesopic light levels, both the rods and cones are actively contributing pattern information. What contribution the rod information makes to pattern vision under these circumstances is unclear.


title: Retina

text: The response of cones to various wavelengths of light is called their spectral sensitivity. In normal human vision, the spectral sensitivity of a cone falls into one of three subtypes, often called blue, green, and red, but more accurately known as short, medium, and long wavelength-sensitive cone subtypes. It is a lack of one or more of the cone subtypes that causes individuals to have deficiencies in colour vision or various kinds of colour blindness. These individuals are not blind to objects of a particular colour, but are unable to distinguish between colours that can be distinguished by people with normal vision. Humans have this trichromatic vision, while most other mammals lack cones with red sensitive pigment and therefore have poorer dichromatic colour vision. However, some animals have four spectral subtypes, e.g. the trout adds an ultraviolet subgroup to short, medium, and long subtypes that are similar to humans. Some fish are sensitive to the polarization of light as well.


title: Retina

text: In the photoreceptors, exposure to light hyperpolarizes the membrane in a series of graded shifts. The outer cell segment contains a photopigment. Inside the cell the normal levels of cyclic guanosine monophosphate (cGMP) keep the Na+ channel open, and thus in the resting state the cell is depolarised. The photon causes the retinal bound to the receptor protein to isomerise to trans-retinal. This causes the receptor to activate multiple G-proteins. This in turn causes the Ga-subunit of the protein to activate a phosphodiesterase (PDE6), which degrades cGMP, resulting in the closing of Na+ cyclic nucleotide-gated ion channels (CNGs). Thus the cell is hyperpolarised. The amount of neurotransmitter released is reduced in bright light and increases as light levels fall. The actual photopigment is bleached away in bright light and only replaced as a chemical process, so in a transition from bright light to darkness the eye can take up to thirty minutes to reach full sensitivity.


title: Retina

text: When thus excited by light, the photoceptor sends a proportional response synaptically to bipolar cells which in turn signal the retinal ganglion cells. The photoreceptors are also cross-linked by horizontal cells and amacrine cells, which modify the synaptic signal before it reaches the ganglion cells, the neural signals being intermixed and combined. Of the retina's nerve cells, only the retinal ganglion cells and few amacrine cells create action potentials.


title: Retina

text: In the retinal ganglion cells there are two types of response, depending on the receptive field of the cell. The receptive fields of retinal ganglion cells comprise a central, approximately circular area, where light has one effect on the firing of the cell, and an annular surround, where light has the opposite effect. In ON cells, an increment in light intensity in the centre of the receptive field causes the firing rate to increase. In OFF cells, it makes it decrease. In a linear model, this response profile is well described by a difference of Gaussians and is the basis for edge detection algorithms. Beyond this simple difference, ganglion cells are also differentiated by chromatic sensitivity and the type of spatial summation. Cells showing linear spatial summation are termed X cells (also called parvocellular, P, or midget ganglion cells), and those showing non-linear summation are Y cells (also called magnocellular, M, or parasol retinal ganglion cells), although the correspondence between X and Y cells (in the cat retina) and P and M cells (in the primate retina) is not as simple as it once seemed.


title: Retina

text: In the transfer of visual signals to the brain, the visual pathway, the retina is vertically divided in two, a temporal (nearer to the temple) half and a nasal (nearer to the nose) half. The axons from the nasal half cross the brain at the optic chiasma to join with axons from the temporal half of the other eye before passing into the lateral geniculate body.


title: Retina

text: Although there are more than 130 million retinal receptors, there are only approximately 1.2 million fibres (axons) in the optic nerve. So, a large amount of pre-processing is performed within the retina. The fovea produces the most accurate information. Despite occupying about 0.01% of the visual field (less than 2° of visual angle), about 10% of axons in the optic nerve are devoted to the fovea. The resolution limit of the fovea has been determined to be around 10,000 points. The information capacity is estimated at 500,000 bits per second (for more information on bits, see information theory) without colour or around 600,000 bits per second including colour.


title: Retina

text: When the retina sends neural impulses representing an image to the brain, it spatially encodes (compresses) those impulses to fit the limited capacity of the optic nerve. Compression is necessary because there are 100 times more photoreceptor cells than ganglion cells. This is done by "decorrelation", which is carried out by the "centre–surround structures", which are implemented by the bipolar and ganglion cells.


title: Retina

text: There are two types of centre–surround structures in the retina – on-centres and off-centres. On-centres have a positively weighted centre and a negatively weighted surround. Off-centres are just the opposite. Positive weighting is more commonly known as excitatory, and negative weighting as inhibitory.


title: Retina

text: These centre–surround structures are not physical apparent, in the sense that one cannot see them by staining samples of tissue and examining the retina's anatomy. The centre–surround structures are logical (i.e., mathematically abstract) in the sense that they depend on the connection strengths between bipolar and ganglion cells. It is believed that the connection strength between cells is caused by the number and types of ion channels embedded in the synapses between the bipolar and ganglion cells.


title: Retina

text: The centre–surround structures are mathematically equivalent to the edge detection algorithms used by computer programmers to extract or enhance the edges in a digital photograph. Thus, the retina performs operations on the image-representing impulses to enhance the edges of objects within its visual field. For example, in a picture of a dog, a cat and a car, it is the edges of these objects that contain the most information. In order for higher functions in the brain (or in a computer for that matter) to extract and classify objects such as a dog and a cat, the retina is the first step to separating out the various objects within the scene.


title: Retina

text: As an example, the following matrix is at the heart of a computer algorithm that implements edge detection. This matrix is the computer equivalent to the centre–surround structure. In this example, each box (element) within this matrix would be connected to one photoreceptor. The photoreceptor in the centre is the current receptor being processed. The centre photoreceptor is multiplied by the +1 weight factor. The surrounding photoreceptors are the "nearest neighbors" to the centre and are multiplied by the −1/8 value. The sum of all nine of these elements is finally calculated. This summation is repeated for every photoreceptor in the image by shifting left to the end of a row and then down to the next line.


title: Retina

text: The total sum of this matrix is zero, if all the inputs from the nine photoreceptors are of the same value. The zero result indicates the image was uniform (non-changing) within this small patch. Negative or positive sums mean the image was varying (changing) within this small patch of nine photoreceptors.


title: Retina

text: The above matrix is only an approximation to what really happens inside the retina. The differences are:


title: Retina

text: Once the image is spatially encoded by the centre–surround structures, the signal is sent out along the optic nerve (via the axons of the ganglion cells) through the optic chiasm to the LGN (lateral geniculate nucleus). The exact function of the LGN is unknown at this time. The output of the LGN is then sent to the back of the brain. Specifically, the output of the LGN "radiates" out to the V1 primary visual cortex.


title: Retina

text: There are many inherited and acquired diseases or disorders that may affect the retina. Some of them include:


title: Retina

text: In addition, the retina has been described as a "window" into the brain and body, given that abnormalities detected through an examination of the retina can discover both neurological and systemic diseases.


title: Retina

text: A number of different instruments are available for the diagnosis of diseases and disorders affecting the retina. Ophthalmoscopy and fundus photography have long been used to examine the retina. Recently, adaptive optics has been used to image individual rods and cones in the living human retina, and a company based in Scotland has engineered technology that allows physicians to observe the complete retina without any discomfort to patients.


title: Retina

text: The electroretinogram is used to non-invasively measure the retina's electrical activity, which is affected by certain diseases. A relatively new technology, now becoming widely available, is optical coherence tomography (OCT). This non-invasive technique allows one to obtain a 3D volumetric or high resolution cross-sectional tomogram of the fine structures of the retina, with histologic quality. Retinal vessel analysis is a non-invasive method to examine the small arteries and veins in the retina which allows to draw conclusions about the morphology and the function of small vessels elsewhere in the human body. It has been established as a predictor of cardiovascular disease and seems to have, according to a study published in 2019, potential in the early detection of Alzheimer's disease.


title: Retina

text: Gene therapy holds promise as a potential avenue to cure a wide range of retinal diseases. This involves using a non-infectious virus to shuttle a gene into a part of the retina. Recombinant adeno-associated virus (rAAV) vectors possess a number of features that render them ideally suited for retinal gene therapy, including a lack of pathogenicity, minimal immunogenicity, and the ability to transduce postmitotic cells in a stable and efficient manner. rAAV vectors are increasingly utilized for their ability to mediate efficient transduction of retinal pigment epithelium (RPE), photoreceptor cells and retinal ganglion cells. Each cell type can be specifically targeted by choosing the appropriate combination of AAV serotype, promoter, and intraocular injection site.


title: Retina

text: Several clinical trials have already reported positive results using rAAV to treat Leber's congenital amaurosis, showing that the therapy was both safe and effective. There were no serious adverse events, and patients in all three studies showed improvement in their visual function as measured by a number of methods. The methods used varied among the three trials, but included both functional methods such as visual acuity and functional mobility as well as objective measures that are less susceptible to bias, such as the pupil's ability to respond to light and improvements on functional MRI. Improvements were sustained over the long-term, with patients continuing to do well after more than 1.5 years.


title: Retina

text: The unique architecture of the retina and its relatively immune-privileged environment help this process. Tight junctions that form the blood retinal barrier separate the subretinal space from the blood supply, thus protecting it from microbes and most immune-mediated damage, and enhancing its potential to respond to vector-mediated therapies. The highly compartmentalized anatomy of the eye facilitates accurate delivery of therapeutic vector suspensions to specific tissues under direct visualization using microsurgical techniques. In the sheltered environment of the retina, AAV vectors are able to maintain high levels of transgene expression in the retinal pigmented epithelium (RPE), photoreceptors, or ganglion cells for long periods of time after a single treatment. In addition, the eye and the visual system can be routinely and easily monitored for visual function and retinal structural changes after injections with noninvasive advanced technology, such as visual acuities, contrast sensitivity, fundus auto-fluorescence (FAF), dark-adapted visual thresholds, vascular diameters, pupillometry, electroretinography (ERG), multifocal ERG and optical coherence tomography (OCT).


title: Retina

text: This strategy is effective against a number of retinal diseases that have been studied, including neovascular diseases that are features of age-related macular degeneration, diabetic retinopathy and retinopathy of prematurity. Since the regulation of vascularization in the mature retina involves a balance between endogenous positive growth factors, such as vascular endothelial growth factor (VEGF) and inhibitors of angiogenesis, such as pigment epithelium-derived factor (PEDF), rAAV-mediated expression of PEDF, angiostatin, and the soluble VEGF receptor sFlt-1, which are all antiangiogenic proteins, have been shown to reduce aberrant vessel formation in animal models. Since specific gene therapies cannot readily be used to treat a significant fraction of patients with retinal dystrophy, there is a major interest in developing a more generally applicable survival factor therapy. Neurotrophic factors have the ability to modulate neuronal growth during development to maintain existing cells and to allow recovery of injured neuronal populations in the eye. AAV encoding neurotrophic factors such as fibroblast growth factor (FGF) family members and GDNF either protected photoreceptors from apoptosis or slowed down cell death.


title: Retina

text: Transplantation of retinas has been attempted, but without much success. At MIT, The University of Southern California, RWTH Aachen University, and the University of New South Wales, an "artificial retina" is under development: an implant which will bypass the photoreceptors of the retina and stimulate the attached nerve cells directly, with signals from a digital camera.


title: Retina

text: Around 300 BCE, Herophilos identified the retina from dissections of cadaver eyes. He called it the "arachnoid" layer, from its resemblance to a spider web, and "retiform", from its resemblance to a casting net. The term arachnoid came to refer to a layer around the brain; the term "retiform" came to refer to the "retina".


title: Retina

text: Between 1011 and 1021 CE, Ibn Al-Haytham published numerous experiments demonstrating that sight occurs from light reflecting from objects into the eye. This is consistent with intromission theory and against emission theory, the theory that sight occurs from rays emitted by the eyes. However, Ibn Al-Haytham decided that the retina could not be responsible for the beginnings of vision because the image formed on it was inverted. Instead he decided it must begin at the surface of the lens.


title: Retina

text: In 1604, Johannes Kepler worked out the optics of the eye and decided that the retina must be where sight begins. He left it up to other scientists to reconcile the inverted retinal image with our perception of the world as upright.


title: Retina

text: In 1894, Santiago Ramón y Cajal published the first major characterization of retinal neurons in "Retina der Wirbelthiere" ("The Retina of Vertebrates").


title: Retina

text: George Wald, Haldan Keffer Hartline, and Ragnar Granit won the 1967 Nobel Prize in Physiology or Medicine for their scientific research on the retina.


title: Retina

text: A recent University of Pennsylvania study calculated that the approximate bandwidth of human retinas is 8.75 megabits per second, whereas a guinea pig's retinal transfer rate is 875 kilobits per second.


title: Retina

text: MacLaren & Pearson and colleagues at University College London and Moorfields Eye Hospital in London, in 2006, showed that photoreceptor cells could be transplanted successfully in the mouse retina if donor cells were at a critical developmental stage. Recently Ader and colleagues in Dublin showed, using the electron microscope, that transplanted photoreceptors formed synaptic connections.


title: Retina

text: In 2012, Sebastian Seung and his laboratory at MIT launched EyeWire, an online Citizen science game where players trace neurons in the retina. The goals of the EyeWire project are to identify specific cell types within the known broad classes of retinal cells, and to map the connections between neurons in the retina, which will help to determine how vision works.


title: Linear independence

text: In the theory of vector spaces, a set of vectors is said to be ' if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be '. These concepts are central to the definition of dimension.


title: Linear independence

text: A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space.


title: Linear independence

text: A sequence of vectors formula_1 from a vector space is said to be "linearly dependent", if there exist scalars formula_2 not all zero, such that


title: Linear independence

text: This implies that at least one of the scalars is nonzero, say formula_5, and the above equation is able to be written as


title: Linear independence

text: Thus, a set of vectors is linearly dependent if and only if one of them is zero or a linear combination of the others.


title: Linear independence

text: A sequence of vectors formula_10 is said to be "linearly independent" if it is not linearly dependent, that is, if the equation


title: Linear independence

text: can only be satisfied by formula_12 for formula_13 This implies that no vector in the sequence can be represented as a linear combination of the remaining vectors in the sequence. In other words, a sequence of vectors is linearly independent if the only representation of formula_14 as a linear combination of its vectors is the trivial representation in which all the scalars formula_15 are zero. Even more concisely, a sequence of vectors is linearly independent if and only if formula_14 can be represented as a linear combination of its vectors in a unique way.


title: Linear independence

text: If a sequence of vectors contains the same vector twice, it is necessarily dependent. The linear dependency of a sequence of vectors does not depend of the order of the terms in the sequence. This allows defining linear independence for a finite set of vectors: A finite set of vectors is "linearly independent" if the sequence obtained by ordering them is linearly independent. In other words, one has the following result that is often useful.


title: Linear independence

text: A sequence of vectors is linearly independent if and only if it does not contain the same vector twice and the set of its vectors is linearly independent.


title: Linear independence

text: An infinite set of vectors is "linearly independent" if every nonempty finite subset is linearly independent. Conversely, an infinite set of vectors is "linearly dependent" if it contains a finite subset that is linearly dependent, or equivalently, if some vector in the set is a linear combination of other vectors in the set.


title: Linear independence

text: An indexed family of vectors is "linearly independent" if it does not contain the same vector twice, and if the set of its vectors is linearly independent. Otherwise, the family is said "linearly dependent".


title: Linear independence

text: A set of vectors which is linearly independent and spans some vector space, forms a basis for that vector space. For example, the vector space of all polynomials in over the reals has the (infinite) subset as a basis.


title: Linear independence

text: A person describing the location of a certain place might say, "It is 3 miles north and 4 miles east of here." This is sufficient information to describe the location, because the geographic coordinate system may be considered as a 2-dimensional vector space (ignoring altitude and the curvature of the Earth's surface). The person might add, "The place is 5 miles northeast of here." This last statement is "true", but it is not necessary to find the location.


title: Linear independence

text: In this example the "3 miles north" vector and the "4 miles east" vector are linearly independent. That is to say, the north vector cannot be described in terms of the east vector, and vice versa. The third "5 miles northeast" vector is a linear combination of the other two vectors, and it makes the set of vectors "linearly dependent", that is, one of the three vectors is unnecessary to define a specific location on a plane.


title: Linear independence

text: Also note that if altitude is not ignored, it becomes necessary to add a third vector to the linearly independent set. In general, linearly independent vectors are required to describe all locations in -dimensional space.


title: Linear independence

text: If one or more vectors from a given sequence of vectors formula_33 is the zero vector formula_4 then the vector formula_33 are necessarily linearly dependent (and consequently, they are not linearly independent).


title: Linear independence

text: To see why, suppose that formula_36 is an index (i.e. an element of formula_37) such that formula_38 Then let formula_39 (alternatively, letting formula_40 be equal any other non-zero scalar will also work) and then let all other scalars be formula_41 (explicitly, this means that for any index formula_42 other than formula_36 (i.e. for formula_44), let formula_45 so that consequently formula_46).


title: Linear independence

text: Because not all scalars are zero (in particular, formula_49), this proves that the vectors formula_33 are linearly dependent.


title: Linear independence

text: As a consequence, the zero vector can not possibly belong to any collection of vectors that is linearly "in"dependent.


title: Linear independence

text: Now consider the special case where the sequence of formula_33 has length formula_52 (i.e. the case where formula_53).


title: Linear independence

text: A collection of vectors that consists of exactly one vector is linearly dependent if and only if that vector is zero.


title: Linear independence

text: Explicitly, if formula_54 is any vector then the sequence formula_54 (which is a sequence of length formula_52) is linearly dependent if and only if alternatively, the collection formula_54 is linearly independent if and only if formula_58


title: Linear independence

text: This example considers the special case where there are exactly two vector formula_59 and formula_60 from some real or complex vector space.


title: Linear independence

text: The vectors formula_59 and formula_60 are linearly dependent if and only if at least one of the following is true:


title: Linear independence

text: If formula_71 then by setting formula_72 we have formula_73 (this equality holds no matter what the value of formula_60 is), which shows that (1) is true in this particular case. Similarly, if formula_75 then (2) is true because formula_76


title: Linear independence

text: If formula_77 (for instance, if they are both equal to the zero vector formula_4) then "both" (1) and (2) are true (by using formula_79 for both).


title: Linear independence

text: If formula_66 then formula_81 is only possible if formula_82 "and" formula_83; in this case, it is possible to multiply both sides by formula_84 to conclude formula_85


title: Linear independence

text: This shows that if formula_81 and formula_83 then (1) is true if and only if (2) is true; that is, in this particular case either both (1) and (2) are true (and the vectors are linearly dependent) or else both (1) and (2) are false (and the vectors are linearly "in"dependent).


title: Linear independence

text: Moreover, if exactly one of formula_59 and formula_60 is formula_4 (while the other is non-zero) then exactly one of (1) and (2) is true (with the other being false).


title: Linear independence

text: The vectors formula_59 and formula_60 are linearly "in"dependent if and only if formula_59 is not a scalar multiple of formula_60 "and" formula_60 is not a scalar multiple of formula_59.


title: Linear independence

text: Three vectors: Consider the set of vectors formula_101 formula_102 and formula_103 then the condition for linear dependence seeks a set of non-zero scalars, such that


title: Linear independence

text: Continue the row reduction by (i) dividing the second row by 5, and then (ii) multiplying by 3 and adding to the first row, that is


title: Linear independence

text: which shows that non-zero "a" exist such that formula_109 can be defined in terms of formula_110 and formula_111 Thus, the three vectors are linearly dependent.


title: Linear independence

text: Two vectors: Now consider the linear dependence of the two vectors formula_110 and formula_102 and check,


title: Linear independence

text: This shows that formula_117 which means that the vectors "v" = (1, 1) and "v" = (−3, 2) are linearly independent.


title: Linear independence

text: where formula_124 can be chosen arbitrarily. Thus, the vectors formula_125 and formula_126 are linearly dependent.


title: Linear independence

text: An alternative method relies on the fact that formula_127 vectors in formula_128 are linearly independent if and only if the determinant of the matrix formed by taking the vectors as its columns is non-zero.


title: Linear independence

text: We are interested in whether for some nonzero vector Λ. This depends on the determinant of formula_131, which is


title: Linear independence

text: Otherwise, suppose we have formula_135 vectors of formula_127 coordinates, with formula_137 Then "A" is an "m"×"n" matrix and Λ is a column vector with formula_135 entries, and we are again interested in "A"Λ = 0. As we saw previously, this is equivalent to a list of formula_127 equations. Consider the first formula_135 rows of formula_131, the first formula_135 equations; any solution of the full list of equations must also be true of the reduced list. In fact, if is any list of formula_135 rows, then the equation must be true for those rows.


title: Linear independence

text: Furthermore, the reverse is true. That is, we can test whether the formula_135 vectors are linearly dependent by testing whether


title: Linear independence

text: for all possible lists of formula_135 rows. (In case formula_148, this requires only one determinant, as above. If formula_149, then it is a theorem that the vectors must be linearly dependent.) This fact is valuable for theory; in practical calculations more efficient methods are available.


title: Linear independence

text: If there are more vectors than dimensions, the vectors are linearly dependent. This is illustrated in the example above of three vectors in formula_150


title: Linear independence

text: Let formula_151 and consider the following elements in formula_152, known as the natural basis vectors:


title: Linear independence

text: Let formula_152 be the vector space of all differentiable functions of a real variable formula_156. Then the functions formula_157 and formula_158 in formula_152 are linearly independent.


title: Linear independence

text: for values of formula_164 We need to show that formula_165 and formula_166 In order to do this, we subtract the first equation from the second, giving formula_167. Since formula_158 is not zero for some formula_156, formula_170 It follows that formula_165 too. Therefore, according to the definition of linear independence, formula_172 and formula_158 are linearly independent.


title: Linear independence

text: If such a linear dependence exists with at least a nonzero component, then the vectors are linearly dependent. Linear dependencies among form a vector space.


title: Linear independence

text: If the vectors are expressed by their coordinates, then the linear dependencies are the solutions of a homogeneous system of linear equations, with the coordinates of the vectors as coefficients. A basis of the vector space of linear dependencies can therefore be computed by Gaussian elimination.


title: Linear independence

text: A set of vectors is said to be affinely dependent if at least one of the vectors in the set can be defined as an affine combination of the others. Otherwise, the set is called affinely independent. Any affine combination is a linear combination; therefore every affinely dependent set is linearly dependent. Conversely, every linearly independent set is affinely independent.


title: Linear independence

text: Consider a set of formula_135 vectors formula_176 of size formula_127 each, and consider the set of formula_135 augmented vectors formula_179 of size formula_180 each. The original vectors are affinely independent if and only if the augmented vectors are linearly independent.


title: Linear independence

text: Two vector subspaces formula_181 and formula_182 of a vector space formula_183 are said to be if formula_184


title: Linear independence

text: More generally, a collection formula_185 of subspaces of formula_183 are said to be if formula_187 for every index formula_188 where formula_189


title: Linear independence

text: The vector space formula_183 is said to be a of formula_185 if these subspaces are linearly independent and formula_192


title: Vanity Fair (novel)

text: Vanity Fair is an English novel by William Makepeace Thackeray, which follows the lives of Becky Sharp and Amelia Sedley amid their friends and families during and after the Napoleonic Wars. It was first published as a 19-volume monthly serial from 1847 to 1848, carrying the subtitle Pen and Pencil Sketches of English Society, which reflects both its satirisation of early 19th-century British society and the many illustrations drawn by Thackeray to accompany the text. It was published as a single volume in 1848 with the subtitle A Novel without a Hero, reflecting Thackeray's interest in deconstructing his era's conventions regarding literary heroism. It is sometimes considered the "principal founder" of the Victorian domestic novel.


title: Vanity Fair (novel)

text: The story is framed as a puppet play, and the narrator, despite being an authorial voice, is somewhat unreliable. The serial was a popular and critical success; the novel is now considered a classic and has inspired several audio, film, and television adaptations. It also inspired the title of the British lifestyle magazine first published in 1868, which became known for its caricatures of famous people of Victorian and Edwardian society. In 2003, "Vanity Fair" was listed at No. 122 on the BBC's The Big Read poll of the UK's best-loved books.


title: Vanity Fair (novel)

text: The book's title comes from John Bunyan's "Pilgrim's Progress", a Dissenter allegory first published in 1678. In that work, "Vanity Fair" refers to a stop along the pilgrim's route: a never-ending fair held in a town called Vanity, which represents man's sinful attachment to worldly things. Thackeray does not mention Bunyan in the novel or in his surviving letters about it, where he describes himself dealing with "living without God in the world", but he did expect the reference to be understood by his audience, as shown in an 1851 "Times" article likely written by Thackeray himself.


title: Vanity Fair (novel)

text: Robert Bell—whose friendship later became so great that he was buried near Thackeray at Kensal Green Cemetery—complained that the novel could have used "more light and air" to make it "more agreeable and healthy". Thackeray rebutted this with Evangelist's words as the pilgrims entered Bunyan's Vanity Fair: "The heart is deceitful above all things, and desperately wicked; who can know it?"


title: Vanity Fair (novel)

text: From its appearance in Bunyan, "Vanity Fair" or a "vanity-fair" was also in general use for "the world" in a range of connotations from the blandly descriptive to the wearily dismissive to the condemning. By the 18th century, it was generally taken as a playground and, in the first half of the 19th century, more specifically the playground of the idle and undeserving rich. All of these senses appear in Thackeray's work. The name "Vanity Fair" has also been used for at least 5 periodicals.


title: Vanity Fair (novel)

text: The story is framed by its preface and coda as a puppet show taking place at a fair; the cover illustration of the serial installments was not of the characters but of a troupe of comic actors at Speakers' Corner in Hyde Park. The narrator, variously a show manager or writer, appears at times within the work itself and is somewhat unreliable, repeating a tale of gossip at second or third hand.


title: Vanity Fair (novel)

text: London, 1814. Rebecca Sharp ("Becky"), daughter of an art teacher and a French dancer, is a strong-willed, cunning, moneyless young woman determined to make her way in society. After leaving school, Becky stays with her friend Amelia Sedley ("Emmy"), who is a good-natured, simple-minded young girl, of a wealthy London family. There, Becky meets the dashing and self-obsessed Captain George Osborne (Amelia's betrothed) and Amelia's brother Joseph ("Jos") Sedley, a clumsy and vainglorious but rich civil servant home from the East India Company. Hoping to marry Sedley, the richest young man she has met, Becky entices him, but she fails. George Osborne's friend Captain William Dobbin loves Amelia, but only wishes her happiness, which is centred on George.


title: Vanity Fair (novel)

text: Becky Sharp says farewell to the Sedley family and enters the service of the crude and profligate baronet Sir Pitt Crawley, who has engaged her as a governess to his daughters. Her behaviour at Sir Pitt's house gains his favour, and after the premature death of his second wife, he proposes marriage to her. However, he finds that Becky has secretly married his second son, Captain Rawdon Crawley, but Becky very much regrets having done this; she had no idea that his father's wife would die so soon after. Sir Pitt's elder half sister, the spinster Miss Crawley, is very rich, having inherited her mother's fortune, and the whole Crawley family compete for her favour so she will bequeath them her wealth. Initially her favourite is Rawdon Crawley, but his marriage with Becky enrages her. First she favours the family of Sir Pitt's brother, but when she dies, she has left her money to Sir Pitt's eldest son, also called Pitt.


title: Vanity Fair (novel)

text: News arrives that Napoleon has escaped from Elba, and as a result the stockmarket becomes jittery, causing Amelia's stockbroker father, John Sedley, to become bankrupt. George's rich father forbids George to marry Amelia, who is now poor. Dobbin persuades George to marry Amelia, and George is consequently disinherited. George Osborne, William Dobbin and Rawdon Crawley are deployed to Brussels, accompanied by Amelia and Becky, and Amelia's brother, Jos.


title: Vanity Fair (novel)

text: George is embarrassed by the vulgarity of Mrs. Major O'Dowd, the wife of the head of the regiment. The newly wedded Osborne is already growing tired of Amelia, and he becomes increasingly attracted to Becky, which makes Amelia jealous and unhappy. He is also losing money to Rawdon at cards and billiards. At a ball in Brussels, George gives Becky a note inviting her to run away with him. But then the army have marching orders to the Battle of Waterloo, and George spends a tender night with Amelia and leaves.


title: Vanity Fair (novel)

text: The noise of battle horrifies Amelia, and she is comforted by the brisk but kind Mrs. O'Dowd. Becky is indifferent and makes plans for whatever the outcome (for example, if Napoleon wins, she would aim to become the mistress of one of his Marshals). She also makes a profit selling her carriage and horses at inflated prices to Jos, who is seeking to flee Brussels.


title: Vanity Fair (novel)

text: George Osborne is killed at the Battle of Waterloo, while Dobbin and Rawdon survive the battle. Amelia bears him a posthumous son, who carries on the name George. She returns to live in genteel poverty with her parents, spending her life in memory of her husband and care of her son. Dobbin pays for a small annuity for Amelia and expresses his love for her by small kindnesses toward her and her son. She is too much in love with her husband's memory to return Dobbin's love. Saddened, he goes with his regiment to India for many years.


title: Vanity Fair (novel)

text: Becky also gives birth to a son, named Rawdon after his father. Becky is a cold, distant mother, although Rawdon loves his son. Becky continues her ascent first in post-war Paris and then in London where she is patronised by the rich and powerful Marquis of Steyne. She is eventually presented at court to the Prince Regent and charms him further at a game of "acting charades" where she plays the roles of Clytemnestra and Philomela. The elderly Sir Pitt Crawley dies and is succeeded by his son Pitt, who had married Lady Jane Sheepshanks, Lord Southdown's third daughter. Becky is on good terms with Pitt and Jane originally, but Jane is disgusted by Becky's attitude to her son and jealous of Becky's relationship with Pitt.


title: Vanity Fair (novel)

text: At the summit of their social success, Rawdon is arrested for debt, possibly at Becky's connivance. The financial success of the Crawleys had been a topic of gossip; in fact they were living on credit even when it ruined those who trusted them, such as their landlord, an old servant of the Crawley family. The Marquis of Steyne had given Becky money, jewels, and other gifts but Becky does not use them for expenses or to free her husband. Instead, Rawdon's letter to his brother is received by Lady Jane, who pays the £170 that prompted his imprisonment.


title: Vanity Fair (novel)

text: He returns home to find Becky singing to Steyne and strikes him down on the assumption—despite her protestations of innocence—that they are having an affair. Steyne is indignant, having assumed the £1000 he had just given Becky was part of an arrangement with her husband. Rawdon finds Becky's hidden bank records and leaves her, expecting Steyne to challenge him to a duel. Instead Steyne arranges for Rawdon to be made Governor of Coventry Island, a pest-ridden location. Becky, having lost both husband and credibility, leaves England and wanders the continent, leaving her son in the care of Pitt and Lady Jane.


title: Vanity Fair (novel)

text: As Amelia's adored son George grows up, his grandfather Mr Osborne relents towards him (though not towards Amelia) and takes him from his impoverished mother, who knows the rich old man will give him a better start in life than she could manage. After twelve years abroad, both Joseph Sedley and Dobbin return. Dobbin professes his unchanged love to Amelia. Amelia is affectionate, but she cannot forget the memory of her dead husband. Dobbin mediates a reconciliation between Amelia and her father-in-law, who dies soon after. He had amended his will, bequeathing young George half his large fortune and Amelia a generous annuity.


title: Vanity Fair (novel)

text: After the death of Mr Osborne, Amelia, Jos, George and Dobbin go to Pumpernickel (Weimar in Germany), where they encounter the destitute Becky. Becky has fallen in life. She lives among card sharps and con artists, drinking heavily and gambling. Becky enchants Jos Sedley all over again, and Amelia is persuaded to let Becky join them. Dobbin forbids this, and reminds Amelia of her jealousy of Becky with her husband. Amelia feels that this dishonours the memory of her dead and revered husband, and this leads to a complete breach between her and Dobbin. Dobbin leaves the group and rejoins his regiment, while Becky remains with the group.


title: Vanity Fair (novel)

text: However, Becky has decided that Amelia should marry Dobbin, even though Becky knows Dobbin is her enemy. Becky shows Amelia George's note, kept all this time from the eve of the Battle of Waterloo, and Amelia finally realises that George was not the perfect man she always thought, and that she has rejected a better man, Dobbin. Amelia and Dobbin are reconciled and return to England. Becky and Jos stay in Europe. Jos dies, possibly suspiciously, after signing a portion of his money to Becky as life insurance, thereby setting her up with an income. She returns to England, and manages a respectable life, although all her previous friends refuse to acknowledge her.


title: Vanity Fair (novel)

text: Amelia, called Emmy, is good-natured but passive and naïve. Not very beautiful, she is frequently ignored by men and women but is well-liked by most men who get to know her because of her personality. This popularity is then resented by other women. She begins the work as its heroine ("selected for the very reason that she was the best-natured of all") and marries the dashing George Osborne against his father's wishes, but the narrator is soon forced to admit "she wasn't a heroine" after all as she remains soppily devoted to him despite his neglect of her and his flirtation with Becky.


title: Vanity Fair (novel)

text: After George dies in the Battle of Waterloo, she brings up little George alone while living with her parents. She is completely dominated by her increasingly peevish mother and her spendthrift father, who, to finance one of his failing investment schemes, sells the annuity Jos had provided. Amelia becomes obsessed with her son and the memory of her husband. She ignores William Dobbin, who courts her for years and treats him shabbily until he leaves. Only when Becky shows her George's letter to her, indicating his unfaithfulness, can Amelia move on. She then marries Dobbin.


title: Vanity Fair (novel)

text: In a letter to his close friend Jane Octavia Brookfield while the book was being written, Thackeray confided that "You know you are only a piece of Amelia, my mother is another half, my poor little wife "y est pour beaucoup"". Within the work, her character is compared and connected to Iphigenia, although two of the references extend the allusion to all daughters in all drawing rooms as potential Iphigenias waiting to be sacrificed by their families.


title: Vanity Fair (novel)

text: Rebecca Sharp, called Becky, is Amelia's opposite: an intelligent young woman with a gift for satire. She is described as a short sandy haired girl who has green eyes and a great deal of wit. Becky is born to a French opera dancer mother and an art teacher and artist father Francis. Fluent in both French and English, Becky has a beautiful singing voice, plays the piano, and shows great talent as an actress. Without a mother to guide her into marriage, Becky resolves that "I must be my own Mamma".


title: Vanity Fair (novel)

text: She thereafter appears to be completely amoral and without conscience and has been called the work's "anti-heroine". She does not seem to have the ability to get attached to other people, and lies easily and intelligently to get her way. She is extremely manipulative and, after the first few chapters and her failure to attract Jos Sedley, is not shown as being particularly sincere.


title: Vanity Fair (novel)

text: Never having known financial or social security even as a child, Becky desires it above all things. Nearly everything she does is with the intention of securing a stable position for herself, or herself and her husband after she and Rawdon are married. She advances Rawdon's interests tirelessly, flirting with men such as General Tufto and the Marquis of Steyne to get him promoted. She also uses her feminine wiles to distract men at card parties while Rawdon cheats them blind.


title: Vanity Fair (novel)

text: Marrying Rawdon Crawley in secret was a mistake, as was running off instead of begging Miss Crawley's forgiveness. She also fails to manipulate Miss Crawley through Rawdon so as to obtain an inheritance. Although Becky manipulates men very easily, she is less successful with women. She is utterly hostile to Lady Bareacres, dismissive of Mrs. O'Dowd, and Lady Jane, although initially friendly, eventually distrusts and dislikes her.


title: Vanity Fair (novel)

text: The exceptions to this trend are (at least initially) Miss Crawley, her companion Miss Briggs, and her school friend Amelia; the latter is the recipient of more-or-less the only kindnesses Becky expresses in the work, persuading her to marry Dobbin in light of what Becky comes to appreciate to be his good qualities and protecting Amelia from two ruffians vying for her attentions. This comparative loyalty to Amelia stems from Becky having no other friends at school, and Amelia having "by a thousand kind words and offices, overcome... (Becky's) hostility"; 'The gentle tender-hearted Amelia Sedley was the only person to whom she could attach herself in the least; and who could help attaching herself to Amelia?'


title: Vanity Fair (novel)

text: Beginning with her determination to be her "own Mamma", Becky begins to assume the role of Clytemnestra. Becky and her necklace from Steyne also allude to the fallen Eriphyle in Racine's retelling of "Iphigenia at Aulis", where she doubles and rescues Iphigenia. In lesser contexts, Becky also appears as Arachne to Miss Pinkerton's Minerva and as a variety of classical figures in the works' illustrations.


title: Vanity Fair (novel)

text: Rawdon, the younger of the two Crawley sons, is an empty-headed cavalry officer who is his wealthy aunt's favourite until he marries Becky Sharp, who is of a far lower class. He permanently alienates his aunt, who leaves her estate to Rawdon's elder brother Sir Pitt instead. Sir Pitt has by this time inherited their father's estate, leaving Rawdon destitute.


title: Vanity Fair (novel)

text: The well-meaning Rawdon does have a few talents in life, most of them having to do with gambling and duelling. He is very good at cards and billiards, and although he does not always win he is able to earn cash by betting against less talented gamblers. He is heavily indebted throughout most of the book, not so much for his own expenses as for Becky's. Not particularly talented as a military officer, he is content to let Becky manage his career.


title: Vanity Fair (novel)

text: Although Rawdon knows Becky is attractive to men, he believes her reputation is spotless even though she is widely suspected of romantic intrigue with General Tufto and other powerful men. Nobody dares to suggest otherwise to Rawdon because of his temper and his reputation for duelling. Yet other people, particularly the Marquis of Steyne, find it impossible to believe that Crawley is unaware of Becky's tricks. Steyne in particular believes Rawdon is fully aware Becky is prostituting herself, and believes Rawdon is going along with the charade in the hope of financial gain.


title: Vanity Fair (novel)

text: After Rawdon finds out the truth and leaves Becky for an assignment overseas, he leaves his son to be brought up by his brother Sir Pitt and his wife Lady Jane. While overseas, Rawdon dies of yellow fever.


title: Vanity Fair (novel)

text: Rawdon Crawley's elder brother inherits the Crawley estate from his father, the boorish and vulgar Sir Pitt, and also inherits the estate of his wealthy aunt, Miss Crawley, after she disinherits Rawdon. Pitt is very religious and has political aspirations, although not many people appreciate his intelligence or wisdom because there's not much there to appreciate.


title: Vanity Fair (novel)

text: Somewhat pedantic and conservative, Pitt does nothing to help Rawdon or Becky even when they fall on hard times. This is chiefly due to the influence of his wife, Lady Jane, who dislikes Becky because of her callous treatment of her son, and also because Becky repaid Lady Jane's earlier kindness by patronising her and flirting with Sir Pitt.


title: Vanity Fair (novel)

text: The elderly Miss Crawley is everyone's favourite wealthy aunt. Sir Pitt and Rawdon both dote on her, although Rawdon is her favourite nephew and sole heir until he marries Becky. While Miss Crawley likes Becky and keeps her around to entertain her with sarcasm and wit, and while she loves scandal and particularly stories of unwise marriage, she does not want scandal or unwise marriage in her family. A substantial part of the early section of the book deals with the efforts the Crawleys make to kowtow to Miss Crawley in the hope of receiving a big inheritance.


title: Vanity Fair (novel)

text: George Osborne, his father (a merchant, of considerably superior social status to Dobbin's grocer father, albeit self made, and ironically a mere corporal in the City Light Horse regiment of which Dobbin senior, by this time an alderman and a knight, is colonel), and his two sisters are close to the Sedley family until Mr. Sedley (the father of Jos and Amelia, and George Osborne's godfather, from whom the latter takes his middle name of 'Sedley') goes bankrupt following some ill-advised speculation. Since George and Amelia were raised in close company and were childhood sweethearts, George defies his father to marry Amelia. Before father and son can be reconciled, George is killed at the battle of Waterloo, leaving the pregnant Amelia to carry on as well as she can.


title: Vanity Fair (novel)

text: Raised to be a selfish, vain, profligate spender, handsome and self-obsessed, George squanders the last of the money he receives from his father and sets nothing aside to help support Amelia. After marrying Amelia, he finds after a couple of weeks that he is bored. He flirts with Becky quite seriously and is reconciled to Amelia only a short time before he is killed in battle.


title: Vanity Fair (novel)

text: The best friend of George Osborne, Captain William Dobbin is tall, ungainly, and not particularly handsome. He is a few years older than George but has been friends with him since his schooldays, even though Dobbin's father is a fig-merchant (Dobbin & Rudge, grocers and oilmen, Thames Street, London - he is later an alderman and colonel of the City Light Horse regiment, and knighted) and the Osbornes belong to the genteel class and have become independently wealthy. He defends George and is blind to his faults in many ways, although he tries to force George to do the right thing. He pushes George to keep his promise to marry Amelia even though Dobbin is in love with Amelia himself. After George is killed, Dobbin puts together an annuity to help support Amelia, ostensibly with the help of George's fellow officers.


title: Vanity Fair (novel)

text: Later, Major and Lieutenant Colonel Dobbin discreetly does what he can to help support Amelia and her son George. He allows Amelia to continue with her obsession over George and does not correct her erroneous beliefs about him. He hangs about for years, either pining away over her while serving in India or waiting on her in person, allowing her to take advantage of his good nature. After Amelia finally chooses Becky's friendship over his during their stay in Germany, Dobbin leaves in disgust. He returns when Amelia writes to him and admits her feelings for him, marries her (despite having lost much of his passion for her), and has a daughter whom he loves deeply.


title: Vanity Fair (novel)

text: Amelia's older brother, Joseph "Jos" Sedley, is a "nabob", who made a respectable fortune as a collector in India. Obese and self-important but very shy and insecure, he is attracted to Becky Sharp but circumstances prevent him from proposing. He never marries, but when he meets Becky again he is easily manipulated into falling in love with her. Jos is not a courageous or intelligent man, displaying his cowardice at the Battle of Waterloo by trying to flee and purchasing both of Becky's overpriced horses. Becky ensnares him again near the end of the book and, it is hinted, murders him for his life insurance.


title: Vanity Fair (novel)

text: Thackeray may have begun working out some of the details of "Vanity Fair" as early as 1841 but probably began writing it in late 1844. Like many novels of the time, "Vanity Fair" was published as a serial before being sold in book form. It was printed in 20 monthly parts between January 1847 and July 1848 for "Punch" by Bradbury & Evans in London. The first three had already been completed before publication, while the others were written after it had begun to sell.


title: Vanity Fair (novel)

text: As was standard practice, the last part was a "double number" containing parts 19 and 20. Surviving texts, his notes, and letters show that adjustments were made – e.g., the Battle of Waterloo was delayed twice – but that the broad outline of the story and its principal themes were well established from the beginning of publication.


title: Vanity Fair (novel)

text: The parts resembled pamphlets and contained the text of several chapters between outer pages of steel-plate engravings and advertising. Woodcut engravings, which could be set along with normal moveable type, appeared within the text. The same engraved illustration appeared on the canary-yellow cover of each monthly part; this colour became Thackeray's signature, as a light blue-green was Dickens's, allowing passers-by to notice a new Thackeray number in a bookstall from a distance.


title: Vanity Fair (novel)

text: "Vanity Fair" was the first work that Thackeray published under his own name and was extremely well received at the time. After the conclusion of its serial publication, it was printed as a bound volume by Bradbury & Evans in 1848 and was quickly picked up by other London printers as well. As a collected work, the novels bore the subtitle "A Novel without a Hero". By the end of 1859, royalties on "Vanity Fair" had only given Thackeray about £2000, a third of his take from "The Virginians", but was responsible for his still more lucrative lecture tours in Britain and the United States.


title: Vanity Fair (novel)

text: From his first draft and following publication, Thackeray occasionally revised his allusions to make them more accessible for his readers. In Chapter 5, an original "Prince Whadyecallem" became "Prince Ahmed" by the 1853 edition. In Chapter 13, a passage about the filicidal Biblical figure Jephthah was removed, although references to Iphigenia remained important. In Chapter 56, Thackeray originally confused Samuel – the boy whose mother Hannah had given him up when called to by God – with Eli, the old priest to whose care he was entrusted; this mistake was not corrected until the 1889 edition, after Thackeray's death.


title: Vanity Fair (novel)

text: The serials had been subtitled "Pen and Pencil Sketches of English Society" and both they and the early bound versions featured Thackeray's own illustrations. These sometimes provided symbolically-freighted images, such as one of the female characters being portrayed as a man-eating mermaid. In at least one case, a major plot point is provided through an image and its caption. Although the text makes it clear that other characters suspect Becky Sharp to have murdered her second husband, there is nothing definitive in the text itself. However, an image reveals her overhearing Jos pleading with Dobbin while clutching a small white object in her hand. The caption that this is "Becky's second appearance in the character of Clytemnestra" clarifies that she did indeed murder him for the insurance money, likely through laudanum or another poison.


title: Vanity Fair (novel)

text: "The final three illustrations of "Vanity Fair" are tableaux that insinuate visually what the narrator is unwilling to articulate: that Becky... has actually been substantially rewarded – by society – for her crimes." One of the Thackeray's plates for the 11th issue of "Vanity Fair" was suppressed from publication by threat of prosecution for libel, so great was the resemblance of its depiction of Lord Steyne to Marquis of Hertford. Despite their relevance, most modern editions either do not reproduce all the illustrations or do so with poor detail.


title: Vanity Fair (novel)

text: The style is highly indebted to Henry Fielding. Thackeray meant the book to be not only entertaining but also instructive, an intention demonstrated through the book's narration and through Thackeray's private correspondence. A letter to his editor at "Punch" expressed his belief that "our profession... is as serious as the parson's own". He considered it his own coming-of-age as a writer and greatest work.


title: Vanity Fair (novel)

text: Critics hailed the work as a literary treasure before the last part of the serial was published. In her correspondence, Charlotte Brontë was effusive regarding his illustrations as well: "You will not easily find a second Thackeray. How he can render, with a few black lines and dots, shades of expression, so fine, so real; traits of character so minute, so subtle, so difficult to seize and fix, I cannot tell—I can only wonder and admire... If Truth were again a goddess, Thackeray should be her high priest."


title: Vanity Fair (novel)

text: The early reviewers took the debt to Bunyan as self-evident and compared Becky with Pilgrim and Thackeray with Faithful. Although they were superlative in their praise, some expressed disappointment at the unremittingly dark portrayal of human nature, fearing Thackeray had taken his dismal metaphor too far. In response to these critics, Thackeray explained that he saw people for the most part as "abominably foolish and selfish".


title: Vanity Fair (novel)

text: The unhappy ending was intended to inspire readers to look inward at their own shortcomings. Other critics took notice of or exception to the social subversion in the work; in his correspondence, Thackeray stated his criticism was not reserved to the upper class: "My object is to make every body engaged, engaged in the pursuit of Vanity Fair and I must carry my story through in this dreary minor key, with only occasional hints here and there of better things—of better things which it does not become me to preach".


title: Vanity Fair (novel)

text: The novel is considered a classic of English literature, though some critics claim that it has structural problems; Thackeray sometimes lost track of the huge scope of his work, mixing up characters' names and minor plot details. The number of allusions and references it contains can make it difficult for modern readers to follow.


title: Vanity Fair (novel)

text: The subtitle, "A Novel without a Hero", is apt because the characters are all flawed to a greater or lesser degree; even the most sympathetic have weaknesses, for example Captain Dobbin, who is prone to vanity and melancholy. The human weaknesses Thackeray illustrates are mostly to do with greed, idleness, and snobbery, and the scheming, deceit and hypocrisy which mask them. None of the characters are wholly evil, although Becky's manipulative, amoral tendencies make her come pretty close. However, even Becky, who is amoral and cunning, is thrown on her own resources by poverty and its stigma. (She is the orphaned daughter of a poor artist and an opera dancer.) Thackeray's tendency to highlight faults in all of his characters displays his desire for a greater level of realism in his fiction compared to the rather unlikely or idealised people in many contemporary novels.


title: Vanity Fair (novel)

text: The novel is a satire of society as a whole, characterised by hypocrisy and opportunism, but it is not a reforming novel; there is no suggestion that social or political changes or greater piety and moral reformism could improve the nature of society. It thus paints a fairly bleak view of the human condition. This bleak portrait is continued with Thackeray's own role as an omniscient narrator, one of the writers best known for using the technique. He continually offers asides about his characters and compares them to actors and puppets, but his cheek goes even as far as his readers, accusing all who may be interested in such "Vanity Fairs" as being either "of a lazy, or a benevolent, or a sarcastic mood". As Lord David Cecil remarked, "Thackeray liked people, and for the most part he thought them well-intentioned. But he also saw very clearly that they were all in some degree weak and vain, self-absorbed and self-deceived." Amelia begins as a warm-hearted and friendly girl, though sentimental and naive, but by the story's end she is portrayed as vacuous and shallow. Dobbin appears first as loyal and magnanimous, if unaware of his own worth; by the end of the story he is presented as a tragic fool, a prisoner of his own sense of duty who knows he is wasting his gifts on Amelia but is unable to live without her. The novel's increasingly grim outlook can take readers aback, as characters whom the reader at first holds in sympathy are shown to be unworthy of such regard.


title: Vanity Fair (novel)

text: The work is often compared to the other great historical novel of the Napoleonic Wars, Tolstoy's "War and Peace". While Tolstoy's work has a greater emphasis on the historical detail and the effect the war has upon his protagonists, Thackeray instead uses the conflict as a backdrop to the lives of his characters. The momentous events on the continent do not always have an equally important influence on the behaviours of Thackeray's characters. Rather their faults tend to compound over time. This is in contrast to the redemptive power the conflict has on the characters in "War and Peace". For Thackeray, the Napoleonic Wars as a whole can be thought of as one more of the vanities expressed in the title.


title: Vanity Fair (novel)

text: A common critical topic is to address various objects in the book and the characters' relationships with them, such as Rebecca's diamonds or the piano Amelia values when she thinks it came from George and dismisses upon learning that Dobbin provided it. Marxist and similar schools of criticism that go further and see Thackeray condemning consumerism and capitalism, however, largely overstate their case. Thackeray is pointed in his criticism of the commodification of women in the marriage market, but his variations on Ecclesiastes's "all is vanity" are more personal than institutional. He also has broad sympathy with a measure of comfort and financial and physical "snugness". At one point, the narrator even makes a "robust defense of his lunch": "It is all vanity to be sure: but who will not own to liking a little of it? I should like to know what well-constituted mind, merely because it is transitory, dislikes roast-beef?"


title: Vanity Fair (novel)

text: Despite the clear implications of Thackeray's illustration on the topic, John Sutherland has argued against Becky having murdered Jos on the basis of Thackeray's criticism of the "Newgate novels" of Edward Bulwer-Lytton and other authors of Victorian crime fiction. Although what Thackeray principally objected to was glorification of a criminal's deeds, his intent may have been to entrap the Victorian reader with their own prejudices and make them think the worst of Becky Sharp even when they have no proof of her actions.


title: Sundeep Kishan

text: Sundeep Kishan (born 7th May 1987) is an Indian actor and producer who works in predominately in Telugu and Tamil cinema.


title: Sundeep Kishan

text: Sandeep Kishan was born in a Telugu family in present-day Chennai. He is the nephew to the cinematographers Chota K. Naidu and Shyam K. Naidu. He moved to Hyderabad in 2008 to pursue a career in films.


title: Sundeep Kishan

text: Kishan began working as an assistant director to Gautham Vasudev Menon for a year, before starring in "Sneha Geetham".


title: Sundeep Kishan

text: After working with him in "Prasthanam" (2010), director Deva Katta referred Kishan to his friends Raj Nidimoru and Krishna D.K., which led to Kishan's casting in his first Hindi venture, "Shor in the City". He also did the Telugu movie "DK Bose" (2018), opposite Nisha Aggarwal.


title: Sundeep Kishan

text: Kishan stated that he was very happy since his 2015 released film "Tiger" was appreciated by actors like Ravi Teja, Allu Arjun and R. Narayana Murthy. He has also acted in "Venkatadri Express" and "Ra Ra... Krishnayya". After "Maanagaram", Sundeep signed for his next project titled "Nenjil Thunivirundhal", directed by Suseenthiran.


title: Sundeep Kishan

text: Kishan owns a restaurant chain in Hyderabad named "Vivaha Bhojanambhu." He also started a saloon business Express Unisex in Vijayawada.


title: Fluid dynamics

text: In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including "aerodynamics" (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation.


title: Fluid dynamics

text: Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, such as flow velocity, pressure, density, and temperature, as functions of space and time.


title: Fluid dynamics

text: Before the twentieth century, "hydrodynamics" was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability, both of which can also be applied to gases.


title: Fluid dynamics

text: The foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of mass, conservation of linear momentum, and conservation of energy (also known as the First Law of Thermodynamics). These are based on classical mechanics and are modified in quantum mechanics and general relativity. They are expressed using the Reynolds transport theorem.


title: Fluid dynamics

text: In addition to the above, fluids are assumed to obey the continuum assumption. Fluids are composed of molecules that collide with one another and solid objects. However, the continuum assumption assumes that fluids are continuous, rather than discrete. Consequently, it is assumed that properties such as density, pressure, temperature, and flow velocity are well-defined at infinitesimally small points in space and vary continuously from one point to another. The fact that the fluid is made up of discrete molecules is ignored.


title: Fluid dynamics

text: For fluids that are sufficiently dense to be a continuum, do not contain ionized species, and have flow velocities that are small in relation to the speed of light, the momentum equations for Newtonian fluids are the Navier–Stokes equations—which is a non-linear set of differential equations that describes the flow of a fluid whose stress depends linearly on flow velocity gradients and pressure. The unsimplified equations do not have a general closed-form solution, so they are primarily of use in computational fluid dynamics. The equations can be simplified in several ways, all of which make them easier to solve. Some of the simplifications allow some simple fluid dynamics problems to be solved in closed form.


title: Fluid dynamics

text: In addition to the mass, momentum, and energy conservation equations, a thermodynamic equation of state that gives the pressure as a function of other thermodynamic variables is required to completely describe the problem. An example of this would be the perfect gas equation of state:


title: Fluid dynamics

text: where is pressure, is density, and is the absolute temperature, while is the gas constant and is molar mass for a particular gas. A constitutive relation may also be useful.


title: Fluid dynamics

text: Three conservation laws are used to solve fluid dynamics problems, and may be written in integral or differential form. The conservation laws may be applied to a region of the flow called a "control volume". A control volume is a discrete volume in space through which fluid is assumed to flow. The integral formulations of the conservation laws are used to describe the change of mass, momentum, or energy within the control volume. Differential formulations of the conservation laws apply Stokes' theorem to yield an expression that may be interpreted as the integral form of the law applied to an infinitesimally small volume (at a point) within the flow.


title: Fluid dynamics

text: All fluids are compressible to an extent; that is, changes in pressure or temperature cause changes in density. However, in many situations the changes in pressure and temperature are sufficiently small that the changes in density are negligible. In this case the flow can be modelled as an incompressible flow. Otherwise the more general compressible flow equations must be used.


title: Fluid dynamics

text: Mathematically, incompressibility is expressed by saying that the density of a fluid parcel does not change as it moves in the flow field, that is,


title: Fluid dynamics

text: where is the material derivative, which is the sum of local and convective derivatives. This additional constraint simplifies the governing equations, especially in the case when the fluid has a uniform density.


title: Fluid dynamics

text: For flow of gases, to determine whether to use compressible or incompressible fluid dynamics, the Mach number of the flow is evaluated. As a rough guide, compressible effects can be ignored at Mach numbers below approximately 0.3. For liquids, whether the incompressible assumption is valid depends on the fluid properties (specifically the critical pressure and temperature of the fluid) and the flow conditions (how close to the critical pressure the actual flow pressure becomes). Acoustic problems always require allowing compressibility, since sound waves are compression waves involving changes in pressure and density of the medium through which they propagate.


title: Fluid dynamics

text: All fluids, except superfluids, are viscous, meaning that they exert some resistance to deformation: neighbouring parcels of fluid moving at different velocities exert viscous forces on each other. The velocity gradient is referred to as a strain rate; it has dimensions . Isaac Newton showed that for many familiar fluids such as water and air, the stress due to these viscous forces is linearly related to the strain rate. Such fluids are called Newtonian fluids. The coefficient of proportionality is called the fluid's viscosity; for Newtonian fluids, it is a fluid property that is independent of the strain rate.


title: Fluid dynamics

text: Non-Newtonian fluids have a more complicated, non-linear stress-strain behaviour. The sub-discipline of rheology describes the stress-strain behaviours of such fluids, which include emulsions and slurries, some viscoelastic materials such as blood and some polymers, and "sticky liquids" such as latex, honey and lubricants.


title: Fluid dynamics

text: The dynamic of fluid parcels is described with the help of Newton's second law. An accelerating parcel of fluid is subject to inertial effects.


title: Fluid dynamics

text: The Reynolds number is a dimensionless quantity which characterises the magnitude of inertial effects compared to the magnitude of viscous effects. A low Reynolds number () indicates that viscous forces are very strong compared to inertial forces. In such cases, inertial forces are sometimes neglected; this flow regime is called Stokes or creeping flow.


title: Fluid dynamics

text: In contrast, high Reynolds numbers () indicate that the inertial effects have more effect on the velocity field than the viscous (friction) effects. In high Reynolds number flows, the flow is often modeled as an inviscid flow, an approximation in which viscosity is completely neglected. Eliminating viscosity allows the Navier–Stokes equations to be simplified into the Euler equations. The integration of the Euler equations along a streamline in an inviscid flow yields Bernoulli's equation. When, in addition to being inviscid, the flow is irrotational everywhere, Bernoulli's equation can completely describe the flow everywhere. Such flows are called potential flows, because the velocity field may be expressed as the gradient of a potential energy expression.


title: Fluid dynamics

text: This idea can work fairly well when the Reynolds number is high. However, problems such as those involving solid boundaries may require that the viscosity be included. Viscosity cannot be neglected near solid boundaries because the no-slip condition generates a thin region of large strain rate, the boundary layer, in which viscosity effects dominate and which thus generates vorticity. Therefore, to calculate net forces on bodies (such as wings), viscous flow equations must be used: inviscid flow theory fails to predict drag forces, a limitation known as the d'Alembert's paradox.


title: Fluid dynamics

text: A commonly used model, especially in computational fluid dynamics, is to use two flow models: the Euler equations away from the body, and boundary layer equations in a region close to the body. The two solutions can then be matched with each other, using the method of matched asymptotic expansions.


title: Fluid dynamics

text: A flow that is not a function of time is called steady flow. Steady-state flow refers to the condition where the fluid properties at a point in the system do not change over time. Time dependent flow is known as unsteady (also called transient). Whether a particular flow is steady or unsteady, can depend on the chosen frame of reference. For instance, laminar flow over a sphere is steady in the frame of reference that is stationary with respect to the sphere. In a frame of reference that is stationary with respect to a background flow, the flow is unsteady.


title: Fluid dynamics

text: Turbulent flows are unsteady by definition. A turbulent flow can, however, be statistically stationary. The random velocity field is statistically stationary if all statistics are invariant under a shift in time. This roughly means that all statistical properties are constant in time. Often, the mean field is the object of interest, and this is constant too in a statistically stationary flow.


title: Fluid dynamics

text: Steady flows are often more tractable than otherwise similar unsteady flows. The governing equations of a steady problem have one dimension fewer (time) than the governing equations of the same problem without taking advantage of the steadiness of the flow field.