And before going into the, let us briefly recapitulate what is the boundary layer. We introduce this concept qualitatively in one of our introductory lectures and just to take it up from there, we take this example that you have a flat plate, the plate is confronting a free stream of fluid with a velocity of say u infinity. Now when this fluid is coming in contact with the plate, what is happening that is what we are interested to see. Clearly when we consider say one section and we go further and further away from the plate, the physical phenomenon tends to get changed. How it tends to get changed, adjacent to the wall, say if you consider a fluid element which is adhering to the wall just as marked in this figure, this fluid element is stationary relative to the plate by the no slip boundary condition. As you go little bit away from the plate on the same vertical section, you see that you may have fluid elements which are subjected to the slowing down effect of the wall but not as severely as the wall adhering fluid layer because this fluid element is somewhat away from the wall. So on the top of it, it has a faster moving layer and at the bottom of it, it has a slower moving layer. So the net effect is that it is slowed down but not to that extent as a fluid element adhering to the wall. So it will have some sort of a velocity. Now if you go further away, you will find that there are fluid elements which are further away from the wall. So they are slowed down by the wall but not explicitly but implicitly. How implicitly they have a layer at the bottom which is moving at a slower pace and which is closer to the wall than the fluid elements which are above that and which are moving faster. So as a consequence, as a competing consequence, what happens is this fluid layer tends to move at a velocity which is somewhat larger than the velocity of the layers which are there at the bottom. In this way, the velocity increases till one may reach a point where we reach a point is a question that we will see that one may reach a point somewhere where the velocity is almost equal to u infinity and let us say that we are happy with the location where it is 99% of u infinity. It is as good as u infinity and beyond that the velocity virtually does not change. Beyond that, velocity remains almost the same. So if you join the tips of the velocity vectors at this location, this is what you get as a velocity profile at this location. These are certain terminologies that we introduced earlier and we could also qualitatively understand that despite not being in contact with the plate, you will see that somehow there are fluid elements which are feeling the effect of the plate. The fluid element directly in contact with the plate is supposed to feel the effect of the plate. Others are not directly in contact with the plate but to some extent within a distance from the wall, they are feeling the effect of the wall and how they are feeling the effect of the wall? It is through the fluid property viscosity which is propagating the message of the wall from the wall towards the outer fluids. Now if we draw the same thing that is the velocity profile description at a section further away from the wall, further away from the inlet, then what we see here the velocity is 0. Then in the vertical sections as you move above what you see, the qualitative picture is same as what is there in the first section but quantitatively what you expect that at this layer the velocity should be even less than what was in the corresponding previous section. The reason is very, very obvious that now more and more fluid is in contact with the plate. So the effect of slowing down by the plate is more and more severe. So you have velocities which are less at a given vertical section location. In this way here also the u infinity condition will be reached but the distance over which it will reach will be somewhat more than the distance over which it was reached in the previous section. Let us say somewhere here the u infinity condition is reached beyond which the velocity remains the same. So you can clearly see that if we want to demarcate the characteristics of the velocity profile, we are having 2 important characteristics. One characteristic is where the velocity profile has a sort of a gradient and another characteristic is that the gradient ceases to exist. If we demarcate these 2, one way of demarcating is to find out to what distance this velocity profile exists. Let us say that to this distance the velocity profile exists for this section and for the next section may be to this distance the velocity profile exists that means gradient in the velocity profile exists. So in this way the plate may be as large as you want along the axial direction but like these 2 representative sections are good enough to have a sort of a physical picture. So if you want to figure out that what happens subsequently of course this physically this type of behavior it continues along the plate. So as if like this effect of the wall gets more and more propagated. Now it is possible therefore to have a region within which the velocity profile has a strong gradient so that the effect of the wall is felt very explicitly and a region which as if does not understand the effect of the wall and to demarcate that one may just draw a locus of these mark points because these indicate the distance from the wall up to which the viscous effects are explicitly felt. So if you draw the locus of such thing this gives a imaginary line. This line is not something is which is existing in the flow field it is just for our conceptualization that we are having such a demarcating boundary. So this demarcating boundary demarcates the flow domain into 2 parts. One part adjacent to the wall where wall effects are important or so to say the slowing down effect of the wall or the viscous effects are important and outside these viscous effects are not important. It does not mean that the fluid outside these has no viscosity simply the effect of viscosity is not manifested in the form of shear stresses because of lack of existence of velocity gradients. So within this layer the viscous effects are sustaining shear stresses and transmitting the disturbance of momentum from one layer to the other. So that is what is called as momentum transfer. So the layer where this physical behavior is occurring is known as boundary layer and outside the boundary layer is in this case is the behavior of the free stream and this line therefore is a demarcation between the boundary layer and the region outside the boundary layer and this is known as edge of the boundary layer. So boundary layer is just a concept which helps us in demarcating the flow domain into parts where the part that we are focusing on that is the boundary layer is an interesting part because that is where the viscous effect is explicitly felt and the entire purpose of the boundary layer theory is to have a detailed understanding of what happens in this layer. So the whole objective is there is a scientific objective scientific objective is we must understand. So if say if we call that this is x direction and the transfer direction is y direction then at a given x there is a thickness of the boundary layer which we call as delta. So we see qualitatively that as x increases delta also increases but the question is how thick or how thin the delta is. So that is one of the very interesting things because that will dictate us that what is the extent within which this velocity gradient exists. From an engineering point of view what is important from an engineering point of view what is important is what is the total shear force or the shear stress that is acting on the wall and to know that you must have a detailed picture of how the velocity gradients are shaping up within the boundary layer. So from an engineering point of view the shear stress is important why because it will give rise to a drag force and based on the drag force one may intelligently design engineering systems where you have interactions between fluids and solids and not only that it is also possible to have a clear idea of what is the extent over which the wall shear stress effects are predominant. So for all these reasons it is important that we emphasize on the characteristics of flow within the boundary layer and the studying of the boundary layer theory has one of those things as important objectives. So we start with the boundary layer theory by considering a simple case and we will adhere to that simple case in a major part of this chapter that is we will consider the density equal to constant and we will consider steady flow. Not that there are no relevance of boundary layer theory for variable density or unsteady flows yes they are definitely but for an introductory course this is what is going to give us a lot of insight on the very very fundamentals of the theory. So as usual what we will consider we will consider that our basic governing equations which we have developed these governing equations will be the basis with which we will start the analysis. So what are the governing equations first the continuity equation is there. So we will not write it in an index notation it is just a 2 dimensional description. So we will consider a 2d flow also with velocity components as u and v. So u is u1 v is u2. So we will write the continuity equation under these assumptions you have so let us write the continuity this is the continuity equation we will write the momentum equations subsequently but let us first start looking into the continuity equation. Now when you look into the continuity equation what we are trying to guess from the continuity equation. First of all we will try to have a picture on the order of magnitudes of different quantities that means what is the order of magnitude of u and what is the order of magnitude of v this is very important because before getting into the exact quantification order of magnitude will give us an idea of the range of values that this may take. So for that one important style of analysis is known as order of magnitude analysis or scaling analysis. So we will look into that scaling analysis magnitude analysis this is a very powerful tool in scientific analysis because we will soon see that without solving the equations we will have a fair idea of the order of magnitudes of various quantities which are governing the physical behavior. So how we do that to do that we basically refer to the governing equations and we refer to the characteristic values of various parameters appearing in the governing equations. So what are the characteristic values the characteristic values are called as so called scales. So when we have say this du dx type of term we are looking for so this term if you write in terms of an order of magnitude the order of magnitude of order of magnitude is given by this type of a symbol this one or maybe order of either way. So order of this one is what order of this one is order of u divided by order of x so to say. So order of u means what is the maximum u so to say and order of x is what is the maximum x. So first of all x is like it is apparently mathematically it is unmounded towards the particular direction and this type of problem is physically like a marching problem that means you start marching along the plate as you march further and further you see that the boundary layer is growing and growing. So if the plate was at infinity maybe it would have grown may not be mathematically exactly in the same fashion because maybe initially it is laminar but after some distance the boundary layer may become turbulent because the Reynolds number here is defined on the basis of the characteristic length scale which is the axial length scale. So as you go further and further as your axial length scale becomes more and more there may be a place where the location where the Reynolds number is so high that slight disturbance will trigger turbulence. So qualitatively the boundary layer will grow but the quantitative manner in which it will grow will change. But whatever it is here we are focusing on the laminar boundary layer. For turbulent boundary layer we will separately have some consideration although not in details but at least we will have some consideration. But irrespective of the way in which the boundary layer grows the growth of the boundary layer will be dictated in a strong way by what is the total length of the plate because the characteristic Reynolds number here that you are looking for for a boundary layer description is this one rho the characteristic velocity u infinity into L by mu or in terms of the kinematic viscosity u infinity into L by nu where nu is the kinematic viscosity. Therefore the characteristic length which dictates the problem is the length of the plate here as an example for internal flows like flow through pipes and channels what we have seen is that it is like say if you have a parallel plate channel is the distance between the plates or the depth that is the important characteristic length scale where whereas here it is the axial distance. So for a flow through a channel or a pipe the axial length is not an important characteristic length but here the axial length is an important characteristic length. So the scale of x is the maximum x and that is L what is the scale of u? u infinity. So the scale of the first term is u infinity by L okay what is the scale of the second term? See it again requires the knowledge of the scale of v and scale of y. What is the scale of v? We do not know directly but let us say that the scale of v we may estimate that the scale of v is the maximum v and which exists maybe at so called at the edge of the outside the edge of the boundary layer. Let us give it a name some name say v infinity there is nothing as such physically v infinity but just for a nomenclature say v infinity. What is the scale of y? Delta the governing equation there are 2 terms these 2 terms are somehow adding an effect that the net result is 0 that means each of these terms should be of same order of magnitude and one term would be of opposite sense than the other so that they get nullified. What do we mean by this? So if this is like 10 meter per second divided by 1 meter this cannot be 1 meter per second divided by 1 meter because then these 2 cannot cancel each other. To cancel each other they must have the same order of magnitude and if they cannot cancel each other then the result cannot be 0. So when we talk about order of magnitude see this is what we are looking for is the velocity 1 meter per second order or 10 meter per second order or 100 meter per second order. If we say 1 meter per second order we do not literally mean that it has to be 1 meter per second it could be 1 meter per second, 2 meter per second, 3 meter per second like that but order of magnitude wise it is like a sort of a single digit meter per second something like that if you express it in a single digit. So if it is order of 10 meter per second it might be say anything between 10 to 99 meter per second of course there is a fuzziness. What is the fuzziness? If it is 99 meter per second I would say that it is much closer to 100 meter per second than 10 meter per second. So this kind of fuzziness exists but the important thing is it is your perception of how you get rid of the fuzziness and get a correct description of the values. The fuzziness that we found is because of the use of the decimal system. So it has nothing to do with the order of magnitude analysis. It has something to do with our way of expressing the order of magnitude. So if we do not have any specific bias towards the system of representing the order of magnitude at least what we can say from this relationship that you must have u infinity by L of the order of v infinity by delta. So that these 2 terms somehow nullify their effect and from here we get v infinity of the order of u infinity into delta by L. So when we have this we should try to have an estimate of the relative estimate of v scale with respect to the u scale and we are seeing that what is the parameter that is governing it is delta by L. So we may have all sorts of possibilities. So in a boundary layer you may have delta much much greater than L, delta much much less than L, even delta of the order of L all those are possible boundary layers. But the theory that we are going to develop we will develop for a special case and we will see that what is the relevance of that special case. The special case is delta by L is very very small that is one of the important assumptions of the boundary layer theory. But again see as if we are forcefully doing delta by L is much much less than 1 when there is a physical problem say you are a person working in an industry you may you are always entitled to say that I do not understand what is delta. So delta by L is much much less than 1 what is the I mean how from the system parameters can I tell that whether delta by L is much much less than 1 or not. So there must be something from the system description system parameter description that should tell us whether delta by L is large or small or whatever. But we will now go through an exercise which will tell us that how the system parameters will be linked with the smallness or largeness of the delta by L. So the important assumption the first assumption that we are making is that delta by L is much much less than 1 that is small okay. Now with this understanding let us go to the momentum equations before going into the momentum equation along the x direction which is the dominating effect here we will first look into the momentum equation along the y direction. So y momentum equation since density is a constant we will divide both the sides of the momentum equation by the density. So that the viscosity will be converted to kinematic viscosity. So we will have again we are not considering any effect of body force but if there is some body force that may be added our job is to estimate the order of magnitudes of various terms. Let us consider term by term this term first term what is the order of magnitude of this. So order of magnitude of u is u infinity v is v infinity by L this one v infinity square by delta this one 1 by rho see this is some pressure scale by some length scale. So the question is what is the appropriate pressure scale that we do not know. So by lot of experiments it was understood that in such cases where delta by L is much much less than 1 the pressure scale is governed by the kinetic energy scale. So that is like almost like an inviscid flow. So that means the pressure scale if we call as delta p then delta p is of the order of rho into u infinity square whether this scale is correct or not we will justify at the end that whatever scale we are assuming may be experiments give that but the question is theoretically it has to be consistent with the remaining part we will do that. But for the time being we assume that the pressure scale is governed by this kinetic energy scale. So rho u infinity square see when we wrote the scale we did not care whether we write half rho infinity square or rho infinity square because objective of the scale is not to give the exact value it is the order of magnitude. So with the half order of magnitude does not change. So rho u infinity square by delta these terms say the first term nu what is this you tell the first term V infinity by L square and this term is nu V infinity by delta square all are order of magnitude because all these terms are like ddx of dvdx ddy of dvdy like that. Now without doing any analysis we can straight forward say that if we assume delta by L is much much less than 1 that is delta much much less than L then this term is much much more dominating than this term because here division by delta square at division by L square that means in the first cut analysis without thinking about anything else whenever delta by L is much much less than 1 the first term here is much much more insignificant as compared to the second term and therefore may be neglected so that is see it does not matter whether it is V or u or whatever it is the denominator that is what is creating the difference and therefore we may have a general conclusion that of whatever variable if delta by L is small that is delta much much less than L and see these terms are like the first term is axial diffusion of momentum disturbance physically and the second term is the transverse diffusion of momentum disturbance. So we say that the axial diffusion term is negligible as compared to the transverse diffusion term that is the way in which we speak this in words. Now so that is one of the important things we will keep in mind not just for this equation but for even the x momentum equation we will apply the same logic. So at least we could get rid of one of the terms in terms of the analysis that well this term will not be important so long as the other term is there. Now let us try to make an assessment of the order of magnitude of the different terms in the left hand side and the important terms in the right hand side. First of all you may estimate V infinity with u infinity into delta by L here also like that u infinity into delta by L.  So we may conclude see order of magnitude of two terms if they are the same adding them the order of magnitude becomes either of them not that you add so if it becomes 2 it makes no difference 2 does not change the order of magnitude. So the left hand side order of magnitude is u infinity square delta by L square and what is the physical meaning of the left hand side these are like inertia terms. So these are like because these are like the if you recall these are nothing but the advective components of acceleration or convective components of acceleration. So these like represent the inertia of the flow keeping that in mind let us write the ratio of the inertia terms and the pressure gradient term. This is of the order of inertia term is of the order of u infinity square delta by L square pressure gradient term of the order of u infinity square by delta. So this becomes of the order of delta by L whole square right. Therefore if delta is much much less than 1c so many simplifications are possible just by this consideration. So this becomes one of the very key considerations that means this is a negligible term. So the first conclusion is that in the y momentum equation the inertia terms are negligible in comparison to the pressure gradient term. Then let us consider the ratio of the dominant viscous term by the pressure gradient term. So that is nu into v infinity by delta square divided by u infinity square by delta. So what is this v in place of v infinity we will write u infinity into delta by L. So nu u infinity delta by L delta square into delta by u infinity square. So that becomes delta by sorry delta is gone 1 by Reynolds number right. So 1 by R EL. So we will also consider that the Reynolds number is large and we will see very interestingly that that is not an additional consideration it should follow from the consistency of this assumption. But for the time being we will assume that because we may only show from the x momentum equation the relationship between delta by L and the Reynolds number. So one of the important consequence of considering the Reynolds number small Reynolds number large will be delta by L small. So delta by L small this automatically implies that the Reynolds number is large. So for the time being let us consider it as an independent assumption. We will not assume that this is the truth because we will show that this is the truth for if delta by L is much much less than 1. So for the time being it is as if and another independent assumption we will see subsequently that it is not an independent assumption it is a relationship with the consideration of delta by L. See I have not I just now I have told that we will show that there is a relationship between large Reynolds number and delta by L is small. So you have to wait till that right. Reynolds number is large so that means that the viscous term by the pressure gradient term is small. That means whatever we will be dominating in this equation if at all something dominates is only the pressure gradient term. So from this the important conclusion is that within the boundary layer so all these order of magnitude estimates we are writing within the boundary layer. So now as if our zoom or the focused attention is the boundary layer. So then that will give rise to this equal to 0 right because all other terms are non-dominating. Therefore this gives rise to a very important conclusion. What is the important conclusion? Pressure is not a function of y within the boundary layer. What are the considerations that we used for this? The considerations that we used for this are delta by L much much less than 1 and Reynolds number large. These are the 2 considerations that we used. Now one of the important things that we should mention at this stage is that this x and y coordinates are generic. For flow over a flat plate you have natural choice of x and y that is that means plate is a flat one so along it you can orient a linear x direction and perpendicular to that y direction. But if you have flow over a curved boundary so if you have say flow over a boundary like this. So then x is written in terms of the curvilinear x and maybe local coordinate which is orthogonal to that. So the curvilinear one will become like a tangential coordinate x at a given point. That means it is as if like a tangential x and a normal y at each and every point. So the x and y direction continuously may shift as you are moving along the curve. So the x and y are generically called as stream wise directions and cross stream wise directions something like that. So these are changing continuously. So in the boundary layer we use x and y coordinate for all the cases not just for flow over a flat plate. Flow over a flat plate we just gave as an example. So it is not the only case where the boundary layer theory will be relevant as we understand because many of the surfaces over which flows occurs in engineering are not like flat surfaces. So you may have curved surfaces like say you may have wings of aerofoils and so on. So obviously if you have any curved surface it is possible to have sort of a different way of describing the coordinates but this x and y coordinates will preserve in the sense that we have just discussed and these we call as boundary layer coordinates. So boundary layer coordinates are this generic x and generic y. So it is not that any orthogonal x, y you choose or any constant x, y you choose. It has a special meaning that is locally at a given point if you want to assess what is happening with the boundary layer then along the surface that means tangential to the surface at that point is x and perpendicular to that is y and that may vary from one point to another if the surface is curved. So from the y momentum equation whatever we get you have to remember that this y is the generic y coordinate that we are talking about. Next we come to the x momentum equation which should hold in some sense the key towards assessment of the behavior within the boundary layer. So x momentum equation is that del u del x plus v del u del y – 1 by rho this one. So let us write the order of magnitude of different terms we have now been familiar in how to write those. So what is the order of magnitude of this? u infinity into u infinity by L. So u infinity square by L this v infinity into u infinity by delta. This one – 1 by rho the delta p scale that is rho u infinity square by L this term okay – is not important for order of magnitude then this term nu u infinity by L square this term nu u infinity by delta square. And again just by previous consideration we know that this term will be much less important than this term. Now let us pay a lot of attention to the pressure gradient term. See what conclusion we got from the y momentum equation? The conclusion that we got is that the pressure is not varying with y and y is the direction along which there is a change in behavior because of the existence of the boundary layer. So that means we can say that the pressure variation within the boundary layer is not important. What it means is that whatever is the pressure gradient that is acting on the flow is because of the pressure gradient that is imposed in the outer stream or the free stream by whatever mechanism. And what is the importance of the outer stream or what is the simplification that you have for the outer stream? The outer stream is like an inviscid flow. So the outer stream is like an inviscid flow that means what? The outer stream is like an inviscid flow that means you may use the equations for inviscid flow for the outer stream. So for the outer stream by using the equations of inviscid flow if you find out what is the pressure gradient, the same pressure gradient is imposed on the fluid within the boundary layer and that means the correct scale of pressure gradient should be rho u infinity square because that comes from the inviscid flow analogy. If you neglect the potential energy part, the delta p will be of the order of rho u infinity square. For example, if you use the Euler equation or maybe the Bernoulli's equation when rho is a constant that is what you get. So the important understanding is that this is consistent with the assumption of the pressure scale delta p of rho into u infinity square. So this, if this conclusion was not consistent that means that assumption was not correct. So that is the first thing. The second thing is p in this problem could be a function of x and y. Now p is not a function of y therefore p is a function of x only. Therefore a very important thing is here we can write this as dp dx. Going one step forward it is dp dx where it is dp dx outside the boundary layer because we have just seen physically that the implication of this one implies that whatever is the pressure gradient which is existing outside the boundary layer, the boundary layer fluid is subjected to the same pressure gradient. So that means in terms of symbols we can write this as good as dp infinity dx where infinity stands for the outer stream that is the notation. So that is about the pressure gradient term. Now we will look into the other terms. So what are the other terms? So the other terms you see when you have v infinity here you can substitute the v infinity in terms of u infinity. So this is of the order of u infinity into delta by L. So you can see that combining these terms each of the terms is of the order of u infinity square by L. So the order of the left hand side which is the inertia term is like u infinity square by L okay. Now what we will do with these orders? We will very soon see with an example but before looking into that example let us summarize the boundary layer equations. So boundary layer equations are simplified versions of the Navier-Stokes equation which we feel that are valid within the boundary layer. So boundary layer equations that is the summary of the equation. First equation is a continuity equation that is very important that has to be there. Again when we are writing the boundary layer equations under the assumptions that we have already described. X momentum equation you feel you may always write the y momentum equation like this but it is not always necessary to explicitly write this because its effect is already inbuilt in the simplification of the x momentum equation. So conventionally when people ask that what are the boundary layer equations usually people refer to these 2. Not that the y momentum equation is not present it is there but its implication has somehow been inbuilt in the x momentum equation. What are the assumptions that we consider for these other than rho equal to constant and the steady the special boundary layer assumptions? Delta by L is much much less than 1 and Reynolds number is large much much greater than 1 and so now the question that we will like to answer is that are they sort of equivalent does one follow from the other or what? They are equivalent to each other or not? That is the question that we would like to answer. When we like to answer the question before answering the question we have to keep in mind that these sort of relationship should come from the description of the system and the description of what is happening within the boundary layer. See the Reynolds number is a system scale description. This does not understand what is boundary layer and so on whereas this is related to the boundary layer thickness. So if we sort of describe an equivalent between these 2 we will achieve our first objective that from the system scale variation we will have an estimate of how thick or how thin the boundary layer may be. So to do that we will take that the special example of flow over a flat plate. So let us take an example of flow over a flat plate. Special case flow over a flat plate. First of all what is our objective is to find out what is this dp infinity dx okay. Let us consider the outer stream. See the outer stream the free stream flow is u infinity. So the velocity field is like u infinity i in the outer stream. u is u infinity i. You can clearly understand that this is an irrotational flow because no gradient in the velocity. So when this is an irrotational flow we have discussed earlier that an irrotational flow may remain irrotational if there are no viscous effects. So outside the boundary layer so this is what is the importance in conceptualizing the boundary layer. Outside the boundary layer viscous effects are not important and therefore the behavior is like an inviscid one. So a flow which was irrotational will remain irrotational eternally because of this effect. Of course we forget about other possible effects which might make the flow rotational like the Coriolis forces and so on those are not important in this context. So if you have that as a situation that means outside the boundary layer you have inviscid irrotational flow. So it will be when you have an irrotational flow outside the boundary layer that means if you have also density equal to constant you may use Bernoulli's equation between any 2 points where the points are located at any location but outside the boundary layer. Within the boundary layer you cannot use the Bernoulli's equation that you have to clearly understand but outside the boundary layer you may by considering these cases. So when you use the Bernoulli's equation that means you have p plus half rho u infinity square is constant in the outer layer. Forget about the potential energy difference that is not important here. If the potential energy difference is important you include that in p and call it a piezometric pressure. Now if you differentiate that with respect to x so this is p infinity outside the boundary layer. So dp infinity dx why we are doing it we require that in the x momentum equation. So dp infinity dx plus rho u infinity du infinity dx is equal to 0. Now we have u infinity as a constant here it is not changing with x. So u infinity may change with x because of what? Because of one of the things is the curvature of the boundary. Because of the curvature of the boundary you may have a gradient of pressure and you may have a gradient of the free stream velocity but for a flat plate there is no effect of the curvature but there is a effect of the curvature if you have flow or a sphere or a cylinder we will look into those examples later. So there the curvature will introduce a pressure gradient and that pressure gradient will imply that there will be also a u infinity gradient so to say. But here you have u infinity gradient is 0 u infinity is constant for flow over a flat plate. So du infinity dx is equal to 0 which implies that dp infinity dx is equal to 0. So for flow over a flat plate dp dx is 0. So when you have dp dx is equal to 0 then you are left with your momentum equation with this one okay. So if you write the order of magnitude of different terms here the left hand side order of magnitude is u infinity square by l and the right hand side is nu u infinity by delta square right. If this equation has to be important then order of magnitude of left hand side and right hand side has to be the same otherwise these terms cannot nullify each other. And therefore the important conclusion that we have from that consideration is that u infinity square by l is of the order of nu u infinity by delta square. So we may write delta square by l of the order of nu by u infinity which implies that delta by l whole square if you divide both sides by l is of the order of 1 by Reynolds number. That means delta by l is of the order of Reynolds number to the power – half. So see this is a remarkable thing because this is a very very important observation or conclusion that we arrived at no cost without solving any equation without going into a computation of the equations numerically or whatever without going into any sort of complication just by looking into the order of magnitude. And we will later on see that why this is so important by toiling very hard at the end we will come up with an expression when we solve the equations we will come up with an equation delta by l equal to some constant c into Reynolds number to the power – half. And therefore disregarding that effect of the constant c the dependence on the Reynolds number will be still the same what we get from such a simple analysis and that is what is one of the important powers of this order of magnitude analysis. So the important conclusion that we get from this order of magnitude analysis is very interesting. What is that? If delta is much much less than l that assumption is consistent with Reynolds number is large because if Reynolds number is large then only you have delta by l much much less than 1. So these are perfectly equivalent that answer now we have given. So that means what are the assumptions under which the boundary layer theory is valid are either delta by l much much less than 1 or equivalently Reynolds number is large. But we will always say that Reynolds number large is a more fundamental way of looking into it because an analyst who does not know the boundary layer will or who does not want to go into the details of the boundary layer thickness will always be interested about the system level parameters. And Reynolds number is a system level parameter if you know the length of the plate if you know u infinity you if you know what is the fluid and its properties you can estimate what is the Reynolds number and based on the Reynolds number you can come up to the conclusion that whether this theory that we have developed is valid or not. So you do not have to really deal with delta by l because you know implicitly that if Reynolds number is large delta by l has to be small. So these are equivalent considerations. The other important consideration which does not come for flow over a flat plate but may come for flow over a curved surface which we will see subsequently is that is this boundary layer going to grow monotonically. The question is that this type of growth of the boundary layer monotonically will not always be the situation it will depend on the pressure gradient in the flow. For flow over a flat plate dpdx is 0 so it will just grow like this monotonically. On the other hand if you are considering a curved boundary there are 2 possible cases where you are having dpdx greater than 0 and dpdx less than 0. We have discussed earlier that dpdx greater than 0 is sort of considered as an adverse pressure gradient in the direction of the flow. It tries to decelerate the flow and it might so happen that the deceleration effect is so strong that the flow actually might take place in a reverse direction close to the wall because wall has an effect of slowing down. And on the top of that as if like there is a dead person and the pressure gradient is shooting a gun on the dead person. So wall is like slowing it down very severely and on the top of that there is an adverse pressure gradient. So the poor fluid element which is very close to the wall cannot sustain all these resistances and may have a reverse directional motion and then this monotonic growth of the boundary layer is disturbed. That situation we call as boundary layer separation. We will look into that in a more physical way subsequently but important thing that we will understand is that if such a boundary layer separation occurs then the boundary layer theory is not valid. So what are the important assumptions for validity of the boundary layer theory? One is like the Reynolds number is large which is of course equivalent to delta by l small and number 2 there is no boundary layer separation. So with this understanding what we will do is we will just go one step forward and see that whether we may calculate the wall shear stress from this description. So let us see what is the wall shear stress. Wall shear stress is what is a very important thing because it is one of the important engineering requirements from the analysis. So what is the wall shear stress? The wall shear stress is mu into the del u del y at the wall y equal to 0. So what is the scale of the wall shear stress? What is the order magnitude of this? Mu into u infinity by delta right. So important non-dimensionalization of wall shear stress is what? We have seen it earlier in an example when we are dealing with Navier-Stokes equation that is the friction coefficient Cf tau wall by half rho u infinity square. So order of Cf is tau wall by rho u infinity square, forget about the half because of the order. So that means this is mu u infinity by delta by rho u infinity square. So Cf the order is mu by rho, mu by rho is nu the kinematic viscosity. See the kinematic viscosity is what is the governing the picture that you can see nu by u infinity delta. Now you know how delta varies with x. So delta square by some local x will be what? Will be nu by u infinity from this one just replace l with x local x. So we are now trying to find out that at some local x that is now our length l. So some x somewhere what is the corresponding wall shear stress. So delta square by x is this one. So we can write replace delta with nu by u infinity into square root of nu x by u infinity. So Cf scales with what? 1 by square root of u infinity x by nu. So this is the length. So Cf scales with Reynolds number to the power –1.5. This is also a very important thing and we will see later on that if you exactly evaluate the expression it will be Cf equal to some constant into Reynolds number to the power –1.5. So dependence of the all the quantities important quantities on Reynolds number may be obtained in this way. So let us stop for this lecture and we will continue in the next lecture. Thank you.


Great, thank you. So let me start off with the definition of predicate encryption. So predicate encryption, in the predicate encryption, the game takes place between four different players. You have Alice, Authority, a cloud server, and Bob. In a first phase, the Authority issues a master public key and gives it to the user Alice. It can also issue different secret keys with respect to different predicates and given to different users. For example, Authority can issue a secret key for a predicate P and give it to the user Bob. Given the master public key, Alice can encrypt an arbitrary message M together with a secret attribute vector X and upload it to the cloud server. Any user can come in and download the ciphertext from the cloud server. The basic correctness properties of predicate encryption say that the user holding a secret key with respect to a predicate P and a ciphertext learns the message M if the predicate evaluated on the attribute vector is equals to one and learns nothing otherwise. The security informally says the following, that an adversary can only learn the message or the secret attribute vector if it's, so no adversary can learn the message or the attribute vector given many secret keys for different predicates P1 up all the way up to P sub N. And the subject to the constraint is that for all of the secret keys that the adversary holds, none of them should be able to decrypt the message individually. That is for all I, PI of X is equals to zero. Okay, let me give you an example of a predicate encryption. So suppose Alice wants to protect her contact information and upload it to a magic.com website. At the same time, she wants to include her criteria as for example, her height, the major that she's tied in as the secret attribute vector. Any user can come into this cloud server and download the ciphertext, but the users are also restricted with a special secret keys with their preferences policies. And using the predicate encryption, you can ensure that the user Bob only learns Alice's contact information if and only if she matches his preference. So to give you a distinction between the predicate encryption and an attribute-based encryption, the key feature is that the attribute vector X is hidden. So what do we know about predicate encryptions? Well, there are a variety of constructions for predicate encryptions for different functions. For example, some of you know about anonymous identity-based encryption. It's a special case of predicate encryption. There are also predicate encryption schemas for conjunctions, range queries, or inner products. And they all can be realized from a variety of assumptions such as bilinear or learning with errors. But unfortunately, all of those constructions fall in the complexity class of NC1. That is the secret keys for the predicates that you can issue are restricted to be from NC1 complexity class. Now, there is a different line of work that realizes a more powerful primitive such as functional encryption for arbitrary circuits. But unfortunately, all of those constructions rely on multilinear maps. And as we know, as of today, there are no good candidates of multilinear maps that we can use. So the standard problem that we're facing is whether or not we can construct the predicate encryption for arbitrary circuits from standard assumptions. And this is what we achieve in this work. So we'll construct the predicate encryption for arbitrary circuits, secure assuming sub-exponential hardness of learning with errors. The schema satisfies the following parameters. So first, it works with the a prior bounded depth. So the depth for the circuits for which you want to issue different secret keys has to be bounded before the setup algorithm. And the second feature is actually a nice feature. And it says that the secret key with respect to a predicate P is of size proportional to the size of the description of the predicate P plus some polynomial overhead in the security parameter and the depth of the circuit. So clearly you have to specify in the secret key the description of the predicate P and that's the best you can hope for. And there is some additive factor of polynomial in the security parameter and the depth of the circuit that you have to pay for. Okay, so now that I told you what predicate encryption is and told you our result, let me tell you how we're going to construct it. First, I'm going to define a new primitive which we call a partially hiding predicate encryption. And then I'm gonna show you a transformation from a partially hiding predicate encryption to predicate encryption.  So let's begin. What is a partially hiding predicate encryption? Well, the interface is very similar to the standard predicate encryption except that the ciphertext now takes two attribute vectors, X and a Y. Now the X is the public attribute vector. Think of an attribute-based encryption type of attribute vector and Y is the secret attribute. That is the attribute as in a predicate encryption. But the correctness is that the user holding a secret key with respect to a predicate P and the ciphertext learns the message M if the inner product of the predicate P applied on the public part X together with Y is equals to one and nothing otherwise. In other words, you can apply the predicate P only on the public part of the attribute and you can only take the result and put an inner product with that with respect to the secret Y. So now we know that what the partially hiding predicate encryption is. It seems like a pretty weak primitive. You can only take an arbitrary predicate and apply it on the public part. Let me tell you how to use it together with a fully homomorphic encryption to construct predicate encryption. The starting point for our construction is going to be a transformation by Goldwasser at all. We're gonna use some of the techniques relying on that paper. So let's recap what it is that we want. We want to encrypt the private attribute X together with the message M. What we have is a partially hiding predicate encryption and the first thing that we're going to do is we're going to take the attribute X and encrypt it with a fully homomorphic encryption. So we're gonna generate a fully homomorphic secret key encrypt X. The result, the result in Cypher to X prime, we're going to treat as a public component for the attribute of partially hiding predicate encryption and the secret key from the fully homomorphic encryption is gonna be treated as a secret component of the attribute of the partially hiding predicate encryption. We're gonna take the Cypher text and the secret key and the message, run it through the encryption algorithm to obtain a Cypher text. Now how we're going to generate a secret key for predicate P? Well, what we're going to do is we're going to generate a secret key for related predicate. We're going to compute a predicate P prime, which is a fully homomorphic evaluation circuit with respect to the predicate P and we're going to feed it through the partially hiding predicate encryption key generation algorithm to obtain a secret key for the predicate P prime. Now put together, let's see what we get when we try to apply the decryption algorithm. By the correctness properties of the partially hiding predicate encryption, running the decryption on the Cypher text and the secret key, the user learns the message M if the inner product of the predicate P prime evaluated on C prime together in a product with a fully homomorphic secret key is equals to one. So what do we get? Well, let's expand it a little bit. C prime was a fully homomorphic Cypher text. Now when we take P prime evaluated on C prime, what we get is a fully homomorphic Cypher text, which is an encryption of the result of the computation P of X. Now, how does this help us? Well, what we know is that there exists a fully homomorphic encryption scheme from learning with errors assumption satisfying the following semantics. So first the Cypher text lives in ZQN, also the fully homomorphic secret key lives in ZQN. Moreover, the decryption algorithm is nothing more than an inner product of a Cypher text and a secret key followed by a rounding operation. Now, as we can see in the partially hiding predicate encryption, I almost get the fully homomorphic decryption algorithm. So I'm almost able to check the result of the computation whether or not P of X is equals to one or not, except that I have to take care for this rounding operation. For this talk, I'm not gonna tell you how to take care of this rounding operation, but it turns out to be very easy by issuing multiple secret keys for very related predicates and you can handle this rounding operation. And a result, you do get what you want. You get the message M. If the inner product of P of X is equals to one. So now we have a transformation from partially hiding predicate encryption to predicate encryption. Let me tell you a little bit how to construct it. And the starting tool for this construction is going to be a hemo-homomorphic trapdoor function introduced by Bonnet et al. in the context of attribute-based encryption. So in that setting, they didn't care about how to hide the attribute whatsoever, but we're gonna use this primitive in a not-blackbox way to achieve what we want. So let me tell you about this primitive a little bit. So it's a family of trapdoor functions satisfying the following semantics. You have an additive and multiplicative homomorphism with respect to the keys. In other words, if you take a function and evaluate with respect to a public key pk sub one on some secret S and add it to another function, evaluate with respect to a different public key pk two on the same secret S, then what you get is a function evaluation with respect to the addition of public keys evaluated on the same secret S. And a similar homomorphism holds with respect to the multiplication. So we know that we can construct this key homomorphic trapdoor functions assuming hardness of learning with error's problem. Now, before I tell you how to use this primitive a little bit, let me modify it slightly to help me. So let me say that it's a family of trapdoor functions parameterized by the public key that consists of two components, a public key followed by some scalar x. And when I refer to the homomorphism, I refer to the component-wise homomorphism. That is when you take a function evaluated with respect to the public key and some scalar x one evaluated on a secret S, then you can add it to another function evaluated with respect to the different public key pk sub two on a different scalar x two. And what you get is a component-wise addition of the public keys and the scalars. And the similar operations hold with respect to the multiplication. So you can multiply it to different functions, evaluate it with respect to different public keys and different scalars, and you have component-wise multiplication. Okay, so let's put it together. Let's suppose we fix a master public key as a collection of regular public keys pk one up to pk sub n. And we'll fix an attribute vector x, which consists of many scalars x one up to x sub n. We can take an arbitrary predicate p and evaluate it over the collection of these functions applied with respect to different public keys individually. So we take the predicate and applied of the collection of this function with respect to the attribute vector x. And a result through the homomorphic properties of this function, what we get as a result is a function evaluation, which is evaluated with respect to the homomorphically derived public key. So it's p evaluated with respect to mpk followed by p evaluated on the attribute vector x evaluated on s. Moreover, we can invert the result of the computation and get s if we're given the corresponding trapdoor function. So if we're given f evaluated with respect to the homomorphically derived public key followed by the result of the computation p of x. Okay, so as I told you, this primitive was used to construct an attribute-based encryption which doesn't require any hiding whatsoever on the attribute vector, but let me tell you a little bit more on how it works. So let's expand and see how these homomorphic operations are performed. Suppose I have a very simple predicate that consists of a multiplication and addition. And suppose I have just three bits at the input followed by attribute vector x, which is consist of x1, x2 and x3. I can take the function evaluations corresponding to the input bits, x1, x2 and x3. I can now multiply them homomorphically to get f evaluated with respect to the multiplication of public keys followed by the multiplication of scalars evaluated on s. And I can do the same thing for the addition. And the key observation that we make is the following. First, you can perform the multiplication knowing only x1 or x2. So in other words, you only have to know one of the two inputs corresponding to the input wires of the multiplication gate that you're evaluating. You don't have to know the other one. Moreover, you can perform the addition without knowing any of the inputs at the wires corresponding to that gate. So you can take the function and add them with respect to different public keys and different index bits. And you don't have to know what these index bits are to perform the homomorphic addition. And as a result, you don't have to give all of these vector scalars as in your ciphertext. So you don't have to give x1, x2 and x3. You can only give x1, then you can perform the homomorphic multiplication and then you can perform the homomorphic addition. So what can we do with this? Well, it turns out that if you have this kind of a property, then you can construct a partially-hidden predicate encryption simply for inner products as a step one. So let's suppose we fix the public key to consist of two different public keys. One, they're gonna correspond to the public components of the ciphertext that we're gonna give in clear and the other one is gonna correspond to the secret components. Now in the predicate that I care about is simply an inner product of x and y. And my goal is to make sure I can perform this homomorphic computations without giving you y. So only giving you x and clear. So as I told you, you can actually do this. So for every multiplication, when you want to multiply functions evaluated with respect to x and y, you only have to know one of the inputs. And this is the input that I gave you in clear. This is the x. So you can multiply those functions together and more you can add all of the functions at the output gate without knowing any of its inputs. So you don't have to know what x1 times one y is in clear. And the result what you get is a function evaluated with respect to the homomorphically derived public key, which is an inner product of the public, a master public key corresponding to the public component and the master public key prime corresponding to the secret component followed by the result of the computation that is the inner product of x and y. Okay, now we're not happy with just inner product and we said that we wanna evaluate an arbitrary function p with respect to the public component. The question is, can we do this? Well, yes we can because as I told you, if we were given the entire input in clear, then you can apply arbitrary computations over it, arbitrary additions and multiplications and you can feed the result as the result of the computation into the multiplication gate. So what do we get? Well, we get a inner product partially hide and predicted encryption on steroids. So suppose we fix the master public key to be the same, consists of two components, one corresponds to the public part and the other one corresponds to the secret part and the predicates that we can now evaluate consist are more expressive. So it's a predicate p prime evaluated on x which is a collection of predicates p1, p2 and p3 applied separately on the public component x. You can apply arbitrary predicates because you're gonna know the input x in clear and then you can take the result and taken in a product of that with a secret y. So we do get exactly what we wanted, we can evaluate arbitrary predicates on the public component and taken in a product with a secret component and I don't have to give you the secret component in clear. Okay, so now I gave you this overview. Let me just summarize how the construction is going to look like in terms of components. So your master public is divided in two parts, the public, the master public key corresponding to the public part and the secret part for this key homomorphic trapdoor function. Your ciphertext also consists of two components. One set of components is the key homomorphic trapdoor function evaluated with respect to the public parts of the ciphertext and the public parts of the master public key followed by the secret part. So f with respect to the master public key prime evaluated on the secret component y applied on s. You also have to give the public component x and clear and finally you take a one time pad of the secret s with a message m. Now what is the secret key? Well the secret key is going to be a trapdoor function for the function defined with respect to the following public key. It's a homomorphically derived public key with respect to the p prime evaluated with respect to the public part mpk in a product with a secret part mpk prime followed by one. And one here really indicates the intention of the predicate encryption that is allow the user to decrypt when the result of the computation is equals to one. So when the user performs the decryption, the user simply applies the predicate p prime to the collection of homomorphically given components corresponding to the public part and then takes them in a product of that with a secret part and put together if the result of the computation is one then given the correct trapdoor function the user can invert and recover s. Once the user recover s he can recover the message m and achieve their desired result. Okay, so in this talk I only gave you a high level overview. We're not using this key homomorphic trapdoor function in a black box way. We actually have to do a non-black box construction. There are many setups that didn't go into the security prove the correctness and so on and so forth. So I'm gonna refer you to the paper for this. So finally let me mention some open problems. So open problem number one is the construction of function encryption for circuits from standard assumptions. So such as learning with errors. And let me know that the predicate encryption is only one step away from this. In particular we're able to compute an arbitrary predicate p on the secret component of the attribute vector and the catch is actually in the security definition. In the function encryption you are required to guarantee security even if when you allow to decrypt completely and learn the message. Whereas in our case you will only guarantee security with respect to the users that are not allowed to decrypt the message m. There are some additional interesting open problems such as removing the depth dependency in all of these constructions. We have a bootstrapping theorem for fully homomorphic encryption and we don't know any similar bootstrapping theorems with respect to attribute based on predicate encryption. And finally I wanna mention the construction of predicate encryption for different unbounded models of computation from standard assumptions such as Turing machines, RAM programs, and so on and so forth. Thank you.


Hello guys, my name is Hector Cantele, I'm the founder of Magis, which is a personal training academy for soccer players located in Memphis, Tennessee here in the United States. Before I met Ben and before I joined, he mastered my program. My business was doing alright. I was doing okay at the time. It was just me. But definitely I didn't have the tools to make my business more safe or more secure, if that makes sense. I felt like I was doing things out of the box, out of the blue. I was not following a system. I didn't know how to share the message to the parents that, hey, you know, this is something serious. It's different than any other personal trainer out there. But overall, you know, to be honest, I felt like I was doing okay. I was doing good because of my relationships that I've built here in Memphis. I'm a former University of Memphis soccer player, so that definitely helped me get clients and knowing people and all that. I didn't have any issues. The biggest issue for me was not having a system that would work for the academy long term. Then fast forward to May of 2019 or May or June around that time. That's when I decided to join the mastermind program for Ben. Actually, he gave me, I think it was like a one or two week trial of the program for free. So I hoped he needs a few calls for free, kind of like learn of what he was doing, what he was talking about. So as soon as I log into the trial, after the first day, I noticed, okay, shoot, I need to jump into this and get involved because he actually knows what he's talking about. And I did. And after I joined the program, shoot, I mean, he had all the systems in place, you know, payments, you know, agreements with the players, with the parents, I mean, everything, how to set up a camp, how to run clinics, follow-ups, how to talk to parents. He even had the script of a conversation that you should have with the parent over the phone before a player commits to it. So that was exactly what I was looking for because I needed something that would give more prestige to the academy. And the system allowed me to do that. The payment set up the way he set up the payments, same thing. You know, the script to talk to parents, same thing. I mean, since 2019, right now we're in 2021, I still follow his script. And that's what that's what has allowed me to close so many clients. And now fast forward to now 2021, you know, when I started with him, it was just me. Now I have three other coaches that work in the academy. You know, back in the day, I used to coach, you know, I used to have 10 to 12 sessions a week. So 10 to 12 players per week. Overall, now because we've grown so much, we coached 60, 65 kids on a weekly basis. We just finished the summer. We had three summer camps where we had in one camp, we had 16 players in the other camp. We had 19 players and in this last camp, we had 17 players. We do at least four, six clinics per year. I mean, the growth has has been exponential in all aspects. And that's been thanks to his mentorship, thanks to the system that has he has in place. To me, that's the most important thing when it comes to any type of business, but especially this one, you need to have a system in place that works for you because that's what that's what is going to allow you grow and, you know, help more kids and, you know, grow your business even more. So definitely this Ben's program, I would recommend it to anybody else, you know, anybody out there that wants to any personal training, soccer, football, golf, tennis, any sport, whatever it is, because what he has in place, it's unbelievable.  And his system works. I'm the living proof that it worked again. I was lucky enough to have a good connection. So my me starting out my business was not that hard. But what allowed me to have an exponential growth was his system. And just to give you an idea, my first year in business, I started half towards May, June of 2018. I did like $18,000 that year. Then the following year, 2019, I joined Ben's program. We made $60,000 in revenue. Then last year, we made $120,000 in revenue. We had a hundred percent increase in revenue. And this year, so far, right now, we're at $120,000 for the year so far. And we're expecting to reach $200,000, $215,000 for the year. So almost, not almost, but not fully 100% in revenue, but still exponential growth. So definitely if you're thinking about it, either should you join or not, something about it is worth it. Every penny is worth it. I'm at the point that I know the things or most of the things that he has in the program. And I still am part of the mastermind program because we have a group chat in there like where we can ask any type of questions. And every once in a while, I have questions and his answers are what helps me move forward with my business. So definitely recommend it, you know, Ben, super thankful for your help in every aspect. And yeah, see you guys soon. Bye bye.


I'm used to not giving any fucks, but in that environment I have to work up to being no fucks with an audience. Most energetic person on the planet.   What's good? Welcome. Good morning. Happy Friday. Friday feels. Feeling good Friday.  I slept good. What's up? Twice this week I was in bed before 9. Yesterday I was just exhausted, could barely fall asleep, but I slept well once I fell asleep, bro. Yesterday was one of those days, came home, just said, you know what, I'm just gonna eat and go to sleep. I went to the gym, ate, slept. Super early. You gotta do that sometimes. And that's what I did. What's good? Yo, yesterday I got the funniest comments of all time on my, I'm gonna read them at the end. Some comments on my bombing and my open mic. It was so funny. I appreciate all the love and support, and that's what I wanted to talk about today. First though, first though, if you hit the notification bell today. Oh, man. What they told me, the people from high above, they said today, if you hit the notification bell, the next board game you play, you win instantly. So if you're about to see some family this weekend, I think it's a perfect timing. If you're gonna play some Monopoly tonight, if you want to crush your grandmother, hit that notification bell. She doesn't have a chance, and that's it. So thanks that, and the Naughty Gang, bro, if you're part of the Naughty Gang, that's like people that show up to a show, a comedy show. You're like, man, if I, if nobody showed up, then I'd be standing up here. I couldn't even practice. Like, not only is it that you can't, you don't have a show and you have no one to talk to, but you can't even practice.   You can practice by yourself as much as you want, but it's nothing like the game. And that's a perfect segue for what today's about. You gotta be about it. Stop talking about it. Be about it. What's really cool about this channel and what I've learned about myself is that I love being that positive energy, being that inspiration and motivation. And it's so fun when I'm dunking and I'm like, for example, rehab, stick into my rehab plan, stick into it and really focused on it and not letting myself jump ahead and staying patient on my goals. I see other people able to stick to their goals and shout out to Chase Kilganon too. He's been in rehab for a long time and he just came back and he's killing it already. Like he looks so good. He's got high expectations, high goals for himself. But while he was doing his rehab, we talked a lot because he was like doing tiny little details and I was in the same boat, like working on really little things. And he has such high expectations. I know he wants to push himself to the limit and he was just doing really well and in another setback. So watching him go through his rehab and really sticking to the plan helped me. So I know that it's like an energy force that we're all a part of. And now the coolest thing about the bombing and posting it and talking about it and you guys besides you guys being like, yo guys, the funniest comments. I couldn't stop laughing all day. As you saw, I'm already laughing about it. I mean, I learned so much. It's like, and it's so easy to get over it nowadays because of all the support I get and how much I can learn from it. It's just easy. It's easy to get over it quick.  It's funny to me because it went worse. Now it's just even fun. If it went well, it wouldn't be as funny. Anyway, people were saying, yo, so proud of you. That takes balls, but I couldn't watch it. Or they're like, yo, great shit, but I couldn't even make it through half of it. I was just dying because I was like everybody. I was like, yo, good try. Like great effort. Anybody who asked me like if they can watch it, they're like, yo, did you post it? I'm like, yeah, good luck. Like brace yourself. So really funny stuff. But here's the real, the coolest part. I keep saying it. Here's the coolest part about it. Talk about the motivation, talk about the patience, all the motivation quotes and something.   I posted about it on my Instagram. Definitely check out that whole post I wrote about it. But my point is so many people mentioned that they're going to go try something now or they did go try something. So I'm not going to mention it because I don't know if when he's going to post it, but Barth tried something Connor Barth jumps by Chi. My boy Chi, you guys know Chi. He said he's about to go do something. I forget. I think dunks or something where he wants to try something. But the fact that I even just went on stage and took that action helped others take that action. So it's that next step. So two things about it. The one thing is that just like the motivation where it's like I'm staying disciplined. I'm doing this. My discipline feeds into your energy of discipline. You should go that guy's doing it, but not only are we just like talking about them beings staying focused, but now I took that action and stepped I didn't just say step out of your comfort zone.  So that's really cool because that helps me do it even more. But also it's just amazing to see that it has that much of an impact that makes it so easy. If I know that it's going to help other people do the same thing and then I'm like, oh shoot, that person's doing it too. Now it makes me want to do it more. It's just that energy going back and forth back and forth to give and take, give and take.  So that's what makes it so easy. It makes it so easy to go out of your comfort zone because we have social media these days. I have the best support group in the whole world and it has a lot more meaning than just feeling like shit.  That would have been one thing. I still would have done it because I know I loved it. But then like on top of it, I get to like I get other people to improve their lives. It's too easy. I can't not do it now. Now it's like the force is like, how could I not do it again? Anyway, because I ate so much shit and I grew so much from it and I learned the thing is the one of the moment I think I said one and two and I don't even know if I said the second thing. That's how it goes. That's how it goes. But what I learned from being up there is that there's some things I'm comfortable talking about and there's some things I'm not comfortable, but I know they're funny, but I know I've said them like with my friends and things like that, but I'm not comfortable not giving a full zero fucks on stage yet, which is not like me. I'm used to not giving any fucks, but in that environment I have to work up to being no fucks with an audience, you know, especially people I don't know. I'm like, oh, that's what happened in my head. I'm like, oh, they don't know me. They're not going to get this. It's going to come off weird. So I let it's just like it's the same thing you talk about. I talk about all the time is the only fear I had was the fear of the other people. I wasn't afraid if I thought about myself and that's what I was visualizing the whole time is like myself and I knew that like I would make them laugh, but I couldn't even be comfortable being myself and then I was awkward and then it was it just was the wrong mentality. Anyway, I already wrote more jokes that I'm I already think about saying comfortable and I wouldn't feel because there's a part of me that was like, I don't know how they're going to get this, but I don't care and I don't care, but I do when I'm on stage and I do feel like the fear of like, are they going to get this and the judgment? You know, but I shouldn't care. So I'm going to get to that point, but I didn't know that until I'm on stage that I'm not at that point yet. You think you're at that point, but until you take that action, take that action and you do so much and now since that I felt like I jumped in a fire and I like like got cooked. I want to do it more.    It makes you want to go do it. It's it's hard to explain, but it was great and I really appreciate everybody. Is there anything else that I wanted to mention? But yeah, take the action. I haven't been that mortified in a long time and it was crazy how good I felt after it and next day and like all the things I learned about myself and like my level of awareness went way higher and that feeling of failing or that feeling of being in front is now a known so I can I can use it just like when you're like if you see like a scary door, it's scary to go behind the door, but as soon as you open the door and you think and you know what's behind it, it's not scary anymore. But that's like that subconscious. I always talk about, but I'm telling you, you got to take action. So if you took action because of my standard, I would love to know because that thing is that's freaking fueling and that's what's cool is that that's why I get the weird vibe with some people that are just purely motivational. Like that that's the difference is not saying on the best.  You can't just talk about stepping outside your comfort zone and friggin working on yourself and go into the fear because behind the fears is you got to do it. You really got to do it because there's nothing like being in it. All right. I've said it a thousand times. Yeah. So that's that. And that's, um, let's do some on this day in history. Oh, by the way, new dunk life shirt, new logo, baby, new dunk life mug. Look at his baby. Hold on. Let me see it. Hmm. Coffee takes extra good. So if you want this, it's on the website. They only have the white. Oh, Jesus. Almost hit my computer. I just spilled it.  I got the black one because I'm the CEO. Uh, see if you want the white mug. Go ahead. My speed phase is on sale. Pitching September 13th. Oh my God, it's Friday the 13th. Oh, I'm wearing all black. Okay. 509 BC, the temple of Jupiter and Rome's capital, capital line hill is dedicated on the eyes of September. What does that mean? 335 Church of Holy s why we go back in time? Wait, no, that's 509 BC. Oh, wow. 335 Church of Holy seph sephal trade consecrated in Jerusalem. I don't know words. Okay. Speaking of words, I started the new book Carl Jung undiscovered self. I read, I want Carl Jung. I don't even know how to describe him.  He's really deep and a lot of people who I listen to like Jordan Peterson and different things talk about Carl Jung as this really high level intellectual and I want to dive into it. I've tried his books in the past and they're so difficult. So I looked up how to find, uh, how to start with Carl Jung and I found this great. Um, I don't know if it's that great, but I found this review that said if you want to know about Carl Jung himself read this book and it listed a book and then said, if you want to know about his theories, read this book and I said, if you want to know about yourself, read this one called the undiscovered self. So I picked up the undiscovered self. It was like a dollar 80 on Amazon because I had like five dollars off for some reason. So it was like literally a dollar. I got it yesterday. I started it. It's so difficult. I read the first three pages and it talks about plebiscites and how that and I don't even know. That's just what I remember. And so I think I'm going to go really slow through it and just look up each word. I don't know because there's a like in a sentence. There'll be like three words. I don't know like the criteria of the plebiscites makes it for a ubiquitous and I made that last one up. But the first two were real words in one sentence and I'm like my brain froze just like I'm on stage.  It just froze and it was tough. It was like my SpongeBob. Did I say this already? I love that. I love that visual though. SpongeBob scrambling for like things in my brain. So yeah, really hard. So I think if I like study the book and read those hard words and try to understand what he's saying, it'll take me forever. So I don't know if it's that like great for like bed time reading, but I do want to try and see how it goes. But I really am trying to start diving into Carl Young. So this is my dissent and I've heard it's scary because you learned it by yourself. And I know that that fear is everything. So yeah, and the other thing I was going to say is that I think I'm going to try to learn from other people that have read him that maybe can articulate what he's talking about more. So I kind of have like a warm up because it's hard.   So then when you learn about his definition of that, then you read his books. You know what he means. It's tough, but I'm trying. I'm going to try to dive into it. So stay tuned for that book review. Should I do a top five book review? Let me know. Should I do a new series of dunk something like a dunk phase? I did dunk phases. What was the other one? I was a dunk longevity or certain plant styles or how to find your style. Those are ideas I have. You got to stay tuned. You got to hit that naughty bell. So we have a show together. Dude, I can't wait to post a good set of the comedy thing. I don't think I'm going to record, but I don't think I'm going to post until I have a good full set because I don't want to keep posting things because you got to work out jokes. You know, you can't just I'm a comedian now. I can say it. I did it. It's so fun. So you got to work things out. I'm not a comedian yet. I guess I don't know when are you comedian when you when you headline once? I don't know. You got to work things that you got to practice. Um, so I'm not going to just like post the thing. If it goes well, maybe I'll do it once, but like I'm going to record a lot, but I'm not going to post much, but I will post when I do a headline. And when I come to a city near you, just come and watch today in film and TV 1969. Scooby-Doo. Where are you? They views on CBS in the US. Oh, Scooby-Doo today today. Music Beatles release yesterday. That's a funny sentence 1965. The Beatles release yesterday. Thanks for yesterday because without yesterday, we couldn't figure out what happened today in sport. Just in 15. Evian, champion, Wormel's golf. Lillian, New Zealand wins. Okay. Lexi Thompson, youngest major. Okay. Cool. Okay. So now let's go to my bombing video and read some comments. Okay. This is great. Sorry about that ad. This is this was my favorite because it sounded so innocent. Veggie shark. Honestly, appreciate you appreciate you putting this out there and hope you do well, but this is so cringe. I couldn't watch. I know dude. Imagine me in the moment doing that knowing it. I'm doing that.  But I mean, I'm okay. Look, I'm okay.  Probably is it is it dripping? Yeah. Average bro. It's better than you think. It's not amazing, but not horrible. It's your first time. So people will never get the courage to do it once hats off.  I got to work on it, but I'm excited to. So I thank you. Um, uh, I stopped a little over halfway. Couldn't watch it anymore. She says I didn't know if it was if I was going to make it through the whole thing at first, but decent. So funny. Dude, I'm sorry. You guys are the best for trying. It's so funny. I should have made it like bomb challenge. Also, please Google YouTube this. Maybe I'll attach it at the end. I think it was Nicole TV were relatable Queen part two. It has like 2 million views. It's this girl doing a little thing with weird edits and it cracked me up. I couldn't stop watching yesterday. It's like tick tock on steroid sort of cause it's long. She's really funny. Um, but yeah, what was it about this? That's it. Have a great day. Get your mug. If you want it out. That hurt my ears and new new shirts. I'm going to be dunking soon. I dunk. I feel really good. Lastly, hamstring feels really good. I woke up today less pain and I strength trained it yet yesterday. So it's going well. I took it really easy yesterday. Lightwork light rehab. I was tired. So I just kind of like got my body moving and lifted some weights but kept it light and my lower back and hamstrings are getting stronger and it's slow and I'm not on the strength phase, but I'm just doing a lot of reps to get blood flow and strengthen it slowly but surely and jumping with my focus and I can't wait to dunk on my whole life today. To tell you. So now I'm looking in them ever like that mother.


My name is Havid Janiyon Masha. I am an associate professor of unmanned robotic systems at the research group of mechatronics here at Saxion University of Applied Sciences. I'm originally from Ethiopia. I came here to do my master's at the University of Twente and I actually did my PhD also at the University of Twente in the field of robotics and mechatronics. My specialization is on aerial robots which people actually commonly know them as drones. In general, as an international student basically at the very beginning there would be a cultural difference when you come to here at the Netherlands. The culture may be a bit different from the culture that you are actually used to when you are back home but the people here are really open. They are very friendly and even to top it off that most of the people actually speak English. On one hand it is a bit petty in a sense that it may not necessarily encourage you to learn Dutch but on the other hand you feel at home because it's really easy to communicate with everybody, pretty much everybody, whether you go to the supermarket, whether you go to the university departments and stuff like that. And the most important thing here actually when you come to the Netherlands and especially if you are as an international student doing your study at the university or University of Applied Sciences is that there is no hierarchy. So there is no really a hierarchical difference between the professors and also the students. So the students are really welcomed by all the professors and they are open. So anybody is actually encouraged to, everybody is actually encouraged to ask questions. It makes it really easy and working environment even the teaching learning environment more cohesive and very encouraging for international students. One of the things that actually distinguishes from scientific universities are the research and role to the educations that we give here are more applied oriented. So it's not really fundamental research or fundamental theoretical education more on the applied. So basically our role here at the University of Applied Sciences is actually to bridge the gap between the scientific universities and the industries. So basically we are the technology innovators, technology developers. So you get a lot of possibilities here at Sakshi University of Applied Sciences in order to not only learn the abstract part but also to actually develop new things, even have a good feeling about the existing systems using all the components that are actually available. It's not only theoretical but it's more a combination of a theory with a practice. For example the things that you actually see here, one of the research lines that we have at the research group of mechatronics is the unmanned robotic systems which I am the associate professor for and it covers the unmanned robotic system, covers the unmanned aerial robots which are commonly known as drones, ground robots and even including underwater robots. In most of the projects that we do together with companies, together with other international partners be it in European projects or national projects or regional projects that we actually collaborate with industries, regional industries or national or maybe even European wide industries, we involve students and students are actually involved in every of the activities, the applied research activities that we actually carry out and students, this is an innovative recharging station for aerial robots. It is really new and innovative which actually enables to use aerial robots 24-7 by being able to recharge it and this is what has been also developed not only by researchers that we have but researchers and lecturers together with students and the students get the possibility to actually work on real innovative activities. It's really important to emphasize here that the students actually get also the opportunity to work closely with companies so that at the early stage they get the experience, the working ethics and also the professionalism that they actually are supposed to have when they join industries. That's actually the unique thing that we provide here at Section, the unique opportunity compared to other institutions.


In a previous episode we talked about how jQuery augmented the API of DOM elements to make the developers life easier when it came to the management of CSS classes. Another popular API that jQuery offered was Closest, which returns the closest ancestor of the current element which matches the selector. For example, this is very useful in having a reference to a certain button or input field and wanting to get a hold of the enclosing form element. It is so useful in a fact that the web platform now offers the exact same function directly on DOM elements. All modern browsers offer this API and it can be polyfilled with a couple of lines of code if necessary. The fact that so many APIs of jQuery influenced the web platform itself shows that the jQuery authors were real web developers that solved real problems. What would you like to know more about? Let us know by leaving a comment. I hope you have enjoyed this trip down this jQuery shaped memory lane. If you like supercharged and the short micro tip videos, you should subscribe because there's gonna be more in the future. In the meantime, though, you can just catch up on all things supercharged right here.