Hello, Amal Jyoti, college Kerala, you have a question please go ahead. So my question is that sir in the integral analysis 20, so it was given that the total energy balance for most of the cases for mass conservation and momentum conservation, we have written it as d by dt of a control volume or the property in a control volume is equal to the whatever property in minus property out plus energy source. So in that in this special case why did you consider the heat transfer and work transfer separately or can we do it in such a way that the heat transfer and work transfer can be included in the energy in and energy out so that it the equation becomes much similar to the energy balance equation. Why did we actually is there any anything common to the heat and work transfer so that we are not including it in energy in and energy out. Yes, so the question is related to the energy balance statement which is projected right now on the board. The question is if or why I should say the heat transfer and work transfer terms have been separately written out and whether they can be included in the E dot in that is energy in and energy out terms on the right hand side. Actually your question is really relevant and it is very correct and good question. The answer is that you are right there is no need to separately write out the Q dot and W dot term rather we can include them in the energy in and energy out. It is just that what I wanted to do was the energy in and energy out terms are specifically related to the fluid flow that is going in and out of the control volume. So as the fluid gets in and out of the control volume by virtue of its motion it carries with it certain energy in and out of the control volume whereas the energy so the heat transfer term is slightly different in the sense that it is considered to be a conduction type heat transfer which is happening because there exist temperature gradients within the flow. So that is the reason I wanted to separate it out from the energy terms that are associated with the fluid in and fluid out. Similarly, when it comes to the work term the work term is actually in an integral situation will usually include a shaft work in addition to the surface forces work as well. So in case of an integral analysis these are some special situations of shaft work and that is the reason I wanted to write them separately but in general you are absolutely right there is no reason to write this separately. However, when you calculate then E dot in and E dot out you have to explicitly include these terms as well. Okay sir thank you in and out. Shivaji University, Kolhapur you have a question please go ahead. Sir my question is is there any possibility of change in density with respect to time in incompressible flow? So the question is if there is a chance of change in density with respect to time in an incompressible flow? The answer is incompressible flow you can treat it as an unsteady flow as well. So in that case it is possible. However, if the incompressible flow is a constant density flow like what would happen in case of a liquid flow in that case there will be essentially no change of density with respect to time. Nirmai University you have a question please go ahead. Hello I have question in this slide kinematics 25. Here we have derived the relation between angular velocity and vorticity. So what is the significance of vorticity or how it differs from this angular velocity? So the question is on slide number 25 of kinematics where we discussed angular velocity and vorticity. Specifically what is the physical significance of vorticity and how it is related to angular velocity? Actually the relation is right there on the slide that the vorticity is 2 times the angular velocity. If you go back a couple of slides and if you follow the derivation of the way we have calculated the angular velocity you will realize that eventually it turns out to be one half of the curl of the velocity field which is what has been mentioned in slide number 25. And it is simply a nomenclature that the curl of the velocity field is called the vorticity. In principle on a physical basis both the vorticity and the angular velocity imply the same thing that is if either the vorticity or the angular velocity are non-zero we are dealing with what is called as a rotational flow which means that the fluid particle is spinning about its own axis as it moves. So physically speaking both vorticity and angular velocity imply spinning of a particle about its own axis as it is moving. Relations wise the relation is right on slide number 25 it simply so happens that vorticity is equal to 2 times the angular velocity. I have one more question in slide differential analysis 26 in this first equation is true for incompressible flow and second one we are deriving from first and in second equation it is said that density may not remain constant but in first equation density is constant so that is not clear. Yes so the question is on slide number 26 differential analysis what is getting pointed out and in some sense correctly is that the top equation is written for a constant density or incompressible flow whereas the bottom equation is written for inviscid flow. However it is been written that the density may not be constant so how is it possible is there some sort of an inconsistency. Actually I am sorry this is my fault I should have explained that the second equation which is at the bottom of the slide is not really coming from the top equation. In fact if you go back one slide which is slide number 25 the general equation that you see here which is for a viscous compressible flow from here which is the most general equation so far if you remove the viscous terms for the inviscid flow you will realize that you do get the equation on page number 26 the second one where the viscous terms are now missing. So in that sense the equation for the viscous inviscid flow which is now getting highlighted is actually following from the most general equation which is on 25 by removing the terms related to viscosity. So sorry about that confusion but I hope this answers your question. Yes sir okay thank you. Jaipur engineering college go ahead if you have a question. Sir I want to ask that how you have applied Leibniz rule and divergence theorem in slide number 5. Yes so the question is related to slide number 5 on differential analysis specifically how the Leibniz rule is getting employed for the volume integral and how the divergence theorem is getting employed for the area integral. So if you see the Leibniz rule which was pointed out so the Leibniz rule is presently projected on the screen at the bottom of the screen. Obviously the way it was written here on the slide that you see right now is written for a one dimensional situation whereas what we have in that differential analysis slide is a three dimensional control volume but in principle what we are saying is that when we want to take this differentiation with respect to x inside the integral sign it will result in a partial differential sign which is one thing and then additionally two terms will show up if the limits on the integration are functions of x this is in one dimension. Now going back to the before we in fact go back this partial differentiation will be present when we go back to the differential analysis. However if you notice what we assumed in that differential analysis slide is that the control volume is fixed and because the control volume is fixed its limits which you can consider as the surrounding surface with respect to time which is what the differentiating variable is are actually 0 because the control volume chosen is a fixed control volume. So let me go there and then therefore point out that the only term that remains in that situation is the partial differentiation term. You can see here that the time derivative has been brought into the integral sign but then because the control volume is a stationary control volume its limits that is the integral limits are fixed with respect to time therefore only the partial differentiation remains. So that is the way the Leibniz rule has been employed for the first term. The divergence theorem simply says that the area integral gets converted into the volume integral with the function the vector function which is dotted with the unit outward normal and then integrated over the area surface area that vector function is converted into a divergence form. So the divergence of that vector function then is integrated over the control volume after converting the area integral into a volume integral. So that is precisely what has been done if you see the vector function here it is rho multiplied by vector v. Rho is just a scalar so it goes along with v and then the divergence of rho times v is then integrated over the entire control volume that is how the divergence theorem is applied for second term. In fact if you see that how these Leibniz rule and divergence theorem have been employed for later situations also when we went to momentum equations or momentum balances they are exactly the same. As long as we are dealing with a stationary control volume the limit on the integration as far as the volume integral is concerned is independent of time and therefore only that partial differentiation inside the integral remains there is no additional term as is earlier shown by the Leibniz rule. Similar thing for divergence theorem the vector function which is getting dotted with the unit outward normal is simply converted into its divergence and then that divergence of that vector function is then integrated over the entire control volume after employing the divergence theorem. So that is how the Leibniz rule and divergence theorem work. Thank you sir. Sir, I also want to ask in slide number 10 how have you written the equation number 1 x momentum equation in 2D. Yeah, so the question is on slide number 10 on differential analysis specifically how this top equation has been written. So if you go back to our balance statement so now we are talking about the balance for the linear momentum. So what we have is on the left hand side the rate of accumulation of linear momentum in the control volume on the right hand side we have rate of inflow minus rate of outflow and the rate of increase of linear momentum due to a source which we said the last term that is here the source term is due to the net force acting on the control volume or the material inside the control volume. So now with this if we go to slide number 10 it is simply each of those terms representing what I just described. On the left hand side what we have is the rate of accumulation term which is getting written down here. Right most is the net force term which is the source term in the x direction and the two middle terms are basically the terms that describe let me go back the rate of inflow minus the rate of outflow of the linear momentum in the x direction from the control volume. All you need to realize though is that as I was trying to point out in the lecture look at this particular slide right now here slide number 8. You will realize that there are two linear momentum coming in in the x direction one from the west phase and one from the south phase similarly there are two that are going out one from the east phase and one from the north phase and that is why because there are four linear momentum terms you end up generating two of these terms which represent the rate of inflow minus rate of outflow. If there were only two momentum terms you would generate only one of them that is all that is. So what I suggest is really that all the components required to perform the algebra are written down on slide number 8, 9 and all you need to do is the material from slide number 8 and 9 needs to be substituted in the balance equation and if you simplify the balance equation you will obtain this first equation on slide number 10. Santai Perrier-Vellore you have a question please go ahead. Sir good afternoon sir. I have a question in kinematics question 15 here the velocity getting changed to from V2 V plus dou V by dou x into del x here actually what is the physical significance of partial derivative difference? Yes so the question is related to slide number 16 on kinematics where what we have shown is how the velocity is varying over this fluid particle over a distance delta x and delta y. So the specific question is as we go from point A to point B the velocity in y direction which is V is shown to change from V to V plus partial derivative with respect to x of V times delta x. So what is the physical significance is the question. See the only thing that you need to note here is that we are dealing with a two dimensional field. So if you are dealing with a two dimensional field let me go to the wide board for a for a second. For a two dimensional flow field both u and V velocities will be functions of x and y ok. In general both u and v will be functions of both x and y coordinates. So now going back to our slide what we are trying to show here is as we move from point A to point B we are simply moving in the x direction. So we are trying to find out how the variation of velocity V is along the x direction. Since velocity V is both function of x and y and since we are only moving in the x direction we are required to take the partial derivative of the velocity V with respect to x and then multiply by the delta x term so that the first order Taylor expansion is achieved. So that is the reason we employ partial derivative here. In fact if you note here the other component of the velocity which is u velocity or the x velocity that is also changing from A to B and in a similar manner it has been computed. The reason being that u is also a function of x and y in general in a two dimensional field. Since we are moving only in the x direction we will necessarily need to use partial differentiation with respect to x because essentially that y coordinate is considered to be constant as far as this x movement is concerned. Thank you sir. I have one more question differential analysis 18 when we are taking the momentum above O the shear stress tau xy along in y axis and the x axis corresponding momentum we are considering in differential analysis 19 tau xy minus tau yx it is not clear. So the question is on differential analysis slides number 18 and 19 and specifically the information provided on slide number 19 what are we really talking about here that is the question. So if you go back to slide number 18 which is what is projected right now what you will see is that I have shown all surface forces in terms of the stresses and the body forces that are acting in the x and y direction in our two dimensional fluid particle. So each phase if you see there will be a normal stress and then there will be a shear stress. Now what I have shown in the slide are the expressions for some of these stresses. So there are total of eight surface stresses out of those eight I have on purpose shown only four expressions the remaining four I was hoping that the participants will be able to work out. Addition to that what I have shown here is the body force which is passing through the center of mass. Now the slide number 19 simply says that if we want to take moments of all these forces about the point O what will happen. So since the normal stresses the four normal stresses and the body force are all passing through the point O there will be no moment arm or the moment arm for these forces will be 0 and that is why the moment due to the normal stresses and the body forces about the point O will be 0. The only moment which is non-zero is going to be because of the shear stresses. Now if you see the shear stresses they occur in pairs in the sense that you have a shear stress on the east side then you have a shear stress on the west side that forms an anti-clockwise couple there is a shear stress on the north face and a shear stress on the south face which form a clockwise couple. So if you want to evaluate a net counter clockwise or anti-clockwise couple you need to appropriately calculate the forces and multiply those by the moment arms for these shear stresses and that is what has been done here. So what you see on the top line in slide number 19 is the net counter clockwise moment that will be existing on our fluid particle because of the shear stresses and the net moment will come only because of the shear stresses other stresses will not contribute to the net moment that is what is getting pointed out at the top line. Thank you sir. Periyar Manny Ammai please go ahead if you have a question. Hello sir in the slide differential analysis 19 you have said earlier days that only in nanoscale dimensions we cannot apply the CFD approach but here in the middle of the slide in differential analysis 19 you have said if size is sinking to 0 means how this kind of products would happen since the CFD is not for nano or below 0 dimensions how this can happen sir since this kind of products may not happen. Okay so this is an interesting question the question is on slide number 19 which was being discussed a few minutes earlier and the specifically the question is that in this particular case we evaluate this net counter clockwise moment on the fluid particle and then evaluate or equate that to the moment of inertia multiplied by the angular acceleration and after some manipulations we bring the angular acceleration in the form as is shown right now here on the slide. So now the argument was that since delta x and delta y tend to 0 we say that this angular acceleration seems to tend to infinity. So the question that is getting asked is why are we talking about the particle shrinking to 0 whether we are then violating the continuum situation and the answer to that is that we are not really necessarily going down to the molecular level as what we would perhaps require if you are dealing with a nanoscale situation when we talk about delta x and delta y tending to 0 in this continuum situation it is in the calculus sense in the sense that we are not necessarily shrinking them to a point as the geometrical point that we talk about. We simply say that we are taking these arbitrarily to 0 arbitrarily small however keeping in mind that still we are not breaching the continuum limit so the idea is simply to put up an argument that if you start reducing the denominator to a smaller and smaller value it seems to say that the angular acceleration will keep on increasing and the argument was that this is physically not possible because there is no special case that we are putting up here it is just a general discussion and to simply come up with the conclusion then that the numerator also must be essentially then very very close to 0 in order for the angular acceleration to be finite we use the argument of delta x and delta y tending to 0. You do not please consider that we are breaching the continuum limit we are still within the continuum limit by no means when we say that we take the delta x and delta y to 0 we are shrinking it to a real geometrical point that is never the case in the case of fluid mechanics. It is just that we are taking it arbitrarily small however we are still within the continuum limit the idea was that just to point out if you take it arbitrarily small alpha will be unnecessarily large which does not make any sense on a physical basis that is all. Okay thank you. PVG Pune. Pune Mijerthi Guru please go ahead if you have a question. My question is regarding the slide number 2 on differential analysis that is the balance statement for an elemental control volume the last term on the right hand side rate of increase of mass is there any illustration where the rate of increase of mass will take place that is question number 1. Question number 2 can you please explain the concept of conservative and non-conservative equations for differential analysis. Thank you. Yeah so there are two questions one is related to the general balance statement that has been mentioned on couple of times actually in integral analysis also and here as well. So the question is specifically with respect to the source term and if there is any rate of increase of mass because of a source in a practical situation. So in fact this is the general statement that has been written and I have mentioned a few times that the rate of increase of mass due to a source is essentially not possible unless you have a nuclear type of reaction happening in your flow field where you are actually either destroying or creating mass through the Einstein's relation of E equal to mc square such situations are not really to be encountered in general fluid mechanics. So for mass balance this term will be automatically 0 in general. The second question was related to conservative form and non-conservative form. So let me go ahead one more slide and point out slide number 3 here. The conservative form for example right now we are talking about conservation of mass here what you see right now on your screen at the bottom is the conservative form of the mass balance equation or the continuity equation. I have mentioned a few times that the conservative form you obtain when you employ a balance equation such as what we have been employing on a control volume basically you obtain a relation which says that there is a rate of accumulation and that rate of accumulation is because of the quantity that comes in is different from the quantity that goes out etc. So when you employ such balance equations on control volumes you will generate a conservative form of the equation. On the other hand if you decide to employ a Lagrangian form of approach wherein you will be then following a given fluid particle then inherently you will generate a non-conservative form of governing equation. So let me take you to slide number 6 and right at the bottom of slide number 6 we have obtained the continuity equation using this so called Lagrangian form by following a given fluid particle which always has a fixed amount of mass and then essentially saying that the substantial derivative of the mass content for the given fluid particle is going to be identically 0 and in doing so we inherently generate the non-conservative form of the governing equation. So this is the general guideline if you employ a balance statement you will generate the conservative forms if you employ a Lagrangian approach you will generate a non-conservative form. R K college Rajkot please go ahead if you have a question. Yes sir good afternoon sir in slide number 7 differential analysis I have two questions first is for stream function u is considered positive and v is considered negative what happens if the direction is vice versa if we take v as positive and u as negative what is the reason and my second question is city potential function since the stream function is defined only for two dimension and phi is defined for three dimension. So there are two questions first question is on the stream function as we have defined in slide number 7 on the differential analysis and the question is we define the x velocity as plus y derivative of psi whereas v as minus x derivative of psi so why is that this has been done to be honest with you I think it can be done either way so it does not matter whether you define u as minus d psi d y and then v as plus d psi d x typically the convention that people in fluid mechanics have followed is that u has been defined as d psi d y and v has been defined as minus d psi d x the negative sign whether it is on v or whether it is on u is immaterial the reason is because you take this couple of expressions and plug them back into the continuity equation you will satisfy the continuity equation unconditionally whether you define u as plus d psi d y or minus d psi d y and then vice versa. So in that sense there is no hard and fast rule that this has to be plus and this has to be minus is just that I think some accepted conventions are being followed here that most of the fluid mechanics literature seems to define the u and v velocities in this fashion. The second question was related to something called velocity potential and we have not talked about velocity potential because we have not really dealt with irrotational flows in any detail. So velocity potential is a concept that is normally utilized when we deal with irrotational flows and those are also called as potential flows as some of you will obviously know. So in case of velocity potential you will recall that the velocity vector is defined as gradient of the velocity potential and there also there is a debate whether it should be given a negative sign or not. So let me just switch to wide board for a second. So velocity potential which is phi is a scalar function again and the velocity vector is defined as the gradient of this scalar function and as you correctly pointed out that the stream function is valid only for a two dimensional flow whereas the velocity potential can be defined for a three dimensional flow as well. The reason that we do not really have a stream function for three dimensional flow is because for a three dimensional flow then we cannot really identically satisfy that continuity equation with any kind of definition that the stream function may have. So only in two dimensions that continuity equation which is divergence of velocity equal to 0 can be identically satisfied by the stream function definition. So stream function is indeed restricted to 2D velocity potential can be used for 3D as well. G. H. Risoni College Nagpur please go ahead if you have a question. In differential analysis slide number 28 you are given del u by del x equal to del y by del x of u square by 2 so please explain and in differential analysis 14 sign connection. So there are two questions one is on differential analysis 28 where we were deriving that Bernoulli equation. How this partial derivative with respect to x of u squared over 2 times the x has been obtained. In fact I actually worked out the entire derivation for this so let me try to point out what we were doing in there. So this term here which now you see on your screen let me write it again on the white board is u multiplied by partial derivative of u with respect to x. This is simply written as d dx of u squared over 2 that is all that is. So if I want to bring this u inside the derivative I really need to make this u squared over 2 because let us see what happens now. u squared over 2 if you want to differentiate with respect to x this 2 is a common 1 half is a common fact or constant which will bring out d dx of u squared which is 1 half using the chain rule 2 sorry 2 u du dx and then that 2 and 2 will cancel and you will get your u multiplied by du dx which was what you started with. So this is the only manipulation that has been done that I want to bring this u here inside the derivative sign and when I want to bring that it will have to multiply this u and also divide by 2. So it has to be a d dx of u squared over 2. So you employ the chain rule when you are differentiating d dx of u squared over 2 and then you will get back your original form. So that is what has been done nothing more and your second question was related to 14 slide number 14 sign convention I thought you said sign convention. So let me go to the next slide. So what we said is that we will consider a stress component positive if both the surface normal for the surface over which the stress component is acting and the stress component itself both these guys point together either in the minus coordinate direction or plus coordinate direction and that is how we decide whether a stress component is positive or not. So for example let me draw something on the white board. So let me have a rectangular control volume or a fluid element if you want. Let me look at the left vertical face and the right vertical face. So if I show a stress component pointing in negative x direction on the left face and if I show a stress component obviously these are normal stress components pointing in the positive x direction for the right face. Now if I want to show the surface normal for the right face it is the unit surface normal that is also pointing in the positive x direction and so is the stress component. So therefore since both the stress component and the unit normal are pointing together in the positive x direction we call this a positive stress component. Similarly on the left face now the stress component is pointing in the negative x direction and the unit outward normal will also point out in the negative x direction. So again since both n hat which is a unit normal and sigma are together pointing in the negative x direction we call this a positive stress. See the reason why this convention is getting followed is if you are familiar with strength of materials or solid mechanics if you see these two stresses right now the way they are shown they act as tensile stresses on the on the material. So a tensile stress will be considered as positive in general from the solid mechanics or applied mechanics background and in order to preserve that convention we are also doing the same thing here. So I hope that answers your question. Okay thank you sir. Shastra University, Tanjavur please go ahead if you have a question. For calculating volumetric strain rate in the numerator new volume has taken as for two dimensional flow then new length into new rate. But it is newly formed that system is not actually it is not rectangular shape. So how can we take a new length into new rate that is first question and second question I have some about this pressure while taking pressure for static fluid and for dynamic fluid I have some doubts regarding for taking in momentum equation for dynamic fluid actually pressure should be will be in all direction but it will be cancelled automatically why should we take in component wise that is one more doubt these two are my doubts. The first question is related to calculation of the volumetric strain rate where the volumetric strain rate has been calculated as the new volume minus the old volume divided by the old volume and then divide that by the time interval over which this is happening. What is getting pointed out is that the new volume is not going to be rectangular so why are we treating this as a rectangular volume when we calculate it as a new volume. The point is correct actually what you are pointing out is that everything is happening simultaneously as we have repeatedly said. So all these four effects the translation rotation deformation and the two types of deformation that is they are all happening simultaneously as the fluid particle moves from one location to the other. However for the purpose of analysis what we assume is that since that delta t is very very small these are typically very small in terms of the magnitudes and that is why to a first order accurate expression we calculate those volumetric strain rates by simply assuming that the stretching is such that the new volume is also approximately rectangular otherwise you are absolutely right that it is not really remaining a complete rectangle as time progresses. But these are all results which are accurate to the first order accuracy as we say. However the first order accuracy in case of continuum fluid mechanics has been found to be sufficiently good and that is the reason we accept these results and move on with those. So however technically speaking you are I would say right. Sir, second question is regarding pressure while taking momentum in momentum equation for calculating momentum in for one dimensional flow we have taken v1 v1 plus m1 v1 like that we have taken for momentum calculation momentum in. So in that we have taken momentum due to pressure also p1 v1. So while taking that I have p1 a1 sorry p1 a1 my doubt is pressure acts in all direction if pressure is at position and it will be same in all direction it does not have components in all components if you take any component it will be same. But while calculating why we are taking for x direction and y direction and z direction specifically that is my doubts just I want more clarifications on pressure itself. See I do not know if you are referring to any specific slide from any specific topic the question is related to the pressure and it is variation. So let me try to just answer it in little general terms the pressure is equal is what is getting pointed however it is equal only at a given point. So the Pascal's law as you are quoting here will say that the pressure is equal in all directions but it is only at a given point. So as long as you are dealing with a fluid domain in general the pressure will vary from point to point as you go from one location to the next and that is the reason we have this gradient of pressure as the net force acting due to the pressure differences from point to point when it comes to a moving fluid that is all really to it at a given point I understand that it is equal in all directions. However in general from point to point in a continuum fluid domain the pressure will vary and because of which there is a gradient of pressure which will act as an accelerating force for a fluid particle. So therefore we end up taking gradient of pressure as the net force due to pressure in the momentum equations. Ok sir still I have some clarification because if you take pressure at one point it will be in all directions it will be cancelled as its own but while calculating for momentum we are taking in one component x component and if you are calculating for x momentum we are taking in one direction and similarly in y direction and similarly in z direction but it will be cancelled automatically. So what is the requirement of taking it for as a component wise that is my doubt. Can you please point out any specific slide with which I can have this discussion? I have general doubts sir. Because see the if you are referring to let us say an integral analysis where we are dealing with a with an enclosed completely closed control volume and if it is given as part of the problem statement that you can assume that the pressure is completely uniform in the entire domain then what is going to happen is the pressure contribution from all sides on the control volume will simply cancel each other out and there will be no net effect of the pressure force. So that is that is fine. However in general the pressure is not really constant and it does change from point to point and that is the reason we normally want to include that in our in our analysis especially when it comes to differential analysis we want to include that in terms of the gradient of the pressure which will be acting as an accelerating force for a fluid particle. But in if you are referring to any specific integral control volume type analysis we have simply assumed in some cases that the pressure is uniform for a completely enclosed control volume and in which case it will cancel cancel its contribution. As far as momentum flow etcetera are concerned pressure has nothing to do with momentum flow is always calculated as the mass flow rate multiplied by the velocity of the flow. So there the pressure does not come into picture at all the pressure will come in only in the calculation of the net forces that are acting on a fluid particle or equivalently on a control volume. So there is no connection between momentum flow in and out of a control volume and and the pressure. Yeah Nitte Meenakshi please go ahead if you have a question. The relation established between stress component and strain rate in the slide 21. So by assuming isotropic fluids so will it be the same relations will it be applicable to unsteady flow field that is first question and the second question how easily we can define the stream function without any formulae and please can you elaborate more on isotropic fluids thank you with the example. So the question is at least one of the questions is on these Stokes relations or the constitutive relations and the question was about isotropic nature of the fluids what is meant by that. So isotropic nature of the fluid simply means that the fluid will exhibit a stress and strain rate relationship identically in all possible directions. So technically you can say that this is an assumption it is a simplifying assumption that has been used in the Stokes analysis and even though we have not really discussed the Stokes analysis completely in detail I wanted to simply mention it. It so happens that most fluids of practical importance seem to follow this isotropic behavior quite reasonably and therefore for most fluids these Stokes relations are found to be quite applicable. So most of the practical fluids that you can think of including all gases and more or less all types of fluids do follow this quite well. So that is all about the Stokes relations and isotropic nature of the fluid. Can you please repeat your question on stream function. The relations which we established will it be applicable to unsteady flow feed. So that is the first question which I asked and the other one is how we can easily define the stream function without any formula. Yeah so the question is about stream function and the first part of the question is whether it is applicable for unsteady flows. It is actually it is in fact unsteady it is applicable for unsteady flows as well. Let me go quickly to the whiteboard. So in general really I should have probably mentioned this in my slides. In general the stream function is defined for a two-dimensional unsteady flow as well. So it is just that we did not choose to talk about unsteady flows but in general stream function is indeed defined for an unsteady flow. Even though it is defined for unsteady flow the relations that we have for the velocity components namely u is equal to u is d psi dy and v is equal to minus d psi dx will still remain good whether the flow is unsteady or not. So the answer is yes it is applicable for unsteady flows and the relations still remain exactly the same. You also asked whether we would know stream functions without any specific formula and actually to be honest with you the concept of stream function is fairly restrictive. We do not really utilize the concept of stream function nowadays. It is an old concept and for some specially derived flow type situations you can actually have a close form relation for a stream function. So the idea is that in general the stream function is a useful concept when it comes to plotting the CFD results in terms of stream line plots and so on. So that is all the utilities in my opinion in today's world otherwise it is a really old concept and somewhat restrictive as well. R C Patel, Shirpur if you have a question please go ahead. What is the mean for order density function for 0 for irrotational flow? Then why? It is only used for application for irrotational flow only. It is applicable for other flow also. See the so the question is about the concept of vorticity and if vorticity is 0 we call this as irrotational flow. So what we have done here is we have formulated the expression for the rotational motion of a fluid particle. So what I mean by rotational motion is that as the particle is moving it is spinning about its own axis. So if it is happening in a fluid then we say that the angular velocity is non-zero and by the virtue of the relations that we derived which are projected right now on slide number 25 here we simply establish the relation between this so called vorticity which is defined as the curl of the velocity and the angular velocity. So in that sense both vorticity and angular velocity imply the same thing namely physically spinning of a fluid particle and if the fluid particle for whatever reason let us say is not spinning about its own axis as it is moving in a flow we simply say that it is experiencing 0 angular velocity or 0 vorticity and corresponding to that we say that the flow is irrotational. So there are some classes of flows where the action of viscosity is very very low because of which the velocity gradients are very low and if the velocity gradients are essentially non-existent from the expressions that have been listed at the top you may be able to figure out that the angular velocity will be exactly 0. So in such situations which are typically applicable in inviscid flows you will see that the flow can be treated as irrotational which simply means that a particle that is moving in an irrotational flow does not spin about its own axis and that is what has been shown in the sketch at the right bottom. Can I give an angle please go ahead if you have a question. So I have a doubt in that slide number 22 in a differential analysis here normal stresses sigma which we are going to be providing that minus p plus sigma xs viscous can you please elaborate it. Yes so the question is on slide number 22 on differential analysis specifically on the normal stresses that we talked about and I have mentioned that the normal stress in general will be composed of a pressure contribution and a viscous contribution as well. See in general this is true in a fluid. So normally what we are used to assume is that the normal stress in a fluid or on a fluid particle if you want to look at it that way is simply equal to pressure and pressure being compressive we are going to employ that minus sign for it. However in general what is found is that there is a non-zero contribution from the action of viscosity as well on the normal stresses or in normal stresses. So there is a component that gets added because of viscosity in normal stresses. It may not be that obvious to follow that the viscosity is actually contributing to a normal stress but it does and although to be very honest with you we have not gone into serious details of the derivation of the stokes relations. I have simply listed here what are those and that is what I would like to really limit my answer to that at this point you just note that there is a contribution from the viscous effects as well to the normal stresses and I have simply listed the expressions that exist for the normal viscous stress by coating an analogy with the normal stress that occurs in an elastic solid which follows the Hooke's law. Vijay T. I. Mumbai if you have a question please go ahead. Sir how do you physically interpret stream function? So the question is on how to physically interpret a stream function. There are two ways you can interpret the stream function. One of which was explained in one of the slides. Let me try to pull up that slide. So one of the ways you can interpret this stream function is that when we have a stream line the value of the stream function turns out to be a constant for a given stream line. So psi equal to constant will provide you a stream line field. So if there are various stream lines in a stream line field each stream line will have one value of the stream function associated with it that is one way of looking at stream function. The other way is that if you let me draw a sketch on the whiteboard. So let us say that we are looking at a pair of stream lines for which the value of stream function is psi 1 and psi 2 and let us assume that psi 2 is greater than psi 1. So it can be shown that psi 2 minus psi 1 is actually equal to the volume flow rate between these two stream lines on a per unit depth basis. So this is a very standard derivation which you can find in one of the text books that I have mentioned. But since you wanted to know the physical interpretation of stream function this is the most relevant interpretation from a physical basis that the difference between the values of stream function for two different stream lines will give you the volume flow rate that is flowing between these two stream lines on essentially a per unit depth basis because by definition we are always talking about two dimensions when we talk about stream function. Sir in differential analysis slide number 23 what is the significance of lambda? Yeah so the question is on slide number 23 differential analysis what is the significance of lambda? The second coefficient of viscosity. So let me answer this in a slightly hazy manner in the sense that we have not really talked about the complete detail of the Stokes's development in which the relations between stresses and strain rates are established. However, if you actually follow the complete treatment in one of these advanced fluid mechanics books you will see that the normal viscous stress that a Newtonian fluid will experience will actually contain two constants. So one of the constants is usually determined as the dynamic viscosity. So that is what is getting highlighted here as mu the other constant is usually called as the second coefficient of viscosity and that is what is given the symbol lambda. So unfortunately since we have not really completely talked about how these Stokes's relations are derived I am not in a position to talk about these in a serious detail. However, at this point what you can note is that the normal viscous stress in a fluid when it is expressed in terms of the strain rates will contain two constants in the expression. One of them which is called lambda here is what is called as the second coefficient of viscosity that is all I will right now say and what I will do is at the end of the course I will put up a list of some of these advanced textbooks in which you can actually try to follow the entire Stokes's development and there you will be able to see where this set of couple of constants will appear. I want to know what is the difference between the thermodynamic pressure and mechanical pressure and how we equate equal what is the difference between thermodynamic pressure and mechanical pressure. Yeah, so the question is about what is meant by the thermodynamic pressure and what is meant by mechanical pressure. So thermodynamic pressure is what we call a static pressure in fluid mechanics terminology and static pressure or thermodynamic pressure is something that you measure if you move with the flow so that the sensor with which you are measuring the pressure does not have any relative velocity with the flow. If this is what is happening then we call it as a static pressure. So in case of thermodynamics you talk about usually closed systems let us say a gas inside a container that is stationary and if you include a sensor in such a system there will be an interaction of large number of gas molecules with the sensor and they will transfer their momentum on to the sensor which will be eventually interpreted by the sensor in terms of a pressure. So this is what we call a static pressure and since we are dealing with a static fluid in the case of the closed system as I just talked about in fluid mechanics. In order to replicate this static nature what we do in case of a moving fluid is that we have to move along with the fluid at the local speed of the fluid so that there is no relative velocity between the sensor and the fluid so that the sensor essentially thinks that it is measuring a static pressure. So that is what we call as either a static pressure or thermodynamic pressure. Mechanical pressure as I explained earlier in the lecture is an artificially defined term which is simply as the slide projected right now shows is negative of the average of the 3 normal stresses and in general it is different by a factor that is shown on the third equation here. However, for the case of incompressible flow we have divergence of velocity equal to 0 or the 3 linear strain rates added together will identically be equal to 0. So in the case of incompressible flow there is no difference between that so called thermodynamic pressure and mechanical pressure. If you are dealing with strongly compressible flows and those also involving shock waves and such then you actually have to perhaps deal with these two separately even then typically using this Tuxes postulate which essentially assume that the thermodynamic pressure is equal to mechanical pressure you do not see any serious errors in the analysis. So for all practical purposes the pressure that we know from thermodynamics is what you can treat as the mechanical pressure only in case of strongly compressible flow you may have to deal with something different but usually these situations are very very scarce. So normally you do not have to really bother about it I just wanted to point out if you want to read a slightly advanced textbook you will normally come across such terminologies and at least I thought I should point out to you what is meant by thermodynamic pressure and what is meant by mechanical pressure. I will stop this question and answer session right now.


Thank you for joining us today. The Patient Safety Movement Foundation posted the Imbolic Events Actionable Patient Safety Solution in September. Brandon and I, along with Michael Becker, are co-chairs of that Actionable Patient Safety Solution. And today, I get to interview Brandon Lao about some of the details that you'll find in that apps. I come to this with a personal reason. In 2012, my husband, Yogi Raj Charles Bates, died from a hospital-associated venous thromboembolism. So although you might just see me as a regular interviewer in many cases, today this interview is very personal for me. Brandon, thank you for joining me. Brandon is the faculty in the Armstrong Institute for Patient Safety and Quality. And he is the assistant professor of radiology and health sciences informatics with Johns Hopkins School of Medicine. And Brandon and I are going to address a few questions today that we think might be helpful for both clinicians as well as patients and family members. Brandon, welcome, and thank you for joining us today. Thanks for the invitation. Glad to be here. So let's just start by describing what a venous thromboembolism is. Give us a clinical definition of what we're talking about today. A venous thromboembolism, or VTE, is a blood clot that occurs in the leg as a deep vein thrombosis, or DVT, or in the lung, pulmonary embolism, or PE. Oftentimes, clots start in the legs as a DVT and can break off and travel to the lungs, where it can cut off your ability to breathe effectively. Yeah. And as I know, all too well and very personally, it can be fatal. And even if it's not fatal, it can create lifelong complications. So it's a very, very important topic. And let's dive a little bit more into a few more questions. So let's just start by looking at a typical organizational protocol. What is typically required in the inpatient setting with formal risk assessments? Is there a standard protocol? There isn't really a standard protocol that has been universally adopted by all hospitals, which I think is very unfortunate, because we do have pretty clear guidance of what it takes to effectively prevent blood clots in the hospital setting. And it requires a series of actions to be taken. First, patients need to be risk assessed to determine what their risk level is for developing VTE. Based on that risk assessment, the appropriate prophylaxis, which could be medication or sequential compression devices, boots that squeeze the leg, based on the risk assessment, appropriate prophylaxis should be prescribed for the patients. Nurses should be communicating the harms of VTE and the importance of prophylaxis to prevent blood clots. And then patients would then make an informed decision to accept and have the prophylaxis applied or administered to them. And then we need to be mindful of our surveillance practices to look at when events occur and really use that entire loop to look for opportunities to improve practice within hospitals. That goes into what I was going to ask as the next question, Brandon. And that is, what can change in a patient setting that would require a reassessment after an initial formal assessment has been given? That's a really important question. And it's one that is also frequently overlooked. Patients' clinical conditions change dramatically over the course of hospitalization. Risk factors can develop changes in whether or not patients are eligible to get medication to prevent blood clots change dramatically. One example of that, when we think about medications to prevent blood clots, we're talking about anticoagulants or blood thinners. And a patient could come into the hospital and have lab values that would make it inappropriate to give anticoagulants to them. Perhaps their INR is too high or their PTT is too long or they're actively bleeding if they're coming in for surgery. And these are things that can resolve or change over the course of hospitalization. So if you have somebody who's at very high risk of clotting and they don't get prescribed medication because they have one of these bleeding risk factors which is entirely appropriate, if that bleeding risk factor changes over the course of hospitalization, maybe we need to rethink about what the prophylaxis or prevention strategies are to make sure the patients are protected. So there should be some level of reassessment periodically through the hospital, particularly for patients who might need stronger preventive measures that they just weren't eligible for on admission. You know, just yesterday I was with a friend who had to stop her eloquence for just a day or two in preparation for procedure and then she would go back on that. So it is that fine balance between clotting and bleeding and it's very important to sort of manage both of those dynamics that our bodies, you know, naturally try to resolve for ourselves but sometimes need a little help in doing so. Let's talk a little bit about something you and I talk about I think frequently is this notion of with regard to blood clotting and DVT development, there's quite a lot of beliefs that aren't necessarily evidence-based. Sort of the way that we think about it is sort of like attitude and evidence. And because clinicians are very evidence-based, it often surprises me that a lot, many clinicians I've spoken with operate a little bit more on the attitude side when it comes to DVT. What are some of those common areas where attitude might be a little bit more heavily weighted than evidence with regard to DVT and VTE? Yeah, it's really interesting and I think VTE prevention is definitely one of those areas where there are a lot of misconceptions. Misconceptions that we prescribe prophylaxis for every patient, that we prescribe medication for every patient who's coming into the hospital. And I think that that underscores the importance of a formal risk assessment. I've never advocated that every patient who comes into the hospital should be prescribed in anti-coagulant because you're potentially exposing patients to harm, risks of bleeding. And the goal is really to prescribe appropriate prophylaxis to patients. Pretty much every professional organization in healthcare has standardized evidence-based guidelines for what their patient population should be prescribed for different levels of risk. And I think that it's important that as we consider what prophylaxis should be prescribed, we give greater weight to the evidence rather than the intuition or experience that people may think that they possess when it comes to VTE prevention. So there is some disconnect, at least on the prescribing side, but there is very strong rigorous evidence to support appropriate decision-making. The other end, we've actually seen that a lot of doses of prescribed medication aren't being administered to patients. We've actually published at our own institution that between 10 and 15% of prescribed doses of anti-coagulants to prevent blood clots aren't being administered to patients. And we've done some deeper digging into the reasons for this non-administration. And we've seen that many nurses have a belief that prophylaxis is overprescribed for patients. And we've also seen a belief that if patients are ambulatory up walking around that pharmacologic prophylaxis becomes optional, that ambulation is in itself effective VTE prevention. And I mentioned that we've just had a systematic review accepted for publication that we'll be sure to share with you that shows that there actually has never been a rigorous study to look at the effectiveness of ambulation for blood clot prevention in hospitalized patients. In fact, the idea of ambulation started with a single author editorial in JAMA in 1951 where the author wrote that there may be some benefit to ambulation for blood clot prevention. And now we hear very often in hospital settings that if patients are up walking around, their risk for getting a blood clot is not there. And I can tell you that we have seen patients who are ambulatory who still develop blood clots and the evidence just isn't there for it. It confuses me because when I hear it said I've attended appointments with friends and family members sometimes in hospitals often as well.  But what I can tell you is that my husband, Yogi Raj was one of those cases where he couldn't have walked more. He was up and walking so much that he thought that the leg calf pain that he had was a result of walking so much when in fact that calf pain was actually a DVT developing but it was completely off the radar of his care providers. So I think I can't underscore enough what you're saying there that the science just simply doesn't support that mobility is a replacement for a risk assessment and prophylaxis, whether it's compression devices if it's contraindicated to have a pharmaceutical option but it really, really, it sickens me sometimes to hear that that myth has prevailed. And it's just amazing to me how something somebody says so many years ago could have taken hold to such an extent and it's certainly my hope that we can course correct that belief. Of course we want people up and mobile if it's safe to do so as much as possible so many benefits from that but as a replacement for prophylactic for DVT development it's just simply not there and in evidence at this time. Did I say accurately? You're the doctor. You're the medical professional, not me. I think that that is completely accurate and I do wanna say that I don't think that in saying this, I don't believe that nurses are wanting to cause harm to patients. Oh, goodness me. I think that there is a misconception about the importance of prophylaxis, what the evidence says about proven pharmacologic or mechanical prophylaxis. And I think that in doing this in offering patients an opportunity to walk around while there is no evidence to support that that is effective VT prevention they are trying to reduce discomfort for patients. I mean, getting a shot is not necessarily a pleasant experience. And if I didn't necessarily know that there was no evidence for it if somebody presented to me the option of walking around or getting a shot, I would certainly walk around too. But I think that it definitely underscores the importance of where the myths change people's priorities. And it's important to keep in mind that every year more people die from pulmonary embolism, clot in the lungs than from AIDS, breast cancer and motor vehicle collisions combined. And that's a staggering number. And the idea that I would bet anyone who has experienced breast cancer would say if somebody came and presented me an opportunity to get rid of it with a shot or reduce my risk with a shot I would absolutely take it. And I think that we definitely need to give blood clots the level of priority that they deserve in providing evidence-based preventive care for it. Thank you, that's very well put. And because of our personal experience obviously I'm a strong advocate for that communication. Let's dive into that a little bit further. And I should have started this by just having you describe what that is but I think you just did a good job of, the difference between a DVT, a PE and a VTE or a deep vein thrombosis, pulmonary embolism and venous thromboembolism. Let's talk about what information can be shared between patients and clinicians and also between healthcare settings to improve VTE prevention. What would you recommend in terms of communications that would help us really get to address VTE at the level that it needs to be? Well, I think there are several areas of communication at different levels with different types of clinicians between patients and physicians, patients and nurses. First of all is VTE risk. There are a number of risk factors that will never appear on a scan, will never appear as a lab value. One of those, and one of the biggest predictors of developing blood clots is if you've had a personal history of a blood clot or if a family member has had blood clot. I have personally had multiple members of my family who have had blood clots in the past. And one of the things that I will tell my doctors both in primary care and if I am in the hospital is that I have a family history of blood clots. I know that I'm at elevated risk for that. And it will change decision-making processes around preventive practices. Another thing to know is what the signs and symptoms are. Most patients who are going into the hospital aren't necessarily going in to have a blood clot prevented. They're going in for some other condition, but it is a complication that can develop. And I think that we need to be very proactive in telling patients what the signs and symptoms are to make sure that they're aware of what could be. And I will tell you that a patient knows their body better than anyone else. And if they start to see leg swelling, if they start to feel shortness of breath, they are going to know this before anyone else. And that information will certainly empower them to speak up to their nurses or to their doctors about what it is that they're experiencing. So I think that there's a lot of communication back and forth of sharing what potential risk factors exist. But then also making sure that patients are informed about this condition, which unfortunately doesn't get nearly as much attention as it should, what they should be looking for and when they really need to ring the bell and sound the alarm that something is wrong. I often harken back to our experience in the hospital where my husband went into the hospital for traumatic brain injury and had a craniotomy and that was successfully resolved. If somebody had handed me a pamphlet that just said, these are some common hospital associated conditions that can happen after surgery and hospitalization and VTE would have certainly been on that list because we know it's about the third leading cause of hospital associated deaths in the United States. I would have read that pamphlet and then the symptoms, which were very evident for my husband, I would have been able to point to them and say, hey, do you think it could have been such and such? No, not everybody would be able to do that or even have somebody by their side like my husband did all those 13 days, but that education and that communication process needs to be in place as a standard protocol just to make sure that we are doing the very, very best that we can to educate across the board. Even those PT and OT people who also saw the signs and symptoms were not making the correlation between the evidence symptoms and the development of that blood clot in my husband's leg. Brandon, I wanna just move to our final question here and that's really about going global. Like how do we really help prevent this? And one of the ways that we can help prevent is really, really good reporting. So what are some of the barriers to objective reporting and reporting this all-to-common VTE event post-discharge when somebody actually is in their home setting post-care in Pennsylvania? That's such an important question and the majority of VTE events happen outside of the hospital and we know that VTE is associated not just while you're in the hospital, risk isn't just when you're in the hospital and then when you're discharged, your risk magically disappears. You still have risk associated with your overall health, whatever it is that you were treated, as well as what prevention strategies were taken or not taken while you were in the hospital. So your risk for developing a blood clot still extends after discharge. And one of the challenges that we run into because so many events develop outside of the hospital, we don't necessarily get feedback that a patient has experienced those events. And I think that that's a really important component when you think about continuous quality improvement. We look at within the hospital where patients risk assessed, where they prescribed the right thing, where they administered everything that was prescribed, did they develop an event? And then when a patient is discharged, we don't really know if an event developed thereafter. So we assume at that point that they did not develop an event. They could have gotten suboptimal VTE prevention during hospitalization, but then developed an event after discharge that we wouldn't know about, but we would still consider that prevention practice successful because they didn't happen to develop that event while they were in the hospital. So I think one of the great unknowns that there is fantastic opportunity to improve upon is sharing that information, that post discharge information back with hospitals. And I know that it can be somewhat of a challenge that hospitals don't necessarily want to report adverse events, but I think that it's critical for quality and safety purposes and continuous improvement that we really have that full picture to know what we can do better. And I think that that's the main takeaway that we always have opportunities to improve. There are hospitals that do wonderful jobs at ensuring the prophylaxis is administered, but do they actually ensure that it's the right prophylaxis? Similarly, there are hospitals that might make risk assessment mandatory for all patients, but are they ensuring that all patients are getting every single dose or having their prophylaxis applied as it's written? Are they ensuring the full continuum of VTE prevention to ensure that we're providing defect-free care? And I think that that's really the goal that we should be striving for. Well, let's just give an applause to those administrators who and clinicians who are making sure that those precautions are taken for their inpatient settings as well as their clinical settings. And also just a recommendation to patients and family members to report back and to be in communication to ask questions about the risk for DVT during hospitalization or procedures, clinical and surgical procedures. The communication and the learning from the experiences are what will help us get to the state where this is not the problem that it is today. So, Brandon, thank you again for taking time for this particular recording today. I know it will bring value to a lot of people. And I hope that you have a really, really good rest of the year and thank you very much for your continued research in this important area. Thanks for this invitation. It's always wonderful chatting with you. I hope you stay safe as well. Thank you.


Hello everyone. Welcome to yet another session of our NPTEL on non-linear and adaptive control. I am Srikanth Sukumar from Systems and Control IIT Bombay. So we are well into the seventh week of this NPTEL course and we are already looking into algorithms for designing adaptive controllers for single and double integrators matched and unmatched uncertainty systems. So the sort of algorithms that we are designing are not so far from the kind of algorithm design that will be required to drive autonomous systems such as the satellite in orbit that you see in the background. So what we were looking until last time was the design of the adaptive controller or the unmatched uncertainty case. So this we had essentially looked at how to use back stepping based methods in order to design an adaptive controller for systems that look like this where the uncertainty is now in the state that is different from the state in which the control appears. So this is what is called an unmatched parameter system. So we of course saw the back stepping design and it was a little bit more complicated than what you would expect for the matched case and we also saw that one of the most important or key things to remember is that we had to design two different parameter update laws which is the theta hat and also the mu hat. So we had to design the theta hat and the mu hat that is we had to over parameterize the system.  So in this case although the uncertain parameter was only one scalar parameter we had to design two different estimates for the same parameter so of course then we were able to prove tracking or we did the stabilization problem but as we mentioned the tracking problem also would be a very straightforward extension of this stabilization problem. In fact I would encourage all of you to try the tracking problem. So that's something I would write here. Try the tracking problem on your own. So now we move on to the next set of notes. I mean again please don't mind that it says week eight and so on we have been doing this. So this is merely to sort of organize the material. So we are sort of once we have completed one set of notes for that particular week we have moved on to the next set of notes and we will continue to do this because there is enough material to cover till the end. So we will move on to a slightly different topic or set of topics now. So we want to start off with the unknown control gain problem. So we will look at what is the unknown control gain problem. So this is what we want to do in hopefully the next couple of sessions. We want to look at what is the unknown control gain problem and how do you handle it. And as always we look at the scalar versions of things here and once we have done this unknown control gain problem we will be in a very good position to actually introduce and look at the model reference adaptive control which is probably one of the most popular adaptive control frameworks for linear systems. So that is sort of the outline of what is in this subsequent lectures. So let us start looking at this. So I am going to mark this lecture as lecture 7.5. So this is the fifth lecture of the seventh week. So don't worry that it says week eight in the beginning here and that is mainly for organization sake. All right great. So what is the unknown control gain problem? So until now we have been looking at unknowns that are not in the control term. We have been looking at the unknowns in the drift term. So like in the previous week we had, I mean in the unmatched case also we had the unknown which was connected to the drift term or the state terms. There was never any unknown connected to the control term and that is what changes here. That is what changes here. Now we are again looking at the scalar a single order system just to make our treatment simple. Again we will immediately after look at the model reference adaptive control case which is the entire vector case. So we don't have to worry about missing any details as such or this being not applicable to any real system you are working on. This is in fact with minor little extensions applicable to a lot of vector systems also. All right great. So let us look at this system. So what is this? This is x dot is a fxt plus bu. Now x0 is given to be x0 is the initial condition of course and both the state and the control are in real numbers and the function f is of course taking the state and the non-negative time and giving you a scalar quantity r. And here the important thing to remember is that both a and b are unknown constants. In this case there is an unknown connected to the state which is what we have been seeing until now but there is also an unknown which is connected with the control. So this is new and so then the question is can we solve the same problem? So what is the same problem? The objective is of course that we want the tracking error to go to zero that is we want t goes to infinity e of t goes to zero where e is defined as the tracking error which is the state minus the r. So there may be some non-uniformity in us putting the time arguments with the states and the r and so on and so forth. Please don't worry too much about this notational inconsistency if you may we have just put in the time arguments wherever we think it's not very clear from the context. So when it's clear from the context we don't put any arguments but if it's not we are just putting in some arguments of time. So it's not nothing to worry about if you want to be of course you want to write articles and journal papers and conference papers in the field then you have to be more careful and be very consistent with your notation but here the purpose is instructional and so we are keeping notation so that everybody can follow. So the aim is more that you understand the material then be very precise and exact and consistent with the notation. So the purpose of the notation is so you can follow carefully. So this is the tracking error we always define this tracking error because this is the quantity we want going to zero. So as usual we assume that the signal that we want to track is in fact smooth bounded with bounded derivatives. So this is the standard assumption so it is infinitely differentiable it's bounded and its derivatives are also bounded so it's a very nice signal. We don't want to deal with non-nice signals and so on I mean simply because of that because in most physical circumstances you wouldn't want your robot or any other mechanical or air mechanical system or maybe even an electrical system to follow very bad trajectories. So that's really what's the aim. We also make some assumptions on F. So what is the assumption is that F is bounded if X is bounded. Okay so there are two assumptions one is that the function F is assumed to be bounded if X is bounded. So it's a function of both state and time but we don't worry about time so I would in fact even say that for all t and we also assume that B is non-zero. Yeah that is the gain that is connected to the control is assumed to be non-zero and this is a very reasonable assumption because if this is zero then there is no term here there's no control here so there's no problem to solve honestly we cannot do anything yeah the system becomes uncontrollable right. So this is very I mean this assumption is a very reasonable and a very fair assumption so we don't have to worry about making a very infeasible assumption and so is the assumption on F. Yeah so here so notice that we are assuming that F is bounded when X is bounded we are not saying that F is bounded for all states if X goes to infinity X the function F is also allowed to go to infinity yeah so it's not something like a sinusoid or something it's not something as simple as a sinusoid because if I had something like a sin X right sin X whatever I mean sin X times t or some such thing right this will be always bounded this F will always be bounded irrespective of you know what is X and t because it's inside a sinusoid yeah but we don't care for we are not restricting ourselves to functions like these yeah we are even allowing something like say linear in X yeah even allow something linear in X yeah so the only thing is we have to remember that it should not grow unbounded with time yeah it should not go unbounded with time yeah so that is one thing that we sort of have so I mean it could be something like X sin of t okay something like this is also allowed so this is a fairly you know in general enough set of functions all right it's a fairly general set of functions so I wouldn't worry about whether we are restricting ourselves to a very very simple class of functions or anything like that all right so whenever we talk about the problem setup when we talk about assumptions it's it's it's sometimes or in most times it's critical to understand that the assumptions are not very very restrictive in the system all right great great so now what do we do we again do the standard steps right whatever we are used to doing until now we we compute the error dynamics right so that's e dot is x dot minus r dot right and what is that it's afx plus bu minus r dot okay so this is just obtained by plugging in the dynamics of x great so now when as usually do our design with the known case first this has always been the I mean method of how we have been doing things and so for the known ab case we want to have a very nice target system we've always been doing that too we've been choosing a nice target system and what is a nice target system if I want to drive e to zero asymptotically it's e dot equals minus k e for some k positive and this is the simplest possible linear in fact exponentially decaying system that I can think of right so we've also already worked with the system like this and if you want to do this what would I do I would so what so what we sort of do is we introduce this term the minus k term in the dynamics here yeah I mean earlier we simply chose the control but now we don't directly choose the control just bear with me because of how we want to use the structure okay so what we do is this minus k e is the nice term so I introduce it in the e dot and then I also add a plus k so that these two just cancel out and I have not really changed anything in the right hand side okay so this minus k is there and then I look at all these terms together yeah because I want to sort of cancel this also right what do I do I take the control game being outside okay so this is a trick okay so this is a I mean I would qualify this as a neat trick okay and and lot of nonlinear control design is about such nice neat cute little tricks yeah so don't I mean one should not be feel that this is just you know a magical wand that I'm waving these tricks are based on very good intuitions all right so so it's a neat little trick and a lot of nonlinear control in fact a lot of mathematics is based on doing such neat little things tricks in fact a lot of proofs in mathematics that you will find are accomplished because somebody came up with a trick okay so just because it's we use the word trick is not derogatory yeah if you can actually come up with these tricks often you will be a very very good nonlinear control theorist okay all right so so this is a nice trick we just add and subtract the scale and then this scale is clubbed with this guy yeah how do we club it we take the control game be outside the bracket right and then I look at two different terms there is one term where it is just a one by B because there's no unknown here in these two terms and so there's no unknown here it's just a one by B and there's another additional unknown here so there's an A by B F all right and then there's just a U because the B was pulled out of the bracket all right so this is a sort of rewriting of this term by pulling the B out of the bracket this is also another trick that we pulled the B out of the bracket technically you would have to do it even for the known case because if I prescribe the control out of this my control will have been the denominator right and so we are essentially doing that we just writing it as such okay because we want to rewrite or redefine the unknown parameters that's the A although as of now we are saying that this is a known A B case but we still want to redefine the unknown parameter redefine the parameters of the system okay and how do we do that I now choose A over B as theta one star and one over B as theta two star okay so instead of A and B being my unknowns I have two new unknowns which is theta one star and theta two star defined as such now remember that this is not over parameterization I had two parameters to begin with and I have two parameters now so I'm not doing any kind of over parameterization I'm simply doing a redefinition of the parameters in fact it is very easy to see in this particular case may not always be the case in this particular case it is very easy to see that if I exactly estimate theta one star and theta two star then A and B can be uniquely computed okay A and B can be uniquely computed so not a big deal all right so great great so we understand all this so some tricks and some redefinition of parameters and what do we have I choose my control in terms of these new parameters now because these are known right and that is minus theta one star f to cancel this out and minus theta two star ke minus r dot cancel this out remember I don't need my control to introduce any new term because I already have a nice negative term here so all I need to do is cancel these guys out and once I do that with the control I'm only left with this much yeah so I achieve the target system for the known case yeah this is exactly the same expression of control the control expression is the same if I had started here with the original system this guy right and I had this target system this is exactly the same control you would have obtained I've just rewritten the control in terms of these redefined parameters and this is going to be very helpful to me subsequently okay this is subsequently going to be very very helpful to me that is the whole reason why we are doing this redefinition okay great so this is the control and with this control I know that I'm left with only this so of course I choose a all right so I have not written that here but I will choose a v which is one half e squared and I will get a v dot which is minus k e squared which is negative definite all right great so this is what we've been doing right so this known case doesn't seem to significantly different except for some redefinition of parameters all right great so now we move on to the unknown parameter case so what do we do we take the control law here in equation 1.5 and we replace the theta stars by the hats because this is just what certainty equivalence principle has taught us and this is what we've been using for so long successfully so we don't want to change our successful formula or a success formula right so we replace the theta stars by the theta hats okay and these are of course the estimates right now what happens of course I don't get the desired e dot equals minus k e but I do get that plus b times the tilde terms okay this is I mean this is very easy right I hope this doesn't confuse you right so because if you look at this e dot is minus k e plus b times theta 1 star f plus theta 2 star k e minus r dot sorry plus a control right and now if you plug in this control here okay and you keep in mind this definition yeah it's very easy to see that the hats will give you theta 1 tilde here this hat will give you theta 2 tilde here and that's it this term remains as it is this remains as it is and that's all we have okay now the question is how do I do the analysis now well we try yeah we try our usual idea what do we do we take the earlier Lyapunov candidate which was half e squared and we add to it quadratic terms in the unknowns this is what we have been doing with some of course adaptation gain in each case right so this is what we've been doing right so we want to continue doing something similar right right so then we start to compute the v dot very carefully this contains e e dot and so e dot is simply this minus k e plus b times theta 1 tilde f plus b theta 2 tilde times this right this is based on the control law we have then we have derivatives of these terms which comes out to be minus 1 over gamma 1 theta 1 tilde theta 1 hat dot and minus 1 over gamma 2 theta 2 tilde theta 2 hat dot and we are of course yet to define the update loss we are yet to define the update loss great now again what do we do we have this nice negative term coming from here right but then we have these theta 1 tilde and theta 2 tilde term so of course we try to club these two guys and these two guys that's what we've been doing again right we've just been clubbing the theta tilde terms and using the theta hat dots to cancel out the errors and whatever these terms right so in fact I will write it out because it's not written here so what will this be this will be theta 1 tilde times b e f x t I will take a gamma 1 in the denominator so that I get a gamma 1 here and minus I get a minus theta 1 hat dot okay and similarly I have a theta 2 tilde divided by gamma 2 times gamma 2 b and k e minus r dot minus theta 2 hat dot okay so these are the terms I get in theta 1 tilde and theta 2 tilde okay if I want to and the other term is of course this is equal to minus k e squared so this is of course the nice term nothing to do but I want to get rid of these two terms all right and if I do want to do that I just simply use theta 1 hat dot and theta 2 hat dot here but well look at what happens something rather bad actually theta 1 hat dot is gamma 1 b e f x t and theta 2 hat dot is gamma 2 b e k e minus r dot everything looks nice except the b here yeah this b is bad yeah this b here is bad this b here is a problem it's a problem term why b is in fact an unknown okay and theta 1 hat and theta 2 hat are essentially being designed and defined so that we can identify this b right and so if we are still have an update law right and then notice that this theta 1 and theta 2 appear in the control so it's not like these are for show these are not for show these have to be actually implemented even if you have a microcontroller and a robot then these have to be implemented in the microcontroller these have to be integrated in the microcontroller so obviously these are not for show and they contain the unknowns themselves right so this is a circular problem right so that I'm trying to identify the unknowns using this theta 1 theta 2 which contain the b and the a but then the b appears in their derivative and that's not okay yeah of course this is no adaptive control so this is not an adaptive control that can be implemented okay this is not implemented all right so that's why I put a big cross so whenever I get any adaptive control solutions from unfortunately several naive students and I find that and so the first thing I look for is if the unknown appears in the update law or not yeah I don't have to look at the rest of the solution at all yeah my checking becomes very easy yeah that you know because if you ended up with an unknown in your update law then you made a wrong design your design is something that cannot be implemented yeah so please don't make my life easy please make it hard and design adaptive laws which do not contain the unknown in them okay so this is wrong okay the question is what went wrong and did we design the controller wrong did we design the leapon of candidate wrong but it seemed like we had followed the steps that you know we were doing until now so the problem is that the steps that we were doing until now do not work for this situation yeah so we have to make a modification the modification is not in the control law which is still the certainty equivalence control law so we don't make a change there but we have to make a change in the leapon of function right so here we had chosen a leapon of function which was very much motivated by what how we have been doing things until now with this just a quadratic term and adaptation game here but now we have to do something different we actually have to introduce an absolute value of p here okay we have to introduce an absolute value of p here everything has remains the same and just this new term shows up all right just this new term shows up okay so what happens what happens everything else is same again okay the only difference here right the only difference that we see here is that your because of this absolute value of p which is again a constant remember this is a constant all parameters are assumed constant there's no derivative with respect to it but the second and third terms which were simply these guys in equation 1.7 now have an absolute value of b here so when I again combine this term and this term my update law does contain something about b but it is not the b itself because we use the fact that b divided by absolute value of b is simply signum b yeah I hope you understand this and b is simply equal to absolute value of b times the signum okay so although b doesn't appear now which was wrong signum b appears here okay so so everything has remains the same just that instead of the appearing which was absolutely not available I have signum of b appearing here this I would say is a much more benign requirement okay this is a much more benign requirement okay what is the requirement the requirement is that there is a signum of b that I need to know in order to implement this adaptation law so the amazing thing is that in the entire adaptive control community there is no other known choice of Lyapunov function or control which will help us to not know the value of signum b while implementing this control okay so the only solution that is available in the community is what I have shown you yeah and if you can come up with something that does not require the sign of b either then you have done some magic and you will get a okay all right so this is actually something that's not possible yet okay so if you can come up with something excellent so anyway so with the knowledge of the sign of this control game you can implement this adaptive controller and you will get v dot is minus k e square which is of course now not less than but less than equal to zero all right this is not less than zero this is only semi-definite because there were two other states theta one tilde and theta two tilde and therefore you can complete the rest of the proof with bar bar at slammer and signal chasing in fact you can show that limit as t goes to infinity e of t is zero okay the steps are standard yeah you will just show that v is integrable first you obviously have stability right from here I have of course uniform stability in the sense of Lyapunov no problem beyond that I will be able to show that e is bounded that is l infinity and so I know I can show that v is integrable because it is non-increasing and lower bounded and therefore e is an l2 signal when I integrate both sides so I obtain that e is both l infinity and l2 and then e dot is also infinity can be easily shown and therefore by bar bar at slammer scurrallary you will be able to claim that e goes to zero all right excellent so what have we seen today is how to handle unknown gains on the control control in the adaptive control context so until now we had been looking at unknowns only connected to the states that is in the drift terms but now we have also seen how one can handle controls sorry unknowns in the control vector field okay and this is again something that is very very naturally occurring all right if you have any sensor or sorry any actuator which is misaligned or you don't know the exact position or the orientation for example if I have a spacecraft with a control moment gyroscope as an actuator to help it rotate about its axis right and then I don't know exactly where if it's placed in the center of mass or if it's oriented exactly along the body axis or not then there will always be an unknown control gain okay or you can also think of a quadrotor type situation where the thrust is actually some coefficient multiplied by square root of angular velocity of the propeller now this coefficient may again be an unknown quantity or not a well measured quantity and therefore this also gets into the control gain and this unknown okay so this is a very common problem and we have given an adaptive control solution to control design in this context so we can still achieve tracking all right great so we will stop here and I will see you again next time thank you


Hi everybody Joe for Jasper's case of breaks calm coming at you with another one-and-one basketball break Latest edition just dropped yesterday by Panini ten mocks full case pick your team a number of five Very big. Thank you to everybody here for getting in on the action appreciate it And before we pulled teams out for number blocks. It was Gretchen with last bottom mojo Brooklyn Nets So thanks to her. Thanks to everybody for getting your team straight up. I really appreciate that We've got it down to 18 before we had to do a number block, which is awesome And there are the results of the number block right here Forgot to load the video, but I'll do that after this break Keep that aside next to me right there. There is a fresh case of one-and-one And we've got another pick your team in the store right now. That's down to That is down to 14 teams. It's almost in single digits. I'm wondering if we could get into single digits Maybe even fill it up straight up Otherwise, I'll start thinking about working on a number block for that. Whoa What's this? This must be a case sometimes they are distributors do this. I'll put stuff in a that's why I felt lighter They'll send us loose boxes of stuff And then they'll put other stuff in there that tape looked a little weird, too. All right Fresh case of 2020 2021. That's more like it Here's the here's the list once again. All right. Thanks everybody Are those loose boxes because breakers or car shops send them back? No, I think usually it's it's if if Breakers or shops buy individual boxes. They'll open a case and then they'll and they'll send like let's say someone only wants to buy Four boxes or something like that. It's usually that and then they could sell those to us and they'll let us know like hey We've got four loose boxes left. You want those and we'll like yeah, sure. We'll buy them We'll put them in a mixer or sell them in the shop or something like that All right. First one out of the gate is Ben Wallace eight out of 30 Ben Wallace for the Pistons Tomoya with Detroit and the hit gonna go this way. I think no this way I think and it's Grant Hill nice 68 out of 99 Pistons box first team signatures a personal box for Tomoya nice next up is Charles Barkley 88 out of 99 and it's OB Toppin 30 out of 35 rookie Jersey autographs nice patch nice auto So Nick's are a number block team Nobody wanted them straight up. So that goes to Nick's zero So that's David Nadler.  That would be insane Imagine we get it. We get the downtown. We'll send the rest back. No, they'll they'll buy they'll buy sealed cases back There's pressure to chew them five out of 25 Miami heat wing and then we've got 69 out of 99 PJ Washington, Jr. Jersey and autograph Hornets that's gonna go to Gretchen Let's see if we can find his teammate in here somewhere Is Patrick Williams 23 out of 99 Osward with the Bulls and we've got a CJ McCollum 30 out of 35 dual relic and autograph Trailblazers for Brent Brent picked up the trailblazers straight up. Hopefully Collapsed lung is what CJ McCollum has. It sounds kind of scary. So hopefully he's okay Desmond Bain Bain 22 out of 99 Bain Thomas with the Grizzly. That's my Bain impression. It's pretty spot-on and a redemption and it's Timeless moments Tony Parker That's a Lee Tao with these first No bonus hidden there. All right next. Oh Rex you totally should Can you imagine if you if you were able to get blaster blaster boxes or cello packs at at at your Kroger? All right, we got Andrew Wiggins four out of 30 wiggin out warriors Sean Maddock And the hit. Oh, all right. See you good work. Thank you. Yes, and I'll see you see you Sunday Sunday Actually, I might not be on on Sunday. I think I'm going to a car show on Sunday TV we're gonna see what what this what the schedule is going to be, but I'll see you at some point You too man draft safe All right the hit It's Denny Avija for the wizards jersey and autograph rookie jersey auto 74 out of 99 wizards are a number block team wizards four Brian Ashley, there you go, Brian Remember the old baseball player Billy Ashley, and you just certainly have some upside as well next box Good luck. We got another full case of this in the store. We could possibly Kind of running out of time. We could possibly run that back tonight if you want to We've got a Udoka as a Buike five out of five out of five and under get the world famous train whistle Utah Jazz Q All aboard the big hit express And then we've got a Tyrese maxi nice one out of 35 Almost looks like the uh The french flag right there One out of 35 for the Sixers and Brian s picked up the Sixers straight up Tyrese maxi has been having a nice season three colored jersey and on card autograph a butte Three to go We've got a Rudy go bear 73 out of 99 for Utah and Q Next up is a Emmanuel quickly prime jersey and autograph Nice three color patch and auto four out of 25 Remember nicks are a number block team so nicks four Is for Brian Ashley nice There you go, Brian two to go Downtown jaw moran Things will be great when you're downtown That's what downtown Memphis looks like there's a big grizzly bear the pyramid that's there music of course The firm is there too That's awesome Thomas with the grizzlies Congrats I pulled a Zion in the other case a jaw moran downtown in this case And then we have chris mollin 70 49 out of 75 first team signatures Warriors Sean maddock with golden state and the last one boys and girls Thanks for getting in appreciate it When you're alone and life is making you lonely you can always go downtown markel faults 41 out of 99 That'll be for orlando tamoya. They're not numbered but they are Very short printed.  I'll see you next time for the next one and one