 We were discussing about the Stokes drag on a sphere. So, the manner in which we proceeded is we first try to develop a governing equation in terms of the Stokes stream function and we obtain the solution of the governing equation to get the value of the Stokes stream function. So if you look into this slide it gives you an expression for the Stokes stream function. So you can see that this psi is u infinity r square by 2 sin square theta into a function that function will tend to 1 as r tends to infinity. Because if r tends to infinity then capital R by small r will tend to 0, capital R cube by small r cube will also tend to 0. So this psi will tend to u infinity r square by 2 sin square theta. So that is the expression up to which we derived in the previous lecture. We will take it up from there. Now the next step is once you know what is the value of psi you next calculate what is vr and then you calculate what is v theta. So vr is 1 by r square sin theta del psi del theta. So let us see how you calculate vr it is relatively it is quite straight forward. But just in case you want to get familiar with how to calculate vr 1 by r square sin theta del psi del theta. So that is 1 by r square sin theta into u infinity r square by 2 2 sin theta cos theta into that function which tends to 1 as r tends to infinity. So 1-3 by 2 capital R by small r plus r cube by 2 r cube. So r square r square gets cancelled out sin theta sin theta gets cancelled out. So it becomes u infinity cos theta into that function given in the square bracket. So that is what is vr. So similarly you can calculate v theta I am not going through each and every algebraic step at this moment because like we have worked out enough of the steps to give you a understanding of how to go ahead with it. We have to move a little bit faster in this particular course because we have to cover a whole range of topics. So from now onwards what we will do is that we will make a sort of a compromise that we will work out certain key steps on the board. But many of the steps we will give you the outline and I will of course share with you the detailed solution of each of the steps through the presentation slides which are accompanying this board work. So you will get the full material and you can practice that by yourself. So then you can calculate v theta is equal to minus 1 by r sin theta del psi del r. So and then you can differentiate psi with respect to r. So you will get v theta is minus u infinity sin theta into in the square bracket the r dependence. So you can clearly see that as r tends to infinity v r and v theta that is u infinity cos theta and minus u infinity sin theta these 2 components come as they are. Now next is we need to calculate the pressure field. So to calculate the pressure field what we do with that we consider the r component of the Stokes equation. So this is the Stokes equation. Stokes equation is nothing but the Navier Stokes equation with the left hand side the inertial terms equal to 0. So this equation when you write in terms of the r theta coordinate system the radial component of the equation is given by this. Quite cumbersome I mean there are terms with cot theta and all those things. So it is again you are not expected to remember this particular form or memorize this particular form. So in this particular form you have various terms del v r del r del square v r del r square del v theta del theta del square v theta del theta square all these terms you can algebraically calculate. And once you calculate all these terms you can substitute these terms in the Stokes equation. So you will get an expression for del p del r. So let us go to the next slide. So all the terms when it is substituted you will get del p del r is equal to this one 1 by mu del p del r is equal to u infinity cos theta into 3 capital R by r cube. So now if you integrate it with respect to r then p is equal to minus 3 mu u infinity cos theta into capital R by 2 small r square plus a function of theta which we call p 1 theta. Similar to the r momentum equation you can appeal to the theta momentum equation to integrate p with respect to theta. So the theta momentum equation is written in the bottom of this slide. So you have del p del theta and then you can integrate this with respect to theta so that you get a constant of integration which is a function of r. So now if you see the two forms of p one by integrating the r momentum equation another by integrating the theta momentum equation your observation is that that p is equal to minus 3 mu u infinity r cos theta by 2 small r square plus a constant. Now how do you calculate this constant? You know that at r tends to infinity p tends to p infinity which is the ambient pressure that as you go to infinite distance from the center of the sphere you come to the ambient condition that means c equal to p infinity. So p equal to p infinity minus 3 mu u infinity capital R cos theta by 2 r square. So what fields we have obtained? We have obtained the velocity field, we have obtained the pressure field. Now to calculate the force we also need to calculate the viscous stresses. So to calculate the viscous stresses this tau rr and tau rtheta are the viscous stresses. So tau rr is 2 mu del vr del r. So if you once you know vr it is a matter of differentiating with respect to vr with respect to r to get this particular expression. Tau rtheta this is the shear stress tau rr is the viscous normal stress and tau rtheta is the viscous shear stress okay. So now you can calculate this tau rr and tau rtheta on the top of that you have the pressure distribution. So these are the important stresses which are acting on the system. Now to understand that how to calculate the drag force we will refer to this figure and I will redraw it in the board because it requires certain explanation of how to calculate the forces that is one of the fundamental things. So let us say this is the sphere. Now what you do is at an angle theta let us take a small strip subtending an angle d theta. So this will be the surface of the sphere under consideration, surface of the strip under consideration. So what is the surface area of this? What is the radius of this strip? This is r sin theta right. So this radius of the strip this is capital R so this is capital R sin theta right. What is the width of the strip? This is rd theta. So this is almost like the surface of a cylinder with radius r sin theta and height rd theta. So it is 2 pi r sin theta into rd theta that is the surface area of this small strip. So 2 pi r sin theta to rd theta. Now this is area and the force per unit area. So this is tau rr right and this is the viscous normal stress. Then the hydrostatic stress is minus p. So this into this the shear stress is tau r theta times the area. Now the drag force is what? Drag force is basically if you have the relative velocity direction. What is the relative velocity direction? Let us say this horizontal direction. So the resultant of all these forces in that relative velocity direction. So the plus or minus depends on whether you are considering force exerted by the fluid on the sphere or force exerted by the sphere on the fluid. So accordingly plus or minus sign will change but I mean you can calculate with a proper sign. So tau rr minus p this will come with what? Cos theta right and minus tau r theta sin theta that multiplied by this area. Then integral of that. This evaluated at what r? Small r is equal to capital R evaluated on the surface of the sphere. That is very very important okay. So with this background let us get into the next slide which will sum up the calculations. So what is done in this slide is what we have written in the board just the opposite sign of that is written okay. That gives the magnitude of the drag force okay. So because we are interested in the magnitude of the drag force otherwise it will be oriented in the opposite direction of the flow. So just to get the magnitude of the drag force we are just reverting the sign. So one thing one very interesting thing is that tau rr at small r is equal to capital R is equal to 0. I mean whatever is the expression for tau rr. So if you substitute in this expression for tau rr if you substitute small r is equal to capital R in the square bracket it is 3 by 2 minus 3 by 2 okay. So on the surface of the sphere there is no normal viscous stress okay. So many times see as we see in many cases sometimes ignorance is a blessing in disguise. So if you do not even take tau rr in your calculation you will come up with the same final answer and that will actually reduce your effort. Because you calculate tau rr you calculate the value of tau rr at small r is equal to capital R and then you figure out that it is 0. There is a physics that is there because of which on the solid surface it is 0. But if you are ignorant about the contribution of tau rr completely and think that only the pressure will contribute to the normal stress still you will come up with the same answer. But you have to keep in mind that the normal stress can also be of viscous nature. Normal stress is not just due to pressure okay. So now you calculate tau r theta at small r is equal to capital R and then you calculate the elemental drag force and integrate it from theta equal to 0 to theta equal to pi to get the total drag force. So when you calculate the total drag force you will see that there is one term which comes sin square theta and cos square theta the nice identity sin square theta plus cos square theta equal to 1 can be used. So then you will see that eventually when you are making this calculation you will get the contribution of this term this 3 mu u infinity by 2r into 2 pi r square sin 2 pi r square sin theta d theta integration. So that becomes cos theta so you will get eventually 6 pi mu u infinity r which is the Stokes law okay. So like sometimes the Stokes law is expressed in terms of the drag coefficient cd. So you can calculate the cd as a function of Reynolds number which I leave you live on you as a homework that based on this expression for the drag force you calculate what is the cd the drag coefficient. So what are the assumptions under which we derived the Stokes law we made an assumption that the inertial effects are negligible for low Reynolds number. Question is are the inertial effects truly negligible for low Reynolds number under this scenario let us try to investigate that in some details let us go to the next slide. So now so to do that so let us calculate the viscous force viscous force per unit volume and inertia force per unit volume viscous force per unit volume. So it will let us say we take one representative term del tau rr del r. So let us see what is the tau rr term let us move on to the slides let us yeah let us refer to the slides. So tau rr scales with what tau rr scales with mu u infinity look at the expression for tau rr mu u infinity that is the leading order forget about cos theta I mean cos theta can be of the order of 1. So mu u infinity r by small r square. So del tau rr del r this will scale with 1 by r mu u infinity r by r square. So you can see what is there in the slide 1 by r mu u infinity r by small r square. So this is viscous force per unit volume now inertia force per unit volume 1 representative term we take this rho vr into del v theta del r just one representative term one representative term in the inertia force. So let us get to the vr and v theta expression. So rho vr so vr is of the order of what vr is of the order of u infinity because it is u infinity cos theta into 1- some function of r. So the leading order term is u infinity cos theta therefore vr is of the order of u infinity and del v theta del r. So we go to the next slide del v theta del r is what? So you have v theta as if you look into the slide v theta is- u infinity sin theta into a function of r. So if you differentiate with respect to r the second term will come into the picture. So it will become u infinity of the order of u infinity r by small r square 1 by r being differentiated will become 1 by r square. So inertia force by viscous force this scales with rho u infinity square r by r square by mu u infinity r by r cube inertia force by viscous force. So what is this rho u infinity r by mu into small r by R. This is the Reynolds number based on the radius of the sphere. Normally we calculate the Reynolds number based on the diameter of the sphere. So this is just for concept I mean it makes no difference what we are explaining here. So if you go to that slide where we are calculating the inertia force by viscous force see this is what is summed up. The inertia force by viscous force is equal to Reynolds number remember this is not the Reynolds number based on the diameter but the Reynolds number based on the radius into small r by R. So look at this inference despite the assumption that Reynolds number tends to 0 the inertia force may be large for large smaller by R right. So the Reynolds number may be say of the order of 1 but for large smaller by R the inertia force may be significantly large as compared to the viscous force whereas the inertia force has been completely neglected in making the calculation. So this is one of the significant limitations of the Stokes law despite it is very fundamental nature it neglects inertia force for circumstances where inertia force actually may not be negligible okay. So there have been various corrections to the Stokes law not just by considering this effect but several other effects but this being a fundamental course we will not get into all the details of like how the Stokes law has been corrected and all just one simple correction we do not have enough scope to derive this but this is called as Ossin's very classical consideration in correcting the Stokes law and that is like the Stokes law the Stokes drag modified by a multiplying factor 1 plus 3 by 8 Reynolds number this like if you are interested in looking into like the Ossin's correction and how this has been obtained you can refer to classical literature on this but I mean so far as the scope of our course is concerned we keep ourselves confined to the Stokes law and you can see that typically for very low Reynolds number this correction may not be that significant. So in microfluidics sometimes most of the times we are considering flows where the Reynolds number is typically very low so that this correction may not be that important but whenever this correction is important this has to be given due consideration. So we sum up here the considerations of pressure driven flow not that we are ending our discussions on pressure driven flow but we have discussed various cases of pressure driven flow so some advantages and disadvantages of pressure driven flow we need to discuss at this moment. So the advantages the pressure driven flow why we have started with the pressure driven flow in microfluidics is because it is it gives one of the simplest form of flow actuation methodology in microfluidics. It is easy to integrate with microfluidics chips you basically require some very simple driving mechanisms like the syringe pump for example. So we will have one experimental demonstration soon as a part of this course where we will demonstrate you that how you actuate a pressure driven flow and how you calculate friction factor and all these experimentally. It is easy to integrate with microfluidics chips and it is possible to have a good control over the flow parameters. So that is advantageous but there are disadvantages or I mean sometimes disadvantages is too strong award we may better say limitations. So high frictional losses especially in microfluidics chips and thus reduced efficiency. So we have seen that like even for a fully developed laminar flow through a circular tube the for a given flow rate for a given flow rate the head loss is inversely proportional to the fourth power of the hydraulic diameter. So if you reduce the hydraulic diameter significant head losses significant increases in head loss may occur. Poor reconfigurability and since this is a positive displacement pumping system even small blockages in the pathways can lead to severe damage of the system. So I mean these are some of the effects that are not so good for pressure driven flow so one has to look into other mechanisms I mean which are not involving the movement of mechanical components or like for example there is no piston syringe type of movement which piston cylinder type of movement which is there or there are reduced frictional effects and those can be integrated with a on chip environment in a much more convenient manner. So there are several such possible types of flow actuation mechanisms and we will learn those subsequently. Regarding the science to summarize pressure driven flows offer the most general flow driving mechanism in fluid mechanics. The system of equations for low Reynolds number hydrodynamics in a system where x reference is much much larger as compared to y reference we have discussed about these systems of equations earlier and we have given analytical solutions of low Reynolds number flows for steady and unsteady conditions oscillating wall and pulsating pressure gradient for simple geometries with simplifying assumptions. For more complex situations one has to go for numerical solution methods. So we will now move on to another topic I mean which is not low Reynolds number which is not just restricted to pressure driven flow but that can be used for several other scenarios so that particular topic is known as lubrication theory. So we will give you a brief introduction of what is the lubrication theory and why it is important. So to get a brief motivation of the lubrication theory now I mean this particular terminology has one theoretical foundation in association with one very practical consideration and that is why the name lubrication theory. So the motivation of understanding this theory in engineering is far reaching I mean there are many possible applications but one possible scenario that we consider that you know that in classical mechanical engineering applications you many times use bearings and these bearings are there to support a shaft. So you have a shaft which is rotating and then you have a bearing to support the shaft. Now if you just leave the gap between the shaft and the bearing as it is then there can be direct metal to metal contact and that can give rise to a lot of wire, tear and friction. So to reduce the friction what you can do is you can put a lubricant which many times in industries called as lube oil. So lubricating oil you can put in between the shaft and the bearing so that will reduce the friction. So the situation is that let us maybe discuss the situation a bit in the board before we come on to the mathematical perspective. So let us say this is the shaft and let us say this is the bearing and there is a lubricating oil in between. Let us say that this shaft is rotating with a particular angular velocity in this particular direction whatever direction it is I mean it is not so important to understand the concept. So you typically have and this gap is quite narrow. So you have a narrow gap in which there is a viscous fluid and the 2 surfaces which are separating the fluid are having a relative motion one over the other. If you have the shaft and the bearing related to each other with a certain eccentricity then there is an additional complication that the gap between the 2 is continuously varying. So you have essentially 2 surfaces so if you think of an equivalent planar geometry let us say you have a equivalent planar geometry. So this is a kind of a bearing under rotating condition but you can also have a slider bearing. So I mean you can think it in whatever way it is possible even for a rotary system a good approximation can be a translatory system provided that the gap is very small so that the curvature effect is neglected. So you can think of this situation as a flow in the narrow gap where h is a function of x and one of the boundaries is moving relative to the other. The low Reynolds number hydrodynamics that we have considered so far is little bit more simplistic as compared to this. So the low Reynolds number hydrodynamics that we have considered for flow in a channel and all those things we have not considered one of the boundaries moving relative to the other in general maybe one or two examples we have considered but in the classical case the Poiseuille flow or the Hagen-Poiseuille flow we have considered that like the boundaries are fixed not only that the flow passage is having a constant cross section it is the cross section is not changing. Now what about the situation when the flow passage is a function of x the flow passage is a function of x but it changes with x slowly it does not change with x rapidly this is number one. So h is a function of x although it is a slow variation not a rapid variation not only that you have a length scale along x so h slowly varies with x so h here or h here whatever let us say h here is h0, h0 h at this location. So you can take either of these or these or the average whatever as the characteristic length scale along the vertical because these are the variation along x is small. So h0 let us say is the characteristic length scale along y and let us say if this is the x axis let us say that lc is the characteristic length scale along x. So the second condition is that h0 much much less than lc this is the condition that we already discussed about in details in our discussions on low Reynolds number hydrodynamics we have considered that the y length scale is significantly less than the x length scale and we have not considered any variation in the flow passage not only that we have not made a general discussion for the scenario when one of the boundaries is having a relative motion with respect to the other boundary. So when you take all these in purview the general theory that is developed for this is known as lubrication theory and the name lubrication theory comes from its applications in the context of bearings in tribology. So with this little bit of background we move on to the slide. So the way in which we will move on to the move ahead with this chapter is like we will in general go through the slides but wherever more detailed derivations are needed I will go to the board and work out the in between steps which are there in the slides that is how we will proceed and these slides will be available to you anyway so you can I mean you can go through the material which is discussed using the slides. So the basic idea is the analysis of flow when there are at least 2 length and velocity scales involved with the flow like for example in the 2 dimensional case you have the 2 length scales HO and LC 2 distinctive length scales. If HO and LC are of the same order then you cannot consider that as the as something which is within the purview of lubrication theory because then they are not 2 different scales. You basically have to have 2 different scales one scale significantly smaller than the other. One each of the velocity and length scales are much larger than the other therefore there is a dominant length scale and a dominant direction of flow. So this is now very important then you would say that the same concepts apply in boundary layer theory as well. If you are considering the boundary layer theory in boundary layer theory you have so let us recollect what is boundary layer theory. So you have let us say a flat plate and close to the wall there is a region where velocity gradients exist because of viscous effects. So an outside that region you have inviscid flow so this is the outer stream and this is the inner stream which is the boundary layer and this is like the edge of the boundary layer. So what we see here is that the boundary layer theory not the concept of boundary layer but boundary layer theory remember one thing that boundary layer is what? Boundary layer is a layer adhering to the solid boundary in which viscous effects are felt. So how thick can the boundary layer be? It can be as thick as infinity. On what it depends? It depends on the Reynolds number. If the Reynolds number is large then the thickness of this layer is very small. On the other hand if the Reynolds number is small viscous effects are so highly penetrating that the fluid which is even located at a very far stream feels the effect of the solid boundary explicitly. So boundary layer thickness can be anything from very small to infinitely large but for the case where the boundary layer thickness is infinitely large we cannot use the boundary layer theory. So existence of boundary layer theory and validity existence of boundary layer and validity of boundary layer theory are 2 different issues. While boundary layer will exist for all viscous flows boundary layer theory will not be valid for all viscous flows. It will be valid only for high Reynolds number flows provided there is no boundary layer separation. So now if you are considering a high Reynolds number flow then you have 2 different scales like you have the delta and this as l and the boundary layer theory you can apply when delta is much less than l. So you look at the similarity between this case 1 that we had discussed and the case 2. What is the similarity? The apparent similarity is that here also we have 2 distinct length scales h is much less than lc. Here delta is much much less than l but only to that extent there is similarity. The physics of these problems they are completely different. Typically these problems we are considering in the low Reynolds number regime whereas the boundary layer theory is applicable for high Reynolds number. So let us discuss briefly what is the difference between boundary layer and lubrication flows. The boundary layer theory, the boundary layer theory deals with a very thin region of flow near the solid boundaries where viscous stresses are large. It is dedicated to the cases of large Reynolds number flows and inertia has dominant role in boundary layer theory. The lubrication theory is valid inside the whole flow domain as long as flow and the corresponding scales are dominant in one direction. Inertial effects are usually negligible in the lubrication theory as long as the Reynolds number is not very high. But if the Reynolds number becomes higher and higher inertial effects will be important but those are higher order effects. But in the microfluidics context we are mainly interested about low Reynolds number flow. So when we are considering lubrication theory in the context of microfluidics which is this particular course then we are basically operating in the low Reynolds number regime. On the other hand for boundary layer theory we are typically discussing the situations in the high Reynolds number regime and not the entire flow domain is affected by the solution within the boundary layer. Solution within the boundary layer has some interaction with the solution with the outer stream but the outer stream solution is like not directly associated with the boundary layer solution. So the outer stream solution can be obtained by using potential flow theory. But in the low Reynolds number regime in the lubrication theory when you consider you are applying the theory for the entire domain not only a part of the domain. In both lubrication theory and boundary layer theory length scales and velocities in one direction dominate. This is the only similarity between the two everything else is different as we are going to find out. So let us now proceed to discuss about the lubrication theory. So we will give a brief introduction to the theory. So let us look into the schematic. So let us say that there is a confinement. The confinement is made up of 2 boundaries, 2 walls. There is a bottom wall and there is a top wall. The top wall moves with an arbitrary velocity v prime. See I will explain you a little bit about the notation that we are going to use in this chapter. We will use the prime for the dimensional quantities and without prime the dimensionless quantities. Because like it is the logic of using this notation is that we are mostly going to use dimensionless equations and it is very difficult to write prime all the time. So just for the ease in writing we are going to use that without prime for the dimensionless quantities and with prime for the dimensional quantities. So the top wall I mean it could as well be the bottom wall. The top wall moves with a particular velocity relative to the bottom wall. So this v prime this has components u prime tilde along x and v prime tilde along y. These are the components of velocity of the top wall. So in general the top wall may have a simultaneous motion along x and y. There are 2 important length scales as we discussed. So the length scale along x is lc which is the characteristic length scale and length scale along y which is H0. I mean this what is H0 and all we have discussed to a schematic earlier. And we consider that the ratio H0 by lc is much much less than 1. This is one of the very important hallmarks or essences of the lubrication theory. So basically what we will be doing is we will be utilizing this epsilon as a small parameter and writing asymptotic expansions based on this small parameter. So we have to identify a small parameter. See in mathematics when you have a small parameter the small parameter can be arbitrary. In physics or engineering the small parameter cannot be arbitrary. It has to be a parameter that is relevant to the physical description of the problem. So here we are going to identify a small parameter which relates to the physical dimensions of the problem. So we are going to consider the small parameter as H0 by lc the characteristic length scale along y divided by the characteristic length scale along x. So we are going to discuss about various scales. So the x velocity we assume that the x velocity is of the order of uc which is like uf as an alternative notation that we considered earlier. But here we do not know what is uc. What is uc depends on the physics that governs the problem. Is it a body force that is governing? Is it a motion along x that is governing? Is it a motion along y of the plate that is governing? Whatever is governing based on that your uc will be decided. And accordingly we see the y component characteristic velocity will be decided. The Reynolds number is still calculated based on rho uc lc by mu. So the characteristic length scale is considered as the length scale along x. And non-dimensionalization we are considering the dimensional variable as something with prime and non-dimensional variable is the prime variable divided by its scale. So note that we have not yet specified the details of the velocity scale in the x direction. The velocity scales depend on the nature of the flow that means what is driving the flow? Is it a motion of a plate? Is it a body force? So many of these things. If it is a motion of the plate is it a motion of the plate along the vertical or along the horizontal so many other aspects come into the picture. But these length scales are intrinsic geometrical properties. These length scales are somewhat which are fixed by the geometry of the problem. So we will proceed further with little bit of introduction on the significances of various scales and how these scales can affect the physics of the problem. With this little bit of introduction we will stop here today and we will take it up from here and we will enter into the more details of the lubrication theory from the next lecture onwards. Thank you very much.