 Today we will start with a new topic, thin film dynamics. Now this topic has a bit of relationship with the lubrication theory that we had learnt earlier. In the lubrication theory that we had learnt, we discussed about a narrow gap in which there is a fluid driven by some force and the length scales could be separated in such a way that the gap height is much much smaller as compared to the axial length scale and the gap height could be varying with position and that however the variation in the gap height is such that the gap height by the length scale, the average or the length scale representing the gap height divided by the axial length scale that remains small under all circumstances. So that was the condition that we considered and thin film dynamics as we see here also deals with flows of very thin films whose height is very much smaller than the length. But the difference between the thin film flows and the kind of situations that we addressed through lubrication theory earlier is that they are both the boundaries were rigid. Now here there is a free surface that we are talking about which is the surface, the interface between the liquid and say another medium which may be a gaseous medium. So this thin film dynamics is essentially talking about free surface flows where thin film, thin liquid films are formed and of course we can use lubrication theory in this context as well. But the context in which we use lubrication theory earlier, the difference between that and this is that there we talked about a rigid boundary at the top and here we will be considering a free thin film surface at the top. So one can easily infer that I mean except for this difference but this is a big difference and we will see that what is the implication of this big difference. But except for this big difference one can easily infer that flow of fluid in a thin film will have good resemblances with lubrication flows that is flow in slide for example flow through slider bearings for example. In thin film flows also the velocity in one direction dominates over the other that is there is a dominant direction of flow that is if you recall that in lubrication theory we had the velocity scales uc and vc along x and y where uc is much, much greater than vc. So here also the same thing will hold true that means although there are the possibilities of having both components of velocity but one component of velocity dominates significantly over the other. But there are some subtle differences between thin film dynamics and lubrication theory. In lubrication typically in lubrication I mean it is we should not say that the difference between thin film dynamics and lubrication theory but better to say thin film dynamics and situations pertaining to lubrication because lubrication theory is a theory which can be used even in thin film dynamics as the boundary conditions change. But in typically in lubrication the flow is bounded on both sides by solid walls therefore at both the walls the fluid satisfies the no slip boundary condition. In lubrication theory usually height of the fluid or channel is known as a function of time or space. In thin film dynamics one of the surfaces is usually a free surface as we have pointed out therefore instead of no slip boundary condition the fluid satisfies the no shear boundary condition. We discussed about this boundary condition earlier and we will see that how that we can fit that in the context of lubrication theory. Apart from this the film height is in general not known. This is also a very important thing that like when we are talking about say lubrication in say slider bearings we know the height of the confinement it is already given. But in thin films for example let us say droplet spreading on a substrate. So there if you consider the height of the droplet being a function of time that evolves with position and time in such a way which is not known a priori. So this is an important challenge in the lubrication in the thin film dynamics therefore one has to look into additional boundary conditions and these additional boundary conditions we will discuss in great details which are known as the kinematic boundary condition and normal stress boundary condition. Now in what follows we will discuss the thin film dynamics in much the same way as we did for the lubrication theory. Therefore some algebra which we worked out in details for the lubrication theory we will omit detailed discussions and we have since we have already done those discussions in the lubrication theory and we will straight away use the expressions but wherever possible and wherever the algebra is simple enough I will try to derive that for your recapitulation. Now to understand the thin film theory we will start with the very special case of the thin film theory when the film surface is flat okay. So let us try to consider a situation let us say there is an inclined plane making an angle theta with the horizontal and there is a thin liquid film on the top of it. We can for simplicity we are assuming that this film thickness H0 is a constant. This is not true for a general thin film and we will see that if it deviates from a constant that if it evolves spatio temporally then how to take that in the mathematical formulation but we will first begin with a simple case just to get some essential idea that what is the situation if this film is flat and let us say that this is the direction of G. We can set up the x and y axis x dash y dash dash for the dimensional system and without dash the dimensionless system. So let us write the governing equations for low Reynolds number we can neglect the inertial terms and we can straight away write first the continuity equation assuming 2 dimensional incompressible flow x momentum 0 is equal to plus now there will be a body force. What is the body force along x dash rho g sin theta y momentum what is the body force for the y momentum minus rho g cos theta. Now as you see here we can easily extract the leading order terms from these equations for the situation when the y length scale is much much less than the x length scale. So then which of the terms will remain so this term will remain then out of these 2 terms this term will remain and this term will remain out of these terms here this term will remain and this term will remain the v effect will be much much less as compared to the u effect. So it will be possible to solve this system of equations however we will not make an attempt to solve this system of equations at this stage but we will think of what are the appropriate boundary conditions that is can we close the system of equations. So to understand that okay before going to that let us summarize the various equations. Now in this slide we have not used the dash for the dimensional parameters just for simplicity. So you can see that these equations which I have derived in the board just now these equations are summarized here. So important consideration is that H0 much much less than LC that means we have del del x much much less than del del y and del square del x square much much less than del square del y square and v is much much less than u. Further inertial effects are negligible since the film is very thin so that the left hand side of the Navier-Stokes equation the left hand side is neglected. So you can see the summary of the 3 equations and these equations contain u, v and p I mean either in a dimensional form or in a dimensionless form I mean but you require their boundary conditions. So what will be the boundary conditions for at the bottom wall and at the top wall so we will look into that. So at the bottom wall at y dash equal to 0 what are the boundary conditions u dash equal to 0 that is no slip and v dash equal to 0 that is no penetration. This boundary condition is true at y dash equal to 0 what about y dash equal to H0. Let us say to generalize now we have understood that you can apply a surface tension gradient to modulator flow. Let us say that you have applied a surface tension gradient. So if you apply a surface tension gradient how can you apply a surface tension gradient in the previous class we had learned that it is possible to apply a temperature gradient or a concentration gradient to create a surface tension gradient. Let us say that you have applied you have created a surface tension gradient along x dash. So then we can write that the tau which is there at the free surface is equal to d sigma dx dash. We have proved this in the previous lecture and we will briefly recapitulate how that is possible. So again like in this slide we have omitted the dash. So this is just a representation without dash but these are dimensional parameters. So you can see that we have taken a small element and on this side you have sigma on this side you have sigma plus d sigma and thus stress the tangential stress acting on the dx is tau x y. So if you make a balance then tau x y is equal to d sigma dx. This we proved earlier I am not going to spend much time on this. Now the question is that when you say that tau x y is equal to d sigma dx what are the basic assumptions? There are quite a few assumptions but one of the most important assumptions is that the film is flat okay. So that means this free surface which is the interface between the white coloured background slide and the blue coloured thin film that interfacial region that is the surface which demarcates the 2 regions that is the flat surface. But as we just discussed that there is no necessity there is no guarantee that this surface will be a flat surface. So if the interface is bent or deformed which is almost always likely to happen. So to understand that we will try to look into the tensorial form of the previous description where the tangential direction is not along x but the tangential direction. So let us represent that in the figure. So if you have a curved interface like this where the h local h is a function of x and t then you have directions s and n unit tangent vector and unit normal vector okay. So the s takes the role of x now whatever was the role of x for a flat surface that role is now taken up by s. So now we can write so previously what we could write? We could write that tau is equal to d sigma dx. Instead of that now we will write that the tangential component of the stress is same as the gradient of sigma along the s direction s not x along the s direction. So how will you mathematically write this? So you can write it by invoking the traction vector. Let me erase this. We had seen earlier that the traction vector that is a force per unit area for at a point where the direction normal is n it is ith component is tau ij nj. So we will not give the vector here because we are writing with in next notation. So in tensorial notation this can be thought of as just like this is the stress and this is the normal vector this is the traction vector. So the same things represented in tensorial notation okay. Now sometimes I mean the notations are different at different sometimes it is given a notation like this but commonly in books like in tensorial notation I mean you just use bold faces to represent this. So in the slides that I will be showing this tensorial quantities are shown by bold letters. So in the board I will use some arbitrary notation I mean because different books use different notation but in the slides that I will be showing I will be using bold notation to represent this. So now this n maybe we can show with a cap sometimes unit vector is shown with a cap I mean different books just use different notations. Its tangential component is this is equal to this is what this is a gradient along the direction s grad s okay. So x d sigma dx is replaced by this operator. So how will this operator look how will the grad s operator look? So to understand that let us say let us think of an acceleration vector or any vector a not acceleration just any vector a. So a is vector sum of the normal component plus the tangential component any vector a. So a s is equal to a-an this you can write a-an you can write n dot a what is n dot a? n dot a is the normal component of a n is the unit normal vector and that times the unit vector along the normal direction is the a n vector right. So with this similarity you can write del s is equal to del-n dot del n just replace a with del okay. So we will summarize this discussion through the next slide general shear stress boundary condition. So here we are talking about the gradient of surface tension for which we use an operator which is called as surface gradient. So for the surface gradient this operator has a symbol del s which is equal to del-n dot del n okay. So you can write the traction vector dot s is equal to del s sigma dot s what we wrote in the board. So that is equation number 13 in the slide you can see that this we have already derived. Now this equation what is written here of course we have to keep in mind that like if these are dimensional parameters all these will be tau dash sigma dash n dash and s dash and n and s are the same. So that is not highlighted but these are actually tau dash del dash sigma dash. Now from here one has to derive suitable equations which are representatives of this equation under the assumptions of lubrication theory that will be somewhat simplified versions of this. This is the general one. So for example if you are simulating the tangential stress boundary condition for a problem in CFD this is the boundary condition that you will use. Now a little bit of simplification with neglecting the sum of the small terms will result in a lubrication theory version of this equation. We will take that up subsequently. So now the question is just like we have 1u we also have 1v right. If it is not a fully developed flow v is not identically equal to 0. So there must be a boundary condition for v and the boundary condition for v when the interface so if you see if there is a flat film at steady state v is equal to 0 at the top surface right. However if the interface is moving upwards with time that is the film height is changing with time then v is equal to dh dt right. This is straightforward but this is a special case that v is equal to dh dt when the film is flat. But what happens when the interface is deformed and moving or changing with time. So for that we will use a boundary condition which is called as kinematic boundary condition. The kinematic boundary condition is essentially a generalization of the no penetration boundary condition. So what is the physical meaning of the kinematic boundary condition we will first write what is the kinematic boundary condition and then we will give a physical interpretation to the boundary condition. This is a very very important boundary condition and we should learn this very carefully. So let us say that we have a function f is equal to y- h x t. h x t is the height of the interface as a function of position and time. Now the kinematic boundary condition says that any fluid particle located at the interface must stay at the interface at all times. Any particle any particle means fluid particle any fluid particle located on the interface must be there on the interface at all times. That means the total derivative of f with respect to t must be 0. This is the kinematic boundary condition. So total derivative of f means del f del t f is a scalar. Now we can write an alternative form of this by noting that what is grad f? What is can you tell what is the direction of grad f? Direction of grad f is normal to the surface f. That is the maximum rate of change of f that is along the normal direction of f. So if we divide grad f by its magnitude we will get the unit normal vector n. So if we divide by mod of grad f both sides then this will be n. This boundary condition is called as no penetration boundary condition. Sorry kinematic boundary condition and its special form of and the special form of this kinematic boundary condition is the no penetration boundary condition. So you can clearly see that at a given location if f is not changing with time this term will become 0 and this is like the normal component of v is equal to 0. Then that is the no penetration boundary condition. Let us go to the next slide. So we had made some intuitive derivation. Again I am like giving you a remark that ideally we should have put the dash for all the terms since we had used the dimensional forms not yet the non-dimensional forms. But just for the easiness of writing we have avoided the dash and we have just written the so all these equations without dash. But if you are very particular please use a dash when you convert this to your notes. So the x momentum y momentum continuity equation the boundary conditions the bottom plate u equal to v equal to 0 free surface 2 boundary conditions one is the kinematic condition and another is the shear stress balance. Now a special form of this for flat surface is v is equal to dh dt and tau xy is equal to d sigma dx okay. So this like the general form implies the special form for flat surface and the equation of the film surface is f that is y-h xt equal to 0 and for a flat surface it is y-h0 equal to 0 that means y is equal to h0 right that is the equation of the flat surface. But equation of the curved surface which is evolving with time is equal to y is equal to h xt okay. Now what is missing here? So instead of going through the slide which is a bit heavy we will try to understand this through common logic. So let us come back to the previous slide maybe that will help. How many governing equations are here? 3 governing equations are here right. How many unknowns are here? So very common unknowns which are straight away apparent are u, v and p right. Now there is another unknown which is a subtle unknown what is that? No, no, no theta is like that is the inclination of the plane right. So that is known of course I mean you have a inclined plane and you do not know the inclination of the inclined plane then you cannot set up the problem. H, H is not known. H as a function of x and t that is not known. So now you see that actually there are 4 unknowns. Now you can see here is that there are now these 4 unknowns but each equation for getting each unknown you will require boundary conditions. For getting u how many boundary conditions you require? 2 boundary conditions. For getting p from the y momentum equation how many boundary conditions you require? 1 because it is a first order equation and the continuity equation is also a first order equation so you will require 1 boundary condition. So total 4 boundary conditions are necessary to close the system right. So you have 1, 2, 3, 4. So system appears to be closed but the problem is that H itself is unknown. So you require 1 additional constraint to constrain H to give the variation of H and that constraint we will now look into. So the condition or equation that is missing would give an equation for the film thickness. So why that equation for the film for the film thickness was necessary that leaving the film thickness apart, leaving the film thickness thickness apart we had how many we had the x momentum equation for which there are 2 boundary conditions y momentum 1 boundary condition and the continuity equation 1 boundary condition. So these were requiring 4 boundary conditions which were prescribed but that would have told the system provided H is a constant but H if H is not a constant you require additional constraint and that will then make it a fully well posed problem. So for that we will require a condition which is called as the normal stress balance condition. So we will come to the board to explain that. So when you have an interface we have discussed about the tangential stress balance earlier but what about the normal stress when you have a curved surface. So you have let us say this is S and this is denoted not S we will use a different subscript no subscript let us say here the pressure is P and here the pressure is P air on this side there is liquid on this side there is air. So P-P air this is the pressure difference this is the equivalent normal pressure difference in form of normal stress if the fluid is at rest if the fluid is moving if the fluid is deforming let us say that a liquid droplet that is spreading on a substrate then there is a normal viscous stress also that comes into the picture. So what is the normal viscous stress so this is what the traction vector this is nothing but the Cauchy's theorem. So this is very important whenever we write the pressure difference in terms of surface tension sometimes we omit this even on the dynamic conditions because we will see that in lubrication approximation this may be one order less as compared to this or whenever the droplet is in static condition then this does not play a role this tau is tau viscous that is the tau deviatoric okay. So why this is minus sign because the normal positive normal stress is opposite to positive pressure positive pressure is normal inwards and positive normal stress is normal outwards. So this is equal to what sigma yr again like if you are interested to write the dimensional and all these things better write the dash shear stress anyway tau has to be tau deviatoric right I mean even if you take the I mean even if you consider include the hydrostatic part hydrostatic part cannot generate a shear right. The shear stress has that is what I made this important remark earlier also that when you are talking about shear stress you are conscious that you are talking about the deviatoric stress whenever you are talking about normal stress you are not always conscious that viscous effects can also give rise to normal stress because normal intuition is that only pressure is giving rise to normal stress which is not a correct intuition okay. So this boundary condition closes the system of equations. So we will look into some consequences of this boundary condition once we evaluate the pressure we can get an equation for the interface shape by combining the previous equations 15 and 16 that should theoretically complete our problem. Now the question is that you can close this system of equation without this normal stress balance for a flat film and the reason is quite obvious that for a flat film for a flat surface what is the radius of curvature the radius of curvature is infinity okay. So radius of curvature is infinity means 1 by r equal to 0. So that means you have p dash equal to p dash air if you are not concerned about the deviatoric stress just as a simple case in the y direction you have p dash equal to p dash air the normal direction becomes the y direction. Not only in thin film problems but in any other fluid mechanics problem where the interface shape and position is not known this same procedure is applied. So this procedure that I outlined so 2 important boundary conditions for a curved surface one is the tangential stress balance another is the normal stress balance. What is the origin of this normal stress balance see this is nothing but this delta p equal to sigma into 1 by r1 plus 1 by r2 that particular formula we have applied here. Now the delta p is augmented by the normal deviatoric component because it is under dynamic condition. Now in what follows we will re-derive the thin film equation. So we have derived the equations without referring to the scales that is without referring to the parameter epsilon which is the ratio of the length scale in the y direction y dash direction as compared to the length scale in the x dash direction. So we have not used that as a parameter so far. So we have described the equations but we will now introduce that as a parameter. So to do that see now in this slide we have used the prime notation because we will first write the dashed quantities that is the dimensional quantities and then suitably derive its non-dimensional form. So to recapitulate see the basic difference between the rigid surface and the thin film problem is that this blue surface it is now it is not a rigid surface it is a free surface. It is evolving with position and time, typical length scales so epsilon is equal to H0 by L0 what is H0, H0 is a typical film thickness. Now you can say that if the film thickness varies significantly then I mean a typical film thickness has no meaning because I mean say you have taken this as H0 but somewhere it is the film thickness is 100 times H0 then that also becomes a typical film thickness. So you can take any film thickness as the characteristic length scale along y provided that the film thickness along y is slowly varying that is the variation is not significant okay. So that film thickness divided by Lc is the small parameter epsilon and that small parameter is much much less than 1. So now to summarize the boundary condition so this is a general problem not really a thin film problem but if you have a general problem so we have put this slide because you know this understanding you can apply not just for problems of thin films but any 2 phase flow problem that you are solving. So these boundary conditions have to be satisfied and this is the general form of the governing differential equation we have retained the inertial terms also but you know for thin film with lubrication approximations many of these terms will not be important. So a recapitulation of the various scales so u prime is of the order of uc this is exact recapitulation of what we did for the lubrication theory. I mean we will reproduce some of these just so that I mean you can refresh your memory we will reproduce the derivations of some of these but instead of doing an elaborate derivation we will quickly do some smart back of the envelope calculations to make this derivation. So for example u prime is equal to u prime is of the order of uc v prime is of the order of vc which is epsilon uc where from this comes continuity equation very good p prime is of the order of pc which is mu uc by epsilon square lc say you have forgotten this formula suddenly so how will you retrieve this very quickly so how do you get the pressure scale. So let us write the x momentum equation with the dominant terms so how will you get the dominant terms in the equation you substitute u equal to u0 plus epsilon u1 plus epsilon square u2 p equal to p0 plus epsilon p1 plus epsilon square p2 like that out of that whatever is the leading order terms you collect. So that comes from mathematics but your physical intuition tells that certain terms which should be dominating that must be present in the x momentum. So what are the dominating terms 0 is equal to the pressure gradient along x is one of the dominating terms then out of the 2 viscous stresses components one is del square u del x square another is del square u del y square y length scale being much smaller than the x length scale the del square u del y square should be the dominant right. So instead of writing all the epsilon and all those things you can simply write the dominant term plus the body force term in the lubrication theory discussion that we made earlier this body force term we just kept as general if you recall we kept fx prime fy prime without referring what they are now we have the gravity as the body force in the special problem that we are considering for thin films. So rho g sin theta right there is a mu here mu has to be there okay now this term is what minus pc by xc xc means lc del p del x this is what mu uc by h0 square del 2 u del y2 this is rho g sin theta as it is right. So we have non-dimensionalized with suitable physical scales such that these are of the order of one that means if the pressure gradient is competing with the viscous that gives the scale pc. So pc is equal to mu uc lc by h0 square h0 square is epsilon square lc square right so you have mu uc by epsilon square lc okay. So now you can write this equation in a dimensionless form is equal to 0 is equal to so multiply all the terms by lc by pc. So the first term becomes minus del p del x second term will become del square u del y square because this is pc by lc so multiplying this by lc by pc we will make it 1. So plus rho g sin theta into lc by pc. So this is the dimensionless form of the x momentum equation. We have essentially multiplied all the terms in the dimensional form by lc by pc is it alright there is lc by pc yes okay. So similar thing we can do for the y momentum equation. So for the y momentum equation let us write the dimensionless dimensional form so y momentum 0 is equal to what is the leading order term the v terms are much smaller than the corresponding u terms. So pressure term being important the v terms being much less than the u term the v terms will be much less important as compared to the pressure terms. So minus rho g cos theta will be there this will be p dash y dash okay. So this you can write as 0 is equal to minus pc by h0 del p del y minus rho g cos theta because p scale is pc and y scale is h0. So pc is so multiply both the terms by what h0 by pc so 0 is equal to minus del p del y minus rho g cos theta into h0 epsilon square lc by mu uc this is h0 by pc what is h0 epsilon into lc. So this becomes 0 is equal to minus del p del y minus rho g cos theta into epsilon cube lc square by mu uc okay. So we have simplified the x and y momentum equations considering the lubrication approximation and we will take it up from here in the next lecture. Thank you very much.