 In the previous lecture, we started discussing about thin film dynamics and we started looking into the governing equations and the boundary conditions and we tried to identify that what are the boundary conditions for the free surface and in terms of the tangential stress balance and the normal stress balance and the kinematic boundary condition. Then we tried to derive the dimensionless forms of the momentum equations and let us go to the slide to summarize the various dimensionless forms. So these forms could be derived by expressing u and v, u, v and p in terms of the small parameter epsilon u equal to u0 plus epsilon u1 plus epsilon square u2 and so on. Similarly, v and p and these are the equations. The x and y momentum we derived in the just towards the end of the previous lecture and the continuity equation form is quite obvious that is this one. Boundary conditions, the bottom wall boundary conditions the bottom wall boundary condition is quite straight forward the no slip and no penetration and we will now devote significant attention towards describing the boundary conditions at the top wall. So the kinematic boundary condition how do we express it in terms of the small parameter? So let us try to work it out in the board as much as possible and then we will refer to the slides. So you have this is the kinematic boundary condition. So what is f dash, y dash minus h dash what is grad f dash where i and j are the unit vectors along x and y and y direction and x prime y prime directions are the same. Now let us simplify this. So to simplify this what is del f prime del x prime that is del h prime del x prime right because in f prime h prime is a function of x prime. What is del f prime del y prime 1? So this you can write h prime scales with h0 and x prime scales with lc. So when you write it in a dimensional form a dimension less form it is h0 by lc into del h del x h0 by lc is epsilon. So minus i epsilon del h del x plus j. So what is the normal vector? For getting the normal vector you also require to calculate mod of grad f dash okay. So what is that? It is square root of epsilon square del h del x whole square plus 1 to the power half. So what is n cap in the leading order if you divide this with this and consider that epsilon is a small number then what will be that? It will be minus i epsilon del h del x plus j plus higher order term because in the division this term will tend to 1 as epsilon tends to 0 okay when you divide this with its mod. So now del f dash del t dash so this equation this is actually the original form of this equation the kinematic boundary condition. Now del f dash del t dash is equal to what? Minus del h dash del t dash v dash you can write this as u dash i cap plus v dash j cap right where u dash and v dash are the components of the velocity vector along x and y dot with sorry this is scalar so this vector sign will not be there. So grad f dash is minus i epsilon del h del x plus j this is equal to 0. So now what is the scale of h dash h0 t dash is tc so h0 by tc del h del t plus what is the scale of u dash uc so uc into u0 plus vc into v0 u0 and v0 are the dimensionless 0th order terms for u and v dimensionless because no dash is there at the top and 0 subscript means 0th order so u equal to u0 plus epsilon u1 that because we are writing only the leading order terms. So as I am telling that these things this can be worked out by writing u equal to u0 plus epsilon u1 but now we have we are trying to be smart enough to like a predict what will be the form without always writing that big form expanding and neglecting small lower order terms I mean that kind of insight I mean as we practice more and more that kind of insight should come so that it saves some effort. So this dot with minus i epsilon del h del x plus j equal to 0 what is tc tc is yes lc by uc right so lc and the numerator you get uc so what is h0 by lc epsilon so minus epsilon del h del t plus uc you can cancel from both the terms if you cancel uc from here this will be u0i plus vc by uc is epsilon plus epsilon v0j dot minus i epsilon del h del x plus j equal to 0 require one more step in the camera picture board is almost invisible so I have to shift to the side so minus epsilon del h del t minus u0 epsilon del h del x plus epsilon v0 that is equal to 0 this is alright so you can write v0 is equal to del h del t plus u0 del h del x so this is like what is this like this is like the total derivative of h right so the total derivative of h the total rate of change of h is the v velocity component in the leading order that is what the kinematic boundary condition is giving so we will summarize this discussion of kinematic boundary condition in the form of the slide so you can see that we have calculated the normal vector so if you look into the slide as grad f by mod of grad f so then j minus epsilon del h del x i plus some term of the order of h square which is small so you can see that v0 is equal to del h del t plus u0 del h del x this is the kinematic boundary condition so we have expressed the kinematic boundary condition in terms of the small parameters using the lubrication theory approximation so this is one boundary condition next we will move on to the shear stress boundary condition so in the shear stress boundary condition see we start with the dimensional form so tau dash dot n this is what this is the traction vector in the dimensional form dot s is equal to grad s sigma dash dot s so we have derived this form earlier in the previous lecture at where at y is equal to h h is a function of x and time so del s we have discussed is known as surface gradient operator keep one thing in mind we have not written actually it would have been better if we are written here del s also prime because this is a gradient operator in dimensional form not dimensionless form but just for I mean simplicity in writing we have written it just like the normal grad operator but you have to keep in mind that when it is operating on a dimensional variable we are looking for dimensional derivatives that paradigm we should keep in mind so grad so now I will go through the slide because the algebra maybe a little bit more involved but there is an intuitive part of the algebra like I mean if you once you get more and more experienced in fluid mechanics you will realize that there is some algebra which without doing the bulwark you can get the essence of the algebra and that is through the scales and how you do that I will show you I mean that will help you to do the necessary algebra without doing too much of bulwark all the time so the surface gradient operator del s this becomes I del del x plus del del y into del h del x see this form this is like you have the difference between the flat surface and the curved surface because the curved surface h has a tangential direction that continuously changes as you move along the surface so for that to accommodate that you have the extra term so you had that del del x if it were a flat surface now here you have del del y but it is like a chain rule it is as if like del del y into del h del x where this h is a representative of the variation of the height of the surface as a combined function of x and y so it is it resembles a sort of chain rule del del x plus del del y into del h del x it is an operator that sort of gives the tangential derivative by considering that the tangential direction changes continuously because h is a function of x okay then the tau dash dot n dot s so first we note so we have got the n so it is basically you have to do the bulwark you have to make this tensor product of tau dash tau dash you can write that is the deviatoric component of the stress tensor so that is like mu into del u i del x j plus del u j del x i all those terms you have to assemble then n is the unit normal vector we have already derived what is n so that is what minus i epsilon del h del x plus j that is the n direction what is the s direction we have written i plus epsilon del h del x j how do you get it just let us try to quickly do this I mean these are all basically like vector algebra I mean nothing more than that but just in case you feel little bit uncomfortable I am trying to do all the hidden steps wherever possible so you had n is equal to minus epsilon del h del x i plus j let us say that s let us say that s is equal to a i plus b j where a and b are to be determined then what is n dot s dot product of normal and tangential they are orthogonal to each other so that is minus a epsilon del h del x plus b equal to 0 that means b is equal to a epsilon del h del x not only that s is a unit vector that means its magnitude is unity that means a square plus b square equal to 1 so a square plus b square equal to 1 means a square into 1 plus epsilon square del h del x square is equal to 1 that means considering this as much smaller as compared to 1 you have a is equal to 1 and if a is equal to 1 b is equal to epsilon del h del x so the s vector is i plus epsilon del h del x j okay so let us come back to the slide so tau dash dot n dot s okay so you can express this in terms of mu this assume that it is a Newtonian fluid you can express this in terms of see this is the stress tensor this is the unit normal vector this is the unit tangent vector you just make the product as it is just do the bulwark then when you do the bulwark you will get the corresponding dimensionless terms multiplied by a scale of tau okay so what should be the scale of tau I mean if you work it out you will get mu uc by h0 but you can easily get it from your common intuition like let us let us write this term so tau dash dot n dot s so this is equal to some thing into tau dot n dot s where this tau is the corresponding dimensionless form so tau dash what is the leading order term in tau dash mu del u dash del y dash right so mu del u dash del y dash it is corresponding scale is mu uc by h0 so you can you simply mu uc by h0 will come here okay so you can do that without I mean going through any algebra just the dimensionless form dimensional form and the dimensionless form they are related by this scale this must be true okay now right hand side you have the surface gradient of sigma dash where this is the dimensional surface tension dot s right so how do we get the surface tension gradient question is this so what is the scale of the surface tension gradient is it sigma by lc so let us try to look into that what should be it so to look into that let us say let us assume that the surface tension scales as sigma dash is of the order of sigma 0 that is sigma 0 is the scale of surface tension I mean typically we could have written sigma c but sigma 0 is the common notation used in books so that is why I have written this now that does not mean that the surface gradient of sigma dash will be sigma 0 by lc why for example let us say that sigma dash is equal to sigma 0 into 1 plus beta by lc plus beta x by lc this is a linear term then you can have a quadratic term cubic term etc. So you can tell that I mean are we just doing arbitrary mathematics what is the basis of taking variation of sigma with x like this so let us say that you have imposed a temperature gradient along x right let us say that you have imposed a t versus x linear because the surface tension will vary with temperature therefore sigma may also vary with x linearly if t versus x is linear. So this is a signature of sigma versus x linear so if this is a signature of sigma versus x linear then it is gradient with respect to x is beta into sigma 0 by lc right just differentiate this with respect to x this is the leading order term see in the surface gradient only the leading order term will feature in the lubrication approximation. So this will become so the grad s sigma dash will become sigma 0 beta by lc so these are the two things that we will remember so this we will note down in the board sigma 0 beta by lc let us go to the next slide. So you can write this is the remember this is the order of magnitude of this right so you can write mu uc by h0 tau dot n dot s these are all dimensionless parameters is equal to sigma 0 beta by lc grad s s sorry grad s sigma grad s sigma dot s okay because this is the order of magnitude of the term the remaining is dimensionless dimensional equal to dimensionless times its scale that is the form that is that we are writing so tau dot n dot s is equal to sigma 0 beta by mu uc what is h0 by lc epsilon so this you can write epsilon beta by mu uc by sigma 0 into grad s sigma dot s what is this mu uc by sigma 0 this is the ratio of the viscous force and surface tension force so this is called as capillary number a very important non dimensional number in the context of thin film flows viscous force by surface tension force so we will summarize this part through the slide so you can see the boundary condition equation 13 it is written here tau dot n dot s minus epsilon beta by capillary number grad s sigma dot s equal to 0 where grad s sigma dot s is equal to you can see that chain rule is used del sigma del x plus del h del x into del sigma del y so this we call this is the shorthand notation we use del tilde sigma del tilde is del del x plus del del y into del h del x okay this is the shorthand notation that we use now tau dot n dot s so if we want to evaluate that so tau dot n dot s so the algebra is a bit tedious I have put this algebra in the slide just to tell you just to convince upon you that for every work there is a bull way of doing it and there may be a little bit more tricky way of doing it so if you see here tau dot n we have found out so this is tau and this is n, n is minus epsilon del h del x i plus j right so the components of n are minus epsilon del h del x 1 this we have derived in a previous lecture then you have tau dot n so you have bent the dot product and then tau dot n dot s we have derived what is s also so if you do that you can come up with so many things ultimately in the leading order it is del epsilon del u0 del y now this you can say without any bull work because tau dot n dot s is eventually the tangential component of the stress acting on the surface in its leading order in a dimensionless form so tau if it was in a dimensional form it would have been mu du dy but now so mu into u dimension by y dimension so out of that now if you write it in terms of dimensionless parameter that mu into y dimension and y dimension by y dimension will be absorbed and it will simply become del u del y in a dimensionless form so this is a dimensionless form of mu del u del y okay so with all this algebra this is where you land up with which is which is quite intuitive so the boundary condition becomes that tau dot n dot s becomes del u0 del y minus epsilon beta y capillary number del t the sigma equal to 0 at y equal to h x t you can clearly see that if there is no surface tension gradient at the top surface then del u0 del y is equal to 0 that is the velocity gradient at the free surface of the film is equal to 0 okay so this is tangential stress balance so what boundary conditions we have discussed through lubrication theory kinematic boundary condition and tangential stress balance we are left with the normal stress balance so now we will discuss the normal stress balance this is the final boundary condition that we apply for our analysis before we can solve the whole system of equations normal stress balance so there is a particular note that we have put here and this note is important from a physical perspective normal stress balance is a bit difference from shear stress balance this is simply because while balancing the normal stresses we have to take into account pressure which acts always normally to the interface apart from this we have to include the effects of Laplace pressure that is the pressure difference across the interface caused by surface tension and surface curvature this also we have discussed in addition there is something that we have not yet discussed that we have this is the thin film when the thin thickness becomes very small ultra thin where intermolecular forces of interaction come into the picture then the van der Waals forces of interaction become important and that gives rise to and another additional component of a pressure like quantity which is called as disjoining pressure so that we shall denote that as p dash extra okay and I will show you that where in the mathematical formulation you put it now the disjoining pressure what is the formula and expression for that it is also a function of the film thickness it will typically scale with 1 by h to the power n where n is typically an integer so where h is the film thickness so with the film thickness becomes very small and small this will become this disjoining pressure term will become very very important now expressions for disjoining pressure how they are derived and all those things I mean that is basically within the purview of the theory of the molecular theory of like the corresponding interaction forces and I can refer to you a very good textbook if you are interested that intermolecular and surface forces by israeli israeli value it is a very authentic textbook you can intermolecular and surface forces so you can read that textbook to get the details of what it is but in terms of using it in the continuum approximation I will continuum formulation I will show you where to plug in that term in the thin film analysis so that you can work out that use that expression to work out a thin film problem so you can see here that p dash-p atmosphere-tow dash dot n dot n this is the viscous part that we discussed this is the hydrostatic component is equal to sigma dash by r plus p dash x okay now how do you get this r from n actually from vector calculus you can relate the radius of curvature with the divergence of the normal vector so this 1 by r if r is dimensional then this is the actually delta dash dot n delta dash as I told that we have not written delta dash in the slides so this is the dimensionless sorry this is the dimensional divergence so this 1 by r so now let us see that how we can put various scales here so let us try to put various scales so this term we have tried to simplify by usual scales so let us the final result is given here but I will try to see that like what form we get by putting various scales so p dash-p dash atmosphere-tow dash dot n dot n is equal to sigma dash by r dash or sigma dash by r r is a dimensional radius of curvature plus this one so p dash is pc into p right p dash atmosphere is pc into p atmosphere this is tau dash sorry tau dash is equal to tau c tau c is mu uc by h0 into tau dot n dot n this is equal to sigma 0 into sigma by sigma 0 by r into sigma right plus px is there so this different terms so you will get p-p atmosphere-what is pc just tell what is pc tell from your notes what is pc yes mu uc epsilon square lc so you multiply by or you divide both sides by pc so mu uc by h0 into epsilon square lc by mu uc into tau dot n dot n is equal to sigma 0 by r sigma into epsilon square lc by mu uc okay plus p p dash x into epsilon square lc by mu uc so now let us look into this term mu uc gets cancelled h0 by lc is epsilon so 1 epsilon gets cancelled right then here you can express r in terms of lc you can non-dimensionalize r in terms of lc so you can write rc so if you write this r in this particular form that is also okay but I mean sometimes we write r non-dimensionally we will see in the slide how we write r non-dimensionally but before that you can see that here also the capillary number appears which is more important mu uc by sigma 0 now to non-dimensionalize r if we define a non-dimensional r as r by lc or lc by r whatever then this will become epsilon cube or in other words depends on how we actually non-dimensionalize so better we will keep it in this form so this particular form is sigma by r into 1 by capillary number into epsilon square lc so you can make a suitable choice of the scale suitable choice of the non-dimensional r such that this epsilon square you can absorb through lc by r it becomes epsilon cube so that is what is shown in the slide but that is again it depends on the choice so you can clearly see here so how do you come up with the choice of a suitable scale for r how can you come up with a suitable choice for the scale of r so you have so let us write this as del dash 1 by r is equal to del dash n so what is del dash dot n what is del dash i del del x dash plus j del del y dash dot what is n j minus epsilon del h del x i so this you can convert del del x dash to del del x and you can convert del del y dash to del del y how do you do that so del x dash is equal to x into lc so i del del x 1 by lc plus j s 1 by 8 0 8 0 is epsilon lc then del del y dot n right so 1 1 by lc comes out there in this expression in that 1 by r so that lc will cancel with this lc okay there is a 1 by lc common you can see here that 1 by lc will cancel with this 1 and this 1 by epsilon this will go in the numerator to make it epsilon cube okay so if you come to the slide you can see that p minus p atmosphere minus epsilon tau dot n dot n is equal to epsilon cube by capillary number into del dot n these are all dimensionless sigma this is dimensionless sigma now how do you calculate the term tau dot n dot n and epsilon tau dot n dot n so tau you can express in terms of the velocity gradients the non dimensional velocity gradients n we have shown that how the n comes so tau dot n dot n you can see that it comes of the order of epsilon square so that is why see many times ignorance is a blessing in disguise as I have told earlier and again I am repeating see this term being of the order of epsilon square and this term being of the order of 1 actually it will not matter if you do not have this term because this will be of the order of epsilon square so fundamentally it will be wrong if you do not write that term but that term will not be a dominant term so and not only that del dot n so there is a del dot n why do we require the del dot n del dot n we require because you have this epsilon cube by capillary number del dot n so del dot n del is i del del x plus j del del y dot n is that minus epsilon del h del x i plus j so this becomes minus epsilon del square h del x square okay so then this becomes p s minus p atmosphere is equal to this so this is the expression that relates that is basically the previous expression where that del dot n is substituted so note that in the boundary conditions we have term we have retained some terms involving epsilon or higher powers of epsilon so you might argue that since these are higher powers of epsilon why have we retained this in the boundary terms this reason we have already discussed in the context of lubrication theory what is that reason the reason is thus these terms actually incorporate certain other scales which are yet to be determined like u c for example so look at this term so if you look into this term you will realize that you have the capillary number where you have mu uc by sigma 0 so in that there is a uc unknown so that uc may be as large or as small to make this term comparable to p minus p atmosphere even if epsilon is small because you do not know whether that uc is large or uc is small so you have the scales and these scales see the beauty or physics of these problems is that these scales are not so much describable a priority these scales depend on the physics of the problem so what is the dominating factor is it the body force is it the boundary condition along y direction is it the boundary condition along x direction or whatever is it the surface tension gradient like what will be uc for example if you have a surface tension gradient if you have a surface tension gradient then what will be uc the surface tension gradient then we will dictate what is uc how will you find out uc you have like the surface gradient of surface tension coefficient is equal to the tangential stress and the tangential stress scale is mu uc by h 0 so from there you will get the appropriate uc so depending on what is the dominating factor you will get the scale established from that parameter so when we describe the lubrication theory we do not say what is uc what is vc all these because these are apparently unknown generalized parameters which can only be affixed once you know what is the physical situation governing the problem so the important question is that exactly what we should solve for see there are too many equations there are quite a few boundary conditions so what we actually solve for so we saw in the lubrication theory one of the main unknowns is pressure here also pressure is an unknown along with that the film thickness is not known a priority once the pressure and the film thickness is known all other quantities can be easily deduced hence our aim should be to find out an equation for film thickness using the available equations and boundary conditions we will take it up in the next lecture thank you very much.