 So, in the last lecture, we basically looked at Turing patterns and we looked at how diffusion can actually give rise to spatial instability, spatial patterns being formed provided the diffusion coefficients satisfy a certain relationship okay and the reaction kinetics have some property of an activator species and an inhibitor species. So, that was just to give you an idea about how the general theory of the stability that we are doing can be actually applied to a wide variety of systems okay. So, now today we will come back to fluid mechanics and we are going to look at this problem of Marangoni instability. So, what I am going to do is first talk a little bit about Marangoni instability then we will talk about the formulation of the boundary condition and then we will come and actually solve the problem of Marangoni instability okay. So, do this thing in 2 or 3 different ways. So, you can get a overall picture okay. So, this Marangoni instability, the physical system is the same as what we have seen earlier in the context of the Rayleigh-Benard problem. Now just to refresh, the Rayleigh-Benard problem was one where you had a liquid and what you are doing is you are heating this. So, this is at a higher temperature T high okay and let us say for the sake of argument that this interface this is a solid wall okay that is a solid wall and here it is a let us say this is gas. So, you have a liquid layer resting on a solid wall exposed to an atmosphere of gas. So, there is a small difference in the sense last time when we did the Rayleigh-Benard problem, both the walls were solid or both the walls were free okay. But now all I am saying is the lower fellow is solid, upper fellow is free. So, you have a gas liquid interface here interface and let us say the ambient temperature is constant at some value T ambient. So, the first thing which comes to mind is when I am going to be heating this, there is going to be a temperature gradient which is established okay and beyond a certain Rayleigh number, you will which we derived last time some 650 for a particular boundary condition, so 1700 for some other boundary condition you will see the onset of convection okay. So, as the lower plate temperature increases, we see convection if the Rayleigh number is greater than a critical Rayleigh number and this is provided buoyancy is the one which is actually driving the convection okay. This is when buoyancy drives the convection. You remember in the gravitational term we had rho g and rho we said was varying linearly with temperature and we included that and the first experiments actually done by Bernard, they were actually with a very, very thin film okay of liquid. So, the liquid film was very thin which means the Rayleigh number has this film thickness remember it has the film thickness which is d and is raised to the power cube raised to the power 3 okay. So, Rayleigh number is proportional to d cube where d is the film thickness. The point is even when Rayleigh number is of the order of 1 that is much lower than the critical numbers that we have seen okay, much lower than the critical Rayleigh number convection has been observed. In fact, Bernard who first observed these patterns he did not do it quantitatively. So, we actually do not have any estimate of the Rayleigh number but the idea was that the film is very thin. So, the film is very thin we are suspecting that the Rayleigh number is much smaller okay. So, one can come up come to two conclusions. One is that the theory that we did was wrong in the sense we are predicting Rayleigh number has to be greater than 1700 for the convection okay. So, clearly experiments are showing that you have convection even when Rayleigh number is 1. So, that means something is crazy okay. That means the theory is wrong and we have to go back and relook at it maybe go back and make some more realistic model okay or the other conclusion is that the convection is actually not being driven by buoyancy but by some other mechanism yeah. When if the film is extremely thin then and you are having you are heating it then the interface will deform if convection sets in and if it is thin then that will have it will play a role in determining everything. So, we have not taken that into consideration. We did not take that into consideration in the Rayleigh-Bernard experiment yes. So, if you remember the Rayleigh-Bernard experiment well one analysis was with solid walls okay. So, in that case you do not worry so the question is I mentioned that the resolution of this conflict between the theoretical prediction and the experimental observation is there. In the sense the theoretical prediction predicts 1700 or more okay and experimentally of course 1700 is for 2 rigid walls and what you can do is actually do a prediction when you have one solid wall and one free surface here and you can calculate the critical Rayleigh number and also be still high. So, the question was maybe the assumption of the interface being flat is the problem okay. So, if you allow the interface to deform then maybe the critical Rayleigh number would be much lower okay. So, that is also a possibility so that is one way by which you actually go about. So, is it possible that by allowing the interface to deform the critical Rayleigh number will be lower that is the question okay. So, we had assumed that the interface was flat and got a high Rayleigh number critical. What if the interface is allowed to deform. So, maybe that can result in a much lower Rayleigh number is that the point maybe this will result in a much lower Rayleigh critical okay. So, okay the answer to this question is one has to actually do the calculation to find out the critical Rayleigh number is right but it is going to be still higher it is going to be much higher on the order of 1000s if you actually do the calculation. But the fact that there is the another mechanism other than the density gradient comes because what people want to even when you have experiments being done in outer space on a space shuttle where the effect of gravity is not there okay. People observe national I have not used the national convection they use observed convection. That is even in the absence of gravity even in the absence of this gravitational field which is what is actually giving rise to the convection in the Rayleigh binar problem. You can actually see convection okay which means that there is another mechanism which is actually causing the convection. So, for example how do you do these experiments one is you can go to outer space and do these experiments and people have done that just to negate the effect of gravity they say look gravity is causing this problem. So, I want to grow crystals semiconductor crystals let us say in outer space I do this over there and I do not have you know any imperfection I have very good quality crystals or you can do these experiments by just having a free fall. You just drop your experimental setup for 30,000 feet high in the atmosphere nothing the effect of gravity is neglected and then what you see is you do your experiments. So, of course you have a very short window for doing the experiment but time it actually falls. So, people have done this the fact is that even in the absence of gravity to explain why that there is another mechanism for causing this convection you actually do see convection okay and that is convection occurs in the absence of gravity and it will also occur if the geometry is such that the interface is below that is you have a solid wall on top and you have the liquid film at the bottom I mean liquid at the bottom exposed to another fluid okay even in that case you do see convection patterns. So, these convection patterns are basically present even when the gravitational field is in the opposite direction okay to the temperature gradient. So, that means the gravitational field in buoyancy term is not the one which is causing the convection. So, point is even when gravity is in the opposite direction I mean I think that is possibly the wrong thing to write. So, let me put it this way even when the solid wall is on top and hot and the interface is below it means this is my solid wall and that is my liquid and this is my gas liquid interface okay and that is my gas liquid interface we can see convection. So, you have actually hot liquid at the top hot surface at the top and cooler surface at the bottom okay and that is normally a stable configuration right. So, that is we have hot wall and let us say cold interface. So, here again we see convection. So, basically these are indications that there is something else which is actually causing the convection to occur okay. And so, the idea is what is this thing that is causing the convection to occur and that is that the Marangoni instability is basically being is the other mechanism and is this particular convection is actually being driven by surface tension gradients okay. So, what is causing this? It is surface tension gradients. What I want to emphasize here is that this atmosphere here is at a uniform temperature T ambient that does not mean that the interface here is at T ambient. There is going to be some kind of heat loss. So, I am not fixing the temperature of the interface okay. The temperature of the interface is something which I have to find out okay and it is going to be given by a boundary condition like your Newton's law of cooling which says that minus k dt dy equals h times T minus T ambient. This is at the gas liquid interface okay and the gas liquid interface and this is the typical kind of boundary condition which you normally use right. The conductive flux here must be equal to the convective flux. The point I am trying to make here is at the interface which is you can decide what it is y equal to b let us say. I am not fixing the temperature. I am just saying that T ambient is constant which is infinite amount of atmosphere available. So, that guy does not change. So, everywhere T ambient is constant but T can change okay. So, if temperature can vary at y equal to d in this direction then what can happen is the surface tension can be is usually dependent on temperature okay. So, now let us see what is going to happen because of that. So, supposing the earlier problem of natural convection we included the effect of density as a function of temperature. Now, we are going to include the surface tension as a function of temperature okay and now we are going to see whether that can actually give rise to convection. So, I come back to this figure this T high that is T ambient okay. Now, supposing you give a small perturbation as a disturbance which is actually let us say you know you have this kind of a configuration and you give a disturbance. What is going to happen is the liquid here which is hot okay. Let us say is getting pulled up. I am not going to say that it is being induced by buoyancy but let us say there has been a disturbance which has actually caused the hot liquid at the bottom to rise up. Now, at this point so this guy is at a high temperature. So, this is at a high temperature. The interface remember is at a low temperature. So, at a particular instant of time this guy is at a low temperature and there is a hot packet of liquid which has momentarily come up because of our disturbance. The question we are asking is how is this disturbance going to propagate? If this guy is at a low temperature and let us say the surface tension decreases with an increase in temperature and that is the usual behavior okay. For any liquid the surface tension is a function of temperature and let us say decreases with an increase in temperature which is a normal behavior. So, the surface tension here is going to be temperature is low. Therefore, the surface tension here is going to be high. So, gamma is higher and the hot packet of liquid has come up here. So, gamma is low. At the interface the force is higher compared to here and therefore, there is a tendency for this liquid to be dragged along the interface because here the surface tension is less, here the surface tension is high. This guy is going to be dragged. Similarly, here I mean the hot packet has come up right. Temporarily the temperature has gone up here because let us say one liquid has come up one small packet of liquid has come here. This guy is also low and so this guy will have tendency to come to this side. So, basically what I am saying is the liquid will have a tendency to go towards the colder temperature because of surface tension that is high the force there is high. Now, when that happens because of the equation of continuity because of conservation of mass this guy has to come down. When one packet has gone up this packet has to come down. So, this guy will come down and this guy also has to come down eventually to fill up this space here. So, this liquid will move and you have this kind of a convection pattern. The point I am trying to make here is that I am not talking in terms of density induced convection. I am talking in terms of surface tension induced convection okay. So, if the surface tension is not a function of temperature then this driving force is not going to be there. This guy is not going to pull okay. So, therefore what it means is I need to include the effect of surface tension as a function of temperature in my model and it is going to occur in the boundary condition okay and then we should be able to see convection. So, this is basically what Marangoni convection is all about okay. So, here surface tension decreases with the increase in temperature if we have a hot packet of fluid rising due to a disturbance okay. And this is dragged on both sides by the cold fluid which has a higher surface tension okay and by continuity which is basically conservation of mass the cold fluid comes down and we see a convection. Remember if it had been the other way if the surface tension had actually increased with temperature then you would not have seen this. If the surface tension had increased with the temperature this guy would actually be lower surface tension. This guy would be sorry this guy would have been higher surface tension and the liquid would have been pulled from here and it would have stabilized the flow okay. So, basically what we want to do now is talk in terms of the boundary condition which is going to be applicable here and remember till now we only talked about the normal stress boundary condition. And basically here what is going to happen is the boundary condition that we are interested in is the tangential stress boundary condition okay. So, we need to basically find out how to include this boundary condition and boundary condition has to take into account the gradients in the surface tension which is actually being caused by the gradients in the temperature. Now the surface tension of course is also going to be dependent upon the concentration. So, for example you know that if you add a surfactant the surface tension is going to go down okay. So, similar to this temperature gradient if you also had a concentration gradient and if the concentration increases or the surfactant increases you will have a decrease of the surface tension okay. So, the same kind of a behavior you can expect to see. So, basically when you can actually add surfactants and you can you know have concentration variations inducing convection okay. So, that is something which people have also done. So, in addition to temperature gradients you can also have concentration gradients which can actually cause Marangoni instability. So, it is not just Marangoni instability is a very general thing it is talks about surface tension variations which can be either due to temperature or concentration or anything which can actually cause convection okay. So, if the motion is induced by surface tension gradients we have Marangoni convection. So, what I want to do now is this is just the background for why we have to worry about the surface tension gradient. Now what I want to do is talk about the formulation of the boundary condition at a general let us say liquid-liquid interface. So, normally you are used to dealing with flat surfaces continuity of shear stress okay. So, let us look at certain things. So, when we talk about interfaces one thing is we say that the interface is infinitesimally thin. So, it has got 0 thickness that means it has basically got no mass okay. So, what that means is at the interface we always have a force balance that is to say the net forces we are acting on the system on the interface is going to be 0 because if it is not 0 that means there is going to be some kind of acceleration if the mass is 0 the acceleration has to be infinite okay. So, basically what this means is since interface has 0 thickness negligible mass negligible mass negligible thickness okay the forces acting on an element interface has to be 0 why if it is non-zero that means there is going to be an acceleration and which will be actually infinite because the mass is 0 okay. So, this okay if not we would have let us say infinite acceleration at the interface. So, of course when you talk in terms of a liquid-liquid interface let us say this is liquid 1 and that is liquid 2 okay. So, you have molecules all over the place and you also have molecules all over the place here just to differentiate these molecules I am using 2 different symbols the molecules which are in the bulk okay far away from the interface they are going to be surrounded by molecules of the same liquid this guy is going to be surrounded by molecules of the same liquid. So, whereas the guys at the interface at the bottom they are surrounded by molecules of liquid 2 and on the upper side they are going to be surrounded by molecules of liquid 1. So, what I am saying is the cause actual cause for the forces they are acting on the interface is actually attributed to molecular interactions at the interface whereas here it is completely symmetric and uniform. So, this guy has an atmosphere which is only of this kind of molecule this guy has molecules of liquid 2 partially and liquid 1 okay. So, there is a difference in the environment which the molecules of the interface is going to see and because of which because of this intermolecular forces you actually have this surface tension. So, that is actually the and if you really want to be able to predict what the surface tension is you have to possibly go to a molecular level and come up with some kind of a theory for the description of the surface tension okay. So, point is at the interface we have the molecules having an environment both molecules okay whereas in the bulk far from the interface molecules are surrounded by the same species understand what I am saying this guy will always have molecules of liquid 2 this guy always has a molecule of liquid 1. So, does it really experience any net force whereas this guy partially liquid 1 partially liquid 2 there is a net force on the interface. So, you really want to understand what is going on you need to go to the molecular theory and do this okay. So, just like we have constitutive relationships for the shear stress in terms of velocity gradient people have basically come up with some kind of a constitutive relationship and talk in terms of a net force like a surface tension which is acting on the interface. Now how valid so how valid is this approach of just saying that look there is a net force sigma which we have to use which is acting on the interface the validity will come by using it in developing our theory looking at predictions of the flow behaviour and seeing if it is consistent with experiments okay. So, if it is consistent that means this theory is something which you are happy with and you can use it for practical purposes if it is not consistent then you go back and redefine your theory and come up with introducing new properties for example. So, for example there are many people who sit down and talk in terms of you know viscosity which is a bulk property. So, there are people who talk in terms of interfacial viscosity that is there is going to be a viscosity at the interface which is actually different from the viscosity of the bulk okay. So, depending upon the level of detail you want to get into you will start working and including these effects right now what we are going to do as far as our approach is concerned is we will just say that there is a force which is acting on the interface and this thermodynamically we want to have a consistent picture. So, thermodynamically we have the argument that surface tension is basically looked upon as the work done per unit area. So, if you have an interface with a particular area increase the area of the element that is more energy which is stored. So, the thermodynamic perspective is on an energy per unit area basis but since we are more interested in the dynamics of the system we want to use a force perspective. So, we talk in terms of force per unit area okay. So, basically what I am saying is the thermodynamic perspective is the energy per unit area that is what surface tension is okay. So, if you increase the area of an interface you can try to have a spherical drop and you try to change the shape of the drop. The amount of energy because the sphere has the minimum area you go to necessarily have a larger area okay. So, you actually have to do work in order to change the shape of the drop. So, that is basically your thermodynamic perspective whereas from a mechanical perspective we talk in terms of force per unit length okay. What I want to do is basically write down what I just said earlier that is the net forces after acting on the interface should be 0 okay. So, now let us look at an interface which is boy this is going to be tricky. So, this is my curved interface and that is my normal and that is kind of tangential okay. So, this is my normal to the interface and that is the direction of the tangent. So, this is the view from the side. So, if I want to look at the thing from the top I would have some kind of an interface like this and I am going to take an area element. So, I am talking about an interface that is going to be an area element dA okay. So, this is my area element dA and the outward normal is now outside the board. So, I cannot show that but the tangents are going to be in this direction that is my tangent. So, what I am going to do now is look at let us say T here and T tilde earlier I had used the T1 and T2 but now I am following Gary Lille in his convention. So, I am just going to use whatever he is done okay. So, T is the total stress tensor in the upper liquid and T tilde is the total stress tensor in the lower liquid okay. So, basically what this means is T is minus Pi plus tau the pressure plus the shear stress coming because of the motion okay T tilde is minus P tilde plus tau. So, on this interface I like to write down all the forces that are acting on the system right. So, on this interface where the direction of the normal is given by n okay what is the force balance the force exerted by the upper liquid force exerted by the upper liquid is T dot n the total force that is all the 3 components are there I mean if I have an interface the direction of the normal is n what we said is T dot n tells you what the total force component is right. This is a stress tensor and that is your outward normal which is a vector. So, this is going to give you a vector. So, all the entire force component is there now force exerted. So, this is as you are approaching the liquid from the top the interface from the top similarly force exerted by the lower liquid is going to be given by T tilde dotted n tilde. But remember n tilde is minus of n okay remember n tilde equals minus n and therefore this is minus T tilde dotted with n. So, this is due to the bulk stresses which are acting on the top and the bottom in addition to that if I look upon the surface tension force as a force per unit length along the perimeter along the perimeter of this area element I have a surface tension force okay and that is going to be acting along the length along the tangential directions. So, the surface tension force is given by gamma T dl okay and this is along the perimeter whereas this is along the area element da. So, these guys this is along da this is along da and now I want the net forces to be balanced okay. So, I have integral I am going to put a double integral for my area T minus T tilde dotted n da plus integral over the perimeter must be equal to 0. So, basically this tells you all the net forces are acting on the system I told you I want these forces to actually balance out and give me 0 okay. I have a problem in the sense that this guy is an area integral and that guy is a line integral. So, now if you go back to your Reynolds-San's-Bowth theorem we have the same situation we had a surface integral we had a volume integral we converted the surface integral to a volume integral by using some divergence theorem okay. So, what we are going to do now is because I like to get a boundary condition a boundary condition which is going to be valid for every area element da right. So, what I am going to do now is convert this guy into an area element and then I am going to say that this has to be valid for any da and so every da it should vanish that is the argument okay. So, now for this you need to go to calculus so I am not going to do the mathematics I am just going to tell you what the formula is. So, I am going to convert the line integral to an area integral okay and that means over the double integral over that is fine n dot del or del dot n maybe I will write this as del dot n. So, this is of course those of you are interested you should go and make sure that you simplify the left hand side you simplify the right hand side and show that they are equal okay otherwise you just have to accept whatever I have said. Here this is gradient of s or the gradient along the surface okay. Now, the gradient along the surface this particular gradient is if the gradient is a vector for your operator right it is a grad of something I want the gradient along the surface. So, what I have to do is I have to take the complete gradient and subtract the gradient along the normal. So, that is going to give me the gradient along the surface. So, the gradient of s is written as minus gradient of n times n dot del this is the total gradient this is the normal component of the gradient n dot del tells you what the normal component is multiplied by n tells you what the when this just tells you the magnitude of the projection with n tells you the actual component and so I subtract that from the total gradient I get my surface gradient okay. So, basically the vectorial gradient I am resolving into 2 components which is the normal direction and the tangential direction. And the point that is important here is that this guy the line integral actually has 2 components one which is along the tangential direction and one which is along the normal direction okay. So, clearly what this means is the normal stress balance will have this term contributing the tangential stress balance will have this term contributing okay. So, what I am going to do now is substitute this back inside here and I am going to get double integral over the area as. So, I want this to be true for any area element no matter what which one I choose how small I choose which means this can vanish the integral can be 0 only if at every point it is 0 you understand. This is true for all area elements dA therefore t-t tilde dotted n plus gradient on the surface of gamma minus gamma n times del dot n equals 0. And remember this is a vectorial equation okay. This is a vectorial equation because this is the gradient along the surface this is got 2 vectorial components this is a scalar but this is a vector and t dot n is also a vector. So, basically I got a force balance for every element no and all I have done is written the force balance converted line integral to area integral and made some arguments and I got this but when it comes actually solving a problem I need to resolve this in the normal direction and in the tangential direction and get my boundary condition okay. So what in order to find so this is the force acting on this area element in order to find the normal component I will dot this with n to find the tangential component I will dot this with t and I get my relationship that I want okay. So, to get the normal component balance what do I do t-t tilde dotted with n and I did n dot in the earlier lecture so I will just stick to that gradient of gamma is not going to contribute grad s is not going to contribute because that is in the tangential direction this is doing a dot product with the n okay. So, when I look at this thing for example look here if I do n dot grad s I get n dot n sorry n dot del-n dot n n dot del so n dot del will cancel so n dot grad s is 0 okay. So, that goes off and what I have n dot n which is again unit normal is 1 so I have-gamma del dot n equals 0 that is my normal stress boundary condition and this remember is your curvature term del dot n and gamma is your surface tension okay. If you want to actually get the tangential stress boundary condition what should you do take the dot product with t and you get t dotted with n plus gradient of gamma along the surface dotted with t equals 0. So, what we will do is we will stop right now we will continue from here the next class.