 So, welcome to today's lecture on multiphase flows. What we will do today is discuss another instability problem in multiphase flows in 2 phase flows okay and this particular instability problem is called viscous fingering and it also goes by the name of Safman-Taylor problem after the 2 people who first investigated this problem. So what we will do is we will discuss this problem today. The Safman-Taylor formulation and that is basically what we are going to stick to. We are going to make the assumptions which Safman and Taylor did and then who analyzed the problem okay and formulation of viscous fingering that is the physical phenomena. Now this particular problem has lot of applications especially in the field of oil recovery okay. So, you are talking about enhanced oil recovery. So, as chemical engineers or as engineers of interest in energy we are having certain upstream operations wherein you have all these petroleum reservoirs, screwed reservoirs and one of the objectives is to take out all this oil which is present underneath in a porous media okay. We want to be able to extract it so that then you can come and process it in your refinery and get your petrol, diesel etc okay. So, enhanced oil recovery is an upstream operation and what is our objective to recover the crude oil from the underground reserves okay. So, basically what we are talking about is you have oil which is present in a porous bed okay and so what we want to do is we want to be able to pump it out. So, one way that is normally used is you actually pump in water at high pressure and the water will actually displace this oil and what was originally containing petroleum will now contain water and all the petroleum which was present the crude was present will actually come to the surface okay. So, what we are trying to do is we are trying to displace one liquid using another liquid. So, we displace the crude oil presently porous bed with water okay. So, what it means is a porous bed initially has oil after sometime it is filled with water. Now the problem that we are talking about if you really want to visualize this let us say this is my porous bed okay I am just drawing it horizontal and so I mean I do not want to confuse things but just imagine this is a porous bed. So, these are your soil particles is that everywhere I am just not drawing too many of them and you have oil here present. What I am going to do is I am going to be pumping in water from the left okay. Of course in actual situation we do not have a well defined geometry but since we are interested in the mathematical formulation for understanding we are looking at a very nice geometry rectangular geometry okay and here let us see there is a water. And this is the interface between oil and water okay. Ideally the question that what we are interested in asking is whenever we do this pumping of water to displace oil will this interface for example always remain flat. If it remains flat then the entire oil which is present in the reservoir is going to be recovered is going to be displaced because if the interface continues to remain flat so for when you are doing the pumping there are going to be disturbances. Disturbances can be because of some non-homogeneity in the soil property the density of the soil packing things like that okay in the porosity for example. The question is is this interface going to remain flat as it moves. So, the problem is I am pumping in water continuously is the interface going to be flat. If it is flat then we have all the oil recovered all the oil will be displaced and you have 100% recovery okay. If yes then 100% oil recovery is done but supposing the interface does not remain flat what is likely to happen is you give a small disturbance and the interface starts growing like this. So, when you get this is after the disturbance has actually evolved in time. So, first you give a small disturbance which may be sinusoidal but what is likely to happen is if the system is not stable in the system is stable the small disturbance is going to disappear and you get back your flat surface. Supposing this is interface is actually unstable then this interface is going to get deflected okay and if it gets deflected then clearly this when you are pumping in water this guy this is water here and this is oil here the water is going to the tip of this finger as you can see this is all looking like fingers okay and that is the reason we call it fingering and it is actually got to do with the viscosity okay. This is moving at a faster rate and that means it is essentially unstable and this guy will actually penetrate the bed and come out as water okay and what that means is there will be pockets of oil which are going to be left trapped inside the reservoir which you have not recovered. So, basically you know incomplete recovery of all the crude oil you understand so that is basically what the implication is so basically what this means is here the velocity of the water tip is fast it breaks through the bed okay and the petrol crude is going to be left behind partially. So, you do not have recovery of the entire petrol crude so one of the things we want to do is we want to try and understand what is it that is actually causing this kind of a phenomena if this occurs actually experimentally you can actually do this. So, all you have to do for example if you want to do a simple experiment you can just take 2 glass slides puts small glass spherical particles okay you can make your own porous bed and you can just pump you can fill it with one liquid pump another liquid and see if the liquid is actually going to be displaced and if the interface is flat. It is a very simple experiment for you to do because clearly you cannot do an experiment underground right. So, basically what we want to do is we want to do an experiment make a porous bed which is hopefully transparent so that you can see this interface. So, you can make put small glass beads make this bed fill it with oil of whatever properties you want and then pump another liquid and then see under what conditions is the interface is going to remain flat under what conditions is the interface going to actually deform and deflect. So, you can possibly put a small dye to be able to contrast between the 2 liquids where both of them are transparent then you will not be able to see how the interface is. So, things like that so but this is the actual question that we are asking okay and clearly the 2 things are important one is the viscosity and of course the surface tension because you are talking about the interface. So, what Saffman and Taylor did is they studied the problem in the absence of surface tension that is what we will do but then you can always include the effect of surface tension and then you can redo the analysis. This problem is similar in some sense to the Rayleigh-Taylor problem if you remember the Rayleigh-Taylor problem I mean I think that is something which you people have to do we have done all these different instabilities in the class you will be able to see what the similarities are and the dissimilarities are that helps you understand things better. In the Rayleigh-Taylor problem if you remember you have 2 liquids one on top of the other and the 2 liquids were stationary okay and then you ask the question when is the interface going to be stable and unstable okay. So, in that particular case we consider the situation of inviscid liquids here I want to include the effect of viscosity because it turns out that is an extremely crucial parameter it turns out that the viscosity ratio between these 2 fluids is the one which actually decides whether the interface is flat or not flat okay and what we like to do is prove this mathematically. Over there we included the effect of surface tension but we let go of viscosity here we are doing the opposite we are going to keep viscosity but right now to begin with we are going to let go of surface tension. So and then we can see and the only thing is that it was a vertical thing here also we can put vertical in fact we will have vertical geometry here also and of course that the base velocity was 0 and we did the linearization here the base velocity is not going to be 0 because we are actually having a continuous pumping of one liquid over the other. So, having said that what we want to do is rather than analyze this problem with a base velocity which is a uniform velocity we will do the analysis of the problem in a moving reference frame okay. So, basically what I am saying is the base velocity whose stability I am interested in measuring let me come here this is a question we are asking the base velocity is uniform across the cross section okay and let us say that is V now what is going to happen what is the base state the one which you are talking about is the one where the interface is flat and then we give perturbations and then we are asking whether it is stable or not stable okay. So, the base state is going to be one which is having a flat interface and continuously pumping one liquid. So, what is going to happen this interface is going to keep on moving. So, actually your base state is not a steady state you understand because the interface keeps moving. So, in order to make the base state a steady state one way to do is just analyze the problem in a moving reference frame that is if the base velocity is V you move along with the liquid. So, at every point of time it looks like you are at a steady state okay. So, the interface keeps moving at a fixed velocity okay and if we analyze this frame okay and so the moving reference frame will be having a velocity V okay capital V again and if I am moving along with this it will look like the interface is stationary okay. So, you know for all practical purposes this is really a multi phase flow problem because you have a solid bed interstaces and everything and then you have 2 liquids right. So, this must be the grand climax of this course having so many different phases. Now we have not done flow through porous meter so far but you always used Navier-Stokes equations and one of the things which we want to do is in a flow through porous media and that basically that is what it is right flow through porous media this is characterized by what kind of a relationship for momentum balance the one which relates pressure gradient to the velocity field. Can we use Navier-Stokes equations of course you can use Navier-Stokes equations but what is typically used in porous media to describe pressure drop and velocity the Ergen equation or a simplified form of the Ergen equation would be the Darcy's law okay. So, basically what we are going to do is we are going to use Darcy's law which relates velocity the gradient of pressure okay. Now if you really see many people do a lot of extensive research in porous media okay and depending upon the level of detail you are interested in you will decide to you know go into detail and start doing the analysis of the problem. So, one approach one approach is extremely detail is you take a porous media and the porous media is basically going to be an interconnected channel right basically the liquid is going to flow through some interconnected network of channels or pores and if you are interested you can really go down to finding out what this geometry of the channels is it could be a random network and then in each of these channels you should write down the equation of continuity equation of motion and then you solve for the velocities and then you actually predict the behavior of the flow that is one approach clearly what that means is you need to possibly model the entire thing as a section of channels and find out the velocity field. The advantages or one of the simplification is that since the size of the channel is very very small your Reynolds numbers are going to be very low and for all practical purposes you can use the creeping flow limit low Reynolds number limit you can neglect all the inertial terms and you only have the pressure term and the viscous term okay. So, that is one approach where you go to the detail level. So, we can go down to the level of the network of pores or channels solve the Navier Stokes equations with equation of continuity and get the flow field and this is of course extremely intensive right. This is shall we just say painful who is going to sit down and go to every pore and then do this network right. Because most of the time what we are interested in is some kind of an average information right we want to find out. So, just like you do use the continuum hypothesis and talk in terms of a velocity and a particular point which means it is the velocity of a collection of molecules in the neighborhood of that point right. So, we are going to use something like a similar averaging approach to actually discuss describe and that is what Darcy's law does. Darcy's law when he is talking about u being related to gradient of p okay being linearly related to gradient of p because you are in the creeping flow limit okay. He is talking about an average velocity and so what is this averaging? This averaging is being done over length scales which are larger than the length of the pore but not so large that the variations of velocity from one point to another in the system are lost you understand. So, that is you have to do an averaging of velocity over a bunch of pores so that you get some average information of the velocity. But if you take the averaging over the entire length then you will get a uniform value across the entire length but then you are losing the information of changes along the length of the channel okay. So, you want that information to be present. So, you have to choose this averaging domain averaging size the length scale over which you are averaging properly and that is basically what we do in continuum also. In continuum hypothesis when I am writing down the equation of continuity equation of momentum you choose your length scales or your volumes so that it is not too big and it is not too small. It should be sufficient to have enough number of molecules so that you can actually get a good average. If it is too big then you would not get the velocity variation okay. So, basically that is what we are doing so we are not going to be using this approach because this is just not worth it because you are going to get information which is possibly not useful. Who cares what the velocity is in each pore? You only want to know if the interface is going to be deflected or not. So, I do not want to waste my time trying to get this detailed velocity field information okay unless your grade in the course depends on this then you have to okay. So, what we do is and your grade does not depend on that. So, you can be sure okay. So, we average over a length scale which is larger than the pore size okay but lower than the size of the system. So, it might contain several pore sizes okay. So, basically your average velocity is the velocity average over this 4 or 5 pore sizes okay and idea is you take more it is not going to change the average value. So, this average information is over several pore diameters but if it is too many if it is if the length scale is too large then the velocity variations will not be captured. So, for example if you have a porous bed that is it I am not going to draw more of this. But if you are interested what is the actual velocity field in each of these pores that is my first approach and but if that is this approach I am talking about but I am going to say look I am going to take this length scale take several pores into account I am interested in the average velocity here okay. Then I can get the average velocity at this portion I look at this region here get the average velocity. This way if there is a variation in velocity from one section to another I am able to capture it. If I take the entire thing and then do the averaging I will get one velocity everywhere and then the variation in the velocity I am losing okay. So, I want to get the variation in the velocity in this domain. So, I am going to have to keep it sufficiently small to do that I do not want to go too small because then I do not have enough pores to do the averaging okay. So, it is a kind of a tricky business and that is basically what the theory is okay for Darcy's law okay. And that is yeah if the entire cross section is used then we get one average and lose information on the local variations. So, what I like to do is I am not going to derive Darcy's law but I am going to give you some feel for how Darcy's law can possibly come from the Navier-Stokes equation. Because once you understand that then it is we can proceed further okay. There are some small subtleties which I like to mention. So, let me say an approximate derivation Darcy's law not a derivation actually okay. Now, what is the Navier-Stokes equation that you write du by dt plus u dot del u equals minus gradient of p plus mu del squared u plus rho g that is your Navier-Stokes equation and clearly inside every pore the Navier-Stokes equation is valid okay. But remember we are talking about porous bed where the size of the pore is this or order of some microns. So, the Reynolds numbers are going to be very, very low okay. So, essentially what that means is if the Reynolds numbers are very, very low you have the initial terms are very, very negligible. So, this is basically 0 for all practical purposes and that is your creeping flow limit okay. So, since pore size is of the order of microns the left hand side which has initial terms is set to 0 okay. Low Reynolds number i Reynolds number tends to 0 and then what am I left with? I am left with this equation 0 equals minus gradient of p plus mu del squared u plus rho g okay. So, now let us forget the fact that means actually you have 3 velocity components so let us just look at one velocity component. So, for each velocity component it is going to be of the form for each component your equation is going to be of the form. Let us say dp by dx if I bring it to the other side equals mu del squared u so of course the velocity vector plus rho gx. Now del squared u and like you have seen in the case of the laminar flow for example the flow is going to be laminar inside these pores okay. Del squared u is therefore going to be given by some kind of one dimensional velocity field where you have something like a parabolic velocity profile. So supposing you have a second derivative here which is what you will get in the y direction minus dp by dx is mu d squared u by dy squared. You have a very thin channel and you have a second derivative inside this channel. One way for you to do this is to approximate the second derivative by a simple finite difference scheme okay. You can so this is flow through a small channel which is very thin just use a second derivative approximation numerically what would you get ui-1-2 ui-1 okay. So that is what I am going to do. I am going to basically use the fact. So this is the y direction and this is the x direction d squared u by dy squared and rather than take you know 10 grades and 20 grades let us just take 3 grades 2 will coincide with the wall 1 will be at the center okay. So this is u1, u2 and let us say u3. What is d squared u by dy squared evaluated at y equal to the center it is u1-2 u2 plus u3 divided by delta y squared okay where delta y is this distance yeah. Now because of the no slip boundary condition u1 and u3 are going to be 0 okay. So basically what I have done is I mean this is a 3 point finite difference that is what I am using and from the no slip boundary condition these guys become 0. So u1 equals u3 equals 0 from no slip and now if I were to basically use this information over there what am I going to get dp by dy-rho gx sorry dp by dx dp by dx yeah-rho gx equals mu times this thing here mu times-2 divided by delta y whole squared times u2 okay. Now basically this is of course a constant and what I could do is I can bring in this gravitational field also as a gradient. I can write this as d by dx of rho gxx okay and I can combine these 2 guys and I can get d by dx of some modified pressure the thing that I was describing the other day equals-of-u and remember that is basically what I wrote Darcy's law is the pressure gradient is the linearly related to the velocity and that is basically what I am showing you here and this is the pressure gradient which includes the effect of the gravitational field is linearly related to the velocity okay. Of course this is not a formal proof but that is basically the idea that you have and Darcy's law was actually found by experiments actually and then you know people started wondering about how to go about getting this. So basically what I am saying is I am going to write this as d by dx of p-rho gxx equals-2-mu times 2 by delta y whole squared times u2, u2 is some velocity inside the pore which is let us say averaged out okay and this tells me this is d by dx of some kind of a modified pressure equals-mu by ku and this is for one component which I have written. You can write the same thing for different components of velocities in the y and z direction okay and you will get the same thing. So that is basically to try to tell you how Darcy's law comes up okay. A Pm is including the actual pressure and the gravitational field. P is only the pressure. I want you to keep this in mind because tomorrow when I am applying the normal stress boundary condition I am going to be saying things about the pressures are equal okay then I have to use the actual pressure not the modified pressure. So that is an important subtle point which I have to keep in mind. So this is basically the form of Darcy's law. Now I have suddenly put in a k here and the delta y disappeared okay. This k is actually a property of the bed because depending upon whether you are so clearly the pressure drop and the velocity relationship is going to be decided by number 1 the property of the fluid and it turns out that the property of the fluid which is important is viscosity okay because that is the one which gives you the drag force along the walls. So that is being retained and this approximate thing whether delta 2 by delta y square disappeared I just put in a k here. There is something like a proportionality constant that is going to be decided by the material of the bed. Is it cyan? Is it clay? Is it glass spheres? So depending upon that you will actually get a different proportionality constant. So this k is obtained experimentally and is the permeability of the bed. So you will say that so if the bed is very permeable that means the void spacing is very very large. The liquid can flow through very easily. If the pore space is very very low then the permeability is low things like that okay. So that is the proportionality constant which is a property of the bed. Now that is a simple so one thing which I want you to be clear about is that the viscosity plays an important role okay. This is a viscous flow. Viscosity is important. However the fact that velocity is linearly related to the pressure okay is symbolic. So for example many people like to write this as I can write the same relationship as u equals minus k by mu times gradient of Pm and I am trying to follow Gary Lee's notation as far as possible okay. And this can be written as gradient of phi and I am using plus here instead of minus. That is this is remember some kind of a scalar. The gradient of that is basically my velocity okay. I mean I am k divided by mu multiplied by Pm is my phi okay. So this phi is I am using the symbol phi because phi is normally used to denote potential okay. So if you go back to your fluid mechanics what kind of flows are actually going to be given by a relationship of this kind. Whether velocity is a gradient of a potential of course you have these things called potential flows. And if you remember something you must have studied somewhere that potential flows essentially arise when you have an inviscid when viscosity is not there okay. Normally you associate potential flows with inviscid flows. So the point I am trying to make here is this is something like a potential flow because I can view the velocity as the gradient of a potential. But actually I am not saying this inviscid. I am saying viscosity is present. So I mean I am trying to incorporate in Darcy's law formulation we are including the effect of the viscosity okay because that is how I have u equals-k by mu gradient of Pm. But it is also similar in some sense to a potential flow and potential flow is normally associated with inviscid okay. And inviscid I am not going to say inviscid when I am doing my viscous fingering problem because if I say inviscid viscosity is 0, viscosity is 0 then I mean I will not be able to say you know when there is going to be viscous fingering when there is going to be fingering and when there is going to be no fingering. So if I would viscosity equal to 0 I want to keep the effect of the viscosity okay. So I want to I am going to use a fact that is something like a potential flow. But I am going to keep the effect of viscosity. So that is something like a very subtle point which I want to emphasize here. So there are many people who mistake potential flows to be flows which are inviscid. I am saying here we have a potential flow which is not inviscid which is actually viscous. In fact the people who are actually distinguishing between potential flows which have the effect of viscosity included okay. So for example so to summarize what I am saying is Darcy's law has the effect of viscosity okay. It can be viewed as a potential flow. This is an example of a potential flow which is viscous. Normally potential flows are synonymous to inviscid flows. I mean that is some kind of a misconception which people have. A potential flow that means the inviscid it is not necessarily true. It is a potential flow you can also have viscosity effect okay. And that is something which is extremely important which I want you to understand. So now life is a little bit more simple in the sense to just give you the outline of what we are going to do. We have you know 2 equations which have to be satisfied. One is the continuity equation and the momentum equation alright. So we have to satisfy divergence of u equal to 0. That is the continuity equation. And the momentum equation which is my Darcy's law okay. My Darcy's law which is gradient of Pm equals minus u. I mean you can put this k and mu and all that. Where does that come? Multiply it by mu divided by k. That is my Darcy's law in the general form, in the vectorial form okay. So the simple thing to do is and this is the approach we are going to use tomorrow. I am going to take the divergence of this equation. If I take the divergence of this equation I get divergence of u here. I get divergence of del P which is del square P okay equal to 0. So basically what I am saying is the flow field satisfies del square Pm equal to 0. And that is a Laplacian equation which you know everybody knows how to solve because they have done this course in calculus and you know can do by separation of variables whatever blah blah okay. So this is a linear equation which you can solve and that is one of the advantages of this potential flow. The relationship is linear. Then you go back you solve for the pressure field. Once you know the pressure field you can go get the velocity field. You can put those boundary conditions. The other important point is although this has the effect of viscosity in it okay. What is the order of the equation for the velocity in the Navier-Stokes equation? The regular Navier-Stokes equation is second order okay. But when I am going to not use the Navier-Stokes equation but I am going to use Darcy's law. I have the same problem in the sense that I do not have my second order term for my velocity my second derivative term d square u by dx square. Normally I would have d square u by dy square in this case I need to have 2 boundary conditions. I have only the first derivative term in the equation of continuity okay. So again I have the same problem as that of inviscid flow. Inviscid flow what do you do? You say look my equation which was second order the momentum equation has now become first order. So one boundary condition I have to let go of and the boundary condition which we let go of is the tangential stress boundary condition. So here again you can see mathematically it is a first order equation in velocity okay. It is not a second order equation in velocity. So you have to let go of one boundary condition. Again the boundary condition which you are going to let go of is the tangential stress boundary condition. So although you have viscosity present in it you are going to let go of the tangential stress boundary condition because you will not be able to use the, there is an extra boundary condition which you cannot solve for the constant because the order of the equation is actually reduced okay. So that is again something which you have to keep in mind tomorrow when we are doing it although ideally you need the tangential stress boundary condition, the normal stress boundary condition. We are going to let go of the tangential stress boundary condition because mathematically we do not require one boundary condition. The normal stress boundary condition is important because the present difference is the one which is actually driving the flow and I want to keep that guy. If I let go of that nothing is going to happen okay. So that is some small points here. We will do the actual derivation and get the condition for stability and try to understand things tomorrow. Thanks.