 Good morning all, welcome to lecture 4 of multi phase flows. Today we will be dealing with something called as scaling analysis. I am sure like most of you would have gone through what scaling analysis is inserted part of your academia. So today we will just go through a systematic approach about how to use scaling to solve certain problems in fluid mechanics. So I will be first going through what the motivation for scaling analysis is and then I will solve a problem which can be solved using two scales and then I will tell you how you can get certain physics out of the problem using one scaling and certain other physics from the other scale. So I will start with the motivation. Scaling is important because first it helps to reduce the parameter space. So by this what I mean is that if you see any fluid mechanic problem, obviously the flow field would depend on the viscosity density, the pressure that is applied, the length scales, the channel wall, how large the diameter of the pipe is and things like that. So what scaling helps us is to reduce the parameters that are affecting the flow. So you can club all the parameters to essentially lesser number of non-dimensional groups. So for example the famous numbers like Reynolds number, Weber number, they all are group of all these individual parameters that affect the problem but they have been clubbed and then essentially you can determine the flow if you know the Reynolds number. So if you know the Reynolds number to be low then you know that it is kind of a creeping flow limit and if the Reynolds number is high you know that it is kind of turbulent. So the first use of scaling is to reduce the parameter space. So it essentially results in obtaining non-dimensional numbers. The second advantage of using scaling is to say when certain physics are important. Like this what I mean is that say for example if you have a fluid flowing in a pipe then you know that if the Reynolds number is low then the viscous forces are important and inertia can be neglected. You know these things, the understanding of physics when you can say that certain forces are not important and certain forces are dominant that comes after like you know scaling the problem appropriately. So we will see the same thing in today's example that we will be looking at. So scaling essentially helps us to find out again non-dimensional numbers using which we can say you know which of the physics can be neglected under what conditions. And the third important point is to assist experimental observations. By this what I mean is that say an experimentalist is you know doing a particular experiment wherein he sees certain physics happening in it. So once you know the problem and then you calculate the non-dimensional number and then you find out what its value is then you can say that you know to explain the physics of whatever is observing you can certain you can neglect certain forces and you know take care only the other forces which are important. For example if he sees something happening in a low Reynolds number regime then you know that inertia is basically not important it is something to do with viscous forces and things like that. So it basically helps experimental observations to be explained correctly. So once you scale the problem correctly you have these advantages that you know makes your life easy. Now I will be just taking you through what the essential steps are involved in scaling a given problem. So these are fundamental steps that you can use for any problem that you will be encountering in fluid mechanics or any other problem for that matter. So this is a systematic approach which you can just apply once you know you get a feel of it. So the first thing is to obviously write down what the governing equations and the boundary conditions or initial conditions are. So any description of physics in mathematical terms would have a governing equation and associated boundary conditions or initial conditions and the governing equations generally are PDs or ODs or something like that. So the second step involves defining certain unspecified scales by which I mean that you know scaling would essentially be you having a governing equation and boundary condition and you making it non-dimensional. So the scale that you choose to non-dimensionalize the equations the reference scales that you are using to non-dimensionalize the equations could be infinitely. You could choose any velocity to make your velocity in the pipe dimensionless because you could divide it with whatever velocity say if the wall is moving with some velocity you could divide it by that velocity to make it non-dimensional or you could even divide it by earth's velocity for that matter even then it would be non-dimensional. But then the point is you want to scale it in such a way that it is easy for you to analyze the problem. So right now we would just say that you know there are certain unspecified scales that we use to non-dimensionalize the above equations. And the third one is something called as unspecified reference scales. So before going into the details of what a reference scale is, what kind of scaling that we are going to use is called as an order of one scaling. By order of one scaling what I mean is that we are going to non-dimensionalize the boundary conditions, the governing equations and things by certain scales you know when you are non-dimensionalizing it you are non-dimensionalizing it in such a way that they become of order one which means that if you have velocity you choose a scale such that you know it changes between zero and order of one. By order of one I mean that it could be of any order of magnitude around one. So it could be varying from 0.1 to 10 but then not more than that. So it is not exactly one it is around one but not orders of magnitude more than higher than one. So you choose scales to make the governing equations, the boundary conditions and the variables and their derivatives of order one. So this is what makes it a unique scaling. So when I said you know you could non-dimensionalize the governing equations with any infinite possibilities of scales you could choose the velocity of earth, you know moon, anything of a track which is going around but then we are choosing a velocity which would make the equations, boundary conditions, initial conditions, everything of order one. So this would essentially from the physics tell you certain unique scales that you can choose. So that is what we will be looking in today's class. So now I come to the third point which is the unspecified reference scales. So unspecified reference scales are those which basically help you to bound the variables between zero and making it order of one. So for example if you have certain things which are not, certain variables in the problem which are not by default going from zero to one. So making it order of one is very easy because you just have to choose a correct scale to make it reach that order of one. So the maximum value that could reach is of order one but the minimum value should be zero you know it should be bounded between zero and it should go to some order of one value. So referencing basically helps you to make the minimum value go to zero. So for example if you have, this is still the third point. So if you still, if you have a slab, a rectangular slab when which heat transfer is happening and you have an equation which governs the temperature in this slab. So you have boundary condition which would say basically tell you there is a hot you know temperature on top and say some cold temperature at the bottom. And the temperature inside this by diffusion is going to only change between Tc and Th. Now if I would have chosen any scale to non-dimensionalize the temperature it would not start from zero because the lowest value which is there is Tc. So it would go from Tc by some Tc scale that temperature scale to Th by temperature scale. So to make it go from zero to order of one we choose a reference scale. So for example in this case wherein you want the temperature, the non-dimensional temperature to vary from zero to order of one you could define your non-dimensional temperature to be T minus Th T minus Tc by Th minus Tc. So what this tells you is that if I hadn't used Tc as my reference then this would not have gone to zero at the bottom all. So it is basically this reference which helps me to make it go from zero to one. So that is what we say when we have the third step as an unspecified reference scale it basically helps you to go from zero to order of one magnitude. The fourth step would be to non-dimensionalize using the scale factors and the reference scales. Once you do this at this step you basically would have your governing equations, boundary conditions, initial conditions, non-dimensionalized and varying from zero to order of one. And then comes the fifth one which basically helps you to retain important terms. So what this means will be clear to you when we do the example that will be following soon. So it basically just helps you once you do the non-dimensionalization using the correct scales and references there would be certain terms which becomes very small so that you can neglect them. So that you can tell that you know for these conditions you don't have to keep certain terms and things like that. So that is what I mean by retaining important terms in the problem. So the first problem that we will be looking today is fluid confined between two infinite plates and the fluid is driven by pressure plus the top wall is moving. So I have the top wall moving with the velocity u, wall and I have a pressure which is also driving the flow. So I have some pressure here P1, some pressure here P2 and there is a pressure drop which is driving the flow along with the motion of the wall. So this problem can be completely solved analytically but then there are two things which is important. So the flow is essentially driven by two forces and it is being restricted by viscosity. The viscosity is the only thing which is offering a resistance to the flow and the thing that is driving the flow is the pressure which is applied plus the motion of the wall. So you can solve this problem completely but then there are certain problems where you wouldn't be able to keep all the physics together. So you would have to solve the problem in two limiting conditions which means that you would have solved the problem when you inherently assume that the motion of the wall is not important. That is one case and the other case would be the motion of the wall is important. So those are the two extreme limiting cases under which we can solve this problem. One wherein you say that it is basically the pressure which is driving the flow and the motion of wall is not that important and the second case is where it is the motion of the wall which is important or at least we are looking at the regions close to the wall where the motion of the wall is the one which is driving the flow. So those are the two cases that we will look today. So there are two cases, case one wherein motion of wall is not important and case two where motion of wall is important. So first we will take case one wherein we are going to solve this problem for the condition where the motion of wall is not important. So I will write down the governing equations for this particular problem. So you have a fluid which is flowing in the x direction and I choose y to be downwards. So this is y equal to 0 the wall and y equal to some h is where my bottom wall is located and this is not moving so the velocity here is 0. So the governing equation for a one dimensional flow, so the flow is only in one dimension in the x direction and we assume that it is steady which means it is not changing with time and we assume that it is fully developed which means it is not dependent on x coordinate. So whatever velocity field is here is the same everywhere along x. So there are two inherent assumptions, we are looking at steady and fully developed one dimensional flow. So the governing equation would be something like this which comes from the Navier-Stokes equation. So this equation is dimensional which means that all of them have their own dimensions. So pressure has a dimension, the lengths have dimensions which are of length. So this is the first step that we looked here wherein we are writing the governing equations and boundary conditions. So since it is a steady state we do not have any initial conditions it is independent of time we are just looking at what is going to happen after all the transients have died down okay. So we have an equation which is like this and then we have two boundary conditions which is u which is the x component of velocity being u w at y equal to 0 and u equals to 0 at y equals to h okay. So this is step one where we have written down the equations and boundary conditions. So step two would be to look at the unspecified reference scales, the unspecified scales which are used to non-dimensionalize the equations and the boundary conditions that was step two. So I will just say that I have a velocity scale which is us is my velocity scale okay and let us say I have y and x let us say ys is my length scale in the y direction. So it is my y direction scale okay. So this equation can be integrated in the x direction so which means that this term is basically independent of x so it is like a constant. This term can be integrated in x and then you can just put the boundary conditions that the pressure is p2 p1 at x equal to some distance between them being l okay. So when you do that you will get this equation to be where delta p is minus of okay. So that comes by just integrating it in x direction and then putting the limits. So let us say this is my governing equation that I am looking at and the boundary conditions. Now I have got my velocity scales and the length scales written down but they are unspecified that is what I meant by just writing down unspecified scale factors. Now the step three here is to make it referenced to zero. So the variables that we are looking at is u and y, u is already referenced to zero because it is zero at the bottom wall okay and y is also referenced to zero because one of the walls is at y equal to zero. So in this problem we do not have to put any reference scale so step three is basically redundant in this problem. So already referenced already referenced to zero. So yes essentially completed three steps for this particular problem. So the step four is to non-dimensionalize the equation with the given scale factors that we had chosen okay. So I will write it here. So I am going to now non-dimensionalize my equation which is my step four to give me the governing equation would form something like okay wherein my u star is u by us y star is y by ys okay. So I have non-dimensionalized my velocity and length scale lengths using the corresponding scales that I had chosen us and ys and just substitute for u as u star us y as y star us in the governing equation and then I would get this particular equation okay same I should do for the boundary conditions also. So I would get u is u star into u scale is u w at y star into ys being zero okay and the second the other boundary condition is u star into us being zero at y star into y is equals to h okay. So I am just going to rearrange the terms and then write it down again okay. So I will just rearrange everything to give me delta p ys square by l mu us plus dou square u star by dou y star square equals to zero that is my governing equation and then the boundary conditions are u star is u w by us okay at y star equals to zero and u star being zero at y star equals to h by ys. So these are my equations and the boundary conditions. So the basic funda was that we want all the variables to be going from zero to order of one okay and in this particular problem the case one this is case one that we are still solving case one is the one that we are looking at where in the motion of wall is not significant okay. So using scaling we are going to find out the conditions under which we can assume that the motion of wall is not significant. So till now I have just you know orally told that you know we are going to look at a case where the motion of wall is not significant but what does it mean mathematically you know under what conditions can you say that the motion of wall is not significant that is going to come out through scaling analysis and that is what we are going to look at today. So till here it is clear right so I just non-dimensionalize my equations using the scales. Now the case where I am looking at where in the motion of wall is not significant. So the velocity is going to be affected by the change in pressure throughout and the viscosity is the one which is resisting the flow okay so you have pressure which is driving and viscosity which is resisting the flow. So the length scale is very much evident from the physics of the problem as well as from the boundary condition here that ys could be chosen as h so that y star goes from zero to one okay. So that is also from the physics because I know that when my pressure is driving the flow that is going to affect the flow throughout the entire domain okay. If it was the case where in the motion of the wall was significant then it is not that evident that the motion of wall is going to pervade throughout the domain okay but a pressure which is driving the flow would be pervading throughout the domain so we can directly take y star to be h okay. So from here you know that ys to be h so from here you know that if I choose ys to be h y star would become one and that is what like order one scaling also would ask you to do. So you know that y star is going from zero to one now so I choose my ys to be h okay. So if I look at this guy here I could choose my velocity scale to be motion of the wall itself uw so I have an option of choosing us to be uw okay. If I do this I am telling that the motion of wall is important because I am going to divide all my velocity throughout the domain by that velocity of wall. So when you choose a scale you should choose a scale so that you want to get the physics also correctly. So right now when I am saying that I am looking at the case where the motion of wall is not significant I am not supposed to be scaling with the motion of the wall. So I won't choose the velocity of wall to be my velocity scale even though I could do that but then if I do that then it means that the motion of wall becomes significant okay because that would always come through the boundary condition. I will never be able to neglect this term if I choose my velocity scale to be uw because if I choose us to be uw at the wall the velocity is always going to be 1 okay. So what I will be doing now is trying to get a velocity scale which is coming from the problem directly. So I will just look at the governing equation here okay and this is the pressured term which was driving the flow and this was the viscous term which was resisting the flow okay. So what I have done is I know what a vis is I put it here okay. I want to look at the case where the pressure is important so I would retain this term okay I can't neglect this term because right now I am looking at the case 1 where in the motion of wall is not significant. So I retain this term so that gives me making it order of 1 would give me ?p ys is h h2 by l mu us okay I make it of order 1 which basically means that I quit it to 1 so that I get the scale which is us out of the problem. So right now I can just manipulate this to get us is that correct okay I will get us as ?p h2 by l mu okay. So I have two scales for velocity basically because there are two things which are driving the flow the motion is because of the wall being moving and also because the pressure which is driving the flow. So this particular scale the reference scale that we have got here is the one which corresponds to the pressure driving the flow. So because it has a ?p by l so it is the one which corresponds to the velocity scale which is given by pressure driving the flow whereas this particular scale gives you the velocity scale which is coming because of the motion of the wall okay. So in case 1 we will non-dimensionalize the equation using this scale because this is what we are looking at wherein in case 1 the pressure is the one which is important and not the motion of the wall okay. So we will do that here so if I choose us to be given by that ?p h2 by l mu I would see that the first term in the governing equation becomes 1 because we got the velocity scale by making that term as 1. So if you put back velocity scale there you will get that term to go to 1 and you will get ?square u star by ?y star square equals to 0 okay. So this is your governing equation and the boundary condition would be u star being uw by us. So uw as it is and us would be ?p h2 by l mu okay. So this is my velocity at the wall which is at y star equals to 0 and I have the other boundary condition which just means that the wall is not moving at y star equals to 1 because ys was chosen to be h so I have y star equal to 1. So if you look at this particular term here uw by another term here which is us you will see that it is the ratio of two velocities. So the top numerator basically tells you that it is the motion of the wall and this tells you that it is the motion which is driven by the pressure which is imposed on the fluid so this is a non-dimensional number because it is the ratio of two velocities okay. So this can be called as ? we will call this as a non-dimensional number ? which basically is the ratio of two velocities uw by ?p h2 times l mu okay. This is clear so I just took this guy on top and ?p h2 was at the bottom. So this particular number the ? is a non-dimensional number and this is what which tells you when you can you know neglect the motion of the wall and when you cannot neglect the motion of the wall. So if ? is very large which means uw is much high compared to what is the velocity determined by the pressure then it is case two because the motion of wall becomes significant okay when ? is less than one okay that is what corresponds to case one okay which means the motion of wall is comparatively insignificant compared to what is driven by pressure okay. So for case one we basically have that ? is much much less than one okay and this is one because we are looking at an order of scaling one wherein you have everything of order one okay. So this is my equation so I can just replace this boundary condition as ? okay and I can very well solve this problem this is a simple problem which you can just integrate it is actually an ODE in second order so you just integrate and then you can solve for velocity very easily okay. Now we look at the case two we are going to case two you can just see that if you put if you are looking at the extreme case wherein you do not want the wall to be important at all you can just put y u star equal to zero and then you would get a parabolic profile okay but this is possible only because you have scaled it in this way if you had scaled it wrong this would not have been possible you would see that the equations become inconsistent so which we will see in the end. So right now I will look at the case two which basically I am looking at the case where motion of wall is important okay so I have the governing equation to be the same which is my ?p by L was balanced by ? times square u by ? y square okay and the boundary conditions were the same thing again u was uw at y equals to zero and u was zero at y equals to H okay the basic difference is in the choice of scales that we are going to do right now in the previous case we were interested in the motion of wall being insignificant right now we are going to look at the case where the motion of wall is important okay so earlier we had seen that we had two choices of velocity and things like that so we will do something similar so we had the motion of wall and some pressure which is driving the flow okay so what we will be doing is choosing the scales correctly right now so you know that since the motion of wall is important you could directly choose the velocity scale to be given by uw okay so the velocity scale is pretty evident but the length scale is not evident you know the y scale is not very evident because when you are looking at the case where the motion of wall is driving the motion it is not that evident that the motion of wall would permeate throughout the domain you know so there could be just a small region of influence delta only till where the motion of wall is pervading the flow and after that it is the pressure only which is driving the flow okay so we will choose the reference scale for y to be delta which is currently unknown so ys would be delta okay so this is something very important because this is what basically makes your problem different in analysis earlier we had the y scale given directly by H and the velocity was found out from the physics of the problem right now you have velocity scale given by the motion of the wall and the length scale is being found out from the problem so we will just do the same thing that we had done in case one we will non-dimensionalize it using this so delta P by L remains as it is and then we have mu times uw by delta squared and dou square u star by dou y star squared equals to 0 okay and we will just again use we will just bring these terms on to the other side and we will get delta P delta squared by L mu uw plus dou square u star by dou y star squared equals to 0 okay so what was not known to us was delta so we can use again making it order of one to get the value of delta so making that term to go to one we get delta P delta squared by L mu uw equals to 1 and this would give you delta squared to be L mu uw by delta P okay and I could just divide it with H square on either sides to give me delta squared by H squared equals L mu uw by H squared delta P okay so if you see this term which is here there is nothing but the same pi that we had obtained in case one okay so that is exactly same as pi so you could write delta by H to be root of pi okay so the pi which is the non-dimensional number which is the ratio of two velocity scales is not only helping me to find out under what conditions can the motion of wall be neglected it is also giving me the area of influence which is the delta so I know if I can calculate from my experiments if I have an experimental setup wherein I know what the pressure drop is between the two ends of the pipe and I know what the motion of the wall is I can easily calculate what my pi would be because I know what the values are so if I know L mu uw H square delta P for a given experimental conditions I can find out pi and using pi I can determine whether the motion of wall is significant or not significant and using pi I can also find out the fraction of the channel in which the motion of wall is important because delta was the region into which the motion of the wall was important delta by H gives me the fraction of the you know the domain which is being pervaded by the motion of the wall. So that is one important thing and now before finishing I will just and this problem can be easily solved you know solving the problem is not hard it is to understand like you know how to solve it under two different extreme conditions. So before going I will just look at the problem for the condition wherein I would have used the wrong scaling so for example I would look at case one okay wherein the motion of wall was insignificant so I had wall motion insignificant okay so I have my governing equation given by delta P by L plus mu dou square u by dou y square equals to 0 okay so I am going to look at case one now and the boundary conditions where u was uw at y equals to 0 u is 0 at y equals to H yeah. Now what I am going to do is I am going to choose a wrong scale I am going to choose the velocity scale which is u to be uw okay and ys is H okay so I have case one wherein I have purposely chosen the velocity scale to be uw which is not correct because right now I am interested in the case where the motion of wall is insignificant okay so I will just do this and put it up into the governing equation and see what comes up. So we have delta P by L plus mu times uw by H squared coming and then you have dou square u x star by dou y star squared equals to 0 okay and then I rearrange again to get delta P H squared by L mu uw plus dou squared u star by dou y star squared equals to 0 okay and this thing as we had found was nothing but pi right was it pi or is it 1 by pi L u w mu was pi so this term could be replaced by 1 by pi okay so this term was nothing but 1 by pi okay so you have an equation of this form okay and for case one where the motion of wall was insignificant we found that pi is less than less than 1 because pi was nothing but uw by u pressure because that big term which we had is nothing but the velocity scale determined by pressure so this thing for case one is less than one okay so if you put that here then this term is going to blow up so 1 by pi would be something very huge compared to what this term is here this term is of order one because we have non dimensionalized to make it of order one you know so it is no way possible for these two terms to cancel each other and get me zero okay because this guy is of order one and this guy is kind of blowing up so from the final equation that you arrived at you know yourself that you know there is something wrong in the basic assumption that I started you ended up with this equation because of this wrong scaling okay if you had scaled it correctly you would have seen that the this term would be 1 plus dou square u by dou y square in both the cases because in both the cases we chose correct scales so we got two terms which are both but being balanced and both of them are of order one and they could cancel each other but then if you choose a wrong scale the equations themselves will tell you that there is something wrong in the equation and it is not possible for them to cancel okay okay thanks