 Good morning everyone, I welcome you to lecture 5 of Multi-phase Flows. So, today we will continue a bit more on scaling analysis. Last class we had seen a basic introduction to how to use scaling analysis to analyze the physics of the problem under different limits. So, today I will be looking at a transient problem, a problem which you know evolves in time. So, the last class we had a problem wherein the flow was kind of steady. So, there was no time scale involved. Today we are looking at a problem which is unsteady and it evolves in time. So, the problem that we are going to look at today is a fluid which is confined between two walls. The top wall is stationary and the bottom wall is going to be moving periodically given by u x in the x direction u naught times cos omega t and the top wall is stationary. So, and my coordinate system is x in this direction and y in perpendicular direction. So, this is the problem that we are going to look at. So, it is a transient problem. So, at time t equal to 0 the fluid is at rest and we look at the case wherein the bottom wall is set into motion periodically. And right now there are three scales the time scales in the problem which we will be seeing shortly. So, the thing that we will be discussing today is essentially three things. The first one is when we can neglect the initial transients. So, we will be finding out conditions under which we can neglect the initial transients. The second thing that we will be looking at is a case called as pseudo steady state or something called as quasi steady state. So, we will find out conditions under which we can assume that the flow is essentially at an approximation which is called as a pseudo steady state approximation. And the third thing is we will find out the region of influence where the wall motion is significant. So, essentially what I am saying is that in the third condition I am saying that the flow of the motion of the wall is going to affect the flow only till a particular region of influence something called as delta. So, these are essentially three cases that we are going to discuss. The first is wherein I have the motion of the wall set into motion at some time t equal to 0 and we are going to find out the conditions under which we can neglect the initial transients. In the second case we look at the case where we can say that you know the quasi steady state approximation holds true and the third is the case where we will be looking at where the motion of the wall is only influencing till a particular depth. It is called the region of influence delta which we will be finding out for scaling. So, we will go to step one as was discussed in the last class which is nothing but writing down the governing equations in the dimensional form. So, as in any fluid flow problem the continuity equation and the Navier stokes equation are the governing equations. So, we have the two equations given as these. So, we have the continuity equation which is nothing but mass conservation and we have the Navier stokes equation which is nothing but Newton second law applied for the fluid. So, we will write it down in the expanded form for the three directions. So, we fundamentally assume that the flow is one dimensional. So, you only have an x component of velocity. So, we assume that you only have u x, u y and u z are 0. So, this is my u vector or the v vector basically in this problem. So, we have v given by this particular vector. Now, if I substitute this into the continuity equation I would get that my v x is not varying with time varying in space x. So, dou v x by dou x is 0 which basically means that my flow is fully developed whatever I have at x equal to 1 point is the same as it would be at any other x. So, this is my first equation the governing equation the continuity equation. If I substitute v x and you know v y and v z to be 0 in this particular governing equation the Navier stokes equation you will see that very easily we can find out that the governing equation would be dou v x by dou t given by mu times dou square v x by dou y square and this is the x component of the Navier stokes equation. The y equation in the absence of gravity would eventually be very easy and you would get d p by d y equals to 0 which basically means that pressure is not changing in y direction this is in the absence of gravity. So, we are neglecting gravity in this problem. So, this is my equation that I have. So, we can basically neglect this equation it just tells me that pressure is not changing in y and this just tells me that my velocity in the x direction is not going to change in the x direction. So, the basic equation that we are left with is this one which I will call as 1. So, we have finished a step 1 wherein we have got the governing equations in the dimensional form. Now, we go to step 2. Step 2 essentially involves choosing reference scales to non-dimensionalize the problem. So, we choose v x scale to be the scale for the x direction velocity component then we have a T s which is the scale for time and then we have the y direction. So, we choose a scale in the y direction to be given by y s. So, our non-dimensional velocities is given by v x by v x s non-dimensional time is given by time by T s and the non-dimensional y direction length is given by y star being y by y s. So, these are the non-dimensional variables that we are going to use we substitute in step 3. Now, we just substitute these variables into the governing equations that we had got. We neglect the first and the last equation we just directly look at equation 1 because that is the one which is helping me to find out how my v x is going to change in time. So, I write down rho multiplying v x scale v x star by dou T star. So, I have a time scale coming in here multiplying by mu times v x into dou square v x by dou y star square. So, these are the governing equations that we get once we have non-dimensionalized. So, what we have accompanying these governing equation is the initial and the boundary condition which basically means my velocity is 0 at time t equal to 0 and my velocity v x is 0 at y equals to h where h is the distance between the two plates and my v x is u naught cos omega t at y equals to 0. So, these are the initial and boundary conditions which are used to solve this problem that we have. So, non-dimensionalizing the boundary conditions and the initial conditions we get v x into v x scale equals to 0 at t times equals to 0 this is the initial condition we have. Next we have the boundary condition at y equals to h which is given by v x star times v x s equals to 0 at y star times y s equals to h. And then the third condition we have v x star times v x s equals u naught cos omega t star times t s at y star times y s equals to 0. Now, next we just rearrange the equation and the initial conditions and the boundary conditions. So, this was our v x and this height is given by h. So, we just rewrite the equations and we get rho times y s square by mu times t s dou v x star by dou t star equals dou square v x star by dou y star square. So, I just rearrange the governing equation got all the coefficients on to the left hand side to get a term here which multiplies my dou v by dou t. Now, I do the same thing for the initial condition and the boundary condition. So, I would get v x star equals to 0 at t star equals to 0. We get v x star equals to 0 at y star by y star equals to h by y s and we get v x star equals u naught by v x scale cos omega t s y star equals to 0. So, this I get just by rearranging the terms in the initial condition and the boundary condition. So, one time scale that is very evident from the problem is the time scale associated with the periodic motion which is given by omega. So, omega basically tells me the frequency at which the plate is moving. So, we have a time scale directly given by omega and the other problem the other time scale that involves is the motion of the wall should be diffusing into the fluid and reaching the entire the domain that the fluid is present in. So, we have a time which is called the viscous time scale which is basically the time scale associated with the motion of the wall being transported because of viscous forces throughout the domain. So, one time scale that we very evidently have is the periodic motion of the wall. The other time scale of the problem is the problem is the time scale which is associated with the motion of the wall being transported throughout the domain. So, these time scales we can directly get using the scaling analysis. So, we look at the equation which is the governing equation 1, the initial condition is given by 2, we have the boundary conditions given by 3 and 4. If I look at 3 and 4, we directly can see that using the order of 1 magnitude analysis we can make choose our ys to be h. So, that y star goes from 0 to 1 and then from here we see that we directly have a velocity scale which is given by v x s being u naught. So, that you know v x star goes from 0 to order of 1. So, we can choose one particular scale scales. So, this is case 1 where I am choosing my y scale to be h, my velocity scale to be u naught. So, I am going to say that my velocity is going to vary from you know y equal to 0 to h. So, there is the motion is being driven by the wall throughout the domain. So, that comes by choosing the scale y is equal to h. So, and then I have my velocity scale which is given by v x equal to u naught which tells me that my velocity of the wall is important in this problem. So, I have retained both the physics I am telling that my velocity of the wall is important and that the flow will be affected throughout the domain when I choose y s is h. Now, we will just substitute these into the equation. Before that we know that we have to choose a scale for time 2. So, we have a choice for time scale from the problem here. We could look at equation 4 and we could make this particular term of order 1 and get a scaling for the time directly which basically means that I could choose my time scale to be given by 2 pi by omega which means the frequency of the wall motion is the time scale that I am choosing. So, that can be called as something as TSP wherein is the time scale associated with the periodic motion of the wall. So, I have one time scale which is given by 2 pi by omega. Now, once I have chosen once I have chosen y s as h v x s as u naught I could substitute these into the governing equation 1 make this term 1 you know then I would get a time scale if I just substitute v x here the y s here. So, which means that I could get one more time scale which I would right now call it as TSP which can be made by just substituting y s y s is h and v x s is u naught and I am making that term which is the coefficient in equation 1 to be 1. So, that means rho y s square by mu times T s can be made 1 if I do that I will get a time scale associated with this particular scaling to be h square by mu by rho or basically h square by mu. So, what we have essentially done right now when we have said that this particular term rho y square by mu T s is 1 is that we have said that d v x by d t is important in the problem and the right hand side was basically the viscous term you know you got it from the second derivative velocity coming in the Navier stokes equation. So, that was the viscous term and this is the transient term. So, right now when we have found out a time scale by equating these two terms what we are saying is that the time scale is the one which is associated with the effect of the motion of the wall to penetrate throughout the domain h through the viscous forces. So, that is why I have called this as T s v which is nothing but the time scale associated with the viscous transport you know. So, the fluid has transported the motion of the wall throughout the domain and this time scale is the time scale associated with that transport. So, how much time it would take you know for that motion of the wall to be transported throughout the domain. So, that gives that is given by this particular T s v. The another time scale which is present in any transient problem is the time scale associated with the observation time scale which I will just call it as T 0. So, we have a time scale associated with just the problem being transient. So, which means that this could essentially vary from 0 to infinity. So, I could start observing the flow from any time t equal to 0 to an infinite time. So, this is a problem this is a time scale which is present in any transient problem. The other two time scales that we have observed the periodic one and the viscous time scales were obtained because we have a problem which is periodically being driven. So, that is the time scale associated with the periodic motion of the wall and then the viscous time scale is the time scale which is corresponding to the you know motion of the wall being transported throughout the entire domain. So, we could scale the problem using any of the three time scales and we will see that you know we could get different physics out of the same problem using the different time scales. So, I will just go to equation 1 again. So, I have a choosing by S to be H I have rho H squared by mu by time scale is the time scale which I would take to be the one associated with the periodic motion of the wall. So, I would get something like this. This is multiplying dou V x star by dou t star equals to mu times equals to dou square V x by dou y square. So, what we see here the term that is multiplying dou V x by dou t star is nothing but you know we could rewrite this particular term as H squared by mu which is mu by rho by 2 pi by omega. So, this thing H square by mu was the time scale which was associated with the viscous transport in the problem which we had got here T SV. So, we could rewrite this term again as you know T SV by T SP dou V star by dou t star giving dou square V x star by dou y star square. So, if you look at the left hand side of this equation we have to T SV by T SP. We could neglect this term only under the condition that T SV is much less than T SP. If this condition is satisfied then we could essentially neglect the left hand side and then we would get an equation which was dou square V x by dou y star square equals to 0. Now, if you see the governing equation it is a simplified form of the actual governing equation that you had started with you had a transient term right now under these conditions we know that there is no dou V by dou t which essentially means that the problem is independent of time at least in the governing equation. So, this particular case is the quasi steady state approximation you know. So, you can have a quasi steady state approximation only under the condition that the viscous time scale is very less compared to the periodic motion of the wall and then this can be easily solved like you know the second derivative of V x with y is 0. So, you could just integrate this you know. So, you will get V x okay and to solve this problem we have the two boundary conditions which where V x star is 0 at y star equals to 1 and V x star is cos 2 pi T star at y star equals to 0 okay. So, if you use these boundary conditions you could easily solve and get the velocity profile directly okay. So, we have looked at the problem and found out the condition under which we could use the quasi steady state approximation and get a simplified solution to the flow okay. The second case is essentially we will be finding out the condition under which we could say that the transience of the problem have been died down okay. So, that essentially corresponds to a time scale T0 be much much much greater than the time scale associated with the periodic motion. So, I had the three time scales in the problem. So, the T0 was the observation time scale. So, I have gone till what time you know I have waited till what time is the one which basically tells you the T0 is basically the physical time associated in the problem. So, if I have waited long enough compared to the time scale of the oscillation then I could say that the initial transience have you know kind of died down. So, this condition or T0 is greater than 2 pi by omega okay. So, this particular condition is the one which helps you tell under these conditions the initial transience can be neglected. So, the case 3 was the case where we are going to find out when the motion of the wall is confined to some region of influence okay. So, we have the governing equation and the initial and boundary conditions as was given in the case 1 which we just looked at. So, what we are going to say is that in the case 1 we had chosen Ys to be the entire flow domain which was going from 0 to H. So, we had chosen a length scale in the y direction to be H. So, here right now what we are essentially saying is that the flow or the motion of the wall is confined only till a region of influence. So, in this condition we will choose a Ys which is given by some delta which right now is unknown okay. So, I have chosen a scale Ys which is given by delta my Vx scale is the one which is the motion of the wall. So, I have that given by U0 okay and the time scale I am going to choose the time scale associated with the periodic motion okay. So, that is nothing but 2 pi by omega okay. So, I am right now very close to the wall okay. So, there is a small region of influence where the motion of the wall is felt which is given by delta and I do not know what this delta is right now. So, what we will be essentially doing is finding out using scaling analysis the value of delta you know. So, we can find out what region will be influenced by that motion of the wall okay. So, we have the same equation which is given earlier it was doh Vx by doh T okay. We have the okay this was the governing equation that we had initially the transient term being balanced by the viscous term. So, we have right now this was the dimensional equation we use those scales which were given there to non-dimensionalize this particular equation. So, we get rho times U0 here times doh V star by doh T star. So, the time scale was 2 pi by omega. So, we have 2 pi by omega here okay. This is given equals to U times U0 which is the scale used for Vx doh square Vx star by doh Y star square okay and Y star is scale with delta. So, we get delta square times Y star. So, I just have used the corresponding scales and non-dimensionalized my equation okay. So, if you see U0 gets cancelled okay. So, if I rearrange my terms I would get rho delta squared mu by 2 pi by omega that correct I think mu comes below into doh Vx star by doh T star equals doh square Vx star by doh Y square okay. So, I just rearranged my terms. So, what I am saying is that right now I am interested in the condition wherein I am looking very close to the wall and the portion of the wall is confined to a region of influence and when that is happening when the motion of the wall is confined only to a region of influence the problem is always going to be transient problem. So, I have to retain the doh Vx by doh T in the problem okay. So, the way you retain is by making this term which is associated with it to be 1 okay. So, you make the term which is associated with doh Vx by doh T 2 pi by omega equals to 1. So, only if this is of this term is of order 1. So, only if I make this guy 1 will this LH is also become order of 1. So, from here I can see that I can get my delta you know solved which is nothing but 2 pi by omega by is that correct root under. So, I have taken it on to the right hand side okay. So, if I look at this delta. So, I could divide delta with H okay to get me delta by H I just divide delta on the left hand side with H. So, I have to again divide H square here because this term is under the root. So, I will get delta by H to be 2 pi by omega okay and here I would get H squared and there was a mu here okay. So, I have delta by H given by this term. So, if I look at this guy I could rewrite it as 2 pi by omega by H square by mu to the power half okay. So, this is nothing but delta by H being given by a time scale which is periodic 2 pi by omega by PS discuss. So, which essentially tells me that the region of influence which is the fraction of the region of basically delta by H is how much of it you know the fraction of the channel which is influenced. So, delta was a region till which the motion of the wall was affecting and H is the total height of the domain. So, delta by H is basically the fraction in which the flow is being affected is directly given by the time scales ratio you know the time scale of the periodic motion by this one which. So, if the time scale associated with the motion of the wall is very low which means if TSP is less which means that the plate oscillates very fast then you will see that the region of influence would be small okay. So, if TSP is less than TSP then H would delta would be small. So, which means that if I have a fast moving plate the region of influence would be very much close to the wall if it is slowly moving then I have enough time for the fluid to respond you know because discuss time scale is given by the H square by mu once I have fixed my H and I have chosen my fluid H square by mu remains fixed okay and then I could make the entire delta more you know occupy the entire region or confine it to a small region choosing my omega correctly. So, if I choose very small omega the motion of the wall would be confined to a very small region and if I choose large omega then the fluid has enough time to actually you know respond throughout the domain it could be carried out by viscous forces okay. So, the case 3 problem which was the region of influence problem you could use these scales and get the equation directly. So, under these scales you would get that the problem is transient and you would get the governing equation to be dou Vx by dou T star equals dou square Vx by dou y square okay and when you are looking at this problem you could give the initial condition instead of initial condition you could give a condition which means the motion of the wall would remain exactly same after you know one cycle of oscillation. So, you could instead of using the initial condition which was the wall being at rest you could use a periodic condition in time which means that u or the velocity in the x direction at t okay is the same as velocity after some 2 pi by omega you know. So, this would essentially in the non-dimensional frame because my scale right I have chosen as 2 pi by omega would be Vx at T star would be same as Vx at T star plus 1 and then my boundary conditions I already have which means my wall is not moving the top all is not moving at y star equals to 0 okay and then I have the other condition which is Vx star that was at infinity at H because the top wall is not moving y star is infinity basically at that point because this would come as ? by H okay because basically I have a condition y this is the condition at y equal to H so in the non-dimensional frame it would be y star by y star into H by ? so y was H so y star into ? is H so y star would be H by ? so and ? is very small right now so it basically corresponds to a fact that it is infinity so I have a velocity which is going to 0 as I move away from the wall and Vx star would be the one which is the periodic motion which is the given by cos 2 pi T star at y star equals to 0. So, you could easily solve this problem and then get the velocity profile for these conditions.