 Alright, so what I want to do for the next 10 minutes or so of this lecture is to show you a few videos that will hopefully better illustrate some of the things we have been talking about. Alright, so you will just have to look at the screen for a while. So as the title suggests, we look at a few applications of multi-physics flows and see exactly how some of the things we have been looking at in this lecture translates in the real world. So this is an image in a microchannel of a stratified flow in an aqueous two phase system. So what you will have to do is if you look carefully at those bands that you see, each band is in fact a microchannel and the dimensions as you can see are about 150 micrometers. So that is extremely small width, maybe a few hair widths. Then if you look closely in each width, you will see that about half the channel is dark in black color and the other half is a sort of a lighter white and it is that white portion which is one fluid and the dark portion which is another fluid and you have a perfectly flat interface between the two of them. And the two different fluids are given here at 50 microliters per minute, polyethylene glycol and sodium citrate and this was used in some extraction experiments. So there is a video as well and you will not say anything much because there is no interface deflection at all. So the fluid just flows straight through and what we need to look at when we look at this video is to see whether the assumptions that we made in the in class actually hold. So for example, is the interface really steady and flat or is it fluctuating all the time. So we see in this experiment at least the interface is flat, it is steady and we can possibly be confident that at least that assumption was right. So we can look at the models that we used. This was the model that we looked at in class where we had two infinite plates in the x direction in this case, flow in the z direction and we were looking at how the velocity profile was in the y direction and we had two fluids phase one and phase two and then applying the simplifications to the Navier stokes and the boundary conditions you would have been able to derive the velocity profiles which are given here. Note that the viscosity ratio here is mu and alpha s is the hold up or the thickness ratio. So then after getting the velocity profiles we discussed about calculating the flow rates that is q1 and q2 by integration and then to find the flow rate fraction which is the ratio of the flow rate to the total flow that was sent into the channel. So here in fact we show the relationship between flow rate fraction and the hold up which or the thickness ratio and you see that the viscosity ratio plays a role in this whole relationship. Of course as we expected delta p is no longer to be found because that was a simple magnitude parameter that should have been lost when we took ratios of the flow rates and you remember I would said that it is only when the viscosity ratio is 1 and the hold up is 0.5 that I would expect the flow rate fraction to be 0.5. What happens when that does not hold is what we can look at now in this slide quickly. Before that the model that we considered one of the biggest assumptions was that it was an infinite plate or that we did not look at the third dimension which most obviously is going to be there in a micro channel and that third dimension will in many practical situations be of the same order as the height of the channel that we were looking at. So in this work some of the work we had done we looked at the 2D model as well so that we could you know relax the assumption of the infinity in one of the directions. So we got an aspect ratio naturally which was the width of the channel in this stratified direction that is in the y direction which we had which we previously had and d which was the now new added finite dimension in the problem. So that gave me an aspect ratio and you would expect that if h were much much smaller than d then I should be able to work with the assumption that I had before. In other words if d was much larger then it would tend towards the infinite problem that we had solved. So if lambda is 0 essentially it goes back to the previous problem for any other finite lambda we would have to use the rectangular channel. So you see that here we have partial differential equations and these are not reducible any further. So this is clearly why we had looked at the 1D problem in the first place because it gave you significant mathematical simplicity while retaining considerable physics but in some of the problems you might have to come to this stage where then you will have to deal with the mathematical complexity as you may and then of course but the good thing is you already have some insight from the simpler problem. In turns out that you can get analytical solutions and then once again calculate the flow rate fraction. So here you see a flow profile. This is part of your assignment to calculate these different profiles flow profiles. You can look first at the more familiar figure on figure B where you see that the maxima is in fact in fluid 2 and it turns out that though not shown here that the viscosity of fluid 1 is twice fluid 2. So it makes sense that the maxima is there. On the figure A you see the cross sectional view of the velocity contours. So each coloured surface represents a velocity of a particular value and the max velocity is on the white colour and 0 is the blue colour. So you see that the maximum velocity is located somewhere within fluid 2 near the centre of the channel and it just a slice along the centre that is shown here as a velocity profile. Alright so now of course what every modelling exercise has to deal with is how well it compares with experiment and that is what we have shown here succinctly in this figure where you can see the data points from 2 experiments. These experiments have been carried out by Matsumoto and I can give you his reference later. So he did some experiments where he looked at exactly what we are plotting here essentially he pumped in different flow rate fractions of 2 fluids and then he using images tracked where the interface was located or the thickness ratio and what he found was that when the viscosity was somewhat similar about half each other the relationship moved more or less along a diagonal. But when the viscosity ratio is very different he got large deviations from a diagonal and the diagonal is important because it just means that phi s is equal to alpha s or in other words if you pump 2 fluids in a certain fraction you would expect the thicknesses to have the same fraction. But we see here that that is totally false if they have very different viscosities and that makes sense because the problem favours the fluid with lower viscosity because that will give it lower pressure drop if you want to think of it in terms of energy minimization. So this becomes very important when you are trying to estimate properties of the flow estimated pressure drop using empirical relations where you want to know what the hold up is and there is no simple way to estimate it from the flow rate fraction unless you do the kind of calculations where do. And of course what we were able to see is that our model or the simplified models we are looking at predict this experimental data really well. So you see the blue case follows its set of data and the red dash line follows its set and there is a considerable difference between the two and the fact that the model captures it tells us that the reason for this difference really is in the viscosity ratio and the physical mechanism of that viscosity ratio effect is what our model is capturing which basically involves the shear stress at the wall that is about the only thing that the viscosity is doing that makes the fluids different. So both fluids are moving along their own walls and the more viscous fluid exerts a much greater drag. So naturally it has difficulty to flow the less viscous guy gets pumped through an easier way. Finally here we can look at 1D versus 2D cases and see what the importance of the aspect ratios. So let us just look at the first figure where I have viscosity ratio 1 to equal viscosity fluids for simplicity and I have shown different figures, different relationships here for different aspect ratios. So this is a square channel and then this lambda equal to 3 is where my depth is smaller than the width between the two plates. This case is closer to my one dimensional approximation where d is larger than h and this is the 1D calculation it is a light pink line. So in the first case you hardly see a much difference between these four different results the four aspect ratios and from the 1D case. So I guess I could say that the one dimensional case serves as a good model here and I do not really need to go to the square channel because no matter what the aspect ratio is it does not have much of an effect on this relationship. But when you come once again to different viscosities you start seeing that the aspect ratio has a significant role to play. So the especially along flow rate fractions of 0.5 you have some considerable difference between the thickness ratios depending on what my aspect ratio of the channel is and importantly what we should verify is that whatever be these cases they should all tend towards the one dimensional case if my aspect ratio goes towards 0 which means as d becomes much larger than the width between the two plates h and that is what we see here that when lambda is 3 it is the furthest away from the line pink line which is the 1D case and as my aspect ratio becomes smaller I approach it asymptotically. So the same thing you can see here for mu equal to 5 is just that the relationship has sort of reversed. So what this tells us also is that if my channel is long and rectangular or all standing up I can have different thickness ratios and merely by orienting my channel or orienting the fluids in that channel in the right way I could have some control over what my hold up is for the same flow rate fraction. It also tells us when we would be able to use the one dimensional model with some with better accuracy. So we will stop here for today and we look at some of the other applications of multi phase flows in the next class.