 In the previous lecture, we started discussing on the basic mechanics of flow on a rotating device and we started by considering the derivative of a vector in a moving reference frame. So to summarize that, we will go to the slide that what we have done so far. So we have considered that there is a capital X, Y, Z which is a inertial reference frame and small x, y, z which is a frame which rotates at an angular velocity omega with respect to capital X, Y, Z. So we came up with the final expression that acceleration with respect to small x, y, z is d square r not dt square with respect to capital X, y, z plus acceleration with respect to small x, y, z plus omega dot cross r plus 2 omega cross v relative plus omega cross omega cross r. So this is what we derive. So we can write this in a shorthand notation a capital XYZ is equal to a small x, y, z plus a relative. A relative is shorthand way of writing all these terms r0 double dot plus omega dot cross r plus 2 omega cross v relative plus omega cross omega cross r. Now these terms will actually give rise to some extra terms in the Navier-Stokes equation. So how to do that? We will just recall the Reynolds transport theorem which we applied for momentum conservation from which the Navier-Stokes equation was derived. In fact the Navier equation was first derived and then the corresponding constitutive behavior was imposed. So if you see like if you write dn dt where n is m into v small x, y, z the small n parameter is capital N per unit mass which is v with respect to small x, y, z. So dn dt is d dt of the system and n is rho into v small x, y, z that is small n is v small x, y, z. Now d dt of this is the velocity is acceleration and acceleration of small x, y, z we can write as or acceleration with respect to small x, y, z we can write as acceleration with respect to capital x, y, z minus a relative where a relative is the collection of these terms which are there in the right hand side. So then rho into a capital x, y, z dv integral of that this is the force acting on the control volume right by Newton's second law. Then there is a correction term force on the control volume minus integral of rho a relative dv that means you can use the Navier-Stokes equation where you replace the force by force minus mass into acceleration. So this mass into acceleration is the inertia minus mass into acceleration is the inertia force or the pseudo force. So what are the inertial forces or the pseudo forces? This is due to the linear acceleration of the small x, y, z reference frame. This is the angular acceleration of the small x, y, z frame. This is the this is because of the velocity of the fluid relative to the rotational reference frame and this is the so-called centripetal or centrifugal depending on whether you are expressing it in terms of force or force in the rotating reference frame or in terms of acceleration with respect to the inertial reference frame. So you can see that if you are considering forces there are several types of forces which are coming into the picture. These additional terms come into the Navier-Stokes equation because of the rotation that we have seen. There was one additional term because of the translation that is d dt of r0 d2 d2 d2 of r0 but that particular term is because of translation translational component of acceleration that is not here. So you can see that you have a centrifugal force, a Corioli force 2 omega cross b relative and Euler force rho omega dot cross r. So these are the forces as signatures of the rotational platform and you can see how these forces are directed. So the rotational force the centrifugal force if you were sitting here it is radially outwards then you have the Corioli force like this which is transverse. So if you want to achieve mixing between 2 fluid streams you can use the Corioli force to a good effect because if there are transverse streams which are not trying to mix well then using the Corioli force you can make the transverse streams to mix. So this can be used as a good technology and Euler force depending on the direction of omega dot the direction will be there. So then in addition there are interfacial forces. One is a surface tension force perimeter into sigma into cos theta and the viscous drag force. Viscous drag force how do you find out what is the viscous drag force? So let us try to quickly do it. So if there is a channel with height h say a rectangular channel then u by u average what is tau wall? This is fully developed flow tau wall mu so this is y tau wall is minus mu du dy at y is equal to h by 2. So this is minus mu 3 by 2 into minus 2 y by h square by 4 at y is equal to h by 2. Then what is the drag force on the length l tau wall into so let us say that this plate is of length l. So on this plate what is the drag force tau wall into b, b is the width into l. If you consider the 2 plates tau wall into 2 b into l. So if you calculate that let us go to the slide you will get the viscous drag force as this one. So different forces we can quantify so based on these forces we can revisit the burst frequency. So centrifugal force surface tension force viscous drag force out of that drag force does not come into the picture for the burst frequency calculation. The reason is pretty clear because at the burst frequency fluid just starts flowing and there is no viscous force prior to that. So burst frequency is calculated from the balance of centrifugal force and surface tension force. So whatever we first interpreted in terms of a pressure gradient the same thing you can get as a balance of forces as evident from this slide. The final expression is of course same as what we derived in the earlier slide. Now you can see valving. Valving we have already shown that you can use the centrifugal force as a valving parameter that is by making the fluid rotate at a certain frequency if your centrifugal force overcomes the surface tension force for a hydrophobic channel the fluid will start rotating. So you can use this for valving. Mixing of fluids for that what you can play is the rotational speed and the interplay between the centrifugal force and the corioli force and aliquoting that means basically distributing the fluid through these different small tributaries. So you can use the rotational force versus corioli force interplay. So we will mainly discuss about like the capillary flow dynamics and mixing on a rotational platform. Now how is the CD fabricated? So again let me bring out that sample because that will help you to understand. So if you look at this it will appear to be like a single piece but there are actually 5 pieces here. So I will go back go to the slide and show what are these 5 pieces. So this 5 piece is 5 layer is not the only approach of making the CD. You can use a 3 layer approach also. So in the 3 layer approach you can use photolithography techniques to fabricate channels. So what is the photolithography technique? I will discuss about this in details when we discuss about fabrication. So one of its advantages is that you can make very intricate structures on the CD but the problem is it is not a low cost fabrication process. So in a low cost paradigm we use a 5 layer approach using a tabletop CNC machine. So what is done? So 5 layers are fabricated. So you can see these layers 1, 2, 3, 4, 5. These 5 layers are stacked together just like you make a burger. So very very similar. So these 5 layers are stacked together. So the layer number 1 it is a top plate typically thickness 1.2 millimeter. I mean this is not a magic number. This is based on some experiment that we have done in our lab. So you should not take that all cases it should be 1.2 millimeter. This is just to give you an estimate of what are the typical dimensions. 2 and 4 are pressure sensitive adhesive layers of 100 micron dimension. And typically you can use vinyl cutters to cut. These are like cutting plotters to cut microchannels on this pressure sensitive adhesive layer. So those are the layers on which microchannels are cut. Then you must have chambers which connect the microchannels like reservoirs and all. So there is a middle plate which is made which is the plate number 3. It contains all the chambers and typically 1.2 millimeter dimension. And 5 a bottom disc just for helping the alignment of all this by using pressure. Basically you press fit this entire system to make it a compact disc type of arrangement. So I will show you maybe I will try to show one video that how this is made. And I sincerely hope that the video works. So this is the schematic you can see that now there are various tools you can see the drilling process drilling of holes on the CD. These are based on a simple tabletop CNC machine. Then you mill various chambers. So the video is actually self-explanatory. You can see what is happening. Various chambers are being milled. Then channels are milled. You can see that the channels are basically connecting the chambers. Top layer with inlet and outlet holes. Top adhesive layer. Middle layer with all the chambers. And bottom adhesive layer. And then the bottom layer that is the layer number 5. And then pressure applied on all the sides. This will make it pretty clear how it is made. And this is just an example where we wanted to test the Diagonal's malaria on a compact disc platform. But that is a different story altogether. Now this is like how this entire setup is working. So you can see that there is a programmable motor that will basically run at a desired RPM. Some student in our lab is struggling to set it up. You can see. And then this starts spinning. And that is more or less how it functions. So pretty simple basically. I mean you can implement it. And then you can insert the fluid through a hole. And fluid reaches the outer chamber. So that is how a capillary filling process can take place. So we will continue with this discussion. So the experimental arrangement. You can see there is a computer and CCD camera to acquire the image. A detector and computer to obtain the data and also feedback of the motor unit. A AC servo motor for rotating the CD. There is something. This is very important. There is something called as stroboscope. So what it does is basically there is a light. There is a reflective strip on the location of the CD where you want to see the flow, what flow is taking place. So light flashes on that and then it gets reflected. And once in a cycle, this process is analyzed by the stroboscope. So once in a cycle when you do, the same point is actually repeated. Because after one complete rotation, the same point comes back to the same place. So essentially you are basically trying to diagnose or you are trying to observe the features or observe the flow at one particular location when it comes back after one rotation to the same place. And this is done by using a system called as stroboscope. So there can be several microfluidic structures and valving. So this is for example, you can make hydrophobic valve. Hydrophobic valve is a thing. You can use the CD as a valve only if it is hydrophobic. Because if it is hydrophobic, surface tension will try to oppose the flow and the centrifugal force will try to create a radially output. So hydrophobic valve may be made by a constriction in a chamber made of hydrophobic material or by the application of hydrophobic material to a zone in the channel or with by the application of a hydrophobic material made with structured vertical walls. So these are just some technological mechanisms by which you can make hydrophobic walls. So typically in practice the CDs are made by injection molding of polycarbonate or by the 5 layer manufacturing process that we have shown and a rendered hydrophilic by oxygen plasma treatment. So because I have earlier told that by oxygen plasma treatment you can make a surface hydrophilic. So if you make it hydrophilic then you can make it selectively hydrophobic by using the techniques that we have mentioned. Inkjet printing is commonly used for the introduction of the hydrophobic material at the valve position where you want to make it hydrophobic. And the parts are capped with PDMS to form the fourth wall of the channel. So basically the moral of this discussion is that if the entire thing is hydrophilized then depending on where you want to implement the hydrophobic valve there you can make the hydrophobic material treatment. Now some theoretical aspects. See CD is a beautiful platform because I mean I can tell you from my experience I mean as researchers and as research advisors we have to sort of interact with so many types of students research students PhD students who come to work with us. Now I mean interest of different students are different. Some student will say that I want to develop a technology I do not care about fluid mechanics. So I do not I want to develop a technology which can be used. So for example I want to develop a technology for detection of a disease. I say fine you use the lab on a CD platform. Some other student next day comes and says that I do not care about all these technology. This is only done by people who do not understand science. So for me the ultimate thing is solving equations. So I only want to solve nice equations and even if I do experiments I want to do experiments only to bring out fundamental science. I do not care about what people will use and or what will be the application and you can use the CD for that also. So what are the issues that you can address for fluid mechanics on a CD. Somebody who does not care whether you can do blood test with malaria, typhoid, dengue, whatever disease but cares about yes I want to solve Navier-Stokes equation. So here are the where are the things what you can do. Capillary instabilities, complex rheology. See when you are dealing with blood. See as a professor in mechanical engineering it is very tough for me to make my students work with blood samples because they will say it is a biology work. So when somebody comes to me for work with blood I will commonly say that think of the blood as a fluid with complex rheology and start with beautiful complicated equations. Then they will be sort of addicted to blood and once they are addicted to blood then you can make them do whatever work they want. So complex rheology of blood, contact angle modification that is like selective hydrophobization or hydrophilization, meniscus shape deformation and dynamic evolution, starting the Corioli effects and starting the effects of pulsating rotation. If you alternate the directions of rotation of the CD what will happen? So there are I mean if you think of like fluid dynamics on a CD there are so many unresolved questions that like I mean there are quite a few PhD thesis that one can do out of these considerations. So I will talk about briefly some research work because these are all research work I will not be able to sort of justify the topics by starting with the fundamentals because it may take a time but I will just try to give you an idea that what kind of research perspectives are there and this is based on that is the PhD work done by one of my former PhD students. So capillary dynamics on a CD. So basically we have discussed about capillary dynamics earlier capillary filling problems. Now here the capillary filling problems are augmented with the rotational forces. So how the rotational forces affect the capillary filling problem? So typical design specification channels with thickness 100 micron with 1 millimeter and length 23.05 millimeter. I mean from this you can clearly understand that this is definitely not the length that we try to design initially. So 0.05 millimeter nobody will try to design but this is based on what eventually we got after our manufacturing. Reservoir radius is 17.55 millimeter 6 channels were used to see the repeatability of the experiment and surface was plasma treated to render hydrophilic. So the reduced order model again we are coming back to the capillary filling problem. So you can see that see we have done the capillary filling problem earlier with the left hand side the inertial effects this L is like the S parameter that we had that is the instantaneous displacement of the capillary front within the channel. Surface tension drag force we have added one force that is the centrifugal force. So how do you calculate the centrifugal force? Centrifugal force is what? If you take a small element of mass dm then the centrifugal force acting on that is dm into omega square r. So what is dm? dm is rho into h into b into dr where h is the height of the channel b is the width of the channel right. So this integral of that from initial radius to the final radius is the total centrifugal force. So that you can put here in your capillary filling problem. Some assumptions are taken for the results that are presented here that it is a Newtonian fluid incompressible flow, low Reynolds number flow, entrance effects are neglected and corioli effects are neglected. Corioli effects are actually not negligible at high rotational speed. So these experiments are for low rotational speed. So how to model the drag force? So this is like what we have done is we have considered it just like a pressure driven flow where the pressure gradient is augmented with the centrifugal effect right. But I mean we have used this with an approximation but technically this expression is wrong. Can you say why is this wrong? That is why I have put a question mark here. See recall that the velocity profile that we derived in a rectangular channel it was for fully developed flow right. So for when you can consider fully developed flow when the velocity is not a function of x right but centrifugal force is a function of x. So how can velocity be not a function of x? So technically you will not get a fully developed flow okay because you can see that here you have an x dependence. The centrifugal force is a function of x right. So this is the sort of patch up between the fully developed flow solution to get something analytical with an approximation that approximately you can get the same expression that is how we calculated the drag force. And then we corrected it with different flow regime this I have already discussed. So I will not get into the details. Then dynamic contact angle this issue also I have discussed while discussing about the capillary filling problem and so dynamic contact angle. And then we made a reduced order model. Now how do we know that the results from the reduced order model are good enough? Of course one verification is experimental another verification is you do a full CFD problem without considering that the mass in the channel as a lump mass. So for that we used an approach called as volume of fluid approach which is relatively well known technique in computational fluid dynamics. So what it essentially does it uses a volume fraction parameter in the 2 phases the volume fraction parameter takes 2 different values. So the volume fraction may be 0 in say 1 phase and may be 1 in another phase and across the interface it has a transition. So now across the interface there is a surface tension term. So there is a body force term in the momentum equation. So you can see that there are 2 important modifications. One is because of the effect of the rotating reference frame these 2 terms come. Because we rotated the CD at a constant omega that is why the omega dot cross r that term is not there. Otherwise there would have been a third extra term that omega dot cross r. So but there are 2 extra terms and then there is a force to model surface tension. So you can model the surface tension. So this force is acting only on the control volume which belong to the interface. So basically surface force is equivalent represented in terms of body force term at the control volume. So I mean of course it is not a CFD course. So it is beyond my scope to discuss about this but just to tell you that how we validated it. So the boundary conditions which we used the inlet gauge pressure and outlet gauge pressure the inlet gauge pressure as this and the outlet gauge pressure as 0. That is the basically the ambient. So some results so you can see the semi analytical experimental and numerical semi analytical is the reduced order model. Experimental is of course what you observe from experiment and numerical the CFD volume of fluid that we used. So you can see that there are deviations from experimental results as you if you consider the simulations the semi analytical as well as the numerical. So you can see actually there is a very interesting thing. You are basically acquiring an image once in a rotation right. So there is a time lag between acquiring 2 consecutive images. If the disk is rotating at a high speed then what happens then this time lag is short and that gives you more number of consistent experimental data. So that is one thing but there are other experimental uncertainties like surface condition fabrication external perturbation vibration so many other things. So that means that even at high rotation speeds there is no there is not an exact match that is not possible. So what are the various balancing parameters like which forces dominate at what stage? At early stages the surface tension and viscous forces balance. So then viscous forces start dominating and centrifugal force because it increases with r so as the fluid advances through the capillary more and more it becomes more and more important. So the centrifugal effects initially that may not be that dominating but as the fluid is progressing through the channel the centrifugal force is trying to play an interesting role. So these are typical experimental results. So you can see that the capillary front. So there is a particular algorithm by which I mean you can do the image processing using the experimental images. There is a sobel edge detection algorithm by which you do this and you can compare the numerical with experimental results for the interface location y as a function of x. So these experiments we did for oil water interface in a microfluidic channel. The next research problem that I will discuss and it is a very important problem is how to generate bubbles on a CD. Like bubbles and droplets are very important because you can do digital microfluidics with this and I mean there are whole lots of applications with droplets and bubbles. So you can see here that there are so this is the typical design. So you have oil and you have gas and these interact at the junction to generate bubbles. So and there are different inlets and outlets and this structure is designed on the CD and it is fabricated by using the same technology that I had demonstrated by the video. Now just to let you know why droplets and bubbles are important. You can study multiphase phenomena. Droplets and bubbles reduce the contact with solid walls as compared to continuous flows. There is a substantial reduction of volume, development of digital microfluidic systems, capability of local manipulation, large surface area may be exploited for reactions and of course scientific charm. I mean many times I mean this is what is important. When Newton first observed I mean Apple was falling from a tree and he was observing it. He never knew that rocket science will be based on that. I mean it was so many times you know that we are fascinated or driven by applications but it is also not a very broad outlook by which one can look into science. Science itself, studying science itself can give rise to lot of fun and lot of satisfaction and many times we are driven to study science because of that and droplet based microfluidics can may not be of any exception. So you can what you can do with bubbles and droplets you can generate bubbles and droplets you can manipulate and you can transport. So there are different other methods I have just summarized that what are the other methods of generating bubbles. Like most commonly method used method is the hydrodynamic generation. So basically you can have shearing flows you have in which you have co-flowing streams so or cross flowing streams cross flowing streams are there given in the next slide. So I will come to that co-flowing streams you see here that there is a dispersed phase and there is a continuous phase. So because of shear now when these phases interact basically droplets are formed. This continuous and dispersed phase they may be flowing cross with each other earlier they were flowing in the same direction in the previous slide. Now they flow in the cross direction now whether it will be droplet or bubble it depends on what are the phase constituents individual phase constituents like if one of the phases is a gas you will typically get bubbles. You can use elongation flows two liquids can come in contact at the tip of the inner micro channel and liquids flow together through a contraction. So as if the liquid is stretched and then because of shear it can break into droplets. So there are other mechanisms also like selective withdrawal. I will not go into the details of all this because this is just to let you know that so many other technologies are available for generating droplets and bubbles and you can use geometrical structures also to make droplets or bubbles and there is a whole lot of very interesting literature on how do you make use of geometrical structures to form droplets or bubbles. Now in a rotational paradigm whether droplets or bubbles will be formed or not it depends on the scaling of various forces. So I will try to discuss about the scaling of various forces and one of the key parameters here for bubble generation on a CD is the interplay of centrifugal force to surface tension force. The ratio if it is typically of the order of one then you know these two forces actually compete with each other. So let us try to derive that how do you get these scaling centrifugal force by surface tension force. So centrifugal force per unit volume is what rho omega square r is some sort of r average because I mean you have a so in the channel the r of this and r of this is different. So some r average let us say this is what force per unit volume right. So then you have to multiply this with the volume. So volume is delta r into h, h is the height of the channel and delta r into h into b width right and what is the surface tension force sigma into so perimeter. Perimeter is 2 h plus if this is b then what is the perimeter 2 h plus 2 b this 2 and all these factors are not important when you consider scaling but so you take the you divide both the numerator and denominator by b and consider the limit as h by b tends to 0 divide by b. So this term goes away this term will tend to 0 this term will be 2 or of the order of 1. So you look at the slide now let us go to the slide centrifugal force to surface tension force you can see rho omega square r average delta r into h by sigma that 2 factor is not important it is just like I mean scale will not matter I mean for scale it will not matter. Then capillary number viscous to surface tension force this we have already discussed in thin film flows there is non dimensional number Reynolds number ratio of inertial force to viscous force of course all of you know about it. Then this is called as rosby number inertial force to corioli force. So how does it come? So inertial force is like of that of the order of what? Inertial force per unit volume how do you calculate inertial force per unit volume? So let us try to come to the board. Inertial force term is what rho u del u del x this type of term. So rho u square by some L reference per unit volume. What is the corioli force? Corioli force is rho omega cross v relative right rho omega u. u is the velocity at which fluid is flowing with respect to the rotating reference frame so that v relative is u. So this gets cancelled so then that rho gets cancelled. So u and L reference is typically the r average I mean you can take any important length scale of your choice but along the direction x it is like the r is like the relevant length scale. So you can get the inertial force by corioli force. Wave one number similarly inertial force to surface tension force and this phi is a parameter is a transverse force due to angular acceleration as opposed to the viscous force. So how do you calculate it? What is the force due to angular acceleration? Rho omega dot cross r. So order of magnitude is rho omega dot r average per unit volume. What is the viscous force per unit volume? Viscous force remember per unit volume is delta yj del xj. So order of magnitude of tau by the corresponding length so order of magnitude of tau is mu u by h that divided by delta xy del y okay. So this is for the angular acceleration. So you can see that. So this is something which is very interesting. For beta much much less than 1 surface tension dominates right because beta is centrifugal by surface tension and the gas thread retracts from the junction. So these are experimental pictures taken in our lab. When beta much much greater than 1 the gas experiences the strong centrifugal force and yields a 2 phase z. This is the third figure and when beta is of the order of 1 the gas thread at the junction is not dynamically stable and it bubbles and it breaks into bubbles. So that is this one. This is called as dripping to jetting transition. So through this you have from dripping to jetting transition. So we have made some experiments like the time for the formation of the dimensionless time it is linear with 1 by beta and so technologically what is important is what is the time that you require to form the bubbles and what is the length of each bubble. So we have parameterized those in terms of the parameter beta. Then we made some experiments where you can see in the inset of the figures that the rpm was varied as a function of time like a square wave for example. So by varying rpm you can control the space between the bubbles and you can control the size of the bubbles. Now for those students who would not care about like how blood flows on CD and all those things but are interested on doing instability dynamics on a CD. So what you basically do to understand the instabilities because you see instabilities can give rise to break up for example. So instabilities means what? You have perturbations these perturbations will amplify. So to do that what you do you part of the parameters u, v and p and h in the thin film by considering a thin liquid film you can perturb the thin film equations with u as u0 plus an amplitude as a function of y into e to the power ikx plus sigma t. So why such a form? So first of all this k is called as wave number e to the power ikx. Now why you take e to the power ikx? Because in the x it can be unmounded but you ultimately make it bounded by giving the form e to the power i theta because e to the power i theta no matter what is theta it will give rise to cos theta and sin theta components which are constrained between 0 to 1 okay magnitude wise. Then so that is why the form e to the power ikx that is the perturbation form e to the power sigma t. So why sigma t? So basic logic is we want to see whether the perturbation amplitude whether the perturbation grows with time amplifies with time. So when it will amplify with time so just I will quickly figure out in the board. So sigma is a real part and an imaginary part. So sigma is a complex number what is written here? They are sigma. So if sigma r is equal to is greater than 0 what does it mean? That means the perturbation is getting amplified with time because it is e to the power or quantity which is greater than 0 right. So this means unstable. Sigma r less than 0 is stable and sigma r equal to 0 is neutrally stable or marginally stable. It is a sort of at the in the borderline between stability and instability. Now for that case you can have sigma i which is not equal to 0 and which is equal to 0 and you can have sigma i not equal to 0 both are possible. So when sigma r equal to 0 if sigma is also equal to 0 then you will get basically sort of patterns in the flow. So you will get a pattern flow structure but if sigma is not equal to 0 you will basically see some oscillatory structure in the flow. So that means these all these parameters if they are studied then they will give us an indication of the instability of the system subjected to certain perturbations. So of course I mean it is beyond the scope of this course to discuss about the fluid dynamics of instability but what I wanted to impress upon you that studying instabilities on a rotational platform is itself a big topic in fluid mechanics and anybody who is interested to do series based research can do this. So let me see this movie will of course not run but let me see if I have this movie. So you can see the simulation of the entire phenomenon in a volume of fluid platform the last example that I will talk about is mixing on a CD and mixing on a CD is critical and mixing on a CD is important because I told you earlier that mixing in a microfluidic platform is challenging because of typical load Reynolds number. Now you can use the rotational forces on a CD typically the corioli force to create add to give rise to additional side wise thrust and that can help in good mixing. So various scaling of forces so first of all to get an order of magnitude of the velocity u one can make a scaling of centrifugal force to viscous force to get what is the typical u order of magnitude that you get okay. So what you do is basically you write what is the centrifugal force expression you write what is the viscous force expression centrifugal force expression will contain omega and viscous force will contain u. So from that u you will get u as a function of omega square right is very straightforward. Then corioli force to centrifugal force so corioli force to centrifugal force ratio this is what is the key parameter for mixing right because corioli centrifugal force tries to move it move it radially outwards corioli force tries to give it a transverse directionality. So you can see now you may say that in the corioli force there is a u right omega cross u or v relative where that u has gone basically this u has been substituted so you can write this entire thing in terms of omega. So you can write the balances of other forces but in terms of the non-dimensional numbers like the Reynolds number the Rosby number and all those things but essentially this parameter gamma which is the ratio of corioli force to centrifugal force will start playing a dominating role in applications where mixing is important. So you can have various regimes when gamma is much less than 1 mixing is diffusion dominated when gamma is of the order of 1 then mixing is corioli force based third point very interesting when you have gamma greater than 1 the mixing is purely determined by fluid dynamics instabilities and see this is such a beautiful platform that if you have good mixing on a CD you can do rapid diagnostics with a blood sample and that you can do with instabilities which is a purely fluid dynamics of fluid mechanics perspective. So there are different regimes the mixing length as a function of gamma. So first you have diffusion based mixing then corioli based mixing and then instability based mixing. So and different mixing regimes are characterized by the mixing length and the standard deviation based on the intensity of various images that you so if there are two fluid streams then when they mix they will come to a uniform intensity but if they do not mix well there will be a difference in intensities across various parts of the fluid element that means the standard deviation of intensity can give rise to an indicator of the efficiency of mixing. So we are done the numerical studies on mixing for these platforms also and some analytical estimates on the corioli force based mixing and instability based mixing again we have done the perturbation analysis and based on the perturbation analysis we have figured out. So this is the perturbation equation which is the governing equation for the parameter sigma that and it gives rise to an instability map the Reynolds number the Rosby number and we can get that based on this map in which regime if you go it is unstable. So this is the neutral stability curve beyond this it is unstable and below this it is stable. So you can utilize this particular regime which is beyond this curve to get instabilities and trigger good mixing using this instabilities and that can make you achieve rapid reactions on a rotational platform. There are other alternative strategies we have discussed about strategies in which you use the interplay of the centrifugal force and the corioli force but there are other alternative strategies what are the strategies you can use pulse you can use pulse mixing that instead of having the CD rotate in one direction you make it directionally rotate in different directions. It is a common sense that if you do that it will be basically mixing the fluid elements more vigorously and you can also use multiphase flows on a CD to achieve good mixing. So these are the things that we have tried and we have been partially successful in these attempts. So to conclude lab on a CD is an elegant platform for performing microfluidic operations it is effectively a rotational platform with interplay of centrifugal corioli oiler force as well as interfacial tension and viscous interaction. These are the important forces capillary dynamics on a CD involves an accounting of the above forces for dynamical filling of micro channels engraved on a CD. Microbubbles may be elegantly generated on a CD by dynamically tuning the rotational speeds and provides a simplistic design basis taking advantage of the competing influences of surface tension centrifugal and oiler forces of course if it is rotating with varying rotational speed to produce microbubbles of desired specifications on demand. CD may be used as an efficient mixing platform by effectively exploiting the interplay of corioli and centrifugal forces. Three distinctive regimes for mixing can be realized namely diffusion based mixing, corioli based mixing and instability based mixing. Just keep in mind that these are not the only operations that you can do on a CD but you can do a whole lot of other operations and using these operations you can develop important applications that can be good for not just scientific studies but also good for societal needs. Thank you very much.