 We were discussing about surface tension driven flows and we brought into discussion one very important aspect that is surface tension driven flow of blood and to understand that how do we model the surface tension driven flow of blood we started discussing a little bit about the rheology of blood that is the constitutive behaviour expressed in terms of some mathematical model. So we suggested that there could be several complex models of blood rheology but at the same time for simplicity we prefer to use a model like this where k and n are the 2 parameters consistency index and behavioural index. Now these indices unlike many other fluids are not constant for blood and these indices sensitively depend on the composition of the blood. So see I am trying to give you a broad picture. Let us say that somebody is handling of disease blood sample that means a blood sample that carries the signature of a disease. Now there are several diseases in which because of the disease there will be alteration in the blood rheology and these parameters k and n will change and if these parameters k and n change the fluid flow characteristics will change. So if you have a micro channel or a micro capillary through which blood is being transmitted and the flow characteristics of the blood are changing then that might give an indication of the existence of a certain disease in the blood sample. So that is many times critical and let us see now that what are the parameters on which this will depend. So as I pointed out in the previous lecture that the t stars can be neglected under certain situations so t is equal to ks to the power n where this t and s we have discussed these are stress and rate of deformation dietic of dietics. Now this k and n these are again dependent on several parameters. One of the important parameters is called the hematocrit fraction. Hematocrit fraction is essentially the volume fraction of the red blood cells. So it will depend sensitively on the volume fraction of the red blood cells because we see here that the k depends on the h, h is the hematocrit fraction. So in some way see these are not in medical applications see medical applications are very complicated. So you do not always expect that equations and expressions will be derived from first principles. So these are the expressions for k and n which are given here these are not derived from first principles but these are curve fitted from a huge number of experimental data collected over the years. So I mean in that spirit you need to understand I mean there is no like first principle proof of like I mean where from these expressions come. So we will we see here that some coefficient c1, c2, c3 appear in this context and before getting into what is what should be c1, c2, c3 on what parameters those will depend let us quickly revisit the blood properties. So human plasma is transparent plasma is the watery part of the blood that is transparent slightly yellowish with a typical density of typical 1035 kg per meter cube. So it is like you can understand that it is just like water density it contains a solution of plasma proteins in an aqueous medium proteins contain about 7% of the total plasma volume and can be classified into 3 major groups albumin, globulin and fibrinogen. We discussed about fibrinogen and it is importance in the context of the constitutive behavior that if you take away the fibrinogen the t star component will go away. Plasma also contains the emulsified fats or lipids cholesterol free fatty acids hormones dissolved oxygen dissolved carbon dioxide. So this is the plasma composition now whole blood is plasma plus the blood cells. So blood cells will contain red blood cells erythrocytes red blood cells are also called as erythrocytes. So you can see like how the red blood cells typically look in these pictures WBCs or the white blood cells which are also called as leucocytes and platelets in an aqueous solution. The red blood cell count is approximately 5 million per millimeter cube which is about 40 to 45% by volume of the whole blood. As I told you that we define a parameter H which is hematocrit if you express this hematocrit by volume percentage then it is about 40 to 45% but you can well understand that it is not constant. It will vary from one individual to another individual it will vary from one diseased condition to another diseased condition. So based on the hematocrit it is possible to like describe the rheology of the blood sample. Now just to give you some idea typical dimensions of RBC like 7.8 micron in diameter. See I mean long time back when we were discussing about some introductory application introductory ideas regarding applications of microfluidics we told that biological cells are very important constituents in microfluidics and you can see that the dimension of the red blood cell see it is around say roughly 10 micron I mean I am not going into the intricacy of difference between 7.8 and 10 just say roughly 10 micron. So if it is so you see that like it is its dimension is comparable to dimensions of many microfluidic channels. So microfluidics can capture the variations over the length scales of these kinds of cells very nicely and these are typically 2 micron thick and 88 micron cube in volume its function is transport of oxygen I mean it is one of the main functions and less number of RBC will lead to a disease called as anemia. Size of WBC is significantly larger all the order of magnitude wise it is still the same but size of WBC is 16 to 22 micron for monocytes but I mean it is much reduced for lymphocytes and granulocytes I mean these are different types of WBCs so monocytes will be significantly larger but lymphocytes and granulocytes these are typically less in size WBC protects bodies from disease and normal WBC is to RBC concentration is equal to 1000 but if there is abnormal rise in WBC number that sometimes is an indicator of a disease called as leukemia. Platelets are smaller than RBC and WBC and their diameter are about 2 to 3 micron and platelet is to RBC is 1 is to 10. So you can see that RBC is the dominating type of blood cell but other blood cells have also have very significant functionalities platelets and WBCs are actively ordinarily not enough to influence the blood flow characteristics. So like one has to understand that here we are studying the or here we are going through the different types of blood cells not because we are interested as such about the biological characteristics of the blood cells but we are interested to see how they influence the fluid flow, how they influence the fluid flow. So in terms of influence on the fluid flow platelets and WBCs do not have that kind of very significant influence as RBCs have however platelets play an important role in forming blood clots that may severely interfere with the flow. And depending on formation of blood clots there can be serious conditions with in a human body which are known as strokes. So it is not an explicit control of the rheology of blood but by implicitly controlling the formation of clots it is possible that platelets can also significantly alter the blood flow. Now we have seen that it is the hematocrit that plays a significant role towards altering the blood rheology and it is the volume fraction of red blood cells. And in describing that how the consistency index and behavioral index depend on hematocrit 3 important parameters C1, C2 and C3 came into the picture. So in this particular slide we will discuss that what are the parameters or what are the factors on which C1, C2, C3 depend. So C1 depends on plasma globulin concentration and on hematocrit. So that is why you see these are not universal constants that is what I want to impress upon you that these are not universal constant. These constants will vary from one individual to the other one diseased condition to another diseased condition. See C1 increases with total plasma-albumin fraction. What is albumin? That is a protein in the blood plasma. So if there is a change in concentration in albumin then C1 will change. So see that many times you go for blood test and you go for detection of a disease or sometimes quantitative evaluation that is say blood composition test. So this is an indicator that instead of going for the blood composition test explicitly there could be another implicit way that the flow carries the signature of the blood composition through the rheology. So it might be possible that by relating the flow with the blood rheology and that with the composition of the blood it might be possible to argue about the possible constitution of the blood. C2 also increases directly with TPMA and both C1 and C2 show more sensitivity to the viscosity of the blood plasma than to any other factors related to blood cells. Now what about C3? C3 is relatively independent of plasma chemistry and appears to be remarkably constant for a given animal species. So this is a constant which is relatively independent of the blood chemistry. As C3 decreases so does the average hemoglobin concentration. So it is not so much species dependent but it is dependent on the average hemoglobin concentration and increase in shear rate will deform softer erythrocytes or red blood cells and more non-Newtonian the fluid becomes harder the cells more Newtonian the fluid is because increasing the shear rate has no effect on the viscosity of the hardened red blood cell suspension. So this is a very very important point to discuss like many times like people loosely ask a question and I have seen I mean like many interrogators troubling other people by asking that is blood a Newtonian fluid or a non-Newtonian fluid? It is actually not a very straightforward answer it depends on many things like for example it depends on the hardness of the red blood cells. So if the red blood cells are soft then they can be very easily deformed. So you can see then an increase in red blood shear rate will deform soft erythrocytes or the red blood cells and more non-Newtonian the fluid will become. However if the red blood cells are hard then the shear rate has no effect on the viscosity effective viscosity. So viscosity is a constant the apparent viscosity will not change and that means it will be like a Newtonian fluid. So whether it is Newtonian or non-Newtonian may depend on all these parameters and you can see now you see that the hardness of the red blood cell is a critical issue and hardness of the red blood cell since it dictates the rheology and hardness of the red blood cell also carries the signature of certain diseases. So it is possible to detect a disease by using the rheology of blood because I mean the hardness of the red blood cell many times is significantly altered by virtue of certain disease conditions. The parameter C1, C2 and C3 for a given sample of blood depend on the chemistry of the plasma specifically it is globular protein composition chemistry of the red blood cell specifically it is hemoglobin concentration number of that is the volume fraction of the red blood cell that is the hematocrit fraction size and degree of cellular aggregation of red blood cell and the red blood cell shape geometry and deformability mass density of red blood cell plasma and mass density of the composite whole blood. Now you see that there are so many parameters involved and each of these parameters if these are altered then the rheology of the blood will be altered and the flow through the micro channel flow of blood through the micro channel will be altered. So in a disease condition one can have unusual hydrodynamics because of the existence of certain disease in the blood. Now one important thing to understand is that in one way by which flow and shear rate and flow and rheology are coupled in two ways it is a two way coupling is that if you have altered flow you will have the altered shape of the red blood cells if you look here that depending on this is a shape diagram of the red blood cells. So you can see that with change in flow velocity the red blood cell shape will change and the red blood cell shape when it changes it might alter the flow rheology and that might in turn alter the flow. So there might be a two way coupling of course these two way coupling will be strong only if the shape change is very significant. If the shape change is not very significant this kind of two way coupling will not be there. At abnormally high flow rate and in constricted region of circulatory system red blood cell may deform and burst. This is like a severe condition of unusual hydrodynamics that if you have abnormally high flow rate and in a very constricted region I mean if the flow passage is very narrow then the red blood cell cannot sustain the shear and it will rupture and it can be a life threatening condition. Change in effective viscosity with volume percentage of red blood cell is manifested in several diseases like anemia or polycythemia. Anemia as we discussed is a decrease in normal number of red blood cells or less than normal quantity of hemoglobin in the blood. Polycythemia is a disease state in which proportion of volume that is occupied by red blood cell increases beyond the normal amount. So in the two extreme conditions you can have the disease which is related to the hydrodynamics of the red blood cells. I mean the disease is not directly related to the hydrodynamics but the disease is related to the rheology and the rheology concerns hydrodynamics. So we can relate this disease with the hydrodynamics. Now so far we discussed about how the composition of the blood so to say affects the blood rheology. Now the same blood sample when it flows through a tube, tube means it may be a capillary tube or a microfluidic channel whatever when it is flowing through a tube if we reduce the diameter of the tube what happens? So the normal intuition is that so if you have a sample, if you have a blood sample then if you have a channel of say or a tube of say 4 millimeter, 3 millimeter, 2 millimeter in size then you will have a effective viscosity of the blood sample. If you reduce the dimension from say 4 millimeter to 0.4 millimeter, 1 order less. What do you expect? Do you intuitively expect the viscosity to increase? By viscosity I am saying loosely the term viscosity it is apparent viscosity or effective viscosity. If it is not a Newtonian fluid we cannot call it viscosity. So is the viscosity expected to increase or decrease? What does our intuition say? So if you have a blood vessel which has a lower dimension then our intuition will say that actually it might increase the viscosity of the blood sample, effective viscosity because a smaller diameter means a greater constriction and the constriction will not allow the blood flow, blood to flow easily. But in reality what happens is something that is significantly different from that and this effect is something which is very very interesting and we will discuss a little bit. This effect of course it is very difficult to give a detailed explanation of this effect that what happens as you reduce the diameter of the blood vessel or the diameter of the capillary. But let us say let me draw 2 different diagrams. One this green colored channel is a larger diameter. Let us say this is the 4 millimeter and this is the 0.4 millimeter one. The black one is the 0.4 millimeter diameter. Now let us make a schematic of the blood cells there. We are not trying to resolve the shape of the blood cell, just trying to draw something like this. It is little bit enlarged, not in proportion just for clarity nothing more than that. So let us say that let us consider the center of this. Now here at the wall what is the expected velocity? If it is in contact with the wall, if it was in contact with the wall the expected velocity would have been 0 because of the no slip boundary condition. On the other hand at the center line there is a non-zero velocity and because of this relative tangential velocity there will be a rotation of the blood sample, blood cell. Because of the rotation in the blood cell there is a lift force that is I mean if you have a body even in a potential flow if you have a circulation, flow around the body with a circulation there is a lift force and this is something which is very common I mean this kind of lift force I mean you can see in sports balls also. So if you give a circulation to a tennis ball or a cricket ball in a certain direction the ball will tend to be floated up in the upward direction that is the top spin. So this kind of effect in fluid mechanics is called as Magnus effect. So now if you have this kind of rotation there is a tendency of a lift force. Now with a lift force if the lift force is good enough then the cell will be detached from the solid boundary and it will move towards the center line. Now can you tell that given the same flow rate in which case the blood cell will experience a stronger lift force in the case 1 or case 2. In case 1 the reason is that here the center of the blood cell is closer to the center line where the maximum velocity is there. On the other hand here it is far off from the center line. So that means the stronger the lift force what will be the effect the blood cell will very easily join the bulk flow of the center line leaving behind a cell free layer adhering to the solid boundary. Now the region close to the solid boundary is primarily responsible for the viscous behavior of the fluid. Now there if the blood cell gets depleted that is there is a depletion or cell free layer form then actually that will reduce the effective viscosity of the blood sample and this blood cell when it joins the center line will actually add to the inertia of the system because it has its own mass and therefore it will try to have a favored transport of the fluid instead of creating a resistance. So if you reduce the diameter below a threshold limit then actually the effective viscosity of the blood sample will go down instead of going up. This effect is known as Farhuse Lindquist effect and it is a very important and interesting effect in medical science it has been studied in medical science for a very long time and this effect also is important when we are doing fluid mechanics experiments with blood samples. So that has to be taken into consideration. So let us go to the slide to give a little bit of description of this. So streamlines that are closer to the center line of the channel represent fluid velocities that are faster than the ones associated with fluid streamlines close to the wall. This gradient in velocity across any suspended particulates say RBC start the cell spinning right. We have qualitatively discussed all this these are just like recapitulations in such a way that the side of the cell facing the channel center line moves in the direction of flow whereas the side of the cell facing the wall of the vascular channel moves in a direction against the flow. This leads to a drift in the particulates away from the wall resulting in the existence of a region constituted of the bulk fluid but devoid of the blood cells that is called as skimming layer. This implies that near the walls where the viscous effects are the most predominant the effective viscosity becomes less and the axial pressure gradients do not have to work as hard as it would have otherwise required to drive the flow if the blood cells were densely populating in the near wall region and hence the whole fluid appears to be less viscous okay. So again I am repeating this is a very very important phenomenon in medical sciences I mean in medical sciences whoever is studying hematology is giving a lot of attention on this specific behavior. So if you see here that if you look at this graph just to give you an idea that relative viscosity I mean it is a viscosity relative to a reference state. So this relative viscosity it is not important that you look into the numbers but see the fall. So if you take the tube diameter say 4 millimeter from 4 millimeter to 3 millimeter if you reduce your tube diameter you will see that there is not much change. But if you reduce below 0.5 millimeter that is what 500 micron. If you reduce below 500 micron you will see that there is a dramatic drop in the relative viscosity. See these are all natural ways by which nature helps in moving a fluid through a capillary and like it is not there for all fluids but this is I mean possibly nature has made the blood in living beings in such a way that the blood rheology adapts to itself to augment the flow characteristics as you reduce the diameter because as you reduce the diameter otherwise many other effects are adverse. So there should be certain favorable effects which come into the picture and this is one of the favorable effects. So with these considerations see these understandings were existing for a long time in the text of the blood hematology but these understandings were not commonly described in studying the behavior of the blood flow through microfluidic channels. So in my early days of research in microfluidics in 2005 I wrote a paper when like I brought it out that how all these parameters are actually influencing the blood flow through a microfluidic channel. So the paper was describing dynamics of blood flow through a microfluidic channel based on all these considerations and these are the summarizing features of I mean what was revealed. Now blunting of velocity profiles this we have not discussed significantly or elaborately but blunting of velocity profiles due to an axial accumulation of red blood cells as well as a streamline RBC alignment strongly influences the motion of fluid into the channel. This is something to do with the lift force that the blood cells experience as you reduce the diameter of the channel this force becomes more and more important. Influence of capillary forces increases with time. This is something which I want to discuss a little bit because this is just a summary but I want to discuss a little bit about this. Let us say that you have a capillary and there is a meniscus and the contact angle is theta. So what is the driving force on the capillary due to surface tension just due to surface tension. So perimeter into sigma into cos theta this we have discussed elaborately. Now we have to understand as we discussed that this theta is not the static contact angle value but this theta is the dynamic contact angle value. So typically for slow movement of a capillary front over a wetting surface this tan theta will scale with capillary number to the power one third. This we also we have discussed this is called as Tanner's law. So capillary number is mu u by sigma viscous force by surface tension force. So now as the fluid enters the channel forces come into the play. This driving I mean this surface tension if the contact angle is like this and if the capillary front is like this it is driving the flow in the forward direction but there is a opposing viscous resistance. These 2 forces will always be important and inertia force depending on the regime it may be or may not be important. So now if you see this driving force will try to drive the capillary but the viscous force will try to oppose. So if this is so then there might be a critical juncture when the viscous force is so strong that that might stop the flow of the fluid in the capillary but typically we see that in small capillary the static height being large the static lift the total static displacement being large it is possible that the fluid gets traversed by a large distance I mean it is not a short distance after which it stops. So what makes the fluid to traverse a remarkably large distance despite a good amount of resisting force. In the resisting force as we understand that in microfluidic or nanofluidic channels the resistance force will be quite significant. Despite that the fluid traverses a significant distance which is not something very intuitive. So something very interesting happens. So what is that? So just try to roughly look into this formula. So if the viscous force is increasing that will tend to reduce the velocity. If that reduces the velocity that reduces the theta. If theta is reduced the driving force which is proportional to cos theta this cos theta will increase because as theta decreases the cos theta increases. So you can see that this is like a natural justice to the capillary transport. So what is happening is the viscous force is trying to arrest the flow but the surface tension force is almost trying to go beyond its capacity to self-adjust to increase it in such a way that the flow can be sustained. And because surface tension is a very important force over small scales this is one of the mechanisms by which you can sustain a flow through surface tension over small scales. So this is one way by which you can sustain blood flow also. It is true for any general fluid so blood being of no exception you can sustain also blood flow through the same way. Actual accumulation of red blood cells leaves behind a lower viscosity plasma skimming layer near the micro channel wall which otherwise is a region of high rates of strain. Consequently the apparent viscosity of the whole blood sample decreases and the effect of reduction becomes significantly prominent as the size of the red blood cell approaches the hydraulic diameter of the microfluidic channel. So size of blood cell being red blood cell being not so much variable so it is basically the size of the micro channel. So you reduce the size of the micro channel you can get this effect very significantly. So this effect of reduction in the apparent viscosity of the blood as you reduce the diameter of the channel this effect may not be significantly failed for all sorts of blood vessels. It can be only failed for blood vessels which are micro capillaries but not all sorts of blood vessels so we have to remember this carefully. So we will summarize the discussion so far. We have started discussing this chapter with the concept of equilibrium contact angle that measures the relative effect of the surface forces acting at an interface when a droplet is sitting on a solid surface. This can be measured experimentally. We have derived Young's equation and Young's Laplace equation from energy minimization considerations. Vertical capillaries exhibit capillary rise or depression depending on the surface tension of the liquid used. Capillary flow dynamics equations can be derived from Navier-Stokes equation with simplifying assumptions using Lucas-Washburn analysis which yields short time and long time scale solutions. We have derived these solutions. This lump system approach can also be used to model the capillary flow in a circular capillary. Now there are certain limitations that render it is useless to explain the exact contact line mechanics and dynamic evolution of the contact angle. So exact contact line as we discussed that the lump model is good enough to give the essential capillary filling characteristics but the exact description of the contact angle is missing from the lump analysis. The concept of virtual mass overcomes some of these limitations and gives a physically realistic picture of capillary filling dynamics and why we needed the concept of virtual mass to avoid infinite initial acceleration as time tends to 0. So by incorporating all these effects and also the effect of different regimes in the velocity profile not just a fully developed regime, we came up with a model of lumped model of capillary filling dynamics of blood and we use that model for specific case study and some of the results we discussed. So far we have discussed about surface tension driven flows but we have not discussed so much of how do we modulate surface tension because I mean there is of course a straight forward mechanism by which we can alter the flow by having surface tension as a force but many times we can alter the flow by having gradient of surface tension as a force. So how can we modulate surface tension that will be our next agenda to discuss. So we will discuss about again micro flow actuation but in the context of modulating surface tension. So micro flow actuation can be possible by various means. So we have discussed about how flow can be actuated by pressure gradient and this has philosophy very much similar to how flow can be actuated by pressure gradient in large scale systems through micro channels also the concept is very very similar. Capillary effects direct use of surface tension forces or their differentials due to thermal or solutal gradients that is one possibility. Rotational forces that is one typical scenario is electrokinetics which is basically combination of electrostatics and hydrodynamics. We will discuss about electrokinetics I mean after a few lectures. Magnetic forces exploitation of electromagnetic hydrodynamic effects which is also called as EMHDFX. Rotational forces fluid flow modulated by centrifugal and corioli forces again we will discuss about this through the microfluidic device lab on a CD. Sound we will not discuss that particular effect in this course but it is possible to have micro flows through acoustic streaming generated by pressure waves and light as an effective actuation of micro flows by light induced surface tension modulation. So that also we will discuss. Micro flow actuation mechanism through surface tension. So generate gradients in surface tension force so we have discussed about generation of flows using surface tension so that I am not repeating but the additional concept that we will learn through the subsequent discussions in this lecture is generate gradients in surface tension force. So how do you generate gradients in surface tension force? You can generate gradients in temperature, gradients in concentration using electrical field. You can modulate contact angle through design hydrophobisation or hydrophilisation of solid substrates, a combination of these effects and so on. So we will start with that how to modulate surface tension with temperature. This is also called as thermo capillary effect. So let us go to the board and we will try to discuss what it is. For most of the fluids the surface tension is a strong function of temperature okay. So this is true for most of the fluids. For many fluids again as we increase the temperature the surface tension decreases. This is a very common characteristics of fluids of many fluids and this has something to do with the physics and chemistry of fluids over molecular scales. So let us not get deep into that but we first appreciate that the surface tension is a function of temperature. For many fluids the surface tension versus temperature is linear. But for many fluids you can get even higher order variations like quadratic types of variations and so on. But the linear variation for water for example it is a linear variation. Now let us say that there is a free surface which is flat okay. So this free surface which is flat let us say that this is the free surface of a weld pool just as an example that is you are doing some welding process and this is the free surface of a weld pool as the material gets molten. Let us say this is a high temperature this is a low temperature region molten both places are molten. So just as a physical example let us say that there is a solid work piece and you are heating it by some energy source. Let us say laser or whatever. So this wherever heating is made there is a low surface tension because surface tension for that particular fluid we are assuming that it will decrease with increase in temperature. So here surface tension is low sigma as compared to here where the surface tension is high right. So a fluid element which is located on the free surface will be stretched to move from low surface tension to high surface tension and that differential surface tension will drive a flow in this way. So similarly on the other side also same type of flow will occur and because of its inertia and to maintain the continuity you can have a circulation loop like this. So this type of flow where there is a flow due to surface tension is called as thermo capillary flow or in convection terminology this is called as marangoni convection. Now how is this convection possible? This convection is possible because the gradient in surface tension induces a stress in the fluid. How does the gradient of surface tension induce a stress in the fluid? It will induce a stress in the fluid by a mechanism or by a mechanism by virtue of which the free surface is free of shear. That means roughly speaking the shear due to viscous effects is balanced by the shear due to surface tension gradient. So that the free surface is free of shear. So how do you that means how do you get the shear due to surface tension gradient? If you get the shear due to surface tension gradient that is same as the shear due to viscous effect and these 2 effects must nullify each other so that the free surface must be free of shear. So we should also try to assess that what is the stress that is there due to the gradient of surface tension at the free surface. So we will take it up in the next lecture as we go ahead with how to modulate surface tension. Thank you very much.