 We will continue with our discussion on surface tension driven flows to summarize we have discussed about various equations that describe the motion of a fluid in a capillary in a narrow capillary and we have considered both cases one with inertial effects and another without inertial effects. All those we have discussed and we have been discussing about certain limitations of the simplified model that we have developed so far. So we will continue with the limitations of the model we have discussed about one limitation and we have introduced the concept of the added mass or the virtual mass. Next we will discuss about the sanctity of the fully developed velocity profile consideration. So if you look at this schematic you will see that first there is an entrance region of length l1 where the flow is hydrodynamically developing. This region may be very short but it will be present then there is a fully developed region which is also the equivalently the Poiseuille flow regime and then there is a third regime which is normally not encountered in a channel which is entirely filled with the same liquid. But here there is a channel which is partially filled so that there is a meniscus. Now the velocity profile at the end close to the meniscus should adjust to the shape of the meniscus right. You can clearly see that this the shape of the meniscus which is here does not conform to a parabolic velocity profile right. Therefore there is a third regime which is called as the surface traction regime or meniscus traction regime of length l3 where the fully developed velocity profile gradually adjust to the shape of the meniscus okay. So to summarize there are 3 regimes entrance regime or the developing regime, the fully developed regime and the surface traction regime. Now what regimes will be present? See it depends on what is the total length of the fluid column. Let us say this total length at some instant this x is equal to x1 okay. Now if x1 is less than l1 plus l3 then the fully developed poiseuille flow regime will be absent all together because you have to remember that there must be a developing region how small maybe it will be there and there must be a meniscus traction regime that is a must because the velocity profile has to adjust to the shape of the meniscus. So depending on the instantaneous length all the 3 regimes may be present but at least the entrance regime will be there and if x is greater than l1 but less than l1 plus l3 then l2 will be 0 and there will be meniscus traction regime but no poiseuille flow regime. Now when we have considered the drag force or the viscous force there we have only considered the fully developed poiseuille flow regime for evaluating the drag force in our calculations. But if you take into account the existence of these 2 additional regimes then that will alter the drag force calculation. How it will alter the drag force calculation that cannot be worked out numerically analytically but through numerical calculations one can do some curve fitting and say that if dfd star dx be the gradient of the drag force with a star means fully developed then the actual one is basically a 1 plus a function of x by hydraulic radius where the function of x by hydraulic radius can be fitted by a polynomial. So this is basically not a fundamental analytical work. This is what is this? This is basically curve fitting of numerical data. So doing numerical simulations in the for the cases where all the 3 regimes are present then that can be suitably normalized in this particular form where the gradient of axial gradient of drag force is equal to a factor where the factor is greater than 1 times the gradient of the drag force for a fully developed flow. The question is how do you calculate the gradient of drag force for a fully developed flow? Let us come to the board and explain that. So let us say there is a circular capillary of radius r. Now you take a small control volume of length dx. If the drag force acting on this is dfd then what is dfd? What is the elemental drag force on this small element? tau wall into 2 pi r dx. So this is I mean this dfd this can be expressed in a general case. So dfd dx is tau wall into 2 pi r. This is true no matter whether the flow is fully developed developing whatever. Now the calculation of tau wall will depend on what is the velocity profile. So if it is fully developed then u by u average is equal to 2 into 1- small r square by capital R square. This is fully developed flow of a Newtonian fluid through a circular tube. So tau is equal to –mu du dr that is equal to –4 mu u average by into r by r square. So tau wall so this – becomes plus tau wall is equal to this tau at small r is equal to capital R. This is equal to 4 mu u average by r. So if you substitute that here then it is fully developed. So in that case dfd dx is equal to 8 pi mu u average. So accounting for the additional regimes the total resistance is these times are magnifying factor where the magnifying factor depends on where actually you are present and the hydraulic radius or the diameter or the actual radius hydraulic radius is equal to same as the radius for the circular tube but why we introduce the concept of hydraulic radius is you can use the same concept for tubes with other cross sections not necessarily circular. The third issue that we discussed is the dynamic evolution of the contact angle. Now this is a very interesting thing. We always say that there is a no slip boundary condition at the interface between the fluid and the solid. Even in a classical scenario I am not talking about nano channels with hydrophobic interactions all those complicated cases that we have considered. Think about a channel. Let us come to the board to explain this. Think about a channel. Let us come to the board yes. Think about a channel classical scenario there is a meniscus. Now after some time this meniscus comes here. So if there is no slip here then how can this meniscus move like this along the surface. So somewhere there is a singularity at the fluid solid interface that allows this system or this kind of anomaly to be resolved. So how that is possible? So let us look into that. Points on the interfacial line arrive at the contact line with a finite time span. Therefore one must pose an effective slip law that relieves a force singularity condition by ensuring that a finite force is necessary to move the contact lines of a fluid irrespective of the no slip boundary condition at the channel wall. So like you can pose a slip law that takes into account the movement of the contact line on the surface. That has nothing to do with how do you derive the velocity profile using the classical no slip boundary condition. So now how will the contact angle understand that this capillary front is moving? The contact angle also must be sensitive to that. Now there is a departure of the contact angle from its static equilibrium value mainly due to viscous bending of the interface near the contact lines. So now how does it vary? Again like it is not so trivial to discuss this at this level. I mean we will be in a position to make some analysis on this once we learn thin film flows which we will learn after we study the surface tension driven flows. But for the timing assume this and we will understand the consequence of this that the contact angle theta typically scales with capillary number to the power one third. What is capillary number? Capillary number is the ratio of viscous force and the surface tension force that is called as capillary number. So you can see the definition is written in the view graph that capillary number is equal to mu u by sigma. So now the contact angle therefore is a function of u and in the mathematical formulation that we had presented there we used a constant contact angle. But in a true sense the contact angle should vary dynamically because the contact line velocity is also varying dynamically. So there have been various theories which have looked into this issue of dynamic contact angle. I will not get into the theory but I mean typically lubrication theory based analysis have been very commonly used we have discussed about lubrication theory. Now this entire domain has been classically divided into 2 regions. One is the outer region. In the outer region you have the balance between the pressure gradient and the viscous forces and then you have the inner lubrication region as the capillary and the viscous forces balance okay. Now this inner region where the surface tension and the viscous forces balance this region when it comes very close to the wall it is a very fine sub region of the inner region where intermolecular forces now come into the picture. And this intermolecular forces typically originating out of van der Waals forces of interaction this kind of intermolecular forces means that there is an excess pressure beyond the capillary pressure and that excess pressure comes in the picture that is called as disjoining pressure. So that is typically a manifestation of intermolecular forces of interaction that makes the terms to be augmented that makes the pressure terms the capillary pressure term to be augmented with another pressure like quantity which is called as disjoining pressure. So taking all these things into account it may be possible that for wetting fluids a relationship between the like tan theta versus capillary number and lambda is a surface roughness length scale these relationships have been this is a relationship that has been obtained in the literature. The whole purpose of showing this relationship is that like notionally it also follows the simple the scaling of the Tanner's law like capillary number to the power one third behavior. Now so far so good we have understood the limitations of the simple model but now we will apply this simple model for a special case that is flow of blood through a micro capillary which is like a very big issue in not only in micro fluidics but also in medical sciences and bio fluidics in general. So we will see let us say that we are analyzing the blood flow through a rectangular channel. So like when we are considering blood flow through a rectangular channel you know whatever we did for the circular channel a similar thing thing will be there in the mathematical modeling paradigm but you know the expressions for the perimeter area all those things will be different and you may argue that in human body there is no rectangular channel right. So why are we interested about blood flow in rectangular channel see we are interested about blood flow through both rectangular and circular channels. Circular channels are for the context of human understanding the blood flow dynamics in a human body and rectangular channels are for artificial lab on a chip devices in which you make pathological test for blood samples. So I mean it has relevance in the I mean it one need not waste a lot of time to convince ourselves that yes I mean this is a very interesting and important problem the capillary flow dynamics of blood. So like you can see here that is like this x1 is like s the parameter s. So ddt of m you can see here we have considered the added mass we have discussed. So this is the corrected model not the basic model but the corrected model. So you have the added mass here ma which for a rectangular channel has this expression then this is mass within the channel that is density into the height into width into x1. So that is basically density into volume and then this is dx1 dt is basically the velocity of the capillary front. Right hand side this is the surface tension force p into sigma into cos theta where p is the perimeter minus the drag force drag force is this one tau xy into pdx right integral of that. So this is a common framework in which you can you can pose almost all problems of capillary filling dynamics but the question and the very important question is that what is very special that we think about blood. Now typically I will briefly talk about the constitutive behavior of blood like actually it is a very important and interesting topic by itself that is how blood as a fluid differs from other fluids. I will discuss a few points about this but the first and foremost point to understand is that although not in all circumstances but in many circumstances depending on the shear rate of course blood behaves like a not like a Newtonian fluid. So that means blood will not obey the Newton's law of viscosity. Question is if it does not obey the Newton's law of viscosity then what does it obey. So there are different models of constitutive behavior of blood. So the different models like I mean I would name certain models like the Casson model, the power law model, the several viscoelastic types of models because you know blood has deformable blood cells. So it has some partly viscous and partly elastic behavior. So there are wide ranges and different types of model. Now for simplicity and for capturing the very essential physics in this elementary in this basic course we will only consider the power law model. The reason is that although the power law model does not represent all the important characteristics of blood but to some extent it represents the behavior quite satisfactorily over the ranges of experiments using which the parameters for the constitutive behavior are chosen. So let me elaborate this point. So typically for a generalized power law you can write in this way where this T is called as stress tensor dyadic. So that is where these 2 double marks are there. So I mean we will not get into the details of like what is a tensor dyadic and all those things but we will be in a position to simply write the expression for a one dimensional scenario. This S is a rate of deformation tensor dyadic and this T star this is like a yield stress tensor dyadic that means it is a very common scenario that blood or in general or non-Newtonian fluid may not start flowing until and unless a critical stress is applied. So that is like it gives an yield stress like behavior which is very commonly seen for solids many times for certain fluids this yield stress type of behavior can be observed. So now like what typically in blood what is there? So in blood there is plasma and that will contain its proteins and then there are blood cells red blood cells white blood cells platelets these blood cells are there. So that makes blood a very complex suspension and this yield stress like behavior in blood is because of a protein called as fibrinogen which is present in the blood. So now if you see that this if you see that what is the value of this? The value of this is not very high and not only that you will see that until and unless you start the blood from zero velocity condition to a flowable condition this may not be that important. This may be important when you start the blood from a zero velocity to a finite velocity condition. Otherwise if the blood is already in a moving condition then this may not be that important and typically we will not so much we will put not so much attention on this term just for algebraic simplicity but it will not bear a lot of significance on what whatever we are going to derive but this is not actually zero but if you take away very interestingly if you take away the fibrinogen from the blood sample then this will be zero. So like in biology these are very common things. So if you say you observe certain behavior and you say that this is possibly because of fibrinogen then that possibly has to be justified that means then you take away the fibrinogen out of it and show that that effect is not there then only that can be it can be prove that yes it is because of fibrinogen. There are of course other proteins like albumin, globulin and these other types of proteins are there. Now so we will not concentrate so much on this. Now for a one-dimensional situation where there is velocity gradient along one particular direction so now this is what what is basically this is what this is basically the rate of deformation gamma dot and for a one-dimensional situation this is du dy so that means we can write tau is equal to k into du dy to the power n. This k and n are two very important parameters like for a mathematician these are just two parameters but somebody who is working with a biological problem to solve then this k and n are just not arbitrary mathematical parameters. So this k is known as flow consistency index and this n is known as flow behavior index. So tau versus n sorry tau versus du dy if you make a plot for different values of n you can make a plot. So the common situation is the Newtonian fluid for which n is equal to 1. So you know that Newtonian fluid with an yield stress that is something which is like a Bingham type of behavior. So there you will have a shift of this so that you do not call as the Newtonian fluid because it will have a yield like characteristic but beyond the yield point it can have a linear stress versus strain. I am not drawing all the plots just to give you some idea. Then there are fluids and blood closely resembles that that is called a pseudo plastic fluid. So what is a pseudo plastic fluid? Pseudo plastic fluid is a fluid for which the apparent viscosity what is the apparent viscosity? You can write tau is equal to k into du dy to the power n-1 into du dy. Why you are casting it in this form is because then this is like an apparent mu just to have the resemblance with the Newton's law of viscosity. So if it could be treated as a Newtonian fluid what could be the possible viscosity that is called as the apparent viscosity and the apparent viscosity will depend on the shear rate. So typically that occurs for n less than 1 that the apparent viscosity is actually decreasing with increase in shear rate. So for n less than 1 so typically what is happening in blood? See blood has blood cells and these cells are soft and deformable. So at low value of shear these cells will, cells may form agglomerates or aggregates and that might increase the apparent viscosity but at high shear rates these soft cells can be deformed and then that will actually increase the flowability or reduce the effective viscosity. However if hardness of the cell is increased say because of some disease the cell has become very hard then this behavior will change dramatically. That means rheology of the blood can carry the signature of a disease in the blood sample. Currently we are trying to device medical diagnostic protocols that will identify disease based on rheology then we do not have to get into a chemical based detection of the blood sample. So this is a fluid dynamic way of detecting diseases. So I am just giving you a perspective that why are we studying all these because eventually we will be doing some mathematics but we have to understand that what is the broad picture that is there. n less than 1 this is pseudo plastic n greater than 1 that is called as dilatant. That means it is apparent viscosity so this is n less than 1 this is n equal to 1 and maybe let us draw with a different color this is n greater than 1. So n greater than 1 examples like say soft tissues I am just giving biological examples I mean because we are talking about that. So soft tissues if you strain them it becomes harder and harder to strain them further and further. So that means their apparent viscosity if you call that as a fluid. See at this level of material for this type of material the sanctity of the definition of a fluid is under question whether it is fluid or solid or whatever we simply call it is a material and the material has some constitutive behavior. So we can see not only that the apparent viscosity of the blood may also be a strong function of time that means you leave the blood sample and you may see that actually you have a reduction in the effective viscosity of the blood as you allow more and more time or as you strain more and more. So it may be possible just I am drawing a hypothetical picture that you have a hysteresis. This is the forward experiment the backward if you do maybe you come along this same line along a different line. So the dependence of the apparent viscosity with time and in fact the reduction of the apparent viscosity with time makes blood a strongly time dependent or gives blood a strongly time dependent rheological characteristic and this particular characteristic is known as thixotropic. So blood is also called as a thixotropic fluid. However the time scale over which we are doing experiments in the microfluidics may not be like large enough to have significant to be significantly influenced by the thixotropy of the blood sample. So we are neglecting thixotropy for our analysis but see there are certain things which we are neglecting but we have to keep in mind that these effects are important. These effects cannot be ruled out. They are not important always but for certain cases they are. Now for n less than 1 what happens as you go for a large value of strain what happens? What will happen to the apparent viscosity? It will be tending to 0 right. So you can see that that is not actually a correct physical behavior. So actually the power law model is not a very mathematically correct model to describe the behavior of for example a fluid like blood. But over the ranges of parameters for which we are doing experiments if these parameters k and n are appropriately fitted we will see how they are fitted then it may be possible that by using this law one can capture a significant level of the physical behavior of blood through this model. Now what we will do is we will use this power law model and try to derive what is the change that means what is the drag force in a circular capillary because of the flow of a power law fluid. So we will basically deal with Navier stokes not Navier stokes equation the Navier equation no more Navier stokes equation because it is non-Newtonian fluid right. So we have to be careful and sometimes loosely because we are always doing mostly with Newtonian fluids we loosely have a tendency to like use Navier stokes equation for anything and everything but we because it is a non-Newtonian fluid we will start with the Navier's equation and not the Navier stokes equation and the Navier's equation in a cylindrical coordinate system because we are using circular capillary. So we will do that I mean I will write the Navier's equation for the axial component of the momentum it is impossible for me to remember it or reproduce it properly. So I will just reproduce it from my notes. So rho we have neglected any body force otherwise there will be a rho into body force there this is z momentum expressed in terms of thus the stress but stress is not related to rate of deformation. So depending on how is stress related to rate of deformation you can further simplify this equation. So we will start with this again this is not Navier stokes equation okay. So now we are considering steady flow so this is equal to 0 fully developed flow that means v r equal to 0 no variation with respect to theta because it is axially symmetric and fully developed flow so v z is not a function of z okay. So the geometry that we are discussing is this is a circular capillary and the axial direction is the z direction then this term is 0 and this term is also 0. So you are left with 1 by r del del r of what is tau z z depends on what? Which tau z z depends on which derivative of v z or which derivative of v z it is v z derivative with respect to z and that is 0 for fully developed flow. So tau r z into r is equal to del p del z now tau r z will be equal to minus k this minus sign is to add the adjust to the coordinate system see when you write du dy then in the in the law in the constitutive behavior that y is normal away from the wall and here this r is normal towards the wall. So just to make an adjustment to that you actually have this minus sign okay so that means you have now we will use this form subsequently now what we will do is we will first integrate this. So tau r z r is equal to now this is a function of r only this is a function of z only therefore each is equal to constant so this will be dp dz. So dp dz which is a constant into r square by 2 plus c 1 right r equal to 0 tau r z is equal to 0 right why at r equal to 0 tau r z should be 0 because of the symmetry in the velocity profile so that means you have c 1 equal to 0. So tau r z is equal to dp dz into r by 2 we have to keep in mind so how do you get wall shear stress just substitute r equal to r and you can see that the wall shear stress will not depend on k or n I mean wall shear stress but this is an illusion the relationship between wall shear stress and pressure gradient does not depend on k and n but wall shear stress will definitely depend on k and n because wall shear stress can be will be fundamentally calculated from the velocity profile. So it is not that the wall shear stress is independent of k and n but the relationship between wall shear stress and dp dz will be independent of k and n okay anyway so we will proceed further tau r z that is equal to in place of tau r z we will write minus k dv z dr to the power n so we can write dv z dr is equal to minus 1 by 2 k dp dz to the power 1 by n into r to the power 1 by n right just for algebraic simplicity we will call it some parameter a so dv z dr is equal to a r to the power 1 by n so then v z is equal to what a r to the power 1 by n plus 1 divided by 1 by n plus 1 plus c 2 the second boundary condition at small r is equal to capital R v z is equal to 0 no boundary condition that means 0 is equal to a capital R to the power 1 by n plus 1 by 1 by n plus 1 plus c 2 so let us write the velocity profile then so v z is equal to a by 1 by n plus 1 into small r to the power 1 by n plus 1 then minus capital R to the power 1 by n plus 1 now what is the next step this a contains dp dz how do we eliminate dp dz we will express a in terms of the average velocity right so what is the average velocity v average or u average whatever name you give v average is integral v z into 2 pi r dr divided by pi r square flow rate by the cross sectional area so that is 2 a by 1 by n plus 1 right 2 a by 1 by n plus 1 right 2 a by 1 by n plus 1 r square integral of this into r dr right so r to the power 1 by n plus 2 minus capital R to the power 1 by n plus 1 into r dr so 2 a by 1 by n plus 1 r square this becomes r to the power 1 by n plus 3 by 1 by n plus 3 minus r to the power 1 by n plus 2 sorry r to the power 1 by n plus 2 sorry r to the power 1 by n plus 3 1 r square will make it 3 divided by 2 right this will become r to the power 1 by n plus 1 if you cancel 1 this r square then you can cancel this 2 divided by 1 by n plus 1 into 1 by 1 by n plus 3 minus this 2 is there a is there so this will become 2 minus 1 by n minus 3 so minus of 1 by n plus 1 right so this will become 1 by n minus 2 r to the power 1 by n plus 1 a by 1 by n plus 1 by 2 into 1 by n plus 3 then minus of 1 by n plus 1 alright then this 1 by n plus 1 gets cancelled 2 gets cancelled so minus a r to the power 1 by n plus 1 divided by 1 plus 3 n then n in the numerator therefore you can write a is equal to minus 1 plus 3 n into b average by n plus 1 by n plus 3 into r to the power 1 by n plus 1 right we will substitute this a in this to express v in terms of b average so v z is equal to n by 1 plus n now in place of a we will write minus 1 plus 3 n into v bar by n r to the power 1 by n plus 1 right we will substitute 1 by n plus 1 into r to the power 1 by n plus 1 minus r to the power 1 by n plus 1 n gets cancelled so v z by v average is equal to 1 plus 3 n by 1 plus n into 1 minus small r by capital R to the power 1 by n plus 1 okay you can quickly check the result by putting n equal to 1 so if it is if n equal to 1 it becomes 4 by this so 2 so v z by v average is equal to 2 into 1 minus smaller by capital R square that is the Hagen Poiseuille flow velocity profile okay so you can see that once you get this velocity profile then you have everything so you can calculate the Walshia stress you can calculate the Walshia stress that is minus k into d v z d r to the power n at small r is equal to capital R and this v is your d s v bar is your d s d t this is d s d t your drag force is equal to tau wall into perimeter into s if it is a fully developed flow it is like over the entire s the same tau wall is there otherwise you have to take a small element d x and integrate over that right if you consider the other regimes that is the developing regime the meniscus traction regime and so on but here we have dealt with mainly the fully developed flow so the drag force this is f d star as for our notation that is fully developed flow drag force this depends on what this depends on the parameter n right so this parameter n is very important to dictate the capillary flow dynamics by dictating the viscous resistance and this parameter n depends on the hematology of the blood sample so how does it depend we will discuss about that in the next class for the time being thank you very much.