 That is we were discussing about the index notations for the stress tensor components and how to relate the stress tensor components with the traction vector. So this is what we came up with. We are not writing the summation notation because as I told that if you are having a repeated index there is an invisible summation over that. So this i is called a free index because it may be whatever 1, 2, 3 like that. So being in the right hand side and in the left hand side in the right hand side only once this is like free but j is a repeated index and therefore it is a dummy index. In place of j you could write k, l, m, n whatever it makes no difference but whatever is this i should also be the same. So this index and this index they should correspond and we also showed that tau ij equal to tau ji. Therefore we can also write this as tau ij nj which is the corresponding form of the Cauchy's theorem by taking the moment balance into account. We will also discuss about some of the alternative notations because in text books different notations and of course the index notation is the clearest one and I would say that most convenient one to use but we will also look into the alternative ones. So when you have this tau 11, tau 22 or tau 33 these are special components of the stress tensor. What do these represent? Normal components of the stress. So in many texts you will see the corresponding symbol as sigma 11 say corresponding to tau 11, sigma 11, sigma xx or sigma x equivalently. So there should ideally be 2 indices but since both are repeated sometimes text omit the repeated one and just use 1x. So sigma x perhaps this is the original symbol that you learnt for the first time when you were learning the mechanics. So but tau or sigma with 2 subscripts would be more convenient and more fundamental notation. So if you have for example tau 12 then in that xy notation you can also write it as tau xy. So always remember 1 for x, 2 for y and 3 for z. That should correspond to any symbolic notation that you have in the book whatever book you are following and whatever notation that we are going to use in the class. Of course we will try to use all sorts of notations in different contexts so that you feel comfortable with the general notation that is used in different places. So what we sum up from the discussion on the stress tensor is that the stress tensor has different components. We can clearly identify which are the normal components. The normal components are the components which are appearing in the diagonal of the corresponding matrix representation. There are off diagonal components which are like so called shear components. Now we have to see that how the characteristic of the fluid is related to the components of the stress tensor and we will subsequently see that how they are related to the components of the stress tensor is going to be strongly dependent on one important property of the fluid which is the viscosity of the fluid. So we will come into the properties of the fluid subsequently but before going into the viscosity we will talk about one of the quantities or one of the properties which has a very important relationship with the stress tensor and that is nothing but the pressure of the fluid. So we will discuss something about the pressure of the fluid. Whenever we start discussing about pressure of a fluid we generally start discussing about fluids at rest. This somehow gives a misunderstanding to people that pressures are quantities very relevant only to fluids at rest. It is not so as we discussed earlier that when a fluid is at rest there is no shear that is acting on it. So only the normal component of force is acting on it and therefore pressure becomes the only relevant surface force in that context and that is why it is easy to isolate all other effects and just focus on pressure if we are discussing about fluids at rest. If we are discussing about fluids in motion obviously pressure does not get irrelevant it may even get more and more relevant but to begin with we will consider that we are discussing about say an element let us say a wedge type element like this. Again I mean you may consider it 3D version by considering uniform width perpendicular to the plane of the board but let us just consider that it is a section of the wedge in the plane. So what we are interested to see is that what are the forces which may act on this wedge shaped fluid element when the fluid element is at rest okay. So we are giving some names of these dimensions let us say this is delta L and maybe this angle is theta. Just like what we did earlier for a general fluid element which may be subjected to normal and shear component here we are going to discuss about a fluid element where there is only normal component of force on the surface. The reason is quite clear we are assuming a fluid element at rest so there is no shear component keeping that in mind so this let us keep in mind this is a fluid element at rest. We will later on see that even if the fluid element is not at rest but internal deformation is negligible but the fluid element is moving like a rigid body then also similar considerations may be valid but for the time being we consider it at rest. So we are identifying various forces which are acting on the surfaces and the volume the surface force and the body force. So fundamentally we will treat any mechanics problem continuum mechanics problem in terms of the forces as a collection of body forces and surface forces which we will try to keep it in either static or dynamic equilibrium. So for surface forces we will first consider say the force on this surface so a normal force and pressure by nature is acting normally inward like the term pressure by that we qualitatively understand intuitively understand it is something which tries to compress the element so it tries to act inward to the surface. So whatever is the outer fluid element that is trying to apply a normal reaction it is like a normal reaction but the normal reaction is inward always. So it is trying to pressing it trying to press it so to say. So what will be this? This will be let us say that we do not know whether pressure will be different along x, y or z. So this is pressure along y so we call it say p y. When we call it p y this is the force per unit area all of you know about that and this multiplied by say delta x into width say 1 is the total force. Similarly and for this let us say that this surface has an orientation such that the direction normal is n so we call it p with subscript n. Why we are keeping these subscripts? We are still not sure that pressure should be varying as we change the orientation. Fundamentally we should be unsure to begin with because we have seen that any force acting on the surface is likely to be strongly dependent on the orientation of the face that is chosen. So there is no reason to believe that pressure should not be whether it should be or should not be that is what we are going to derive. Now this will be this into delta l into there will be no shear component because it is a fluid at rest. There will be a body force which is acting on this. So let us write the body force components say body force along x and body force along y. So what will be the body force along x? Let us again say that b 1 is the body force or say b x is the body force per unit mass acting along x. So this multiplied by the mass of this. So what is the mass of this? So this is like a triangle so half base into altitude that is half into delta x into delta y into the width that is the volume that multiplied by the density is the mass then b x is the body force per unit mass. So this is the total body force which acts along x. Similarly you have total body force that acts along y. I am not repeatedly writing it just to save time. And so we have written all the forces. The resultant force if we write the force equilibrium resultant force along x is equal to the mass of the fluid element times acceleration along x. Well when we say acceleration along x your general idea would be that we are considering a fluid element at rest. So fine let us first consider fluid element at rest. So if we consider a fluid element at rest the right hand side is 0 obviously but we will see that whether the effect of right hand side is there at all or not. For that first let us write the left hand side. So when you write the x component you will have minus p x into delta y then the component of this p n in the direction of x. So you have x axis like this y axis like this. So what will be the component of this p n along so p n into delta l into sin theta then plus there is a body force component along x. So half delta x delta y rho b x that is equal to mass of the fluid element. So delta x delta y half delta x delta y rho times acceleration along x. Of course you can write delta l sin theta as delta y right. So we will make this simplification in place of this we will write delta y. Remember just like we in the last class tried to find out expression for the stress tensor components and traction vector in terms of stress tensor components at a point. So here also we are interested to do that so that we want to shrink the size of these as in the limit as delta x delta y all tending to 0. So we will see what is the consequence of delta x delta y all tending to 0. So if you take that limit then what happens let us see that limit. So if you take that limit as delta y tending to 0 this will be 0. So our limit is delta x delta y tending to 0. This will be 0 tending to 0 these are tending to 0. So you can cancel delta y in all sides that means this you cancel this you cancel this you cancel this you cancel. So what is left you have minus p x plus p n plus half delta x rho b x is equal to half delta x rho a x. So when you take the limit as delta x tends to 0 then obviously this term goes away and this term goes away. Therefore you get p n equal to p x irrespective of whether this is accelerating or not. This is a very very important concept because in your high school physics you perhaps have done the same thing but assuming that it is at rest and that might create a misconception that this will not work if it is moving like a rigid body with an acceleration. And if you see it does not matter even if it has a body force. So even if there is a body force that is acting still this equality is valid. Similarly by considering the force equilibrium along the y direction it will follow that p n equal to p y. So as a conclusion we can say that p n is equal to p x equal to p y which means that we are talking about a quantity which does not sense the direction. So it is insensitive to the direction and it is acting always normal to the surface on which it is being evaluated. So this quantity we call as pressure. Since it is index insensitive or direction insensitive we can just call it without any index and therefore unlike the general stress tensor it is not a tensor because a second order tensor requires 2 indices for its specification you should remember whereas this requires no index for its specification. So of course it is a tensor but tensor of order 0. So it is more easily termed as the scalar. So you can see that although stress and pressure both are expressed in terms of force per unit area mathematically and fundamentally their characteristics are somewhat different and that we have to clearly remember whenever we are discussing about these 2 related terms. There are more involved concepts on pressure which we will come across subsequently. One important concept is that there are 2 terminologies involved. One is called as mechanical pressure another is called as thermodynamic pressure. So first we start with the mechanical pressure we briefly discuss about the concept we do not elaborate but the subtle concept we will try to understand. So when we say mechanical pressure what we mean? What we mean is that see if you are thinking about the normal stress components of the stress tensor you have tau 11, tau 22, tau 33 and if you feel that this P is a representative of the normal stress states of the element that is chosen because we are not considering the shear stress. Effect of normal stress we are representing by P that means somehow there is likely to be a relationship between this tau 11, tau 22, tau 33 and P. The relationship may not be straightforward but since P is same in all directions we can say that there may be a component of or a part of this tau 11, tau 22, tau 33 which is like direction insensitive and that we may say just for the sake of definition of mechanical pressure. So it is a basic definition that the mechanical pressure is defined as the arithmetic average of the 3 normal components of stresses with a minus sign. Minus sign is straightforward to understand because the positive sign convention of these were outwards from the surface whereas the pressure by nature is inverse to the surface. So to adjust that this minus sign is there. So this is also called as something like a hydrostatic component of stress that means you are assuming that it is like it is representative of a state of stress where it is represented by a quantity pressure which acts equally from all directions. So it is as if like a fluid under rest that we are considering. So any state of stress which deviates from this hydrostatic part is known as deviatoric part. So that is what is something which deviates from a hydrostatic state of stress. We will come into the details of these concepts later on whenever we are going to discuss about the equations of motion for viscous flows. But this is just an elementary definition of mechanical pressure. Now when we talk about pressure actually we are not talking about really referring to this mechanical pressure always fundamentally because whenever we talk about pressure think about say you are talking about pressure for gases. You always relate pressure with density and temperature through an equation of state. So pressure from a thermodynamic point of view is something which satisfies the equation of state through the density and temperature. For an ideal gas it is very simple for non-ideal gases it may be a more complicated equation of state but still equation of state is something which relates pressure density and temperature in some mathematical form. So thermodynamic pressure is that pressure P which will satisfy the equation of state. Now the question is is the mechanical pressure going to be equal to thermodynamic pressure or not. So what is the fundamental mechanism that will dictate that whether they are equal or not? Say there is a bubble. Inside the bubble there is a particular pressure, density and temperature. Now you are making the bubble to fluctuate its frequency of formation that is the bubble is changing its state very fast. So what will happen? There will be a particular thermodynamic pressure density temperature suddenly you are changing its state to a new state new pressure density temperature. So in that way say the bubble is suddenly expanding and contracting expanding and contracting like that and it is doing it very fast. Once it is doing very fast the change is not so easily adjusted. So the system requires a at least a threshold time to adapt itself to the change and make sure that it has mechanical pressure which is like a kind of pressure that acts equally from all directions same as what is dictated by thermodynamic state. So thermodynamic state is a change that imposes a kind of disturbance to the system. System requires a time to attain equilibrium so that it will eventually have mechanical pressure equal to thermodynamic pressure and therefore we generally do not distinguish between mechanical and thermodynamic pressure. We say that it is just a pressure but if the change is so fast that the system at intermediate states does not get enough opportunity to attain equilibrium. So that whatever change is in the thermodynamic state the system does not get enough opportunity to adjust to those subsequent changes and then in such cases you will not have mechanical pressure equal to thermodynamic pressure but those are very rare cases. So for most of the practical engineering applications the changes are such that those changes will be adjusted or adapted to by the system in a way that you will have mechanical pressure equal to thermodynamic pressure and therefore whenever we will be talking about pressure we will not be distinguishing the mechanical and the thermodynamic pressure. We will be just calling it as pressure so that is how we will be going about it. Now we have discussed about one fluid property which is pressure whenever we are talking about effects of compressibility there are other related fluid properties which come into the picture in a very related manner and those properties we will look into one by one briefly. So one of the important properties will be density. So loosely what we say that if we have a volume elemental volume say delta v we have the mass of the molecules which are there in this delta v. So we take mass per unit volume all of us like to write limits so we will write limit this as delta v tends to 0 we will think that nicely it should give a limiting definition of what is the so-called density mathematically very nice we will see whether it works or not. So what it says? It says in the limit as delta v tends to 0 that means a limitingly small volume you find out what is the mass of molecules inside so you find that mass per unit volume to get local density at a point that is what this definition is saying whether it works or not we have to come back to the continuum hypothesis to adjudge. So if we remember that in the continuum hypothesis we disregard the molecular nature and we just consider that it is a continuous medium does not mean that there are no molecules but obviously we are abstracted of the molecules and we are just representing that their gross effect. But whenever there are molecules we have to see that what is the number of molecules within this elemental volume. Again if the number of molecules within this elemental volume is very small then because of the statistical fluctuations even uncertainty in one molecule will give a lot of error and will give a lot of fluctuation. So it is critical that what is that elemental volume that you should choose it cannot be too small. What is the smallness? The smallness will come with a length scale the smallness of the length scale here is the mean free path lambda. So as very small volume when we say then that will scale with say lambda cube. Lambda is like a length scale which will correspond to a very small volume. So when the volume is of the order of lambda cube elemental volume then it will have lots of uncertainties in the statistical fluctuations of the molecules because within that length scale you really have uncertainties related to collision. On the other hand if you take this volume delta v very large then also you can calculate a density but it will not be able to capture the local variations it will give a global average. Therefore one has to choose a threshold length scale for calculating this density and how it should be sensitive to the length scale if you make a plot of say the length scale that you choose say we call it Ls and the density that you predict. So you will see that if you choose a very small length scale you will get a variation this type of fluctuation then it will come to a steady one and then if you choose a larger length scale it will be changing like this. So what is the significance of such a plot? These length scales are small enough so that you have really random fluctuations because of the uncertainties. This length scale is fine beyond this length scale you have a variation this variation is because over the system length scale the density is varying from one point to the other. So a correct choice of length scale may be something which can be say in between these 2. So in between these 2 limits therefore if we say delta v tends to 0 that is not fundamentally correct because delta v tends to 0 may make you fall on this regime because delta v tends to 0 means mathematically delta v is as small as possible. So as small as possible will obviously be something smaller than as big as possible. So obviously in that context one has to remember that this delta v tends to 0 should be corrected and how it should be corrected? We should change it as not delta v tends to 0 but delta v tends to some delta v star which is like if we call this as Ls star then maybe it is of the order of Ls star cube. So it is a threshold length scale beyond which you are not having such uncertainties and fluctuations affecting your density calculation. So this delta v star therefore we can say is the smallest elemental volume over which continuum hypothesis is valid. So it is not tending to 0 but tending to a limitingly small volume delta v star over which still continuum hypothesis works. Below this limit continuum hypothesis might not work and therefore this definition will not work because this definition is on the basis of a continuum description of fluid properties like density. So we have talked about density, we have talked about pressure. Next let us talk about bulk modulus. So all of you are aware of the basic concept of bulk modulus but let us just see that how you have defined it. Let us try to define it first in a very loose manner. It is always important to get a qualitative feel and then of course you can have more sophisticated definitions. We will not go into very detailed sophisticated definition of bulk modulus because it requires a detailed understanding of thermodynamic processes and therefore we are not going into that type of definition. So in a loose sense if you are applying say a pressure differential delta p that is expected to give rise to a change in volume of a fluid element. Let us say that change in volume is delta v. So original volume was v. So this is the rate of or this is the total change in volume per unit volume. So this is the kind of a volumetric strain and this is the pressure differential which is responsible for the volumetric strain and you expect that if delta p is positive delta v is negative because if you press a fluid element it should compress its volume should decrease. If you want to give the corresponding fluid property a positive number definition then you should adjust it with a negative sign. Now you can relate the change in volume with the change in density. How is it possible to relate the change in volume with the change in density? So you have say consider the mass of a fluid element. So that is the density into volume. From now onwards whenever we will be discussing about the volume we will be using a symbol not v but v with a strike through because we will be using v for velocity also. Just to avoid that confusion between the symbol of velocity and symbol for volume we will be just distinguishing those in this way. So I will not be repeating the symbol many times but once I will be using this type of symbol you just take it that we are talking about the volume not the velocity. So if you want to say see that what is the relationship between the elemental change of density with the elemental change of volume what you can do simply just you can take log of both sides and differentiate. So if you differentiate keeping in mind that the mass of the fluid element is conserved. So its derivative should be 0. Therefore loosely like if you are following this definition we can relate delta v by v with delta rho by rho. So that we will just absorb the minus sign and it will be like this. So it is like rho delta p over delta rho like this. Now we can relate delta p with the velocity of flow. In a order of magnitude sense the delta p and the velocity of flow this is just like if you consider that there is an equivalent pressure change which is brought about by the change in kinetic energy of fluid which is moving with the velocity u then this is not that they are exactly equal it is just to say that one scales with the other in this way. So you can therefore write a scale of k as so rho let us try to see that what is delta rho by rho scale that means what is the change in density relative to its original density. So if we do that it will be half rho u square by k. Remember one thing that this k by rho it is something which is a very fundamental quantity which you have studied in physics what is this or square root of k by rho if it reminds you more. This is fundamentally sonic velocity sonic speed so to say. What is sonic speed? Sonic speed is not just speed of sound sonic speed is the speed by which a disturbance propagates through a medium and here we are talking about this type of disturbance through the elastic property of the medium so it happens to be the speed of sound. So this is the sonic speed a so where a we call as sonic speed this is a very basic high school physics based definition. So keeping this in view we can write this as half u square by a square. So you can see that the relative change in density is related to a quantity u square by a square. What is this u square by a square? This is a non dimensional quantity that you can see because it is a ratio of 2 velocities. So in the numerator you have u in the denominator you have a. So u is the velocity of flow and a is the velocity of a disturbance which is moving in the medium in which the flow is occurring and these 2 ratios is known as Mach number. I mean ratios of these 2 numbers is known as Mach number. So you have heard about the Mach number like a jet moving with a Mach of this. So higher the Mach number higher is the velocity of flow relative to the velocity of the disturbance with which the disturbance propagates within the medium and therefore we say that it is having a more and more compressible effect. The reason is if you have if you just write this delta rho by rho you see that it will scale with half of square of the Mach number. So higher the Mach number higher is the effect of the change of density relative to its original density. So Mach number therefore is a very important indicator of something which is called as compressibility of a fluid. So what is the what is the signature of compressibility of a fluid? We will say that a fluid is compressible when it has a change in density because of a change in pressure. So in that way all fluids are compressible right because all fluids will have some change in density because of change in pressure but when we say that a fluid is incompressible what we mean is that that effect is negligibly small. So a compressible fluid and an incompressible fluid these are just conceptual paradigms there is no fluid as such which is incompressible but when we say that a fluid is incompressible we mean that its compressibility effect is very very small. Again how small or how large that is something which may be debated. So let us say that we are talking about a change this relative change say 5%. So let us say that if we say that this change is less than 5% we say that it is almost incompressible. So if we want to see that what will lead to that 5% so one may work it out with say 5% means 5 divided by 100. So what would be the threshold Mach number for this roughly 0.3 right. So 0.33 or whatever but roughly 0.3 that means if we say that a relative density change less than 5% is something which we do not consider as a compressibility effect that implicitly means that a Mach number less than 0.3 is something which is not going to give us any serious compressibility effect. So this is important because whenever you are analyzing an engineering flow nobody will tell you that whether the flow is compressible or incompressible as an analyzer it is your responsibility to make a judgment of whether you are going to use the concept of compressible flow or incompressible flow for the analysis of your problem and then you have to be confident that whether a particular analysis methodology is going to work or not. Of course for all flows compressible flow and analysis will work because all flows are compressible but it is like if you have a mosquito you will not like to kill it with a cannon. So if you are ready or if you are having the possibility of doing a relatively simple analysis one should not go for a complex analysis that is what all of us have learnt in engineering that do not go for unnecessary complication until unless it is absolutely required. So whenever compressible flow analysis is not required we should not go for it and this Mach number of flow will give us a guideline of whether we should go for a compressible analysis or not. A couple of other important points or remarks are there regarding this definition of bulk modulus. One is see in this definition we have talked about a change in volume because of a change in pressure or equivalently a change in density because of a change in pressure. But pressure effect of change of density it depends on the type of process. All of you have heard of certain thermodynamic processes like adiabatic process, isothermal process and so on. So given a particular system how the density will change with pressure will depend on the nature of the thermodynamic process. So this definition as such fundamental is not incorrect but incomplete because it does not talk about the thermodynamic process by which you are trying to have this change of state. So there are more fundamental or correct definitions of this in terms of specifying it as say either a reversible isothermal process, reversible adiabatic process and so on. But we are not going into those details here because thermodynamics is not the scope of this particular course but we should keep that in mind because whenever you will be studying thermodynamics again this type of definition will come into the picture and there more detailing will be done in terms of whether it is a reversible adiabatic process, reversible isothermal process and so on. So that is one of the important concepts. The second important concept is as follows. Say you are interested to identify whether a flow is incompressible or not and in that respect there is a subtle difference between the concept of incompressible fluid and incompressible flow. These are very very subtle concepts. So when you talk about an incompressible flow what you mean is that if you have a volume element of a fluid that volume does not change. So incompressible flow means that there is no volumetric strain of the fluid element. There is no change in volume but you cannot directly always relate it with this definition because the change in volume may not always be due to change in pressure directly. It may be because of something else also. There are reasons for which you might have change in volume of a fluid element not because of the change in density due to change in pressure but may be because of change in density due to change in temperature not directly due to pressure. So whenever we are talking about incompressible fluid we are talking about that we are asking ourselves a question that is there a change in density because of a change in pressure. If that answer is that yes it is significant we call it a compressible fluid but not compressible flow definition is something more general compressible flow means a fluid element which if it is going to have a volumetric strain or change in volume per unit volume by whatever reasons it need not be just due to pressure or it may be because of anything then we say that it is a compressible flow. So compressible fluid and compressible flow are related because of course one of the reasons of being a fluid compressible or being a flow compressible is because the fluid itself is compressible. So the density change due to change in pressure is significant but there could be other effects that are creating the change in volume. So this is the concept that we should remember. Next what we will do is we will try to learn about very important property of fluid which is called as viscosity. So when we talk about viscosity we will not try to just learn it in abstraction but we will start with an example. Let us say that you have a flat plate just like the top of a table a flat plate and fluid is coming from far stream just like say fluid is being blown from that side it is coming on the top of the table and going away. So the top of the table may be like a flat plate. So let us say that the fluid is coming with a uniform velocity from a free stream in fluid mechanics usually we give such a symbol infinity with a subscript to indicate that it is a free stream condition. So infinity subscript is like a free stream velocity. So it is coming with a uniform free stream velocity. Now that freeness will be disturbed because of the presence of the plate and let us see that what is going to happen. So when this fluid first comes in contact with the plate what happens first try to understand that say there is a fluid molecule which comes in contact with the plate. So what will the plate like to do with the fluid molecule? Let us consider 2 different examples one is for a gas and another is for a liquid usually whenever we discuss about fluids we are either talking about gases or liquids but sometimes their physical behaviour it is better to discuss distinctly or differently. So let us say that there is a gas molecule as a first example which is coming falling on this plate. So what will happen? There will be first a tendency that the gas molecule is adsorbed on the surface. So once it is adsorbed on the surface then what will happen that it will exchange some of its momentum with the surface. So it will try to have it slowed down and then again it will try to be getting ejected from the surface. So it is like a molecule falling on the surface adsorbed on the surface getting ejected from the surface like this. So in this process many molecules are colliding with this and they are exchanging their momentum with the wall. So if there are very large number of collisions so to say theoretically infinitely large number of collisions then this kind of momentum exchange will bring on an average the molecules in equilibrium with the surface. So if the surface is at rest the molecules will also be at rest. So that will imply that there is 0 relative velocity between the fluid and the solid at the point of contact and this is something which is known as no slip boundary condition. So fundamentally what is the no slip boundary condition? It is 0 relative tangential component of velocity to be more accurate 0 relative tangential component of velocity between the fluid and the solid at their points of contact. We are not talking about the normal component because still the molecule may be colliding like it may have a sort of elastic collision so it may bounce back so it may have a normal component. Now obviously regarding the normal component there are issues like if the molecules are sufficiently large in number and they are at the wall they cannot penetrate and go through the wall. Wall is not having holes so that is called as a no penetration boundary condition then there even the normal component of velocity will become 0 but no slip boundary condition does not talk about that. That is a separate consideration. No slip boundary condition talks only about the tangential component of velocity. So 0 relative tangential component of velocity between the fluid and the solid at the point of contact. Now as I am telling this to you, you are tending to believe that this is always the correct picture and this has happened really for a long time. So for a long time this no slip boundary condition was taken as something which is like a ritual which should not change and the reason was that for many or for most engineering flows it is still valid or it has been experimentally found to be very very accurate. But whenever we are understanding this concept we should ask ourselves a question are there conditions in which the no slip boundary condition may be violated. It is important because in many of the modern day applications of fluid mechanics especially fluid mechanics in small length scales, this boundary condition is something which is put under serious question. So obviously we need to see that or we need to appreciate that this is just a conceptual paradigm. It is not something which is a ritual and which is expected to work always. Let us see, let us try to look into an example with in the context of gas flow that no slip boundary condition does not work. So let us say that you have gas molecules but not very large number of gas molecules. So then what will happen? The molecules will be exchanging momentum with the wall but there will not be very large number of collisions. Because there will not be very large number of collisions the momentum exchange will not be complete. So there will be some velocity of the fluid relative to the solid boundary even if otherwise we tend to believe that there should not be any slip. So that is just because of the rarefied nature of the medium that there is not sufficiently large number of molecules to have a theoretically large number of or infinite number of collisions. On the top of that there may be local strong gradients in density and temperature and that might itself induce motion of molecules of gases over the solid surface. So these are called as phoretic motions. If these are induced by temperature these are called as thermophoresis and these may be induced by any other effect but temperature is one of the common effects by which by introducing a very high gradient of temperature you can introduce local flow of molecules of gases over the solid boundary. So we can see that there may be situations and there are likely to be such situations when the no slip boundary condition is not valid but well in most of the engineering systems that we are talking about the no slip boundary condition will work for gases. Except for rarefied gases or may be gases which are not having sufficiently large number of molecules or gases being subjected to very high local gradients of density or temperature. For liquids it is difficult to believe that the no slip boundary condition will not work because liquids are very compact systems. So liquid molecules will not be insufficient in number to have inadequate collisions with the wall but for liquid molecules there may be slip because of certain reasons. So to understand the picture of what happens in the liquid molecules let us consider a small element of surface like this. The surface may look very smooth on the top but if you look it into a very powerful microscope it will be much much worse than what I have drawn here. So it will have lots of peaks and valleys and what will happen is that the molecules will nicely sit on these peaks and valleys. So some will be entrapped like this and because of a compact nature what will happen whatever is entrapped is not easily being escaped and that will make us believe that yes it will be a no slip boundary condition. At the same time if you are having a very high shear rate which is being introduced on the liquid say a very high rate of shear strain what will that try to do that will try to forcefully dig this out from or take this out from these locations. So then in that kind of a context the liquid molecules may also slip on the surfaces. Otherwise if you have very smooth surfaces say you must have heard about carbon nanotubes these days those are very sophisticated and fascinated technologies to produce carbon nanotubes. So those are very smooth tubes and if you are having liquids in contact with them now obviously there will be the van der Waals forces of interaction between the surface and the liquids but if in such case water flowing through those nanotubes will have very ordered hydrogen bonding and then the motion of that water will be such that it can overcome the van der Waals forces of interactions those are relatively weak in comparison to this strong bonding on in the water to overcome the wall attraction and flow on the top of such surface. So it may actually slip and these are called as highly slipping surfaces therefore we have to keep in mind that no sleep boundary condition is a paradigm which will work for most of the engineering problems that we are going to consider but at the same time we should not take it as a ritual. We should keep in mind that there are situations in which it might be violated but for practical purposes for almost all the problems that we are going to solve in this particular course no sleep boundary condition will work. So let us stop here today and in the next class we will take this up and introduce the concept of viscosity through this no sleep boundary condition that we have discussed today. Thank you.