 So, far we have dealt with various problems concerning pressure driven flows and typically in the low Reynolds number regime and for all these cases we have considered that the no slip boundary condition is valid at the interface between the fluid and the solid boundary. So, this is a classical consideration, but if you ask that where from do you know that the no slip boundary condition will be valid always the answer is not straight forward. I cannot give you just an answer in a single sentence that where from I do know that the no slip boundary condition will be valid. Well one of the possible explanations that is given by people working with classical fluid mechanics is that when people do experiments with classical systems, macro scale systems then it is observed that the no slip boundary condition is valid and seeing is like believing. So, if experimentally things are justified and there are virtually no experiments which are dissatisfying those requirements then that can be considered as like sort of I mean well described boundary condition. At the same time one when goes down the scale, micro scale, nano scale regime it is not quite clear that why the same arguments should hold true and therefore it is important to look into the boundary conditions in fluid mechanics or have a relook into the fluid boundary conditions in fluid mechanics and typically the no slip boundary condition. So, the motivation for this is that we cannot always depend on the no slip boundary condition I mean there are situations some of the situations are described in this particular slide but I mean these are just some typical examples like not all the rough surfaces hindered the flow. I have discussed I mean quite some time back in one of our introductory lectures that if you have rough hydrophobic surfaces then actually there may be a thin layer of low density phase that forms on the surface which acts like a friction reducing cushion. So, what it means is that not all rough surfaces hindered the flow and sometimes this can reduce the drag super hydrophobic surfaces in nature like lota slip can exhibit slip flows. So, it is possible that you can have slip flows with surfaces which are highly hydrophobic that means they have a great phobia for. So, the name hydrophobic is quite self-explanatory hydro means water. So, the surface as a phobia for water it does not like water. So, it is a non-weighting surface. So, the highly non-weighting surface in nature can exhibit slip flows and these super hydrophobic surfaces are not just natural surfaces but you can make artificial super hydrophobic surfaces by using nanotechnology. So, super hydrophobic surfaces can be engineered by nanofabrication technologies and these kinds of surfaces can also exhibit slip. So, while I started with a remark that seeing is believing then if you follow that logic then these things are also believing right. These are also certain observations which clearly state that slip is also a possibility. So, no slip boundary condition is something which may be justified in many cases not necessarily that if we get down from classical regime to the micro scale regime suddenly the no slip boundary condition starts becoming invalid it is not like that. But there may be certain situations when the no slip boundary condition needs to be right needs to be critically assessed before being used. Now before getting into the slip or the no slip we will briefly discuss something which is related to the continuum hypothesis. In continuum hypothesis or in continuum mechanics what we consider? We consider that the fluid is a continuous medium disregarding the discontinuities. So, you typically do not consider individual molecular entities as describing the motion, but it is a collective system that is being described as constituting the motion. So, if you are considering a molecular approach on the other hand you are directly analyzing the dynamics of individual molecules. So, in fact that particular approach is followed in a simulation technique which is known as molecular dynamics. We will discuss about molecular dynamics towards the end of this particular course. Microscopic approach is something which is like sort of a bridge between macroscopic and molecular approach. It does not consider all the individual molecules, but it can represent the statistically average behavior of molecules. So, it is within the paradigm of statistical mechanics or statistical physics. So, that is microscopic approach where you essentially represent statistically average behavior of many molecules and macroscopic approach which is the standard classical approach that we follow in say classical fluid mechanics states. Gross average effect of many molecules that can be captured by direct measuring instruments. So, it creates the fluid as a continuous medium disregarding the discontinuity in the underlying molecular arrangement. So, that is this macroscopic approach is also like a continuum approach. So, the continuum hypothesis works, but so this question is important that we are telling that the continuum approach is a very I mean convenient approach because we do not have to care about the molecular arrangement and all these things. We just have to consider the fluid as a continuous mass, but when is this approach valid? So, the continuum hypothesis works when there are sufficiently large number of molecules in chosen elemental volumes. So, like let me give you an example to illustrate what we mean by this. So, let us say that this is a container and within this container there is a point at which you want to find out the density of gas, okay. This is your problem. So, what you do is that we take a small control volume and basically you in the most fundamental way you try to count the number of molecules what is there inside. So, if m is the mass of each molecule and n is the number of molecules then n into m is the total mass divided by the volume of the elemental volume that is the local density. This is a very elementary way of calculating the local density and you can make this volume very small. Question is how small? If you make this volume very small then this n will also be very small and uncertainties in this n will result in a lot of uncertainties in the calculation of this. Let us say that n is large. If n is large then this molecules are always in random motion. Some molecules are leaving the control volume, this dotted control volume, some molecules are entering that control volume. So, at every instant of time there is a little bit of uncertainty. That uncertainty is a little bit only if the number of molecules is large but if the number of molecules is itself small that is say let us say in this control volume there are 4 molecules on an average. So, if one molecule just goes out there is a 25% error that comes into the picture straight away. So, if you have sufficient large number of molecules that is important but if you have 2 large number of molecules then you do not capture the local picture. Then you have to consider a picture over a larger region in space but not a local picture. So, to get a local picture you have to focus on a small volume but the volume should not be so small that uncertainties for this measurement will become very significant. So, you have to come to a compromise. You have to come to a volume where this small volume this v in the limit as v tends to v star. v star is basically a critical volume, critically small volume below which the continuum hypothesis will not work. The discreteness of the molecular arrangement will become important. Now, let us get back to the slide. So, if you see that the continuum hypothesis works when there are sufficiently large number of molecules that we have discussed. So, that statistical uncertainties with regard to their respective positions and velocities do not perceptively influence the average properties as well as predictions of local gradients. This is also very important. So, whenever you have a property you also have a property gradient. Now, how do you express the gradient? You want to express the gradient through well known rules of differential calculus and these well known rules of differential calculus will work only when you can think it as a continuous function. So, the continuum hypothesis should be justified. The second aspect when the continuum hypothesis works is not well discussed always that is a system is not significantly deviated from local thermodynamic equilibrium. So, this is also a very important consideration. We will discuss about the importance of this consideration when we discuss about gas flows. So, so far in this particular course we have mainly discussed on liquid flows, but like when we come to the boundary conditions and when we consider the applications of microelectromechanical system considerations then it becomes obvious that gas flows are also very important. So, we consider also the gas flows and in fact we begin with the gas flows because the slip boundary condition for gas flows is something which can be more intuitively explained. So, what happens for gas flows? So, this is the physical mechanism that happens when gas molecules collide with a solid boundary these are temporally adsorbed on the wall and are subsequently ejected. So, imagine a gas molecule colliding with a solid boundary they are very temporally adsorbed and they are ejected. So that means there is a possibility that there is an exchange of momentum and energy between the gas molecule and the solid boundary. If the frequency of collisions is very large then what happens? The momentum and energy exchange is virtually complete. So, if there are highly frequent collisions then the gas molecules and the solid boundary they almost perfectly exchange their momentum and energy that means there becomes no relative tangential momentum between the fluid and the solid boundary that is what is the no slip boundary condition that is the classical paradigm. So, that works when the frequency of collisions is very large. However, if you are considering a system which is not highly densed, see frequency of collision between the gas molecule and the solid boundary will depend on what? It will depend on the packedness of the molecular arrangement. So, if the system is a less dense system then what happens? Then you will not have highly frequent collisions and because you will not have highly frequent collisions it is possible that there is imperfect exchange of momentum and energy between the gas molecule and the solid boundary. Now the question is that when we are talking about that it is a less dense system then how so again less dense or more dense these are qualitative terms. We have to say we have to figure out that less dense as compared to what more dense as compared to what so there must be a basis for comparison. So, in a gas system let us come to the board and try to explain this. In a gas system you one of the important length scales is the molecular mean free path which we call as lambda. So, the molecular mean free path what is this? So, if a gas molecule traverses a distance before another collision with either another gas molecule or with the solid boundary whatever the distance that it travels the average distance that it travels is the molecular mean free path. So, if the system is less dense the mean free path is large that it traverses a greater distance before colliding with the molecule but large or small as compared to what so you compare it with the characteristic system length scale. If L is the characteristic system length scale then you compare this lambda with L if this lambda by L is which is also known as the Knudsen number a very important non-dimensional number this is Knudsen number if this is large then what happens is that I mean you must infer that it is a relatively rarefied system that means the molecular mean free path is large as compared to the characteristic system dimension so that means the density of the gas molecules is less. So, gas flows slip or no slip the notion underlying the no slip boundary condition is that there cannot be any finite velocity or temperature discontinuities within the fluid such discontinuities would result in finite velocity temperature gradients and hence infinite stress and heat flux thereby the destroying the discontinuities in no time I mean this is quiet and obvious physical statement thus the fluid velocity must be 0 at the wall if the wall is stationary otherwise it should be same as the velocity of the wall and also the temperature of the fluid must also be same as that of the wall this is what is local thermodynamic equilibrium. However, these are valid only if the fluid adjacent to the solid boundary is in thermodynamic equilibrium the achievement of thermodynamic equilibrium requires an infinitely large number of collision between the fluid molecules and the solid boundary as we have discussed. So the no slip boundary condition holds good so long as the Knudsen number is typically less than 0.001 it is not a sanctity that is not exactly this number this you have to treat this in terms of in the spirit of order of magnitude. So typically if you have Knudsen number as low as this then the no slip boundary condition holds true so if you look into the schematic which is shown at the top of the slide so if you consider typically the Knudsen number between 0.001 to 0.1 you consider that as a slip flow regime so slip flow regime means you can use the Navier-Stokes boundary equation but you have to couple it with a slip boundary condition rather than a no slip boundary condition. Beyond Knudsen number equal to 10 the continuum nature of the flow is lost completely you have to consider the molecular nature of the flow and that is called as free molecular regime. So in the free molecular regime your Navier-Stokes equation will not work you have to go for other molecular simulations like either molecular dynamics or Monte Carlo simulations these types of molecular simulations you have to do and in between the regime is called as a transition regime it is one of the challenging task to figure out what are the flow characteristics of the transition regime where there is a transition from continuum behavior to free molecular behavior. Now when the question arises that there is a possibility of a slip boundary condition then how to characterize the slip boundary condition. Now nowadays the this studies in this area have been quite advanced so there are many good slip models which are existing in the literature and many advanced slip models. But classically the first work the seminal work on slip model was done by Maxwell and this is known as Maxwell's first order slip model. There have been various corrections to this model but this model gives us an essential spirit an essential understanding of I mean how to describe the slip boundary condition at the wall. So we will try to understand this model carefully let us go to the board to do that and then we will come to the slide to summarize our observations. Let us say this is a solid boundary the boundary let us say can move with a velocity u wall uw I mean very common case is uw is equal to 0 but that is not necessary. Then there is a gas layer where the fluid is moving with a velocity ug and this distance is lambda remember that in gas flow the smallest resolution that you can capture is the molecular mean free path below that the resolution has no meaning. If you are writing well known algebraic and differential calculus based equations then below the molecular mean free path there is the resolution has no physical consequence. In many books you will find people write expressions for lambda by 2 lambda by 3 like that please do not consider those as conceptually correct ones because I mean the minimum resolution that you can capture is only lambda. Then let us say that there is another layer here this layer has a velocity u lambda. So this is one layer from the gas. Now we define something called sigma I will write it in the board is equal to tau i minus tau r I will explain the symbols by tau i minus tau w. I subscript means incident I mean incident incident means gas molecules incident on the wall r means reflected. So gas molecules reflected from the wall and w is wall and tau is tangential momentum okay. So tau is tangential momentum I is incident r is reflected w is wall. So let us consider 2 cases. So gas molecules can have specular reflection specular reflection is like regular reflection. So in specular reflection what simplification you can make tau i equal to tau r the incident tangential momentum is same as the reflected tangential momentum. So that means sigma equal to 0 diffuse reflection in diffuse reflection what happens? The reflected molecules capture the property of the wall. So tau r is same as tau wall that occurs in diffuse reflection. So that means sigma is equal to 1. So a real picture is it may not be completely diffuse or completely reflected consideration but diffuse and specular reflection these are some idealized paradigms. So the reality can be something in between these are 2 extremes because the extremes are given by the normalized values 0 and 1 the reality can be something in between. Now let us write what is u g. So in this layer statistically what happens? Half of the molecules can join from the top layer and half from the wall layer right. So u g is equal to half u lambda plus half u reflected okay. So half from the top layer half from the wall reflected layer. What is u reflected? u reflected can have the property of the u lambda or u wall depending on whether it is specular reflection or diffuse reflection. So the fraction or the probability of specular reflection is 1 minus sigma because if sigma equal to 0 the probability becomes 1. So this into u lambda plus sigma into u wall right. Not only that this is equation number 1. We can write another equation which is equation number 2 u lambda minus u wall is equal to what? Let us say the normal direction is eta. This is the definition of a gradient but for that what is assumed? What is assumed? See at the wall the gas velocity is u wall if sigma is equal to 1 right. So it is better to weight this with sigma because this u lambda minus u wall this difference is what? This is the difference between the gas velocity here and the gas velocity here but the gas velocity here is same as the velocity of the wall only if sigma is equal to 1. So when you put sigma equal to 1 you see this is like u lambda equal to u wall plus 2 lambda del u del eta at the wall. So it is the definition of a gradient okay. So this is equation number 2. What we will do now is we will eliminate u lambda from equation 1 and equation 2. So u lambda is equal to u wall plus 2 lambda by sigma del u del eta at the wall. This is equation number 3. Then you put equation 3 in equation 1. Put equation 3 in equation 1. So if you put equation 3 in equation 1 then what will happen? Yes. So u gas is equal to half of this half u wall plus sorry not half because there is a let us write first equation 3 let us simplify and then we will substitute that will make the algebra simplified. So equation 1 u gas is equal to 2 minus sigma by 2 u lambda plus sigma by 2 u wall right and then we substitute. So that this becomes 2 minus sigma by 2 in place of u lambda you write u wall plus 2 minus sigma by 2 into 2 lambda by sigma del u del eta at the wall 2 minus sigma by 2 into this. So 2 into 2 minus sigma by sigma into this. Then plus sigma by 2 u wall. So u gas is equal to what? 2 minus sigma by 2 plus sigma by 2. So that becomes u wall plus 2 minus sigma by sigma into lambda. So you can write basically u gas minus u wall which is the slip velocity this is u slip is equal to something ls into del u del eta at the wall okay. This ls is called as slip length. In this particular case this is equal to 2 minus sigma by sigma into lambda. It has a unit of length. You can non-dimensionalize this with respect to the characteristic length scale of the system. Then this becomes lambda by l which is the Knudsen number. See the Knudsen number is coming in the boundary condition. So this model is a very simple model because it only considers here a first order variation. It does not consider higher order effects. So you have corrections of this model as models with first order with a second order term with a third order term and so on. But that will just give rise to essential mathematical artifacts. But the physics, physical concept will remain the same. That is why this is so classical and we discuss about this in length in the class. So you see that the slip velocity at the wall is equal to what? The slip velocity at the wall is not 0. It is 0. Technically it is 0 only if this lambda tends to 0. So when we say that lambda is equal to is less than 0.001 or something like that. So with respect to the characteristic length scale of the system that may make this to be almost 0. Until and unless the characteristic length scale itself is small. So there are 2 possibilities. The physics is dictated by the Knudsen number which is lambda by L. So either lambda may be large or lambda may not be so large but its effect is compensated by small L. So a rarefied system, a highly rarefied system in a macro scale may be equivalent to not so rarefied system in the micro scale or even less rarefied system in nano scale. Because if this is large and this is also large, the ratio it depends if this ratio is within the regime of slip then slip may occur. Now this may not be that large but this also may not be that large. In that situation also the same Knudsen number can be obtained. So the physics is dictated by the Knudsen number and not the absolute value of lambda. So that is see all the problem that we are discussing here. This kind of problem with slip boundary condition can also occur in the large scale system. But then lambda has to be very very large. It has to be a highly highly very highly rarefied system. But in a small scale system if lambda is even not that high still the Knudsen number can be large because L may be small. So the characteristic length scale may compensate for the not so largeness of lambda. Now a very interesting thing. So we can see here that there is a possibility of slip. Now when we talk about boundary conditions in fluid mechanics, we usually do not discuss about this when we start with undergraduate text of fluid mechanics. Nowadays people have started discussing about this and the discussion is even more relevant for micro and nano technology. And because of this technological boom in the area of micro nano science over the recent years over the past couple of decades this I mean it has I mean it has become an important science but it has also become an interesting fashion that everybody is talking about fashionable technologies, nano technologies, small things and all those things. People are giving funds for research for doing work in this very challenging areas and in this very interesting areas with lots of upcoming applications absolutely fine. Think about the time when Navier lived. At that time see it was right in 1800s. So if I remember the year correctly in 1823 Navier first talked about the slip boundary condition. And in those days there was nothing called nano technology, there was nothing called micro technology, there was nothing called microfluidics and all those things. But see this is what is the insight or foresight of a genius like Navier. What Navier was a mathematical genius. So he did not think about all these that all these kinds of like explanations which are important for micro and nano scale devices because in those days the technology was not that advanced to fabricate micro and nano scale devices that was not possible. So it was not possible and over and above that he was a mathematician. So if you consider this coupled effects like somebody who is from a classical mathematical background may not be familiar with experimental observations I mean conducted by his or her own research group not only that experimentations over micro and nano scale systems were not possible in those days. See I mean this is a passing remark but this remark is very important. Microfluidics is not a new subject you have to understand this very carefully. Microfluidics has been a subject over hundreds of years but the name of the subject was not coined as micro fluidics. I mean people studied it mostly as low Reynolds number hydrodynamics. Now the reason why micro fluidics has attained or has attracted significant level of attention over recent times is the fact that many of the mathematical and physical concepts can directly be tested by experiments because of advancements in fabrication advancements in micro fabrication nano fabrication. So advancements in the fabrication technology had made it possible that one can now very elegantly fabricate micro channels and nano channels and design structures over micro and nano scale systems. So that has become this subject very relevant because now you can make practical devices out of these concepts. But when Navier thought about this what Navier thought is something like this. Navier was not particularly focusing on either gas flow or liquid flow he was mainly discussing about any fluid flow. So what Navier said is that let us say that we are drawing a velocity profile u as a function of eta or y whatever eta is the wall normal coordinate. I mean I am prefering to write eta here because the wall if it has a curvature then the local normal will vary from one position to the other so it may not be a unique y direction. So if you plot the velocity profile Navier said that well where is the guarantee that this will meet this will be 0 at the wall no guarantee. So let us not take that let us assume that there is a slip velocity at the wall. Now how do you characterize the slip velocity at the wall? See in the previous calculations which I just erased for drawing this figure there was something which appeared as ls or slip length I will give you a geometrical interpretation of that that what is that slip length please try to pay due attention to all these things carefully because like these are hearts and souls of some of the calculations in microfluidics. So now when this does not meet the condition of 0 velocity at the wall this may probably if extrapolated will satisfy the 0 velocity if the wall is displaced by some distance from its original position right. If the wall is displaced below at some hypothetical location or the wall is shifted below at some hypothetical location then this velocity profile if extrapolated to that hypothetical wall will satisfy a 0 velocity condition. Now extrapolation extrapolation in physics is a very dangerous proposition because if you have a non-linear variation then the extrapolation may take you to nowhere. So a better way of extrapolation and if the extrapolation is not being carried out over a large region but only a small region here a better way to make an extrapolation is to linearize this system and then extrapolate. So linearizing the system means drawing a tangent to the velocity profile. So I am just drawing doing these things with different colors so that it becomes easy for you to visualize. So this green line is the tangent to the velocity profile the red line is the velocity profile all of you are familiar with what is the velocity profile. So the red line is the velocity profile what we do is that we draw a tangent to the red line and when we draw that tangent to the red line this tangent to the red line meets the vertical line at a distance which we call as the LS. Let us say that this angle is theta. So this angle is also theta and this is what? What is this? This is the slip velocity the difference between the wall velocity and the fluid velocity that is the slip velocity u slip us right. This is the difference between the wall velocity and the fluid velocity at the wall that is the slip velocity. So from this right angle triangle you can say that tan theta is equal to LS by us what is tan theta? Tan theta is the slope of this velocity profile the red line right. So what is the slope of this red line? It is not d u d eta it is d eta d u in y axis eta is plotted and in x axis u is plotted right. So this is so you can write us is equal to LS cot theta cot theta will be d u d eta. So us is equal to LS del u del eta at the wall right. So this us in this particular example is u gas minus u wall and LS in this particular example is 2 minus sigma by sigma into lambda. So it fits with the physics of the Maxwell's model but this was not derived the Navier's slip model this is called as Navier's slip model. This Navier's slip model was not derived by taking the Maxwell's first order slip into account but the Maxwell's first order slip was accounted for by solely appealing to the physics. And now you see that that physics in the first order how nicely it fits with this mathematics you expect it to be fitted only in the first order because this is also a first order model this has no second order terms. Now what was Navier's idea was that if you know the slip length then you can calculate you can give this a boundary condition. What type of boundary condition this is? Is this a Dirichlet boundary condition? Is it a Neumann boundary condition? Is it a mixed boundary condition? Yes it is a mixed type of boundary condition because u at the surface is expressed as a function of the gradient of u at the surface. The analogous boundary condition in heat transfer is what? The convective type of boundary condition with whenever you have the convective heat transfer coefficient. So this is just a mathematical analogy. See many subjects originate from different aspects of physics but you can converge many of the subjects to a common platform of mathematics and that is why it is very important to study mathematics because that gives you a formal basis of treating many diverse subjects through a common platform many diverse physical subjects through a common platform. So it is a mixed type of boundary condition. Now the no slip boundary condition you can see here that the no slip boundary condition can be retrieved from this just by substituting Ls equal to 0. So the no slip boundary condition this is what is Navier's this was what Navier's intuition and this intuition is quite logical that if you write no slip boundary condition you cannot retrieve a slip from that but if you write a slip boundary condition you substitute slip length equal to 0 then you retrieve the no slip boundary condition. So the slip boundary condition is mathematically a much more general paradigm than the no slip boundary condition therefore it should not be disregarded that was Navier's platform of or Navier's perspective of looking into this. Now a big question is what is the value of this? 1 nanometer or 1 kilometer you really have to look into the physics of the problem now see when you are solving a mathematical problem this will never be given to you by the definition of the problem because this is a parameter how do you get the parameter? So for that you need to get into the physics and chemistry of the problem and one important observation is that if you reduce the scales if you go to nano scale the slip may be even much more dominant. In micro scale the slip effect may not be that important but if you go to nano scale the slip effect may be even more important. So when you have the slip length the big question is how do you know what is the value of the slip length? So in nano scale many times the slip length is obtained by molecular dynamics calculations. So you make separate molecular dynamics calculations to get what is the slip length in terms of other parameters then substitute that slip length in a continuum model to get a mathematical description of the physical problem. So sometimes molecular dynamics and continuum calculations are done in conjunction where the parameters of the continuum calculation are obtained from the molecular dynamics simulations okay. So I am just trying to give you a broad basis because these are like very important considerations in modeling microfluidic problems. Now let us get back to the slide and try to proceed with our discussion on Maxwell's slip. So if you see here that Maxwell's first order slip model so u gas equal to u wall plus 2 minus sigma by sigma into lambda by lambda into del u del eta at the wall. You can see that there is an additional term. This term is something which was not introduced by Maxwell but it is a thermal effect. You can see that there is a temperature gradient. I mean in deriving the Maxwell's model we have not considered the effect of local temperature gradient but if you have a local temperature gradient that can drive additional slip flow and the derivation of that is beyond the scope of this course but there is an additional term. This term was not modeled by Maxwell but it was taken up by another famous scientist known as Moluchowski. So this particular model is called as Maxwell's Moluchowski first order slip model. So what is the physical basis of this that if you have a tangential gradient s is the tangent to the surface. So if you have a tangential gradient of temperature that can drive molecular transport and this is called as thermophoresis. So you can have an additional molecular transport because of tangential gradients of temperature. So that can give rise to additional slip velocity at the wall. So we will summarize today's discussion with slip behavior of gases. The summary is that the first set of fluid molecules comes in contact with the plate and these molecules tend to stick to the solid. Molecules of a fluid next to a solid surface are adsorbed onto the surface for a short period of time and then they are desorbed or ejected into the fluid. This process slows down the fluid and renders the tangential component of fluid velocity equal to the corresponding component of the boundary velocity. This is the basis of the no slip boundary condition but this consideration remains valid only if the fluid adjacent to the solid walls is in thermodynamic equilibrium with the solid walls and that requires a highly frequent possibility of collisions. So if the collisions are not that frequent that may not be true. So deviation from thermodynamic equilibrium when the collisions are less frequent they may result in a slip between the fluid and the solid boundary in small channels where the mean free path may be of comparable dimension as that of the channel characteristic length scale. This phenomenon may be more aggravated by the presence of a strong local gradient of temperature or density. Like we have seen that how a strong temperature gradient can drive a flow. Such phenomena can give rise to a net driving force so that the molecules have a further velocity relative to the solid boundary. This kind of phenomena are usually termed as thermophoresis and diffusive phoresis respectively. Now the question is that when this phenomena are taking place for gases we can intuitively argue that for gases this may be possible because the inherent molecular arrangement of gases is not very compact. Gases are not very compact systems but it is not so intuitive to extrapolate this molecular mean free path based explanation for liquids because liquids are usually more densely compact systems. So what happens with regard to the consideration of slip boundary condition for liquids? Is it a possibility? Is it a paradigm? This is a very interesting question and we will address this question in the next lecture. Thank you very much.