 slip boundary condition in fluid mechanics and let us just briefly recapitulate what we did. So we first gave a bit of a perspective or motivation on I mean why we need to investigate on the possibility of slip boundary conditions. Then we discussed about continuum hypothesis and how does it relate to the thermodynamic equilibrium conditions at the fluid solid interface with regard to the boundary condition at the fluid solid interface. Then we assessed the possibility of slip boundary condition for gases and for gases our general conclusion was that like if the number of collisions or the frequency of collisions between the gas molecules and the solid boundary is sufficiently large then there is almost a perfect exchange of momentum and energy between the gas molecules and the solid boundary. So if the solid boundary is at rest the gas molecules in vicinity of the solid boundary will also come to rest and that is known as no slip boundary condition. However if the frequency of collisions is not sufficiently large then this may not hold true and the issue of largeness or smallness of the frequency of collision is well parameterized by the non-dimensional number called as the Knudsen number. So if the Knudsen number which is the ratio of the molecular mean free path with respect to the characteristic system length scale if the Knudsen number is larger and larger it gets larger and larger that means the molecules traverse a large distance before another collision that means on average the number of collisions is less. So that is the hallmark of a relatively rarefied system. So now you can for the same molecular mean free path that is the for the same level of rarefication your Knudsen number will actually be larger and larger if the characteristic length scale becomes smaller and smaller. Therefore if a gas is not very highly rarefied but is put in a confinement that equivalent effect can be a strong rarefication in a larger scale system. So in a smaller scale system the effect of rarefication is magnified because of the reduction in the length scale of the system and that makes the possibility of infrequent collisions more for gases in a confined system than for gases in a in a micro or nano confined system than for gases in confinements which are of larger dimensions. So based on this we discussed about some classification of flow regimes of gases based on the Knudsen number and we made a we made a remark that the no slip boundary condition holds good so long as the Knudsen number is typically less than the order of 10 to the power – 3 but again this is just a rough estimate and this is not a hard and fast number. Then we discussed about a model which was introduced by Maxwell and it is a very classical model known as Maxwell's first order slip model. So for that we introduced a parameter which is called as tangential momentum accommodation coefficient and the tangential momentum accommodation coefficient is basically defined as tau i – tau r where tau is the tangential momentum i is the incident and r is the reflected. So tau i – tau r in the numerator and tau i – tau wall in the denominator where the i is for the incident molecules tau i is the tangential momentum of the incident molecules and tau wall is the tangential momentum of the wall. So you can see that this value of sigma this value of sigma is something when is something between 0 to 1 because there are limiting cases of specular reflection, specular reflection is when sigma when tau i is equal to tau r that is the incident tangential momentum is same as the reflected tangential momentum and then sigma is equal to 0. On the other hand for diffuse reflection the reflected molecules assume the tangential momentum of the wall. So tau r is equal to tau wall which makes sigma equal to 1. So in reality it is neither fully specular reflection nor fully diffuse reflection because the relative proportion of these depend on the condition of the surface. So in reality it is neither of these 2 but something in between. So keeping that sigma as a parameter we arrived at a boundary condition which is the Maxwell's slip boundary first order slip boundary condition which is the difference in the velocity of gas molecule at one molecular mean free path from the wall with respect to the wall molecules. So that particular boundary condition that particular boundary condition was augmented later on by Smoluchowski who introduced the concept of thermal creep that is in addition to the velocity slip you can also have a sort of temperature slip I mean it is not a correct term but just to give you an analogy a slip due to thermal effects. And so the idea is that because of tangential temperature gradients there can be a physical slip of gas molecules over the solid boundary and for that Smoluchowski added a term in the in the presence of a temperature gradient this is this is tangential temperature gradient not normal temperature gradient and this term you can see in the last expression of the view graph that is presented. One important point to mention in this context is that you can use this boundary condition so provided you know what is the value of the parameter sigma. Now again this is like a mathematical exercise that you can substitute any value of sigma between 0 to 1 and you can get some sort of behavior of of the system but in reality how do you get the value of sigma. See we are just we are coming we are converging to a situation when we will realize that it is the it is eventually molecular simulations that can only give us the right picture. So to understand what is the value of sigma see sigma is a parameter it is an upscaled parameter to represent the behavior but where from we will get the parameter. So for that we have to appeal to molecular simulations sometimes statistical sometimes I mean molecular dynamic simulation depends on the platform and hand and depends on the length scales and other physical scales that are being addressed but we have to keep in mind that so far as continuum level calculations sigma is like a fuzzy parameter but this parameter can actually be obtained from smaller scale simulations that needs to be carefully understood okay. So now we will come now another point that we discussed in the previous lecture is that what is Maxwell's sorry what is Navier's slip boundary condition and how can we fit the Navier's slip boundary condition in the context of Maxwell's first order slip then we introduce something called a slip length and the slip length again for Maxwell's first order slip boundary condition is you can see very clearly 2-sigma by sigma into lambda the expression that is given in the last equation. So in that we are not fully confident with lambda but you know I mean if there are if there is some possibility by which you know the Knudsen number of the system then you can have an idea about the average molecular mean free path if you know what is sigma I mean by some molecular simulation or from some other source of information then you can treat the problem like a continuum problem with a modification in the boundary condition so long as the paradigm of continuum is valid otherwise when continuum hypothesis loses its validity all together you cannot use the Navier stokes boundary equation with whatever boundary condition then when continuum hypothesis is valid just remember this very simple thing when the continuum hypothesis is not valid then forget about the Navier stokes equation the Navier stokes equation cannot give you the right picture when the continuum hypothesis is not valid however it may be possible to develop a strategy by which you can use the Navier stokes equation to get some information provided you make some additions and alterations in the Navier stokes equation but that is beyond the scope of this particular course and we are not going to discuss that. Now the most the more interesting issue is about the possibility of slip boundary condition for liquids this is a this is an interesting issue because of several points the first and foremost point is that for gases slip boundary condition is something which may not be very intuitive but relatively more intuitive than liquids the reason is that the gases are not as compact as liquids in terms of molecular arrangement. So if you apply a force and if the gas molecules are located on a solid boundary then it is much easy to dislodge the molecules from the solid boundary because the gases are not very compact and there are and the intermolecular force of attraction is not considered to be that strong on the other hand liquids are highly compact systems. So they have strong intermolecular forces of attraction despite that is it a possibility that liquids can slip on a solid boundary that is the question that we would like to answer in today's discussion. So slip boundary conditions for liquids is it a possibility because of sufficient intermolecular forces of attraction between the molecules of the solid boundary and the liquid it is expected that the liquid molecules would remain stationary relative to the solid boundary at their points of contact okay. So this is something which is expected you know I mean as I repeat this ideology all most all the time that in science this is something which is very important that all of us have certain intuitions and it is not bad to try to explain most of the things through intuitions the intuitions may be correct the intuitions may not be correct but it is I mean we always start with intuition. So because of sufficient intermolecular force of attraction in the liquid it is intuitively expected that liquid molecules would remain stationary relative to the solid boundary. But is it a possibility that the liquid molecules may be dislodged from the solid boundary yes if you apply a very high shear force right if you apply a very high shear force then with the help of the shear force now how high that is a big question. So if you apply a very high shear force then with that very high shear force the liquid molecules may be physically dislodged from the solid boundary even if the boundary is smooth. So it or the boundary is rough whatever if the molecules were stationary and they were not moving by virtue of their interaction with the surface number one, number two their own compact intermolecular arrangement but some other effect may be a strong shear force is trying to dislodge the liquid molecules from the solid boundary. And when they are trying to dislodge the liquid molecules from the solid boundary then there could be a possibility of slip. So in such cases the straining may be sufficient to move the molecules by overcoming the van der Waals forces of attraction. See by the molecules adhering to the solid boundary we will try to adhere to the solid boundary by virtue of the van der Waals forces of attraction. Now if we will discuss more about the van der Waals forces of attraction when we discuss about nanofluidics. Now when the force on a molecule is sufficient to overcome that attraction then the molecule will start moving. So that is a possibility that it may require a very high strain but that possibility cannot be ruled out. There is another theory which argues that the no-slip boundary condition arises due to microscopic boundary roughness. So to understand that what the theory talks about let us briefly go to the board and try to understand. So let us say that you have a flat boundary like this. Now if you I am not going for a curved boundary let us consider a flat boundary. So this flat boundary is what? It is a boundary which is macroscopically flat. Now to analyze this macroscopically flat boundary what we are doing is we are magnifying a portion of the solid boundary. So when we are magnifying a portion of the solid boundary so how do you magnify it? You put it under a very powerful microscope and then what you see is not something which is macroscopically flat. You see something which is as jagged as may be possible and because of the artifacts with the manufacturing process it is very likely that except for a very few exceptionally flat surfaces made out of very controlled manufacturing processes over certain length scales this is what is commonly expected rather than something which under the microscope will also look flat. Now what happens is I mean this is just a proposition that when you have molecules the molecules are trapped in the asperities. They are also at their peaks and they are also at the valleys. Now something if there is a force acting on these molecules that force finds it hard to dislodge this because the already trapped molecules have good intermolecular forces of attraction binding them together. So these roughness elements on the surface are expected to create a hindrance and this is what we are discussing in a static condition even in a dynamic condition what can happen? Even in a dynamic condition if you have say a bluff body and if you have fluid flow past it there will be a drag force this is from the basic consideration of fluid mechanics of flow past bluff bodies. Now roughness elements are like small small bluff bodies. So if you have more roughness elements it is intuitively expected that it will increase the drag. We will see later on whether that intuitive expectation can always be justified or not but this is what is very common that roughness elements can play some role. So going back to the description of the theory that argues that the no slip boundary condition arises due to microscopic boundary roughness since the fluid elements may get locally trapped within the surface asperities. If the fluid is liquid then it may not be possible for the molecules to escape from the trapping because of an otherwise compact molecular packing. Following this argument it may be conjectured however that molecularly smooth boundary conditions should allow slip right. If roughness be the sole consideration or if roughness be the driving factor for triggering no slip boundary condition for liquid then in those cases those may be relatively hypothetical but cannot be ruled out in those cases where the surface is very very smooth in those cases you could possibly have slip in the liquid. So we have to consider from this consideration that no slip boundary condition is just a paradigm. So we should even for liquids it is a paradigm which has commonly been encountered by engineers and scientists in practical scenarios involving larger length scales but that does not mean that it is something which must be the case. So the reason is that in the continuum picture see if you look at this view graph in the continuum picture you have a continuous you have velocity vectors and these velocity vectors are gross manifestations of what is happening at the molecular level and we have significant questions with respect to what is happening in the molecular level and the answers to these questions depend on the nature of the surface, the magnitude of the shear force that is being applied and so many other things. So until and unless those issues are resolved at the molecular scale we cannot confidently tell that yes no slip boundary condition is something that will occur in reality. So this I mean one of the big deviations from the no slip boundary condition see deviations from the no slip boundary condition for gases these have been observed for years. Deviation from no slip boundary condition from liquids for liquids have not been observed so well I mean they have been inferred from some experiments I am again repeating inferred that means it is not that somebody is actually seeing slip of liquids but what is happening is that from the description of the flow rate or from the description of the velocity profile some inferences are coming which is triggering the conclusion that yes there is a possibility of slip at the wall. Now one such inferring example for liquid slip is the classical example I mean it has now become classical but of course a few years back when the carbon nanotubes first came it was I mean it was a relatively new idea that what happens for carbon nanotubes. So researchers have demonstrated that the rate of liquid flow through a membrane composed of an array of carbon nanotubes may turn out to be 4 orders of magnitude faster 4 to 5 orders this is just an estimate then that predicted from classical fluid flow analysis. So that means if you make a membrane made of aligned carbon nanotubes and then you measure the fluid flow through that arrangement then experimentally you get fluid flow which is 4 to 5 times more than what you get through a no slip boundary condition prediction. So this phenomena now researchers I mean every research has to be every research finding has to be explained and sometimes you know I mean I have I mean this is just like I would say that like in a light mood I am saying do not take it seriously I mean I have encountered researchers who say that like I always tell my students I mean that researcher is saying that you give me a graph if the graph in the graph y is increasing with x I can explain if y is decreasing with x that also I can explain. So every phenomena every phenomena has this kind of explanation and this is I mean this can be a joking remark but this is something I mean which talks about little bit of philosophy of I mean the kind of mindset with which people have to do research that you know in every aspect there are some aiding forcing parameters and some opposing forcing parameters. So it is possible that in some experiment under certain conditions the aiding parameters dominate over the opposing parameters in some other experiment the other parameters dominate. So it is not something which is you know philosophically that can be ruled out. So I am not saying that this phenomenon was also explained in that way but that is something which I thought that is not bad to just tell to make the discussion a little bit light. So these researchers attributed this phenomenon to an apparently frictionless interfacial condition at the carbon nanotube wall. Such observations were contrary to common consensus that fluid flow through nanopores having chemical selectivity is rather slow. I mean you expect that it through nanopores and nano channels I mean these are very narrow passages. So you expect the fluid flow through this narrow passages to be very small right that is what is the common intuition. But if you find that through nano channels and nanopores fluid is flowing remarkably fast then that is something which can be used for technology. So that is something which is remarkable. So from fundamental physical consideration what happens? Water is likely to be able to flow fast through hydrophobic single world carbon nanotubes. One of the reasons is that this process creates ordered hydrogen bond between the water molecules but that also occurs for many other cases additional things happen. What ordered hydrogen bonds between water molecules and weak attraction between the water and the smooth nano carbon nanotube graphite sheets that is what is important. That is water being very nicely bonded with hydrogen bond and the very weak bonding between the water and the solid boundary that why it is happening because the graphite sheets the smooth graphite sheets of the carbon nanotube those are very weakly bonded with the water. So not only that the rapid diffusion of hydrocarbons these are qualitatively attributed to the fundamental scientific origin of reduced frictional resistance in such systems. So people gave some explanation but you know that in microfluidics and nanofluidics we are not always working with materials like carbon nanotubes right. So we have to search for examples where we can find out slip as a common phenomenon in microfluidics and nanofluidics. And that also with liquids. So this to understand this particular mechanism we can just revisit the slip boundary condition which we discussed in the previous lecture because we will be using this boundary condition to explain some phenomenon. So you can see that this is just like a magnification of the situation close to the wall the dotted line is a tangent to the velocity profile at the interface and when it is extrapolated then it meets the 0 velocity at a length at a distance slip length ls from the solid boundary that is the definition of the slip length. And as I told you in the previous lecture that this was introduced in an era by Navier I mean when there was no hype on nanotechnology that was in 1823. So I mean in those days nobody has heard of what is nanotechnology but nowadays this is having its most use in the context of nanotechnology. Now I am going to discuss about something which is interesting and important. Now whether it is slip or no slip that also depends on something which is how smaller distance you can resolve. Like think about a scenario let us say that you are capturing the velocity profile experimentally. So experimentally if you are capturing the velocity distribution velocity vector distribution let us say that you are using one of the very sophisticated techniques in the in microfluidics say micro scale PIV micro PIV or micro particle image velocity. So if you are using that then what is the what is your resolution your resolution is not really to the molecular level right or it is not even to a few nanometers so that kind of very small resolution is not there. So let us say that you have a situation when there is a less dense phase I will explain you that when that can be possible there is a less dense phase say this there is a liquid water above this dotted line and there is let us say water vapor just as an example below the dotted line. Now how far is the dotted line from the wall? Let us say that the dotted line is within 10 nanometers from 10, 20 nanometers from the wall okay. Now that cannot be resolved by any even by the most sophisticated modern day flow measuring device I mean it that layer can be resolved just as a layer in a static environment but what fluid flow is taking place in that layer the construction of velocity vectors and all those things this is still an still an impossibility even with so much of advancement in technology. So what the velocity measuring or what the velocity capturing device velocity vector capturing arrangement what it is giving us this is giving us the velocity vectors for simplicity we have drawn it like a straight line I mean it is not necessary I mean you may just consider it to be an arbitrary line. This velocity vector above the dotted line below the dotted line something happens but this line which is drawn below the dotted line that line is not well resolved. So because that line is not well resolved then what happens because that line is not well resolved something interesting can happen. So let me draw a picture in the board to explain that let us say this is the dotted line. So you have a velocity profile here and in the less dense phase you have a velocity profile like this. Now you cannot really resolve this because this is this is a very thin layer. So what you do is that you are basically seeing only the black velocity profile. So what you do is you extrapolate this velocity profile and this extrapolated velocity profile meets the solid boundary a velocity you all which is we call as apparent slip velocity. This terminology apparent slip is what is very important you see this is an illusion why this is an illusion. This is an illusion because the phenomenon is going beyond the resolution of the probing arrangement to get the velocity profile in this thin interfacial layer. So you extrapolate the velocity profile in the outer layer which is the bulk liquid and that is meeting the wall at a velocity which is different from 0 velocity. So this is called as apparent slip. On the other hand had the phenomenon be like this then this is something what we could call as true slip. Now when there is a vapor layer or gas layer it is also possible that you can have a true slip in that condition because you know that for certain rarefication for certain Knudsen number and so on it is possible that even this green line can exhibit a slip boundary condition. But I am not focusing on that because our main objective of this discussion is not to complicate that but to bring out the fact what is the difference between true slip and apparent slip. So let us summarize this that in true slip the velocity of the moving fluid literally extrapolates to 0 at a notional distance inside the wall and is finite when it crosses the wall. This is common for flow over mica sheets I mean this is something which has been observed for flow over mica sheets. In apparent slip the low viscosity component in the fluid facilitates the flow because it segregates near the surface. So it is like a 2 phase type of system with one phase separated a less dense phase separated at the wall and a more dense phase in the outer region. The velocity gradient is larger near the surface because the viscosity is smaller and this classical example is flow over super hydrophobic surfaces like lotus leaf 2 fluid model for apparent slip. Now let us go to the board to explain these 2 fluid model this is a very simple model I mean this model can be followed by anybody who has done undergraduate fluid mechanics. So let us say this is the dotted line which is the boundary between the 2 layers. Let us say this is the channel centre line and this is the wall half height of the channel is h. This distance let us say is delta so this distance is h-delta the axial direction is x and we set up our y axis such that this is y equal to 0 the interface between the let us say this is liquid and this is vapour instead of vapour it can also be a low density system. So our governing equations we are assuming low Reynolds number flow so we will quickly write the governing equations 0 is equal to minus dp dx plus mu l assume u l as a function of everything as a function of y only l for liquid and v for vapour. So if you integrate it d u l d y is equal to 1 by mu l dp dx into y plus c 1 that means u l is equal to 1 by mu l dp dx into y square by 2 plus c 1 y plus c 2. Similarly this equation will give u v is equal to 1 by mu v dp dx y square by 2 into c 3 y plus c 4 oh yes yes dp dx this is dp dx. We are discussing about slip this is slip of writing so this so we have 4 constants of integration and we need to find out this constants of integration using 4 boundary conditions. So what are the boundary conditions number 1 we will write the boundary conditions in one corner and use the boundary conditions to find out the other constants. So at boundary conditions at y is equal to what is the wall y is equal to minus delta right at y is equal to minus delta u v equal to 0 this is no slip. If the gas layer is at a high Knudsen number or in a high Knudsen number regime then it can also be slip there but we are assuming that it is no slip y is equal to 0 what are the boundary conditions continuity of velocity and continuity of shear stress. So at y is equal to 0 u l is equal to u v at y is equal to 0 mu l d u l d y is equal to mu v into d u v d y right this is continuity of shear stress and at y is equal to h minus delta what is the boundary condition d u l d y is equal to 0 this is a center line symmetry this center line symmetry will hold only when everything is physically and geometrically symmetrical that means on the other side on the other wall also if it is a parallel plate channel see the channel is something like this on this wall also you have a thickness of low density layer of thickness delta now if things change in the other wall then that symmetry boundary condition will not hold true but I am assuming just for simplicity in algebra that is what is the case. So now we substitute the boundary conditions at y is equal to minus delta u v equal to 0 so 0 is equal to 1 by mu v d p d x into delta square by 2 plus c 3 delta plus c 4 at y is equal to 0 u l is equal to u v that is c 2 is equal to c 4 at y is equal to 0 mu l into d u l d y into that is mu l into c 1 that is equal to mu v v into d u v d y that is mu v into c 3 and at y is equal to h minus delta d u l d y is equal to 0 so 1 by mu l d p d x at y is equal to h minus delta so h minus delta plus c 1 is equal to 0 if there is a mistake you please in the first condition which condition 0 equal to 1 by mu v d p d x delta square by 2 oh this is minus delta so minus c 3 delta at y is equal to minus delta others are okay so the strategy is very simple this equation will give you what is c 1 so c 1 is equal to 1 by mu l d p d x into h minus delta with a minus sign then this equation will give you what is c 3 mu l by mu v c 1 then if you substitute c 3 in the first condition you will get what is c 4 and c 4 is equal to c 2 okay so very straight forward now so you can get the velocity profile but the interesting part of the story is that you are not actually able to resolve this so what you are doing in experiments in experiments you are drawing a velocity profile like this and then extrapolating this velocity profile using a condition that you are not able to resolve this so you are not even extrapolating to the wall so what you are doing is your resolution may be ends here you are not able to resolve so the wall may be here but for your experimental measurement purpose there is no distinction between this and this I mean we can distinguish this and this for the sake of theoretical discussion but experimentally there is no difference between this and this so it is as if this is like the wall okay as if this is like the wall and as if there is a slip velocity at the wall because clearly if in this arrangement if you draw the velocity profile the velocity here is non-zero right the velocity is 0 only here and it may be interesting to draw the velocity profile properly so you must draw it in such a way that it clearly reflects the proper physics see here the velocity is continuous mu du du y is continuous this is liquid this is vapour which mu is mode liquid mu or vapour mu liquid mu is mode so liquid du du y must be less so this is the slope that you are visualizing here is not du du y but du y du because u is plotting u is being plotted here and y is being plotted here so with all these considerations you should draw this velocity profile carefully so that qualitative physics is being explained by the velocity profile but we are just going for a quantification here so I will not bother you so much on that at this moment so the coming back to the point so if we can now explain that well you have so with c 1 c 2 c 3 c 4 you have got a velocity profile u as a function of y this u as a function of y includes the vapour layer and the liquid layer so now if we can say that u at y is equal to 0 you can find out what is du dy at y is equal to 0 that also we can find out because we know all the constants of integration so now if we write u at y is equal to 0 is equal to ls into du dy at y is equal to 0 then we can get what is this ls right you understand the strategy right so what we are doing we are getting the velocity profile from this model by getting the constant c 1 c 2 c 3 c 4 so we know what is u at y is equal to 0 what is y equal to 0 this cross location is y equal to 0 we get what is u at y equal to 0 that is not definitely 0 we get what is du dy at y equal to 0 so this y equal to 0 we should write at the bottom du dy at y equal to 0 so u at y equal to 0 is equal to ls into du dy at y equal to 0 if we replace this equivalent by a slip length base boundary condition so this y equal to 0 is like a hypothetical wall right it is as good as the wall but we do not know actually where the wall is located so you can find out ls ls will be a function of what what are the parameters on which ls will depend mu l mu v and delta right so you can non-dimensionalize ls with respect to the height of the channel ls by h so ls by h will depend on mu l by mu v and delta by h and I will show you the next view graph which is displayed in the screen that will highlight that will give the expression of ls by h which is written as l by h in this view graph so l is ls right whatever so you can make a note of this final expression which is displayed in the view graph and I leave it on you as a homework to show that this is the expression the steps I have clearly explained it is just a simple algebra that you have to do so you can see that it is a function of mu l by mu v and delta by h okay so clearly when delta by h tends to 0 that is when there is no near wall layer then the slip length is 0 then it becomes a no slip boundary condition so it is sort of a manifestation of the near wall variation. Now this is something which is apparent slip true slip true slip can be realized as what I have discussed earlier that it can be realized if the shear is very high if the shear is very high then liquid molecules despite their compactness in molecular arrangement can be dislodged from the solid boundary. So if you look at the view graph this is very widely cited paper by Thomson and Croyen reported in the famous journal Nature in 1998 that if you have a slip length as a function of the shear rate see this is the molecular dynamics simulation based prediction because very high shear rates may be able to dislodge the liquid molecules from the solid boundary but for that such a shear rate you need a very small confinement because shear rate is what shear rate scales with the shear velocity divided by the length that the gap between the top and the bottom plates if you consider a quiet flow you have 2 parallel plates and you are considering the gap between the 2 plates. So it is basically the relative velocity divided by the gap. So if the gap is until and unless the gap is very small going to a few nanometers the shear rate may not be sufficient to dislodge the molecules and over that few nanometer gap it is very difficult to do actually physical experiments. So people started with molecular dynamics these are also called as synthetic experiments these are simulated experiments. So if you look at this graph you can see that beyond the critical shear rate the slip length starts increasing phenomenally right. So the slip length as a function of shear rate beyond the critical shear rate because the liquid molecules start getting dislodged from the solid boundary you can get phenomenal slip. There have been controversies associated with this model. There have been several corrections put to this elementary understanding but I will not come into the details of that at this level. What could be the other factors that could dictate the slip? There are 2 important factors that I will summarize. One is the surface roughness and we have discussed that why surface roughness is important that if you have a manufacturing process then the manufacturing process is will result in some surface roughness characteristics and that surface roughness characteristics may in turn dictate the fluid flow. So that is why it is very important in microfluidics to relate the manufacturing process with the fluid flow because the manufacturing process of the surface dictates the nature of the surface and that in turn dictates the flow in a very significant manner that it is unlike the case of a laminar fully developed flow where the friction factor versus Reynolds number characteristics are independent of the surface roughness in classical fluid mechanics. The other factor is surface weightability that is we have discussed about some we have made passing remarks on super hydrophobic surfaces and all these. So you can nowadays engineer the weightability of a substrate in engineering this is possible that you can make the surface as weightable or as non weightable as you want. You can pattern the surface with weightability gradient. So many things you can do. So you can play with the roughness you can play with the weightability and combination of roughness and weightability can control the apparent slip behavior. How can it control the apparent slip behavior? This is something that we what with significantly in our research group and this is something which can give rise to some non-intuitive finding. So our common idea is that if we have a rough surface the rough surface will hinder fluid flow right it will not allow fluid flow past the surface quite easily and the reason is quite obvious that if you have a rough surface the rough surface in fluid mechanics will act rough at the surface more it will act like a bluff body and that will have a significant drag force. So that will increase the friction. This is something what is intuition nothing wrong with it but what can also be possible is that if you have a rough hydrophobic surface then in a highly confined environment the roughness and hydrophobicity together can trigger the formation of a less dense layer it can be nanoscale bubbles. This nanoscale bubbles see bubble formation in fluid mechanics can be very interesting. So I can just draw a small schematic to explain that if you have a channel where you have bubbles of this size then that is something which may not be desirable because it can block the flow. In engineering that is very disturbing but if instead of a bubble like this if you have bubbles like this let us say that this dimension this is highly stable and this is around 10 nanometer roughly length scale wise and the bulk length scale is significantly greater than this length scale. So then what happens is that see how we are this created by interaction between 3 parameters roughness, wettability and confinement. These 3 important parameters and interplay of that it is a very involved physics that comes into the picture here and it is I mean believe me or not it is truly beyond my capacity to explain this to you whatever research we have done in this area in an elementary class of microfluidics. So you can refer to the papers from our group published in various journals including physical review letters on like what are the mechanisms by which this can form but I am not going into the mechanism I am just trying to cut the story short and say that this layer of bubbles or even if it is not a bubble it can be a low density layer forget about the bubble to generalize it can be a low density layer. So the liquid which is flowing on the top of this low density layer what is happening to it it is not feeling the effect of the solid boundary it may be very rough here but this roughness is not exposed to the liquid. So the liquid as if is smoothly sailing on the top of a vapor cushion or a low density cushion. So what has given rise to this low density cushion one of the factors is roughness that is why I say that it is a rough that makes it smooth. So roughness in small scale fluid mechanics can be utilized as a blessing in disguise but the whole problem is that although this phenomenon now is well known one of the factors that is hindering its application in common engineering practices that is still a stochastic phenomenon. So controllability repeatability there are so many other issues that are involved. So there are still very open ended areas of research on this particular topic. So we have discussed about the slip and the no slip boundary conditions and we have discussed that for gases the no slip boundary condition may be more common for liquids you can have mostly apparent slip but not true slip true slip only at abnormally high shear rates which may not be realizable in experimental practice but in experimental practice the apparent slip may be common. So thank you very much for your patient hearing with minimum adherence to slip boundary condition. Thank you very much.