 In this course we have discussed so far about various aspects of microfluidics and microfluidics is actually like an ocean. So you could have discussed many other topics and essentially I have tried to confine myself to some of the very commonly addressed topics in microfluidics, but there are several other topics which I did not touch upon, for example, acoustofluidics, several other topics of microfluidics which can be of emerging importance. However, it is also important to understand that nowadays with advancement in manufacturing technology or with advancement in fabrication, it has been possible that one can make channels of dimensions significantly below what could be possible over the past few years. So if you can make channels or if you can fabricate channels which are of dimensions smaller than the microfluidic domain then at some point of time you enter into a domain which is called the nanofluidic domain. While it is important to appreciate that sometimes I mean issues in microfluidics and nanofluidics it depends obviously on the length scale and the physics and the time scale and all these things under consideration that in many cases the microfluidics aspects and corresponding nanofluidics aspects are more or less very very similar, but in certain scenarios you have to take additional effects into consideration when you are thinking of nanofluidics. And I just want to give you a broad perspective that when we are thinking of flow through a microfluidic channel which is of dimension of say 10 micron or so then the physics of that is as good as like flow through a channel which may be of the which may have hydraulic diameter of the order of say 100 nanometer. So, although we have come down from the micro to the nanodomain that does not mean that the physics suddenly gets altered, however there is a critical length scale below which certain additional interfacial interactions tend to become important which were not that much important over the microfluidic domain and that is like a sort of a very critical area or a very critical domain of nanofluidics that one would be interested to consider. Keeping this in view we will discuss some important aspects of nanofluidics I mean I can tell you that nanofluidics is again a huge area of importance and nanofluidics can be studied by various techniques including experimental techniques, theoretical techniques, computational techniques like that. I will make no effort in going through a comprehensive review of understanding nanofluidics because this particular course is primarily dedicated towards understanding microfluidics. But because there are certain areas of common interest and overlap and certain effects which we get in micro channels become even more important like for example slip phenomena may become even more important if you come down to the nanofluidic domain. That means that if you have interest in area of microfluidics then with modern advancements in manufacturing you should attempt to probe certain interesting areas of nanofluidics which could not be experimentally probed maybe even 4, 5 years back and it is a highly emerging subject and there are issues which we will try to appreciate that how certain issues in microfluidics will differ from issues in nanofluidics will differ from the corresponding issues in microfluidics and we will try to see that how we can address these issues. So after we discuss nanofluidics we will discuss a little bit about molecular dynamic simulations so which are commonly used to capture the like dynamics over nanoscopic length scales where molecular effects are important. So we have to understand very carefully that not over all length scales ranging in the nanometer domain individual intermolecular interactions become or the discreteness of the molecular entities become that important but when you come down below a threshold dimension that becomes significantly important. So with this little bit of background we enter into the discussion of nanofluidics so the outline of our discussion will be first we will discuss about some motivation which partly I have discussed the contrast with microfluidics the interactions at small scales the nano versions of some celebrated equations. So nano versions this is a very loose terminology that we have used here so by this what we mean is that modified version to take into account some of the nanoscale interactions. So the Lucas-Walshburn equation if you recall that we use the Lucas-Walshburn equation for capillary filling dynamics. So the capillary filling dynamics modification of the capillary filling dynamics equation and the modified Poiseu-Boltzmann equation and then we will discuss some applications of nanofluidics. So nanofluidics mainly concerns the study of behavior manipulation and control of fluids that are confined to structures having characteristic length scales ranging from 1 nanometer to 100 nanometer. But this is a sort of very loose definition the physics entirely changes as you go from 1 nanometer to 100 nanometer. So you cannot discuss the some common physical basis by which you traverse this kind of length range of length scales. Nanofluidics of course the nano originates from the Greek word meaning dwarf and SI prefix for 1 billion that is 10 to the power minus 9 and the fluidics which originates from a combination of pneumatics, hydraulics and electronics on a foundation of fluid mechanics. So you can see that the wide gamut of activities that can be centered around nanofluidics. Now question is why are we interested about nanofluidics? Is nano just a buzzword because of which we are interested to study nanofluidics? It is not exactly like that. With nano size channel we may tend to approach the molecular dimensions. The concept of continuum no longer remains defined at this scales. Standard analytical expressions at micro scale can no longer be applied for nano devices but we have to understand that what we essentially mean is that we can still apply that for nano devices but not for those cases where the individual discreteness of the molecular entities become important. So then we cannot use so that is below a threshold length scale not necessarily 100 nanometer but if you go down to 10 nanometer or even less then those things become significantly important. Massive increase of surface area to volume ratio with decrease of size this we have discussed in one of our very early lectures and that means that surface effects can dominate to such an extent that it can open up newer areas arenas of transport processes. Close to the wall every flow has a particle based nature and nanofluidics plays its role. This is a very important consideration. So if we are interested about length scales which govern the physical behavior close to the wall then that length scale if that length scale is important for the physical problem under consideration then nanofluidics is important although the system length scale may not be nanometer. So that is also something important so it is not necessary to go down to nano dimensions of the channel to observe some effects of nanofluidics. Nanoflows versus micro flows although governed by a common physics as one scales down from micro to nano the well understood laws of continuum mechanics remain no longer valid. I mean again I am giving you a warning that this is not a hard and fast comment that I want to make. The laws of continuum mechanics may still be valid in the nano domain it depends on what are the length scales and what are the time scales etc that you are addressing. So for example for pressure driven flow through a 100 nanometer channel I mean you can safely pressure driven flow of water through a 100 nanometer channel you can safely use some of the well known equations governed by continuum mechanics with certain modifications but those modifications will tend to be need to be more drastic as you come down to may be 10 nanometer 1 nanometer like that because then actually in a true sense if you come down to 1 nanometer the continuum hypothesis does not work anymore. With the interfacial and the bulk scale having the same order of magnitude laws of differential calculus may be unable to capture the gradients in properties and at this scales one requires a particle based investigation technique like the direct simulation Monte Carlo or molecular dynamic simulations. Some applications nature has been employing nanotechnology over millions of years and effectively mimicking it can create wonders. So this is something very very important in nature you know there are certain observations which if you can make and you try to mimic that observation in the form of an engineering device over nanoscopic scales then actually that can give rise to a very high level of innovation and there are several research groups which actually attempt to do this. I mean there are dedicated researchers who will just observe nature by observing nature they will try to identify some mechanism and then engineers will take that up and try to make devices which mimic that mechanism to the extent possible and this kind of mimicking nature is also known as biomimetics towards the end of this course I will give you one or two examples of some work from our own research group with emphasis on biomimetics. Now in nanoscale the immensely enhanced surface area leads to dominance of interfacial effects greater area available for chemical reactions and potential of large power density and you I mean in one of my lectures towards the end of this course I will show you that how you can use nanofluidics for making a very high power density energy conversion device. So that is basically using the concept of streaming potential which we have already discussed in the context of microfluidics. So in nanoscale you can do more efficient mixing and separation with applications in desalination, DNA hybridization and so many other areas. Of course you have to realize that when we are thinking about mixing in nanoscale you use a different strategy to create vortices in nanoscale you can pattern the surface with nanostructures and therefore patterning of surfaces with small scale structures in the nanodomain is one of the very hot areas of research these days because you can do wonders with this technology in the fluid mechanics you can create nice patterns and vortices in the flow structure to get good mixing. So nanotechnology has allowed one to use to combine fluid mechanics with that and achieve good mixing over nanoscopic scales. Now we will discuss some important physical effects in the nanoscale domain in the context of nanofluidics. Hydrodynamic interactions what we discussed first it is not just important in the nanodomain but it is important in general in the context of fluid mechanics. But in the nanodomain hydrodynamic interactions may play a big role what is hydrodynamic interaction I will basically not go through the detailed mathematical treatment of this because I mean this is very involved and I will just try to give you some glimpse of what it is so that if you develop interest you can read by yourself. Presence or transport of a finite size solid body creates disturbance in the surrounding fluid the surrounding fluid may be at rest or in motion such disturbance gets transmitted to affect the motion of the bodies that are in vicinity of the original body. This effect is known as hydrodynamic interaction why nanofluidics it is very critical because hydrodynamic interaction gets largely altered in presence of confinement. So make the confinement smaller and smaller this becomes more and more non-trivial bulk hydrodynamic interactions that is without confinement effect. So bulk hydrodynamic interactions are commonly accounted by considering following simplifying assumptions. The disturbance created by the solid body is assumed to be a point force this effectively means that the solid bodies can be treated as point sources of drag in the solvent acting at a point typically the center of a spherical body. This disturbance or effectively the point drag force is assumed to be weak enough so that the disturbance created by this force fj drag at rj where rj is a position vector to the velocity field at position ri of another object i. So there is a drag force at point j interest is how does it create the perturbation in the velocity at another point i which is in the vicinity of j and that can be approximated by a linear function for small disturbance. So that linear function is given by this vi prime is equal to minus omega ij into fj drag this is omega ij is the second order tensor. So this is a very common framework by which you do. Now of course if you have confinement then some of these basic assumptions may need to be corrected because then when the confinement is very strong you may have to consider some special type of interactions like van der Waals interactions for example. So when you have to consider those kinds of interactions and we will see later on that how those kinds of interactions are taken into consideration. Now one important effect of the confinement is hindered diffusive transport that means confinement hinders diffusion how is it possible presence of the channel wall restricts the movement of large particles thereby affecting their special distributions within the channel viscous interaction between the particle and the wall intensifies for narrower confinements. One may calculate hindered diffusion of Brownian particles by closed form expressions or analytical expressions under certain assumptions. The molecular dimensions much much greater than solvent molecule dimension wall is smooth on the particle length scale length of the channel is large enough to neglect end effects particle concentration is sufficiently dilute to neglect particle-particle interactions a well defined potential field between a single particle at R inside the channel and the wall exists. But you can see some of these assumptions are violated in many practical scenarios like for example particle concentration in many times is highly concentrated it is not dilute. So then it is very difficult to obtain analytical expressions for the effective diffusion coefficient and one has to go for numerical techniques to do that. The next consideration is very important that intermolecular forces that intermolecular forces over a few nanometer length scales tend to become very important and we will discuss about this intermolecular forces. Wicker than the repulsive electrostatic forces the intermolecular forces that may be attractive in nature are responsible for fluid molecules sticking together water wetting glass water not wetting Teflon and a lot of other stuff that we observe every day. Intermolecular forces are weaker than ionic or covalent bonds and intermolecular forces are responsible for the physical state of a compound solid liquid or gas. Some examples of intermolecular forces van der Waals interactions arise from two kinds of forces which can be mirrored back to electrostatic interactions one is dipole interactions another is dispersion interactions and of course you can have hydrogen bonds. So dipole interactions are electrostatic interactions between oppositely charged regions of polar molecules which are called as dipoles. You can see that the water molecule how it can be treated as a dipole I mean this is well known to you from your basic knowledge in chemistry. So whatever we are going to discuss now will be an interface between fundamental physics fundamental chemistry and engineering. Dispersion interactions dispersion forces are caused by the motion of electrons and increase as the number of electrons increases and dispersion forces are the weakest of all intermolecular forces. I mean these may be weak but over certain scales that you are considering these may play their roles van der Waals interactions exist between any two bodies irrespective of their size or polarity this is very important. They are long range interactions where the interaction energy varies as 1 by r to the power 6 with separation distance of r between the interacting bodies. Three types of van der Waals forces occur namely orientation force which is also called as key sum interaction induction force which is called as Debye interaction dispersion force which is called as London interaction. Now I will go through this in a summarized manner. If you are interested to learn more of in great details about all these interactions I can refer to a very good book which is intermolecular and surface forces by israeli. So if you read that book you will be able to get a whole lot of details which I will summarize here. Orientation force or key sum interaction these force results from the electrostatic interaction between two polar molecules. Example water molecules both having freely rotating permanent dipole moments. So the orientation force or key sum interaction concerns electrostatic interaction between two molecules both of which are polar. This is the keyword both are polar. The interaction is obtained by averaging over thermally excited probability distributions of rotation angles. And attractive force is generated when the dipoles adopt anti-parallel arrangement since this is energetically more favorable and repulsive force when they adopt parallel orientation. And the formula for the van der Waals orientation force is given by this. So I will not get into the much details of the formula because we are not going to derive this. And just in case you require to use such a formula the most important key aspect to keep in mind is that it scales with 1 by r to the power 6. Induction force or Debye interaction how is it different from the previous one? This force results from the electrostatic interaction between a polar molecule with a freely rotating permanent dipole moment and a non-polar molecule. So one polar molecule the previous was one polar with another polar. Now one polar with another non-polar averaged over the thermal thermally excited probability distribution of rotation angles. The electric field from the permanent dipole moment induces a dipole moment in the non-polar molecule as mu induced is equal to e into alpha. Alpha being the polarizability of the non-polar molecule okay. So the non the electric field from the permanent dipole moment can induce a dipole moment even in a non-polar molecule and that is how it gets polarized. So it is eventually a polar-polar interaction but the second polar was induced polar it is not the originally polar. These 2 dipoles interact electrostatically to give the induction force which again scales with 1 by r to the power 6. See all these scales with 1 by r to the power 6 because fundamental is dipole-dipole interaction. Originally there are not 2 dipoles but another dipole is induced in this case. The third one is the most interesting which is called as dispersion force or London interaction. This force results from the induced dipole-dipole interaction of 2 non-polar molecules. So the big question is if they are non-polar how do you have dipole-dipole interaction? So the answer is like this. A non-polar atom has an instantaneous dipole moment which can be associated with instantaneous positions of the nucleus around the atom. The electric field from this dipole polarizes any nearby atom inducing in it a dipole moment. The resulting electrostatic interaction between these 2 dipoles correspond to exchange of virtual photons generating an instantaneous dispersion force. So it is an instantaneous phenomenon but effectively this is also dipole-dipole interaction and then it scales with 1 by r to the power 6. Particle wall electrical double layer interaction. So we have already discussed what is electrical double layer. So you can have a charged interfacial layer in vicinity of the wall as well as in vicinity of the particle and let us say there is a particle which has come in close proximity to a wall. So if the particle has come in close proximity to a wall the particle has its own electrical double layer. The wall has its own electrical double layer. Now because of the interaction between these 2 electrical double layers and when they come very close to each other they may be strongly overlapping electrical double layers. So that can give rise to a net force and one can derive complicated expressions for the potential of interaction of the ideal field of 2. One is the particle of 2 entities one is the particle and another is the wall. And this is a function of the zeta potentials of the wall and the particle surface and the Debye length and the separation of the particle from the wall. Solvation interaction. When a flat surface is introduced into a liquid these are all very intermolecular interactions. So very interesting to study these. When a flat surface is introduced in a liquid which is very common like if you have a say solid boundary and if you are putting the solid boundary in contact with a fluid. The liquid molecules adjacent to the surface rearrange to pack well against it as this helps to lower the overall free energy. This ordering of liquid molecules does not require any liquid-liquid or liquid wall interaction. It is solely determined by the geometry of the molecules and how they pack around the constraining boundaries. Similar ordering also occurs for solvent molecules around the particle or solute molecules in a solution. Forces arising from any disruption of such ordering are known as solvation forces. So I will give you a more detailed accounting or origin of solvation interactions. When one solid surface is brought in vicinity of another one. So you have basically let us say that you have a parallel plate channel. So think the situation in this way you had one plate and you had the liquid molecules surrounding that plate. So there was a particular ordering. Now when another plate comes in close proximity of the first plate to form a very narrow channel then the ordering of this liquid molecules is disturbed. Ordering of the liquid molecules around the first plate that is disturbed and they rearrange to find the most energetically favourable packing density. So it takes energy to disrupt the original ordering of the liquid. Thus the presence of the second surface generates an oscillatory solvation pressure. It is whether oscillatory or monotonic it depends on scenarios. This oscillatory solvation pressure is an example d being the separation between the 2 surfaces. In case the interacting surfaces are atomically smooth. Now there are real engineering surfaces which are atomically smooth. It might be surprising but there are such surfaces like mica is an example of an atomically smooth surface that is atomically smooth. So for such surface the solvation interaction can be approximated by an exponentially decaying cosine function. The expression is given where the result is normalized in terms of the molecular length scale. So d by am. In case the surfaces are randomly rough the oscillatory nature is smeared out and it appears to be a monotonic solvation pressure. So only an exponential function without the cosine part. So the cosine part is there if it is a smooth surface. It is not there if it is a rough surface. Structural interactions. Now layering of solvent molecules in the vicinity of a solid substrate means that the liquid layer exhibits a solid like behaviour with a particular lattice constant. This is a remarkable phenomenon. So you we know that solids have a structure and they tend to if they have a crystalline structure they tend to assume a lattice constant. On the other hand liquids do not have such a structure but close to a solid the liquids tend to assume a structure which is influenced by the structure of the solid. So if the lattice constant of the liquid which is layering on the solid is not equal to that of the solid then the bonds between the solvent molecules in the solvation induced layer structure will be strained. This is because of the mismatch of the lattice constant and the interaction energy resulting from some such straining is known as structural interactions. Origin of the layered structure is caused by the exponentially decaying solvation effects. It is inferred that the structural interaction energy density is expressed in this particular form which is based on the form due to solvation interactions. You see this e to the power –d by a m but it depends on the elastic modulus of the solid form of the solvent. For example when the solvent is water the elastic modulus e is that of ice and the value is like 17 giga Newton per meter square. So like these are these are very non-intuitive physical understanding at least to people like us who have our domain in the engineering scenario this is something which is not very very intuitive to deal with but if you are dealing with nanotechnology these are certain aspects that you should keep in mind. Now so with on the basis of several physical issues I mean I have discussed about one or two physical issues but there could be several more. On the basis of these physical issues or not necessarily on the basis of these physical issues but on the basis of the general scenario in the nanofluidic domain we will try to revisit first the Lucas-Warsburn equation okay. So if you recall that for capillary filling dynamics we attempted to give an accounting of the forces acting on the system by using a lumped mass approach. So if you look into this equation you have the inertia that is the mass into acceleration right hand side the resultant force surface tension viscous and gravity. Now if you neglect the gravity and inertia and then if you also neglect the variation in the contact angle if you consider uniform viscosity neither slip nor stick at the interface then you can arrive at this very simple expression that relates that basically equates the viscous force in a fully developed flow. So we assume also fully developed flow in a capillary the fully developed flow with the surface tension force. So a little bit of different notation that we have used here we have used gamma for surface tension coefficient and eta for viscosity. Now the first effect that we may consider in the nanodomain is slip and how does this equation get affected in presence of slip. So let us go to the board and try to see. So assume that there is a channel a circular channel of radius r and let us say that vz is the axial velocity. So now we are interested to find out the viscous drag force when there is a slip at the one assuming fully developed flow. So if it is a Newtonian fluid 1 by r d dr r d v z dr is equal to 1 by mu dp dz. If it is a Newtonian fluid and fully developed flow other terms are not there you have the pressure gradient term and the viscous term. So now if we integrate this equation so r d v z dr is equal to 1 by mu dp dz r square by 2 plus c1. Now we can apply the boundary condition that at r equal to 0 dv z dr equal to 0 that will give you c1 equal to 0. So you have dv z dr is equal to 1 by 2 mu dp dz into r. So vz is equal to 1 by 4 mu dp dz into r square plus c2. Now what is the other boundary condition? What is the boundary condition at small r is equal to capital R? vz is equal to if we write this is it correct? If we write this vz is equal to ls into dv z dr at the wall of course that is of course is it correct? Actually there is an error in it so I want to pinpoint that error. So when you write v is equal to l dv dn or u is equal to l du dn that n is the outward normal. So if you consider this wall if you want to write vz is equal to ls dv z dn that n is this direction and our r is opposite to that so that has to be adjusted with a minus sign. So you can write 1 by 4 mu dp dz r square plus c2 is equal to minus ls by 2 mu dp dz into r. So what is c2? c2 is minus ls r by 2 mu dp dz minus 1 by 4 mu r square dp dz. Now we are interested to calculate the wall shear stress. So what is tau wall? Again because of the same reason we have a minus sign minus mu dv z dr at small r equal to capital R. So that is equal to minus half dp dz into r from this equation. But dp dz is not known. So we will express the dp dz in terms of the average velocity. So how do you get the average velocity? So let us write vz is equal to 1 by 4 mu dp dz r square minus 1 by 4 mu dp dz r square minus ls r by 2 mu dp dz. We have replaced the c2. So what is v average? Integral vz into 2 pi r dr from r is equal to 0 to capital R by pi r square. So 2 by r square integral of 1 by 4 mu dp dz 1 by 4 mu dp dz r square minus l square minus r square minus capital R square minus 2 ls r into r dr. So v average is equal to 1 by 2 mu r square dp dz then r cube dr is r to the power 4 by 4 minus r square into r dr is r square by 2. So r to the power 4 by 2 then minus 2 ls r into r dr is r square by 2. So r cube by 2. So this becomes minus r 4 by 4 minus 2 ls r into r dr is r square minus ls r cube. So you can cancel this r square. So this becomes r square, this becomes r. So minus 1 by 8 mu dp dz sorry minus 1 by 8 mu dp dz into r square plus 4 ls r. So you can write dp dz is equal to minus 8 mu v average by r square plus 4 ls r. So tau wall is equal to minus half dp dz into r square plus 4 ls r. So you can write dp dz is equal to minus 8 mu v average by r square plus 4 ls r. So tau wall is equal to minus half dp dz into r which is this equation. So 4 mu v average by r plus 4 ls alright. The drag force is tau wall into 2 pi r l. So 8 pi mu v average l by 1 plus 4 ls by r square plus 4 ls r square plus 4 ls r alright. Typically for practical considerations ls is significantly less than r. So this is like 1 plus x to the power minus 1. So this can be approximated as 1 minus x by binomial expansion if this x is small. So this is as good as approximately 8 pi mu v average l into 1 minus 4 ls by r. And this v average is nothing but dl dt. If l is the instantaneous location of the meniscus. So let us get back to the slide to get that picture. So you can see, so whatever we have considered as mu as viscosity in the board the same is there as eta in the slide. So 8 pi eta l dl dt into this correction factor 1 minus 4 ls by r is equal to 2 pi gamma r cos theta okay. So this is the first modification. You can say that this is the very obvious modification. So what is so special about this modification? A very subtle thing is that we do not know what is ls. See it is not a mathematical exercise that you simply assume some value of ls. Typical slip lengths are that may be few nanometers. So you take some 5 nanometer, 10 nanometer whatever and then do this exercise. So this particular formula is of course for small ls by r but even if ls is large you can keep this that in the denominator by keeping instead of 1 minus 4 ls by r in the numerator you have 1 plus 4 ls by r in the denominator that much you can do. But the question is how will you get ls? So to get ls one has to effectively do molecular dynamic simulations. I will not get into the molecular dynamic simulations today but I will show you some results from that which can be used to calculate the slip length. But before doing that I will consider some other approaches that is instead of bringing it through the slip length obtained from molecular dynamic simulations. People have tried several other means to adjust for the observed phenomenon in the molecular domain or in the nanoscale domain. So in the nanoscale domain the Lucas Washburn equation without slip many times fails to capture the essential physics. So instead of taking slip as a alternative measure because slip length is not known people have taken other ways of modifying the Lucas Washburn equation to capture the trends which match with the reality. So one is the inertial so one includes the fluid mass to enter the capillary at t equal to 0 in the LHS. This is the added mass of the virtual mass that we had discussed earlier. And the inertial terms may be included often as an adjustable parameter to explain the dynamics. In the Lucas Washburn equation or the Lucas Washburn model typically we do not have the inertial effects because it is viscous versus surface tension. So inertial term may be used as an adjusting or fitting parameter to match the observed phenomena with the behavior from the equation. So that is one group of researchers have done this. Then dynamic contact angle so we have already discussed that dynamic contact angle may be important question is how important is it in the nanoscale. So typically people have tried to fit the dynamic contact angle as a function of the static contact angle the surface tension coefficient, the viscosity, the slip length and the instantaneous velocity. Or people have used the molecular kinetic theory root to express the dynamic contact angle in terms of the wetting line friction but it is something which is relatively more abstract than the previous approach. But all these approaches have shortcomings we will discuss about that. Then viscous resistance fully developed Poiseuille flow is assumed to obtain the viscous term and modifications are incorporated to include the effect of entrance and the meniscus. The stick or slip associated with the flow also affects the viscous term. So many times this effect has been taken into account by using a modified viscosity. So instead of using the original bulk viscosity the modified viscosity is used as a function of the original viscosity the instantaneous displacement of the meniscus l and l dot. So this has been done in one of the papers and slip length has been used as an adjustable parameter to fit the data but not from fundamental calculations. So this has been reported in one of the earlier papers. Now the what remains as an unaddressed facet of the Lucas-Walburn equation is that if you use these kinds of modifications the modifications in the dynamic contact angle would suggest progressively less value of the contact angle as the meniscus velocity decreases. This we have already seen because like if you recall the dynamic contact angle scales with capillary number to be power one third right. So if you have lower velocity due to viscous resistance you have the contact angle also less as the because theta will scale with capillary number to the power one third. So if theta becomes less cos theta becomes more and the driving force becomes more that means you could see an increasing trend in the l square versus t characteristics. This is what is what is predicted by the standard dynamic contact angle model. But actually we observe this we observe a decreasing trend in l square by l square versus t over the scales that we are considering and not that increasing trend what we see over some elevated length scales. So the question is where is the fallacy and how can we account for that by combining the Lucas-Walburn modified Lucas-Walburn theory with the molecular dynamic simulation results. We will take up this issue in the next lecture. Thank you very