 And in our previous lecture we have discussed about the potential profiles within the electrical double layer derived by using the Poisson Boltzmann equation. Now because of some inherent limitations in the Poisson Boltzmann equation as well as some inherent limitations in the analytical solution procedure we have certain limitations of the obtained potential profiles. So what are those limitations? First is electrical double layer overlap is not considered. So when the value so typically electrical double layer overlap will occur when lambda is greater than h right. But typically even beyond that you can apply the potential profiles that we have derived successfully up to h by lambda less than 4. But beyond that I mean it is difficult to apply. So then the center line condition psi equal to 0 is no longer valid and no analytical solution for psi can be obtained in such a case. The other important factor is that steric effects are neglected. I mean there are so many other important factors we will discuss about those when we will discuss about nano fluidics. But these are some of the important points that is steric effects are neglected. So steric effects means the finite size effect of the ionic species. So in the Poisson Boltzmann description ions are assumed to be point charges and they are assumed to be non-interacting. The Boltzmann distribution predicts unbounded and physically impossible growth in counterland concentration at high beta potential. Because the ions are like point sized entities like they can heavily crowd the surface. Although such a practical large high number of ions at the surface is precluded because of the finite size of the ionic species. And this effect is more prominent as you increase the zeta potential. So one can incorporate finite size effect of the ionic species and that gives rise to a modified form of the Poisson Boltzmann equation which we will see later on. Now so we have prepared a sort of a framework in which we know that there is a electrical double layer because of an electrical double layer there is an electrical potential. But how will this electrical potential be manifested in the form of influencing the fluid flow. So let us take an example. So let us say that you have a channel and you apply an electrical voltage across the channel. So when you apply an electrical voltage across the channel then because the surface has a net charge the bulk let us say the surface has a net negative charge the bulk fluid will have some net positive charge. And because of the application of an external electric field along the axial direction the net positive charge will preferentially move. So ions will move because of the application of electric field this is nothing but electrophoresis. So because of this the ions will move and when the ions will move the ions will drag the liquid water along with that and the water will also start moving. So this is one of the mechanisms. So to understand this mechanism we will refer to an additional body force to mathematically represent that in the Navier-Stokes equation. We will see where from the extra body force comes. The body force comes from the contribution of an electrical force which is expressed in form of an equivalent stress which is called as electrical stress or Maxwell stress which is beyond the hydrodynamic stress which is called as Maxwell stress and a quantity called as osmotic pressure. So we will discuss about this. So the combination of electrical stress and osmotic pressure gradient if that drives the flow then that is called as electro osmosis. So we will see that how that drives the flow but prior to that let us see that how we calculate body forces. So to calculate body forces we have to refer to the very famous four classical equations of Maxwell in the electromagnetic theory. So I will discuss about all these four equations in terms of their physical interpretation but first let us look into their mathematical forms. So the first equation is the Gauss law for electricity which we have already discussed basically. In the previous lecture we have already discussed. Then similarly you have Gauss law for magnetism, you have Maxwell Faraday law and you have Ampere's law with Maxwell's addition. So what are these laws? So you can see that somehow these laws relate like Maxwell Faraday law and Ampere's law, these relate electric field with magnetic field. For example in the Maxwell Faraday law you can see that if there is a time dependent magnetic field that can give rise to an electric field. Similarly in the Ampere's law with Maxwell's addition you can see that if there is a time dependent electric field then that can create a magnetic field. So time dependent electric or magnetic fields can create equivalent magnetic or electric fields and the other symbols like you know that like permittivity of the free space other than that this J is the current density. So when you calculate the total force, total force is the charge density times the electric field plus J cross B, this is called as Lorentz force or electromagnetic force where J is the current density. Now J cross B you can write in terms of the in terms of the equivalent conductivity. So if there is a conducting medium moving with the velocity V then the effective potential gradient or the field is modified by E becomes E plus V cross V okay. So basically the J becomes sigma into E modified which is E plus V cross B. So this is like Ohm's law in the moving reference system and then that cross B. Now we will see we will develop a general framework in which if there is electrical field magnetic field that can be both time varying as well as spatially varying then what is the net body force. How can we calculate from these 4 laws of Maxwell but prior to that we will give some physical interpretation to the Maxwell's equations Gauss law. Gauss law describes the relationship between a static electric field and the electrical charges that cause it. The static electric field points away from positive charges and towards negative charges and these are very elementary school level understandings but it is important at least to have this qualitative feeling. In the field line description electric field lines begin only at positive electric charges and end only at negative electric charges. Counting the number of field lines passing through a closed surface yields the total charge enclosed by the surface divided by the dielectricity of the free space. So recall that originally the all the 4 Maxwell's equations were written for the free space not for any general medium but for free space. So if you look into the equations I mean let me go back to the previous slide you see that you have epsilon 0, mu 0 all these properties of free space are used. Gauss law for magnetism. Gauss law for magnetism states that there are no magnetic charges analogous to electrical charges. So that is why you will see that del dot E although is as a right hand side which is which in general is not equal to 0 but del dot B equal to 0. So instead the magnetic field due to the materials is generated by a configuration called a dipole. Magnetic dipoles are best represented as loops of current but resemble positive and negative magnetic charges in separately bound together having no net magnetic charge okay. So magnetic charge unlike electric charge which can have a net charge value you cannot have a similar thing for a magnetic charge. So in terms of field lines this equation states that magnetic field lines neither begin nor end but make loops or extend to infinity and back. In other words any magnetic field line that enters a given volume must somewhere exit that volume and in more technical terms the sum total magnetic flux through any Gaussian surface is 0 that is mathematically represented by del dot B equal to 0. If there is a vector B which satisfies del dot B equal to 0 then that B is called a solenoidal vector. Now very interestingly there is a difference I mean I do not know whether you are aware of this very fine issues. Electric field is a proper vector but magnetic field is a pseudo vector okay. So there is a concept called a pseudo vector. So vector will transform with rotation in a proper way that is if you have a vector you can transform the vector to a new vector with the help of a suitable transformation and the transformation may be say rotation. So under rotation a pseudo vector will transform in the same way as that of the original vector but under deflection it will flip its sign. Under deflection a proper vector will not flip its sign but a pseudo vector will flip its sign. There are other examples of pseudo vector in mechanics like angular momentum is a pseudo vector. So there are so magnetic field is actually a pseudo vector and if you are interested please read like advanced materials in mathematics on the difference between vector and pseudo vector and what they are. I mean here we do not have an upscope to spend a lot of time on that but it is a like I mean in many senses so far as the rotational effect goes if the rotational transformation of a pseudo vector is just like what is there for a vector. Maxwell Faraday's equation. Maxwell Faraday's equation is a version of the Faraday's law which states how a time varying magnetic field creates an electric field right. This is technologically very very important because by using a time varying magnetic field you can create an electric field. So you can see that look at the corresponding mathematical expression. So you can see that if b varies with time you have a del cross c. So this dynamically induced electric field has closed field lines just as magnetic field if not superposed by a static electric field. This aspect of electromagnetic induction is the operating principle behind many electric generators. For example a rotating bar magnet creates a changing magnetic field which in turn generates an electric field in a nearby wire. Ampere's law with Maxwell's addition that is the fourth Maxwell's equation. Ampere's law with Maxwell's addition states that magnetic fields can be generated in 2 ways. One is by electrical current this was the original Ampere's law and by changing electric field this is Maxwell's addition. So you can see that there are 2 terms in the right hand side of the corresponding equation. So del cross b is equal to 1 is mu 0j plus mu 0 epsilon 0 del i del t. So you can see there are 2 contributions. Maxwell's addition to Ampere's law is particularly important it shows that not only does a changing magnetic field induce an electric field but also a changing electric field induces a magnetic field right. Therefore these equations allow self-sustaining electromagnetic waves to travel through empty space. Now we will calculate the electromagnetic forces and to help you with the derivation I have used different color codes in the derivation so that it becomes clear that from which equation what simplification is made and we will go through this slowly so that you can follow it well. So we will start with the basic expression that dead force is the charge density times electric field plus the Lorentz force j cross b okay. Now this rho e into e you can write remember that we are using the Maxwell's equation initially for the free space. So rho e into e is what? What is rho e? Epsilon 0 del dot e this is what? This is Gauss law that into e and j we can use Ampere's law to express j in terms of b and del i del t. So this becomes del cross b by mu 0 minus epsilon 0 del i del t cross b. So now you can basically separate it into two terms okay. Then next step we will we want to write an expression for del i del t cross b the last term. So we can write del del t of e cross b is equal to del i del t cross b plus e cross del b del t. So that del i del t cross b is equal to del del t of e cross b minus e cross del b del t okay. So that is what is written the last two terms in the expression for f. Next step this del b del t you can write as del cross e using Maxwell's law or Maxwell Faraday's law basically. So now in this equation you see that you have a term del cross b cross b. So del cross b cross b you can use a vector identity for that. So this vector identity which is given in the green color replaces this del cross del cross b cross b okay remaining terms are the same. Now the expression which was there in the previous slide in that expression you have one e cross del cross e another b cross del cross b. So all these are of the form a cross del cross a and for that we have a vector identity okay. So we use that vector identity for both e cross del cross e and b cross del cross b and we get this expression. In that expression you have whatever I have kept in the blue color that is solely because of electric field the remaining terms some are due to magnetic field and one term is a due to a combination of electric field and magnetic field and that is the unsteady term okay. In steady state however you lose the coupling with the e and b because in steady state when this term becomes 0 then you have a collection of terms which are related to electric field and a separate collection of terms which are related to magnetic field. So these terms which are related to this force like remember that in the Navier-Stokes equation when you have a body force due to viscous effects you write for example del tau ij del xj right. So the divergence of the stress so similarly that stress is hydrodynamic stress. Now instead of that you can now so the divergence of the traction vector so to say. So now instead of that here you augment the stress in addition to hydrodynamic stress you have a stress due to electromagnetic effects and that is known as Maxwell stress or electrical stress. So if you write that as a del tau ij mm for Maxwell del xj then this is the expression for the Maxwell stress. So these 2 terms together is written in terms of this vector dyadic this is this symbol is called as the dyadic product so basically what you are doing is so if you have a vector say a say a1 a2 a3 so this is 3 by 1 this is 1 by 3 so the net result is 3 by 3. So this is the meaning of this dyadic product okay. So that is just a shorthand notation of writing the thing then so basically we will not use the shorthand notation we will straight away start with this form f equal to epsilon or not del dot ee plus epsilon or not e dot del e minus half del epsilon or not e square in a medium epsilon or not will be replaced by epsilon in free space it is epsilon or not in a medium it will be replaced by epsilon. So what you can see here is that you have the body force due to electrical field neglecting magnetic effects. Now we are talking about electrical double layer phenomena electric field so we are not applying magnetic field or but you can always say that we are not applying magnetic field so what magnetic field may be induced so the assumption and a very important assumption is that we are neglecting the induced magnetic field as compared to other fields. So if you do that then you have only the expressions with e and you can see that these are the three terms so that was e square basically e square is e dot e. So now you can write this as if you can expand it as and write 2e dot del e okay so 2e dot del and then e because I mean this is basically del e dot e so e dot del e plus another e dot del e so there it becomes 2e dot del e. So this term this 2 and half gets cancelled and then this minus and plus term gets cancelled. So eventually you get epsilon e del dot e okay so and e is what e is minus gradient of the potential so in place of e you can write minus grad phi so this becomes del square this becomes minus del square phi and this becomes minus del phi so total becomes epsilon del phi del square phi minus and minus get cancelled become plus. So this is a force due to electrical stress what about force due to osmotic pressure so we have to first understand that is what is osmotic pressure or what is osmosis. So and we will see that how does it follow from the basic equations of fluid mechanics themselves. So let us say that you have a semi permeable membrane and you have a concentration gradient across that concentration difference. So let us say that here there is low solute there is high solute okay so here there is high concentration here there is low concentration in an effort to equilibrate the concentration how one can equilibrate the concentration one is solute can come from here to here but the other possibility is that solvent can go from here to here right solvent will make the high concentration solute more dilute. So this process by which the solvent goes from the low solute concentration to high solute concentration across a semi permeable membrane this is called as osmosis. So when you have this osmosis then basically there is a driving pressure gradient that drives this transport and this pressure which is the minimum pressure that is necessary to achieve equilibrium of the concentration at the 2 ends at the 2 sides of the semi permeable membrane that pressure is called as osmotic pressure. Now how can we realize this osmotic pressure starting with the Navier-Stokes equation let us see. So let us try to create a physical picture in our mind let us say there is a channel across this wall you have even across this wall also but I am just giving one example across this wall you have electrical double layer formation. So when you have electrical double layer formation you have a potential psi and then along the axial direction you have a potential field phi 0 which is a function of x in general. So psi is a function of y phi 0 is a function of x eventually going to write the momentum equation along the x direction but many times we neglect what is happening for momentum equation along the y direction. So let us write the momentum equation along the y direction with a low Reynolds number or even with a fully developed flow the left hand side is equal to 0. So now let us say this pi is the osmotic pressure the viscous terms are 0 because v is 0 or negligibly small in the y momentum equation this pi is the so called osmotic pressure. So now to generalize this we can write instead of the y direction in any arbitrary direction 0 is equal to minus grad pi plus rho e into e due to edl this e y is e due to edl okay. So physically what is happening because of the electrical double layer formation there is a concentration gradient along the y direction therefore there must be an osmotic pressure acting in the y direction whenever there is a concentration gradient you will expect that there is an osmotic pressure. So the concentration gradient is created because there is number density gradient across the electrical double layer and what is responsible for that is the electrical field within the edl that is responsible for that. So now what is edl minus grad psi and what is rho e minus epsilon del square psi right so minus grad pi which is the force due to the osmotic pressure gradient is equal to minus epsilon del square psi del psi. See this we have recovered from the Navier-Stokes equation we will see that purely from electrochemistry consideration how we can get the same expression for that derivation we will go to the slide. So the osmotic pressure what is I mean whatever is due to this concentration is given by this osmotic pressure gradient is given by first of all the osmotic pressure is equal to n k B T where n is n plus plus n minus. So grad of n plus plus n minus you can write so if you use the Boltzmann distribution n plus minus equal to n 0 e to the power minus plus z e psi by k B T. So you take log of both sides and differentiate both sides basically take the gradients of both sides. So you can write grad n plus by n plus is equal to minus z e psi by k B T and grad n minus by n minus is equal to plus z e grad psi by k B T. So those 2 expressions are written here in the second step and then that k B T and k B T gets cancelled in the numerator and denominator you are left with e z n plus minus n minus that is minus epsilon del square psi from the Poisson equation. So you can see that whatever we got from the Navier-Stokes equation purely from the consideration of electrochemistry the same thing can be recovered. But I would say that as a fluid dynamist I would prefer to go through the Navier-Stokes equation root rather than this root not only that see in this root we can make this derivation easily provided we assume the Boltzmann distribution. In the Navier-Stokes through the Navier-Stokes equation whatever we derived in the board we could derive it without appealing to the Boltzmann distribution. So that is something which is bit more general. Now so we have come up with 2 body forces 2 types of body forces when you have this physical phenomena. So what is the body force? f electro osmosis f electrical plus f osmotic. What is f electrical? Epsilon this is due to the Maxwell stress we derived just before deriving the osmotic pressure and what is f os that is written here minus epsilon del square psi del psi. Now what is del square phi? What is phi actually? Phi is equal to phi not plus psi that phi is the total field. Total field is because of field due to the applied electric field you can remember the schematic there is an anode and there is a cathode and there is an actual electric field applied phi not is the field because of that and psi this is so this is applied and psi is induced induced due to electrical double layer effect. So del square phi is equal to del square phi not plus del square psi right del square phi not is what? Del square phi not is equal to 0 because the applied field will satisfy the Laplace equation. The induced field will satisfy the Poisson equation but the applied field will satisfy the Laplace equation. So del square phi is equal to del square psi. So this equation therefore becomes you can get del square psi as common epsilon del square psi into del phi minus del psi. So this is epsilon del square psi into del phi not. What is this? Minus rho e right Poisson equation and what is this? Minus e external so this is rho e into e external. So by combining the electrical force with the osmotic force the net electro osmotic force is rho e into e external. So here you have already taken into account the pressure gradient due to osmotic pressure therefore when you write the corresponding Navier-Stokes equation do not write the pressure once more. Then it will be duplicate until and unless you have an additional external pressure gradient. Normally pure electro osmosis is a situation when at the entrance and exit of the channel the pressure is the same. So that there is no applied external pressure gradient but you can have combined pressure driven and electro osmotic flow where you apply a pressure gradient in addition to that you have to apply the axial electric field. But even if you do not apply the pressure gradient there will be osmotic pressure but that osmotic pressure is already incorporated in calculating this electro osmotic body force. Therefore you do not have a duplicate representation of pressure in the same equation okay. Now let us summarize this part of the finding. So you can see that when we have written the total body force on the fluid. So this is rho e into e external remember this is when we write rho e it is rho e of the free charges that is what we derived in the previous lecture when we were discussing about the Gauss law. So now let us write the governing equation for electro osmosis. So what is the physical picture that is happening? We will discuss about that by drawing a schematic in the board and then we will solve the corresponding Navier-Stokes equation to get the fully developed velocity profile for electro osmotic flow in a parallel plate channel. Let us say that you have a parallel plate channel and you have an axial electric field E x which is an electric field applied. Let us say this is a positive electrode this is a negative electrode. Let us say that the surface has negative charge and there is electrical double layer phenomenon at the surface. We are interested to write the x momentum equation x is the axial direction assuming a hydrodynamically fully developed flow. So when we are writing the x momentum equation for fully developed flow the left hand side equal to 0. We have we assume there is no additional pressure gradient which is acting on the system. This is the viscous term u is a function of y only for fully developed flow plus there is a body force rho e into E x. Now what is rho e? What is rho e? No, I mean rho e is charge density. Now how do we express rho e in terms of the potential? We use the Poisson equation. So d2 psi dy2 is equal to minus rho e by epsilon. So in place of rho e we have minus epsilon d2 psi dy2. So 0 is equal to mu d2 u dy2 minus epsilon d2 psi dy2. Let us do an order of magnitude analysis of this equation. Let us say that us is the scale of u. So what is the order of magnitude of this? Mu us by ys square. ys is the length scale. If it is a single plate you can use the d by length as the length scale. But we have kept it general because if it is a confinement you can use the height of the confinement as the length scale. So you will see that this length scale will not matter. What is the order of magnitude of this? Epsilon E x zeta by ys square. Therefore if these two terms are of the same order because these are the only two terms. So they must be of the same order. So us is of the order of epsilon zeta E x by mu. Now let us see whether this is positive or negative. This term is positive or negative. Accordingly we have to adjust the sign because order of magnitude does not take care of the sign. So for a negative wall charge what is zeta? Positive or negative? Zeta is negative. So for a positive for a negative wall charge zeta is negative. Now what is E x? So E x is equal to minus of phi 2 minus phi 1 by L where this is section 1 this is 2. E is equal to minus d phi dx. L is the length of the channel. So E x is positive in this because phi 2 minus phi 1 is negative and that adjusted with a negative sign becomes positive. So E x is positive. So the right hand side is negative. Now if you have a negative zeta what is the direction in which you have flow? So remember and try to understand the physics and now explaining the physics through it. So if you have the surface charge as negative then in the bulk you will have some positive and some negative charge but positive will dominate because that system has to be neutral. With the action of this negative electrode this positive charge will be moving in the positive x direction and that will drag the water along the positive x direction. That means that you have us positive under this situation. So you have us positive right hand side you have this as negative. So to adjust this you must adjust it with a negative sign. So this velocity scale minus epsilon zeta E x by mu this is a very well known velocity scale in electro osmosis and this is known as Helmholtz Moluchowski velocity scale. Helmholtz Moluchowski velocity scale. Now this velocity we have got by just by order of magnitude analysis but let us solve the momentum equation to see the complete picture. So now let us integrate this equation once. So if you integrate this equation once so you will get 0 is equal to this will become a constant of integration. So mu du dy minus epsilon E x d psi dy is equal to C 1 right. Now what is the value of C 1? Whenever du du dy is 0 that is at the channel centre line d psi dy is also 0 but you cannot really generalize it. What happens if there is asymmetric zeta potential at the 2 plates then d psi dy is not 0 at the centre line okay. So you have to be careful du dy is equal to 0 and du d psi dy equal to 0 at the centre line both when both hydrodynamic boundary condition wise and the potential boundary condition wise everything is symmetric. So then in that case only you have C 1 equal to 0. Then you integrate this once more. So if you integrate this once more so mu u minus epsilon E x psi is equal to C 2. What is the other boundary condition at the wall? Psi is the zeta potential. Again remember that exactly not at the wall psi is the zeta potential at the shear plane psi is the zeta potential but we use this interchangeably for many practical purposes. So 0 minus epsilon E x zeta is equal to C 2. So that means you have mu u minus epsilon E x psi is equal to minus epsilon E x zeta. So u is equal to minus epsilon zeta E x by mu into 1 minus psi by zeta. So this is the boundary condition is nothing but the Helmholtz-Moluchowski velocity UHS the scale that we had obtained by the scaling analysis. So u by a UHS is equal to 1 minus psi by zeta. Now what will the velocity profile look like? Let us try to draw the velocity profile. Let us say that the electrical double layer is 100 nanometer in thickness and the channel is say 10 micron in height. Just some typical numbers. So that means even if you zoom maybe this much is the region of influence of the electrical double layer. So up to this much you have a gradient in u because you have a gradient in psi. So up to this much you will have a velocity gradient. Beyond this the outside the electrical double layer psi will become 0 and then the velocity will become UHS minus epsilon zeta E by mu which is a constant. So it will be almost a uniform velocity profile except very sharp gradient within the electrical double layer close to the wall. So see how disparate things look very similar. In fluid mechanics you may get a almost uniform velocity average velocity profile for a turbulent flow. Now this has nothing to do with turbulent flow but you see that this is one way by which you can get an almost uniform velocity profile. However this uniform velocity profile will be distorted if you have a gradient of zeta potential. If the zeta potential is not uniform because if the zeta potential is not uniform that will induce a pressure gradient actually and that will distort the uniformity of the velocity profile. So let us summarize the discussion that we have made so far by appealing to the slides. So the velocity profile which I have derived in the board u is equal to minus epsilon zeta E x by mu into 1 minus psi by zeta. So typical values zeta say 25 millivolt this is the just an example epsilon 80 times the permittivity of the free space and mu the viscosity of water 10 to the power minus 3 Pascal's again and typical E 10 to the power 4 volt per meter. So if you take these values see this is the typical order of the velocity that you get typically 10 to the power minus 4 meter per second. So these are important because somebody may ask you that electro osmosis what is the typical order of velocity that you get. Now then to answer that question you have to be careful about the practical numbers which come into the picture. So the typical velocity profile you can see this blue line that this is the UHS and there is a sharp gradient close to the wall. So this is like a plug shape velocity profile. Experimentally if you look if you compare pressure driven flow and electro osmotic flow that this is what it looks. So you can see that electro osmotic flow has almost a uniform velocity profile whereas pressure driven flow has a parabolic velocity profile. Now of course electro osmotic flow will not have uniform profile if you have axial gradients in zeta potential or you have electrical double layer overlap. So there are cases when you will not get uniform velocity profile for electro osmotic flow but we are not considering those cases we are considering those cases only when you get typical uniform velocity profile. So now you see that there is a great advantage of this velocity profile. So when you have a solute in the fluid say a chemical species and say there is a reactant sitting at the wall. So the chemical species moves at equal velocity at all transverse location whereas when you have a pressure driven flow along the center line it moves much faster as compared to the transverse other transverse location. So it is possible that the solute or the chemical species tends to move preferentially along the axis and does not want to go to the transverse location at the wall where maybe other reactants are kept. So this kind of situation is called as dispersion and we will have a discussion on that in one of our next chapters. So some pros and cons of electro osmotic flow advantages. Velocity is not dependent on geometrical dimension. See when we write minus epsilon zeta e by mu no geometrical dimension comes into the picture there. There are no moving components involved so wear and tear and all those things are not there and because you require electric field to drive the flow it is easily integrable with electrical or electronic circuitry in lab on a chip based devices. So these are advantages but these advantages strong dependence on surface chemistry and physicochemical properties of the solution. Stray pressure gradients induced due to surface non uniformities may disturb the plug like profile. And the third point which is technologically a very important point that because you have an electric field in a current carrying conductor which is the ionic system here you have joule heating i square r effect. Because of this joule heating what is possible is that let us say that you have a double standard DNA. The double standard DNA in an electro osmotic flow due to joule heating can be broken into 2 single strands. So there can be some kind of thermal transformation of thermally labile biological samples if the joule heating is not properly controlled. So you cannot use uncontrolled amount of electric field. So to summarize our discussions on electro kinetic so far we have discussed about the basic structure and simple mathematical description of EDL. We have derived the expressions for the ionic distribution as a function of electrical potential. We have demonstrated these principles to explicitly obtain the ionic distributions and potential distributions for various types of boundary conditions. And we have discussed the hydrodynamic implication of an applied electric field in the perspective of Maxwell stress or electrical stress. We have also discussed the osmotic pressure gradient and by using this additional forces we have determined the velocity profile for electro osmosis. The advantages of an electro osmosis electro osmotically driven flow as compared to pressure driven flow have also been discussed. So we will stop here now and we will continue with some other aspects of electro kinetics in the next lecture. Thank you.