 So, what is the form of the energy equation that we derived in the previous lecture? So, to have a look into this, let us have a recapitulation that well, let us recapitulate one or two steps before that. So, first we wrote the Reynolds transport equation for the total energy conservation, then the rate of energy change within the system we expressed in terms of the heat transfer and the work done by using the first law of thermodynamics. The heat transfer involved surface heat transfer and volumetric heat transfer, the work done involved the work done due to surface force and work done due to body force and then in terms of the internal, in terms of the total energy we could write an expression for the first law and combine that with the Reynolds transport theorem to get this differential form of the conservation equation. Then we subtracted the mechanical energy conservation equation from this to get the internal energy, the governing equation in terms of the internal energy. And then in that equation we simplified a term which is because of viscous heating, because of energy dissipation due to work done against the viscous stresses and once we calculated that, we figured out that viscous dissipation term that is given by mu phi in this expression is always going to be positive. So, with this and transferring the internal energy to enthalpy, we have got the governing equation in this form. This is the form that we, I think this is the last form that we derived in the previous lecture before we concluded the previous lecture that rho total derivative of H minus total derivative of P is equal to the heat generation per unit volume minus the gradient of the heat flux vector and plus mu phi which is the viscous dissipation term. Now, we have to appreciate that the in the energy equation we are interested eventually in terms of to express everything in terms of the variation in temperature. Because temperature is the variable that is directly measurable from experimental considerations and we have a particular physical appeal to temperature very intuitively that kind of physical appeal is not existing for enthalpy or internal energy these types of variables. So, our next objective will be to express the expression in terms of enthalpy in terms of temperature and pressure. So, that is what we will do. Now, to do that we require certain thermodynamic basis. I presume that all of you have the thermodynamic basis, but we will discuss in very brief about the concepts that we will be talking about when we go through the necessary steps. So, if you have for example H, H is the enthalpy H is a function of what? H is a function of 2 independent intensive thermodynamic properties provided the substance is a simple compressible pure substance a simple compressible pure substance. So, the you all of you understand what is a pure substance. So, a pure substance is a chemically homogenous substance. Now, what is a simple compressible substance? A simple compressible substance is a substance where the changes in pressure volume temperature these effects are much more important than the other effects for example, electrical magnetic and all any other physical effect. So, we are expressing H as a function of say T and P we could also express it in terms of some magnetization electrical field all these things, but those effects are not important because we are considering it as a simple compressible substance. The implicit thing here is that this is what is nothing, but in thermodynamics called as state postulate that the state of a thermodynamic substance can be expressed in terms of 2 independent intensive thermodynamic properties. Now, the question is where is the guarantee that T and P are independent? Can you give the example of a situation when T and P are dependent like phase transformation for example, let us say you are talking about melting of ice for example. So, if you have a pure substance changing its phase in fact, the phase changing temperature is a single valued function of the phase changing pressure. So, once you specify the phase changing pressure the phase changing temperature will be automatically specified. So, that means we are assuming here implicitly that there is no phase change although we are not writing it somewhere. So, no phase change. So, this is a convenient way of expressing H because we have tried to express H in terms of 2 variables which we can measure in engineering applications. So, D H this is what we can write this is by the simple rule of partial derivative. Now, what is this? This is by definition C p then what we will do is see there are certain variables which we cannot measure directly. For example, I mean variation of enthalpy with pressure. So, there is no measuring device that can directly measure this. So, we will try to express the some variables that can be directly measured. To do that what we will do is we will write this in terms of first something which cannot be directly measured and then express that in terms of other measurable quantities. So, to do that we will use one of the 4 T d s relationships T d s is equal to D H minus V d p. So, I will briefly discuss about the validity of this expression and all. So, to establish this equation it is very straight forward. We start with del Q is equal to D E plus del W. What is this? This is the first law of thermodynamics for a system undergoing any process for a control mass system undergoing any process. So, we will start with this. Now, first we will make one assumption after the other and remember that these assumptions are inbuilt with the derivation of this equation. Again I am repeating that many times there is a tendency that we just plug in a final formula to solve a problem. There is nothing wrong with it provided we know that what are the assumptions and restrictions under which that final formula is valid. So, that kind of I mean as a teacher I always therefore, emphasize on knowing the roots by which a final formula has been derived. So that while applying that you know that what are the conditions under which you apply that rather than taking that as a magic formula to solve a problem for which that formula is not applicable at all. So, see the first assumption that we will make is that the changes in kinetic energy and potential energy are negligible as compared to the changes in internal energy. So, this is the typical thermal problem you know that you have a problem where thermal effects are much more important. So, changes in kinetic energy and potential energy are not that important as compared to the changes in internal energy. So, that instead of DE we can write DI, I stands for internal energy. The next two assumptions are very very critical. The next two assumptions that we are writing is instead of del Q we will write Tds. Well, the next step we will divide all the terms per unit mass. So, we will write them per unit mass. So, we will write in terms of the specific quantities in place of del Q we will write Tds and in place of del W we will write Pdv. So, let us come one by one. So, first question can we write this always? The answer is straight forward no we cannot write this always. So, let us try to assess the situation when we can write this Pdv. When can we write del W as Pdv? When we can we can write del W as Pdv when it is a moving boundary in a quasi static or quasi equilibrium process and only form of work is a moving boundary work. There could be other forms of work for example, electrical work, magnetic work all those works are not considered only the moving boundary work is considered and that also the system is considered to be undergoing a quasi equilibrium or quasi static process. So, the assumption behind Pdv is clear. What is the assumption behind Tds? It is a reversible process. So, now when we say it is a reversible process it also has to be a quasi equilibrium process, but the inverse is not true we have to keep in mind because quasi equilibrium process just means that in terms of the internal irreversibility the internal irreversibilities are not there. So, that the because the system is in quasi equilibrium process what is happening? A system is moving very slowly from one equilibrium state to the other. So, that all intermediate states are in thermodynamic equilibrium. So, it is a sort of a very slow process. So, that makes the process internally reversible, but a quasi equilibrium process could be externally reversible. For example, there could be a finite temperature difference for heat transfer between the system and the surrounding and that will make a process externally reversible. So, external irreversibility may be there even if it is quasi equilibrium right. So, for example, the system boundary is at a temperature of 10 degree centigrade and the ambience is at a temperature of 30 degree centigrade. So, because of this finite temperature difference the heat transfer between the surrounding and the system will become an externally irreversible heat transfer. So, what it means is that when we say that quasi equilibrium we mean internally reversible, but there is no guarantee that it is externally reversible. However, when we say reversible we mean reversible in all respects externally internally whatever. So, once we say that it is a reversible process we automatically ensure that it is an internally reversible process. So, once we write this as TDS automatically we can write this as PDV with the understanding that there is no other form of work other than moving boundary work that we have to be careful. So, under all these restrictions we can write this, but once we have written this this becomes a very powerful thermodynamic expression. This powerful thermodynamic relationship is powerful because this now involves all exact differentials. See when we write del Q or del W all of you understand that these are inexact differentials because these depend on the thermodynamic path. To find out the work done as the system undergoes a change of state from state 1 to state 2 you have to write you have to basically know the thermodynamic path by which the system has gone from state 1 to state 2 to find out the work. Similarly, you need to do that to find out the heat transfer, but to find out the change in volume as the system goes from state 1 to state 2 only the final volume and initial volume is important and not how the system has changed its volume. So, volume therefore is called as a point function or the differential of volume is called as exact differential and volume is a thermodynamic property. So, a property is something in thermodynamics which the differential of which can be expressed as exact differential. So, if its differential cannot be expressed as exact differential that is not a thermodynamic property. So, you can see that in this expression all differentials are expressed in terms of exact differentials right because all differentials are expressed in terms of exact differentials this is a property relationship and the property relationship means that it this does not restrict any process. But here there is a very important conceptual issue you may argue that just now we said that this is valid for a reversible process and now we are saying that it is valid for any process. So, where is the conflict the there is no such as such conflict we have to understand the inner meaning of this statement very carefully. So, let us say that we want to use this expression for getting a change in entropy. So, to get a change in entropy what you need to do you need to integrate this that integration has to be executed along a reversible path like let us draw a schematic to understand this. Let us say this is a any thermodynamic plane this is state 1 this is state 2 this is let us not draw any process just states 1 and 2 state 1 and this is state 2 the system let us say has undergone an irreversible process by which it has changed its state from state 1 to state 2 dotted line because you know intermediate states are not well defined it is a irreversible process. Now you want to find out the change in entropy so what you can do you can construct a hypothetical reversible process between state 1 and state 2 right integrate the TDS expression along that reversible path and once that integration is carried out this history is no more important because it is a point function that S2-S1 will be same for the original dotted line process that is what what we mean by this that this is valid for any process the integration has to be carried along a reversible path but once that integration has been carried out the resultant change can be expressed or utilized to evaluate the change in entropy for any process that is what we mean. So what we will do is we will write let us come back to this expression in place of okay now you the next step is very simple you know I is equal to H-PV so this will be DH-PDV-VDP-PDV so that PDV gets cancelled out so DH-VDP so in place of DH we will write TDS-VDP okay now again see we have tried to get rid of a problem by trying to express things in terms of measurable quantities but apparently we have entered into a deeper problem by expressing it in terms of something which is perhaps the most difficult thermodynamic quantity to measure the entropy. So the situation is that we now need to express the entropy in terms of measurable parameters so just in the same spirit as which we expressed H as a function of T and P let us express S as a function of T and P so we can write DS this plus okay now we can express this del S del P at constant temperature in terms of what we can express this in terms of measurable thermodynamic parameters by using one of the 4 Maxwell's relationships okay. So this is the Maxwell's relationship that we will use remember again like because microfluidics is a multi disciplinary subject I mean you will find that like same scientist this physicist was so strong that they contributed in several ways to different branches of physics and those contributions are many times common or important in several branches of modern day science like microfluidics for example. So Maxwell on the one on one hand discussed about fourth thermodynamic expressions or relationships this is one of them on the other hand Maxwell has 4 equations in electromagnetic theory and we will discuss about those 4 equations those are also Maxwell's equations and very famous equations and we will discuss those in the context of electromagnet magneto hydrodynamically actuated flow in microfluidic devices that is just a passing remark not exactly relevant to what we are doing today. So therefore now let us do the simplification you can write in place of dh Tds plus Vdp so that means this is T del S del T at constant pressure dt minus T del V del T at constant pressure dp plus Vdp is equal to Cp dt plus del h del p at constant temperature dp. Now let us compare both sides if you compare both sides you can write T del S del T at constant pressure is nothing but Cp this is coefficient of dt so to say and del h del p at constant pressure temperature this is what we wanted actually this is our expression of interest this is V minus T del V del T at constant pressure. So this is what we wanted because this is the expression which we were struggling to express in terms of measurable quantities. So with that understanding let us clean up the board a little bit so that we can write dh is equal to Cp dt plus V minus T del V del T at constant pressure dp. The next step is very simple this del V del T at constant pressure is related to the volumetric expansion coefficient. So we will express this in terms of the volumetric expansion coefficient beta. So beta as you recall this is change in volume per unit volume for each degree change in temperature that is what is expressed here. So we can write del V del T at constant pressure is equal to beta V so this is Cp dt plus V into 1 minus beta T dp. So see we have now been successful in expressing dh in terms of dt and dp and these are all measurable parameters. So let us now get back to our energy equation expressed in terms of enthalpy. So we had rho dh dt minus dp dt is equal to Q minus del Qj del xj plus mu phi. So how do we now relate this with this expression? It is simple like we can write this also as delta H is equal to Cp delta T plus V into 1 minus beta T delta P divided by delta T both sides and take the limit as delta T tends to 0. This equation actually pertains to the change that is given by this total derivative because wherever you have the advection as well as change with respect to time at a given location the total change is given by the total derivative. So you have to just I mean notionally adjust this small d with capital D but it is essentially the same change. So we can write rho dh dt as rho Cp dt dt plus if you multiply this with rho rho into V is what? 1 because specific volume is 1 by rho. So plus 1 minus beta T dp dt. We will substitute this expression in the equation for enthalpy. So you have rho Cp dt dt plus 1 minus beta T dp dt minus dp dt plus mu phi. So you can see that this dp dt term gets cancelled out. So we are left with this energy equation rho Cp dt dt is equal to beta T dp dt plus this is the energy equation in terms of temperature that we commonly talk about. Let us recapitulate the physical meanings of various terms in this equation. So this refers to the total change in temperature. This is because of change in temperature at a given location due to change in time and change in temperature as the fluid moves to a new location and encounters a new velocity and temperature field at that new location the combination of that. This is due to what? This is due to pressure work. So normally this term is not important for incompressible flows but I mean it all depends on how strong this beta is okay. So if beta is strong enough this term will be important otherwise this term is not. This is the volumetric heat transfer. This is the surface heat transfer and this is the viscous dissipation term. So that summarizes the physical meaning of this and again just like the Navier-Stokes equation you see rho has come out of the derivative and that is because of simplification with the aid of continuity equation it has nothing to do with constant rho or variable rho okay. Now we have discussed about the energy equation in details. Let us go to the next slide to summarize the energy equation. So what we have done is we have expressed the enthalpy gradient in terms of pressure and temperature gradients and use the thermodynamic relationships with the proper assumptions taken into consideration and that has given rise to the final form of the energy equation. Now one step we have not yet considered is in fluid mechanics when we discussed about the equation of motion we discussed about the special type of fluid which is called as Newtonian fluid. And finally Newtonian and Stokesian fluid with homogeneous and isotropic behavior taken into consideration. Now similarly for heat transfer we also have to consider some special type of fluid. I mean it is not necessary that we always consider that special type of fluid but if we consider a special type of fluid that will simplify the heat transfer equation by further. Because if you look at this equation here the heat flux is an enclosed variable right heat flux is not specified. So how do you solve for temperature from this equation if you do not know what is the heat flux. So heat flux should be expressed in terms of the temperature or its gradients. So one of the very basic laws that governs that is the Fourier's law of heat conduction. So Fourier's law states that heat flux is related is proportional to the gradient of temperature. So this may not be a constant in a sense that you may have different constants along different directions. I mean you may have inhomogeneity, you may have anisotropy, you may have so many other parameters into consideration. So we will not consider all those complications. We will not make an attempt to write it in the most general form. We will say that if along a particular direction x this is proportional to minus del t del x and this is Fourier's law of heat conduction. I will discuss in a moment that what are the conditions under which this Fourier's law of heat conduction is not valid. But see when we write this then we can replace it with a constant of proportionality kx. So when we write this in terms of a constant of proportionality kx then along y it could be ky, along z it could be kz. So I mean there are various possibilities although for simplification in many cases you have kx equal to ky equal to kz. So that you may consider but more importantly what are the restrictions of this equation? I mean anisotropy, isotropy those things can be taken care of mathematically but the physics has to be very carefully understood. So Fourier's law of heat conduction although it does not write explicitly it implicitly means that if you have a thermal disturbance at a point that propagates in all directions at infinite speed. So that means instantaneously the effect is felt everywhere else. Now that may not be the case when the time scale over which you impose the thermal disturbance is very short. Let us say that there is a particular treatment that is going on using femtosecond laser. In modern day medical technology which is very much related to micro scale fluid flow and heat transfer I mean there are various situations when very short pulse width lasers are used for certain treatments. Now if their pulse width is very short see the laser is on, off, on, off, on, off like that. So over a very short period of time when you treat the material, the material may be an engineering material, the material may be a biological tissue. So when you treat the material then the material does not get sufficient time to adjust to that change that means its relaxation time is not yet crossed and a new change has come. So it is not possible for the material so to say to assimilate the change in such a short time and that means that it cannot sort of instantaneously dissipate its change in state to all possible directions at infinite speed and then this kind of law will not be valid. So then you have to use this general form of the energy equation and substitute instead of the Fourier's law some non Fourier behavior that is given by some other constitutive behavior we will not discuss it like this but this heat flux is like the simple tau ij in the momentum equation. You can express tau ij in terms of the constitutive behavior here also you can express the heat flux as a function of the constitutive behavior. The most simple and very general constitutive behavior that we commonly talk about is the Fourier's law of heat conduction. So if you substitute that and you will end up with the equation that is there in the slide that you see the final expression instead of q you just write k grad t and that completes the description of the energy equation. Next we will discuss about the species conservation equation. Again a little bit of background I mean so far we have discussed something which is not very uncommon to the mechanical engineering students because I mean mechanical engineering students commonly learn about fluid mechanics and heat transfer so these equations are pretty well known to the mechanical engineering students may not be derivations to all possible depths but at least the common use of these equations is well understood. Now when we come to species conservation equation this species conservation equation is very commonly studied in chemical engineering and materials engineering and of course in mechanical engineering perspective also it is very important but not very commonly studied. Why it is not commonly studied is because multi-phase multi-component systems are not very commonly studied in mechanical engineering undergraduate curriculum. Now question is that what does it talk about and why do we require a species conservation equation that needs to be discussed very carefully. So now if you think in very simple terms species conservation equation is like a mass balance equation. Now immediately a question will come to your mind that we have already discussed about the continuity equation for mass balance. So why do we require another equation for mass balance continuity equation already gives us mass balance. The answer to this is relatively straight forward. The continuity equation gives the conservation of the total mass but if you are interested about mass of one component in a mixture then individual species masses also have to be conserved. So individual species like the total mass is conserved individual species masses are also conserved with an understanding that some individual species may be created or destroyed due to chemical reaction. So individual mass say there is some entity A say sodium in a mixture. Now you had some original sodium but some sodium may be generated, sodium ions may be generated because of some electrochemical reactions. So that has to be taken into consideration. So keeping that into consideration let us go to the slide. So what we have written the first statement is the pretty straight forward control volume mass conservation. Mass in say the first subscript i is the constituent species i the ith species the mass in minus mass out plus mass generated is equal to the change of mass. So that we have written in terms of rate equations and you can see this m dot dot at the top of all the m's so to indicate that this is a rate equation. So this principle can be applied to a differential control volume in a manner very similar to the continuity relation. The only new consideration is that mass of a species i can be created or destroyed via chemical reactions okay. So I am not going through the derivation of this equation at this level we will do it in a final level. At this level the derivation is same as the continuity equation just except for a source term in the right hand side. The continuity equation del rho del t plus del dot rho v equal to 0 that was there. Now here instead of 0 it will be some r i but that r i depends on the mass generated or destroyed for the species i because of chemical reactions okay. But we have to discuss about one thing very very very carefully that is what is that velocity v i okay. In fluid mechanics when we are talking about the continuity equation that velocity is the velocity of flow. In species conservation equation when we are talking about the velocity ui bar for the velocity vector ui this is the species velocity right this is not the flow velocity because we are talking about the mass conservation of the species not the mass conservation for the flow that we have to keep in mind. So it has to be the species velocity so thus it is the velocity of i with respect to ground that this velocity now this is an abstraction I mean how do you measure say you are doing an experiment how do you measure what is the velocity at which individual species is moving it is not a straight forward thing to do that. So it is the velocity of the species i with respect to ground this velocity is a continuum average that includes the effects of bulk flow and diffusion. So how is the species moving the species is moving because of 2 effects one is because of bulk flow the species is moving other is even if there was no bulk flow the species would have diffused because of concentration gradient. In addition to this in microfluidics where electrical fields are there and the species has charged then the species can move by virtue of electrical field that we have not considered at this stage that we will consider in a later stage as we go to the species conservation equation for charged systems. So this ui bar is the species velocity velocity of i with respect to ground. So let us summarize these equations in the board because we will use this for further simplifications this plus basically this equation we are talking about and one point we have clearly understood that this ui bar is not the velocity of flow conceptually it is species velocity it is not flow velocity. Let us go to the next slide now we define certain properties this is just a definition this is the way in which we define. So the first definition is rho is equal to sigma rho i. So we define the rho as I mean this is just a variable name. So rho is equal to sigma rho i and then we define mass average velocity mass average velocity is equal to sigma ui rho i by rho. So that you know in this equation that we have written in the board we can write rho u bar is equal to sigma of ui bar rho i rho is equal to sigma rho i. Next we move on to the next slide now the question is and it is a very important question that how do we express how do we express this ui bar in terms of u bar and velocity of flow. Eventually we have to express this species conservation equation in terms of the velocity of flow. So how do we do that? So to do that you see that we have defined something J i which is rho i into ui bar minus u bar what is u bar? u bar is the species average velocity and ui is the individual velocity of the species. So very qualitatively this talks about the difference between velocity of individual species and the average velocity of all species. So the difference is called as drift velocity. So the drift velocity is the difference between individual species velocity and the average velocity of all species. So that is given by this J i. So you can write J i this is defined as the drift velocity J i is equal to rho i into ui bar minus u bar. So we can write J i is equal to rho i into ui bar minus u bar. So instead of ui bar rho i ui bar you can write J i plus rho i u bar. So we have del rho i del t plus del dot see this u bar is a vector ui bar is a vector. So J i this is also a vector. Sometimes while writing I miss the vectors symbol I mean this is just like sometimes there is a mistake but you please whenever you write your own notes be careful about notations and write the vector notations properly. So del dot instead of rho i ui bar u this is J i plus rho i u bar is equal to r i. So this is straight forward from there to here. Next let us go to the next slide. So in the next slide what we have done is a very simple thing we have taken the term with the J to the right hand side. So del rho i del t plus del dot rho i u bar is equal to minus del dot J i that del dot J i from the left hand side we have brought to the right hand side plus r i. Now in fluid mechanics we appeal to the Newton's law of viscosity to close the system of equations. In heat transfer we refer to the Fourier's law of heat conduction. In mass transfer we will refer to the Fick's law of diffusion. So these are all equivalent constitutive behaviors because they give rise to similar mathematical paradigm. They may be physically different but mathematically they give rise to a unified concept of diffusion. See when we are talking about the Navier-Stokes equation then when we are talking about the stress we are essentially talking about diffusion of momentum. For example there is a wall and then there is a fluid that was flowing. So when the wall effect is there first when the fluid is in contact with the wall because of no slip boundary condition the velocity is 0 at that location if the wall is stationary. Then there is a velocity gradient because the fluid which is little bit away from the wall does not understand the effect of the wall directly but it understands the effect of the wall implicitly. How does it understand? How does the fluid away from the wall understands that there is a wall? It understands because the momentum disturbance propagates from the wall to the fluid that is by virtue of the fluid property called as viscosity. So viscosity, viscous effect is nothing but a transfer of momentum. So momentum disturbance. Similarly the heat in heat conduction the thermal conductivity takes care of the disturb there is a thermal disturbance how does it propagate in terms of creating a gradient in temperature. So similarly in mass transfer you have a gradient in concentration in heat transfer there is a gradient in temperature that is driving the heat from higher temperature to lower temperature. In mass transfer there is a gradient in concentration that is driving the species from higher concentration to lower concentration and that is given by the Fick's law of diffusion. So you have J i is proportional to minus grad of rho i just like q is proportional to minus grad t that very similar thing and here again we have considered that it is homogeneous isotropic everything so that it is given by constant diffusion coefficient otherwise the diffusion coefficient you can write dx, dy, dz separately along different directions. So that you can do. So remember that this J i J i is related J i is what? J i is the drift velocity. So J i so this drift velocity is the difference between the individual species velocity and average species velocity and that is attributed to diffusion. So that is so when you substitute this J i here as minus d of grad rho i and express rho i in a normalized manner. What you do is that you define a concentration C i is equal to rho i by rho just you do that. So you come up with this equation where instead of rho i you express it in terms of C i. So del del t of so in place of rho it is a very simple thing what you do is that instead of rho i you write rho into C i. So del del t of rho C i plus del of rho u C i is equal to del dot rho d del C i plus R i. So very similar to what the momentum equation energy equation looks but very importantly this u is not flow velocity. Ironically for solving many problems typically with dilute solutions we are dealing with cases when we replace this u u bar with the velocity of flow and you have to understand that the restriction is that we are assuming that the average species velocity is same as the flow velocity that means the species is just like a tracer which is moving neutrally with the flow. So the way in which the fluid is flowing it is allowing the species also to flow in the same way. So if that is the assumption then under that assumption we can write this as this u velocity replace we can replace this with the fluid flow velocity but conceptually this is not same as the fluid flow velocity. The final remark for the species conservation equation is that you have a source term. The source term may be because of various reasons I mean it may be because of chemical reaction it may also be because of other effects for example there can be transport of so if you look at this equation this is unsteady term this is the advection term this is the diffusion term this is an extra term which may be because of some reason for which the species is transported. Let us take the example of DNA. We talked about DNA transport in micro channels in one of our early lectures. So DNA has negative charge. So now if you apply an electric field because of the negative charge in the DNA the DNA will start moving. So that is not advection or diffusion that is because of transport of the charge substance by virtue of an electric field acting on the charge this is called as electromigration. So we will see later on that if you have this electromigration how to take that electromigration into consideration. So that will come in the in the form of another source term in this equation. So if you summarize the various governing equations this is the summarizing slide that we will use for all the equations. See we what are the equations that we have learnt? We have learnt the continuity, the momentum equations that is the Navier-Stokes equations, the energy equation and the species equation. See very interestingly we have prepared this chart just to give you an idea that all these equations can be cast in a general conservative form. So if you see that there is an unsteady term, there is an advection term, there is a diffusion term and there is a source term. So different equations like the continuity equation the phi is the variable per unit mass. So phi is equal to 1 because you are conserving mass, so mass by mass is 1, the diffusion coefficient is 0, the diffusion coefficient is 0, the source term is 0. And then you have the x momentum equation, you have the diffusion, this gamma is called as general diffusion coefficient. I discussed about the physical paradigm that for momentum transfer it is the viscous effect, for heat transfer it is the conduction effect, for mass transfer it is the species diffusion. So you can see that all those I mean mathematically it will be adjusted because like in the momentum equation you are solving for velocity in the energy equation, ideally you should have solved for internal energy or enthalpy but because temperature is a measurable quantity you are measuring in terms of temperature that makes the adjustment of the coefficient gamma instead of k it is k by Cp. So I mean if you cast the equations in a general form you will see that that is how that is what you will get and this is the coefficient gamma for the species diffusion. So this chart will clearly tell you that now if you transfer your philosophy from engineering or physical paradigm to a mathematical paradigm as a mathematician one would ideally like to know that if you look at this equation then what is the mathematical manner in which you can solve this equation. So if you know what is the manner in which you can solve this general form of the conservation equations then you can apply that technique to solve for the continuity momentum energy species all equations or their coupled variations in the way in which you want provided you know how to solve the generic form. So in the computational fluid dynamics that is what is commonly discussed that how do you solve conservation equations of this general form. So we will not go into that stage at this moment but we will try to consider this general form for some analytical solutions to begin with for some special cases which pertain to microfluidics. So from our next lecture we will start discussing on one of the very elementary mechanisms by which we can drive micro flows that is pressure driven microfluidics that is microfluidics where the flow can be driven by a driving pressure gradient. And we will see that how we can express the physical concept in terms of a mathematical paradigm and how we can solve those equations by using simple analytical techniques. Thank you very much.