 Now, let us look at something which Professor Gaithonday talked about yesterday in his opening remark that so called Bernoulli's equation. Yesterday we looked at it from some sort of an energy point of view. If you remember the integral energy equation when we talked about, we could see that for some special situations that integral energy equation appeared to be looking like a Bernoulli equation. So, there is another way of obtaining the Bernoulli equation which is from the Euler equation. So, simply start with this Euler equation and I am using the substantial derivative form sorry. So, substantial derivative of V which is written out explicitly as partial derivative plus that convective derivative operating on the velocity V equal to minus gradient of pressure and plus rho g as it is. Further now consider a 2D steady constant density flow with gravitational force acting in the minus y direction. So, now I am writing out this then in the two Cartesian directions. Since it is steady that partial derivative with respect to time I do not need to care. It is a constant density situation so I am dividing through by density and since the gravitational force is acting in the minus y there is no contribution in the x direction of that. So, if you employ these assumptions and write down the Cartesian forms they will come out in this particular form. So, these still are the Euler equations let us say for special situations like what these three lines say. Now, you recall the equation of a streamline which was written in terms of d x over u equal to d y over v from our kinematics discussion yesterday. So, what is done is we force this Euler equation along the streamline and if you want to do that in Cartesian coordinates what you will say is that using this relation or that is the streamline relation what I will do is I will eliminate v here in the first relation and u in the second relation. So, in particular I am what I am going to do is v will be then u times d y over d x. So, I will go ahead and substitute that for this v here in a similar manner I will eliminate u here in the second equation using the same relation. What we are then saying is that we are forcing the u's and v's to be given by the streamline relation that is the idea. And if you work out the algebra which is perhaps two or three steps at the most you will see that those two relations will end up being in this particular form. So, if you add these two guys what will happen I will have partial d d x of u squared plus v squared over 2 in two dimensional situation u squared plus v squared is nothing but the magnitude of the velocity square correct. So, that is precisely what will come out here why because you have the same addition here. So, d d x of capital v squared over 2 d x plus d d y of capital v squared over 2 d y is simply the total differential of v squared over 2 same thing here if you see it is going to be minus 1 over rho d p the total differential of pressure and then we have minus g times d y according to our assumption that the gravitational force is acting in the minus y direction. We put it together and since we are now saying that we are doing this along the streamline we must then say that the total differential of v squared over 2 that is the magnitude now equal to minus 1 over rho total differential of p minus g d y along the streamline. So, this has to be going with this equation and then you just take any particular streamline take two points 1 and 2 and you integrate this you will get your Bernoulli equation. So, this is really the fluid mechanicians point of view for the Bernoulli equation that you take the Euler equation and you integrate the Euler equation along a along the streamline and you get the Bernoulli equation. So, that way Bernoulli equation is valid only for a in this form if you write it steady inviscid constant density flow along the streamline. So, this is something that I think we must point out let me give another homework question not a problem this is the question thought yeah sorry no no there is no need to include any further simplifications like steady or unsteady or anything let it be in the most general form the way it is. So, yeah coming back to the Bernoulli this is the fluid mechanicians Bernoulli equation in the sense that it is obtained by integrating the Euler equation along a streamline yesterday we obtained this by utilizing that energy equation. So, some people do not like that some people do not like what we did yesterday in the sense that we said that there is an energy equation and which we argued to degenerate to the Bernoulli equation in some special form. So, this guy does not like it I will point out if I quickly find it see these are the things that make the subject interesting because it is fun as to see why somebody does not like something if I can find very quickly the statement here in chapter 3 of there is some example problem 3.6. So, I will write read out Bernoulli's equation which by the way he obtains the way it has been obtained just now namely in the form of equation 348 whatever it is looks very much like the energy equation developed in thermodynamics for a control volume discuss the difference between the two equations. So, that is your homework problem basically. So, he has done something and he says that yeah if you look at the energy equation for a control volume and then employ all those simplifications that we talked about yesterday it does to reduce to exactly this there is a one small paragraph though which I will write read see if you want to agree with it even though several of these assumptions are the same as those made in the derivation of the Bernoulli equation which has steady flow constant density no shear stress. We must not confuse the two equations the Bernoulli equation is derived from Newton's second law and is valid along a streamline whereas the energy equation is derived from the first law of thermodynamics and is valid between two sections in a fluid flow. The energy equation can be used across a pump to determine the horsepower required to provide a particular pressure rise the Bernoulli equation can be used along the stagnation streamline to determine the pressure at a stagnation point of point where the velocity is 0 etc. The equations are quite different and just because the energy equation degenerates to Bernoulli equation in for particular situations the two should not be used out of context this is what he says is that fine or not to think this is this is something that you cannot perhaps answer nobody can answer that confidently that this is agreeable or this is this is not agreeable. So it is really such small points which make the subject interesting like I said it is up to you to really spend time on things like these or just say whatever it is who cares. Now really it is up to you it is up to us each of us how many of you have taken the thermodynamics course the such are the points which Professor Gaithunde will like to discuss and I think that is what makes why these are worthwhile thinking is because it clarifies many other things while you are thinking such things and that helps eventually in the classroom hopefully when you go and say something to the students that you know I think that this is the way it is and you provide some justification it is not like this this fellow is not really provided that much justification so you know I am not willing to on face value accept whatever he has written but perhaps there is some point so it is really up to us to decide yes please absolutely that is that is a really good observation and that is one point and that is one point where you can say that the energy equation is definitely different than however that is absolutely correct point I will point out that if you go and read Gupta and Gupta's energy equation chapter where he actually tailors the control volume in a certain particular manner what he does is he takes a streamline or they take a streamline and choose a control volume which is a stream tube surrounding that streamline and let that stream tube shrink to a really really thin cross sectional region which then essentially is hugging the streamline so that is the way they argue that if you choose that control volume as a stream tube which is essentially yeah so then however they have a different point of view what they say is that the Bernoulli equation then as we remark yesterday by the way is giving you the conservation of mechanical energy so that that is a reasonably good way of arguing what sort of energy are we talking about if you want to utilize that energy standpoint or point of view to describe the Bernoulli however your basic observation is absolutely absolutely spot on that is correct anyway so this is for food for thought okay now if you are dealing with pure fluid flowing yes with the assumption of irrotation flow also we can derive the Bernoulli equation for minus equation absolutely only thing is in that case that there is no restriction of stream line however we have not utilized the irrotationality at all here so for a rotational situation is what in general we are talking about now rotational and inviscid is slightly difficult to imagine however in principle it can exist let us say so if there was a situation like that that it is inviscid as is assumed in the Euler equation but we have not made any assumption on the rotationality meaning that it is in general rotational I can utilize the Bernoulli equation along the stream that is the but you are right if the irrotationality is brought in then the Bernoulli equation is valid between any two points then okay now usually in fluid mechanics in particular if you are dealing with a constant density type situations what you end up doing is that you need to solve for only the velocity field and the pressure field if you are looking it from a solution of a differential equation point of view. So if you write the vector forms all you are looking for is the velocity vector as an unknown and the pressure field as an unknown so in principle you need only two equations to solve which is the continuity equation which is the mass conservation equation and one of these momentum equation let us say the one that is boxed here if we have a viscous incompressible or constant density type situation. So if you are dealing with only that pure fluid flow type situation you do not really need an energy equation unless two situations either you are dealing with a compressible flow where everything is coupled and you have to include along with your mass and momentum the energy equation as well or I can say that I can still deal with a constant density flow but there is an external heat transfer so what is that called this kind of a situation from heat transfer yeah convection if you are dealing with a convection then convection type situation then you have to include the energy equation as well. So what I am doing here is I am not dealing with the complete form of the differential equation in particular I am going to essentially neglect that viscous dissipation work which you are pointing out yesterday and the justification for doing that for me at least is that mostly mechanical engineering situations are relatively low speed we normally deal with low speed constant density flow situation where the velocity gradients in general are not very high the magnitudes of velocity gradients so d u d x d u d y d v d x d v d x etc. If you actually look at a differential expression for that viscous dissipation it involves only the squares of those velocity gradients so if the velocity gradients themselves are not too high it is reasonable on a purely physical standpoint to say that the viscous dissipation part is not really too significant and that is the assumption that I am going to utilize when I derive this so called energy equation so keep in mind that this is a simplified energy equation which seems to be perfectly fine for usual mechanical engineering type situations if you are dealing with again those aerospace type problems very high speed and so on then you cannot do this you have to include the entire business however for our purpose we are going to utilize only this my feeling is that this is the form of the energy equation that Professor Sharma will most likely use in the remaining 3 days for his CFD part so it is okay to go with this right so in particular what we are doing exactly the same as what we did for the other 2 start with the integral form use the Leibniz rule divergence theorem and convert it into a differential form however the right hand side needs to be dealt with slightly better so what are the simplifying assumptions then we are getting into we are essentially neglect the work done by the viscous forces all together in comparison with other terms and as I said for low speed typical ME type flows this is okay essentially what we are saying is that for the purpose of the energy equation we will claim that the flow is behaving as if it is in this okay we say that there is no source and we neglect the body force neglect is a strong word in general sense but the idea about neglecting something is always based upon comparison with something else please keep that in mind I think yesterday also we had pointed this out it is very easy to say that neglect this and go on but usually there is a reasonable amount of justification that goes in when we neglect something so I am saying that here neglect the body forces meaning that the other forces whatever they exist the pressure force and the viscous forces are typically larger much larger than the body force and we also will assume that the heat transfer into the control volume is by purely by conduction all that I am saying is that we have a fluid flow situation where there exist temperature gradients perhaps through this convection type process that there was some heated surface which is transferring heat to the fluid and these temperature gradients within the fluid will make sure that this q dot is by only conduction alright so little more of this vector analysis here is what I have shown is a area element with a unit outward normal n hat and I am showing a heat flux vector the idea is that I want to obtain this q dot which I am now arguing to be or I am assuming to be only by conduction okay so that q dot conduction is what I am going to call is simply now since we did this yesterday a few times I hope that you will follow this so let me see if everyone is agreeing with this I take the dot product of this q double prime vector which is the heat flux vector with the normal unit vector and I provide a negative sign to this integral what is this giving you physically into the the minus sign is basically to to say that this is going into the control volume correct because the weight is shown here that the heat flux is actually going away from the surface then you take its normal component through that dot product because that is the only component which is going to actually bring about the energy out or in whichever way you want and then that minus sign is simply because we want the heat transfer into the series that is how it is underlined okay so that minus sign remains as it is what is q double prime if you are utilizing conduction in vector form in standard form courier's law of heat conduction says that q double prime equal to minus k dT dx if you are talking about an x direction if I want to generalize it I say that a heat flux vector is simply minus k times the gradient of temperature so that is what is written out here so minus k times gradient of temperature is nothing but the heat flux vector through the courier's law of conduction expressed as a vector form I think this was probably done in the heat transfer class and if you are using any book which is reasonable for heat transfer you will get the vector form now this is something that is an area integral okay we are we are basically integrating this heat flux over the surface of the the control volume which I am now converting to a volume integral so that minus and minus anyway get rid of each other how do I do it I have not written anything here divergence theorem so that is it these are the only things that I am using all the time so if you convert the area integral into volume integral using the divergence theorem I get del dot k grad T as it is called integrated over the entire control volume fine here again the the work done I am going to calculate only through the pressure force because viscous forces have been completely eliminated and if you work out the the algebra in the same fashion we basically want the normal component of the velocity in the direction of the pressure pressure is acting inwards unit vector is outwards so that minus again comes in vn is the normal component of this velocity along that n hat because that is what is going to contribute to the work done but I want to write it as a dot product in general so that is simply v dot n and again converting that area integral into the volume integral in the same fashion I have minus of del dot this is the only work done then that I will need to incorporate because the viscous part is been discarded so that is it then you put together everything everything is a integral over the control volume now so I can combine all these guys the only assumption that I will employ here then is that it is a constant conductivity so that this k will come out and that del dot del those who are familiar with the vector calculus a little bit will end up with a del squared and that is it so for assuming a constant conductivity then we obtain a so-called differential form of the energy equation where because we have essentially neglected the viscous terms and the body forces I do not bother about the potential energy at all potential energy is basically going to come through the action of the body force which is our gravity force so since that has also been discarded the total energy is now written only in the form of the thermodynamic internal energy and the specific kinetic energy so this comes in the form of what we call a conservative form as it is we can do a few manipulations to it in particular subtract from this small e multiplied by the continuity which is written out here so again you see the first term here and the first term here this fellow when it gets subtracted from d dt of rho e what is left is rho times d dt same thing you can do for remember this we worked out on a Cartesian basis yesterday for the let me go back very quickly so exactly the same thing I am doing really only thing that now I am directly writing it in the in the vector form but that is fine if you want you can always write the Cartesian form of this that is expanding this del dot rho e v and work out the algebra so if you if you do that instead of then this conservative form as we say we can bring about our non conservative form so you can see is the same operation that we always do to bring the the non conservative from the conservative the right hand side really does not change because it is 0 here so that is another form of the energy equation now we do one more thing because in here I have the total energy which barring the potential energy is now composed of the thermodynamic internal energy and the specific kinetic energy I do not want that kinetic energy in there I want a so-called thermal energy equation I do not want this total energy so if I want to get rid of this the following manipulation is done I take the dot product of the momentum equation with the velocity vector now remember that for the purpose of this derivation we essentially said that we are neglecting the viscous forces and the flow is treated as if it is in visit further there is no body force so I can always use that Euler equation as my momentum equation for this situation which is written out here so V dot of all this if you again work out the algebra it will simply come out as rho times substantial derivative of v squared over 2 and if I subtract from here this fellow that rho substantial derivative of v squared over 2 is what I will get rid of k dash symmetry remains the same what about this guy and this guy it is again that product rule type thing ok write out this completely and you will know exactly what is happening so how will you write me I mean I have been saying this but let me at least once write this del dot PV this is the vector form what is this in Cartesian form these tell me yeah ok I will write the first of yeah exactly so if you follow this then it is very easy to do this algebra in Cartesian coordinates so you can see that this fellow will then be u times dp dx plus v times dp dy plus w times dp dx with a minus sign in it and then from the equation that we wrote here you can get rid of half of those terms finally what you will be left out with is rho times di dt equal to that is substantial derivative equal to k del square t minus p times the divergence of velocity and this is what we will call a differential thermal energy equation because it is written for only the thermodynamic internal energy under the assumptions whatever assumptions have gone in here is what is what we are going to use this for now from thermodynamics if you remember we define something called an enthalpy which is the internal energy plus p over rho or p times the specific volume if you want so this internal energy you can actually replace in terms of the enthalpy and if you do that algebra again what you will see is that another form of this thermal energy equation under the assumptions will come in the form of rho times substantial derivative of h equal to k del square t as it is and instead of this p times del dot v what you will get is a substantial derivative of pressure this is this is something that I think perhaps will be utilized in the CFD part starting from tomorrow so keep only these assumptions in mind that viscous forces are neglected and the justification for that is that low speed flows the gradients are small and typically the viscous dissipation work is involving the squares of those gradients and we do not have to bother about that no sources and no body forces and accordingly everything is simple so this is more or less the end of our differential analysis discussion so let me just summarize this please physically explain that dp by dt because I think that is a little bit difficult thing for this yeah okay so on a on a physical base see here let me actually try to first look at p times del dot v here if I will ask you does that remind you of something from thermodynamics exactly absolutely right this is nothing but the flow work pdv work if you remember from the thermodynamics which is called the flow work when it comes to the fluid mechanics first of all we write everything on a rate basis so that del dot v is essentially the rate basis equivalent of that dv type situation to see it even more clearly can I replace this del dot v from something that we did in kinematics yesterday not continuity just think about that volume yes volumetric strain rate so if I am in a position to replace this del dot v from kinematics using the volumetric strain rate then this term becomes even more clear which work are we talking about it's then exactly that pdv work as you know from thermodynamics now written on a time rate basis because then it this will involve then p times the substantial derivative of the volume okay so in that sense I will prefer explaining this term p times del dot v as the equivalent of pdv work from thermodynamics the substantial derivative of pressure is simply coming through the the manipulation of this I using the enthalpy term and the way to look at it really is that what is it it's a substantial derivative of pressure which means that if I am following a fluid particle if I simply keep monitoring the pressure of that fluid particle as I am monitoring flowing with it I will have some sort of a pressure change associated with the fluid particle which is again because of what why would a pressure which is by the way thermodynamic pressure right here now think about it this is a fluid particle this is a fluid particle that we are essentially following it goes from one location to the next and there is some pressure there is some pressure so we are saying that as it goes from here to here we are following this particle and obtaining a substantial rate of change of pressure why would this substantial rate of change of pressure come at all because of what what physical process can actually cause this the particle can compress or expand that is the only way it can experience a change of pressure inside that particle remember now that we have chosen to call this a fluid particle it is the same fluid material that we are following is the same mass exactly the same mass so unless that same mass is expanding or contracting it will not experience a change in pressure which is what is given by this substantial rate of change of pressure and again fundamentally there is no difference between then the term that is written out as p times del dot v here and the substantial rate of change of pressure because they are both related to which physical phenomenon expansion or contraction of the fluid element which is anywhere related to our volumetric strain rate okay so hopefully that clarifies the presence of this term in the energy equation does it or yes for a substantial derivative generally we say two terms okay one is a convective term and one is a local so if we say about the pressure what just we draw here is a kind of change in the pressure because of convective so how will you say that the local effect I am actually when I write this change in the area might be reflecting because of convective term kind of this thing when I write this I essentially write it as you say using both right so there could be a local pressure change or there could be a convective to me it doesn't matter because now I have decided to not bother about discussing the local and convective separately just maybe that just to give one example that how the pressure terms can be brought in a physical sense point of view actually the physical sense as I just said is really in built in the substantial derivative itself because what I am saying is that imagine that I am talking about this particle P as some sort of a balloon filled with liquid with a thin membrane and as it is going from one location to the next it is either expanding or contracting because some sort of work is getting done through the pressure sensor because it is the same mass that is contained in the fluid particle I am really talking about its own contraction and expansion which is getting resulted into a substantial change of pressure so where is my point was that where is that contraction or subtraction coming from it is simply coming because there is a compressibility involved perhaps in the flow because of which there is a volumetric strain rate because of which there is a change in the volume of that fluid particle as it goes from one location to the next because of which it experiences a change in pressure which is then expressed in terms of the substantial rate of change now it may happen that there is absolutely no local rate of change of pressure at all it may be truly only a convective rate of change meaning that at one location the pressure is always p1 at other location the pressure is always p2 as many times happens in case of pipe flow type situations for example only thing is that here now you may have to think about a compressible flow in a pipe flow because if you think of incompressible flow I still get the p1 minus p2 because of the viscous action here there is no viscosity in our derivation so where is the change in the pressure coming from it may actually be coming from the compressibility effect at all times the pressure at this location is p1 at all times the pressure at this location is p2 so that the local rate of change is 0 however when a fluid particle goes from this location to that location it will experience a change from p1 to p2 because there is a convective rate of change involved so to me it really doesn't matter what is causing the the change in the pressure in the sense that whether it is the convective rate or whether it is the local rate once I decide that I am going to talk about a fluid particle I am essentially following the same set of fluid mass from one location to the other and because of the work that is getting done on it it may expand or contract depending on how the volumetric strain rate is and that is exactly how that substantial rate of change of p comes up so in my opinion there is no difference in the physical interpretation associated with p times del dot v and substantial rate of change of p it is just that you may have to interpret the second term as being applied for one fluid particle as it goes from one location to the next and getting compressed or expanded that's about it is nothing but by the way this expression if you look at this guy here what does it remind you of thermodynamics is this a first law it is first law there is nothing different right all I am writing is essentially the first law of thermodynamics on a rate basis for a fluid particle because it's a statement written in terms of the substantial derivative that's about it there is nothing different and that is why that second term here is nothing but that flow work which you guys were pointing out that is absolutely right it's just the flow work written for a fluid flow situation which then comes in the form of that substantial rate sorry the volumetric strain rate which is expressed through this del dot v that's all so since we are mostly going to deal with mechanical engineering type flows we will specialize to constant density constant viscosity and constant thermal conductivity type flow and the conservation of mass momentum and energy then will essentially get down to these forms that have been written on the slide if you see then we have in principle four unknowns the velocity field the pressure field the temperature field and the internal energy field and it appears that we have only three equations but then that usually that i and t are related through what we what we call a equation of state really speaking or in general one can say that it is called a closure relation where I express that i in terms of a specific heat of some sort times the temperature and usually this specific heat is something that will be assumed to be known so that then the system is closed in principle what we have is four equations one two three and this four in four unknowns which are outlined out here this is what the the CFD people will start begin or begin their analysis from if they are dealing with situations like constant density constant viscosity constant thermal density flows if further the viscous dissipation is negligible and all other things because that is the form or that is the assumption rather I should say that was used to obtain this form by the way one term has been neglected here this p times del del dot v going from thermal thermal energy equation here to the energy equation why is that I will say constant density I have written out here but yeah that's fine is that fine so this is in some sense the summary of what we have talked about which is relevant for the mechanical engineer people I have written it out now in the Cartesian form as well which is nothing different from from what we know any comments though both are sameness ultimately boils down to an structured grade structured grid with a uniform grid spacing yeah many times you do see that essentially is the same thing really I agree with you although the philosophy of the discretization is different it turns out to be the same I agree what I am going to be working out as a example of that finite difference is a is a diffusion equation but with a source term in fact one of the exact solutions that we will work out of the analytical or the navier stokes I am sorry in the next session will be utilized as the model problem for the finite difference example so it's not just standard diffusion equation it is a diffusion equation with a source term from a numerical point of view doesn't make any difference really whether there is a source term or not but the analytical solution of that is then quite involved it involves that separations of variable with the series solution and so on so how many of you here are familiar with separations of variable separation of variable technique and series solutions of partial differential equation in heat transfer you must have done some correct okay so let's move on to this auxiliary discussion now that we have we have seen what the differential governing equations are what is called as a non-dimensionalization of these and for the purpose of this discussion we will simply assume that the flow is steady if the flow is unsteady it's no big deal there is usually another non-dimensional parameter that will show up but for the purpose of our discussion we can do this for study and the idea is something like what you perhaps will remember from your dimensional analysis and model studies when it comes to experiment so when we want to do experiments we actually design experiments on a lab scale and then extrapolate that data to a so-called prototype so the prototype can be a big scale or actually a very small scale can be whatever whatever we do in the lab is the lab scale and then matching certain non-dimensional parameters we say that we are going to extrapolate the results to the prototype and if you if you see these standard textbooks of undergraduate fluid mechanics you will see that there is that Buckingham spy theorem which all of us are familiar with which is one way of obtaining those non-dimensional parameters of interest for a given problem. So there is another way of doing that meaning another way of bringing out the non-dimensional parameters of interest and that is by using this so-called non-dimensionalization exercise of the governing equation. So the idea here is the following that we non-dimensionalize the quantities involved in our governing differential equations using some characteristic values for a given problem. So in particular let us say if I have x as my x dimension then I define a non-dimensional x using an x star for example which will be x divided by some characteristic length. Now this some characteristic length is going to change from problem to problem if you are talking about the flow inside a pipe which is one of the simplest things we normally take the diameter as the characteristic length if we are talking about an external flow over an airfoil or some flat plate type situation we usually take the chord length or the linear dimension along the flow length as the characteristic dimension and so on. But let us say that we have identified some characteristic dimension using which we non-dimensionalize both the x and y dimensions. In a similar manner we can identify a characteristic velocity for our problem and the velocity components let us say we are dealing with a 2D situation. We identify vc as the characteristic velocity and non-dimensionalize u and v with respect to that vc. Now this vc again for example if it is an external flow over an airfoil type situation usually we end up taking the free stream velocity which we normally call u infinity or v infinity as the characteristic velocity. On the other hand if we have flow inside a pipe what do you do mean velocity. So we come up with some sort of a mass average velocity and utilize that as the characteristic velocity. Same thing you can do for pressure and when it comes to the energy equation we need to employ a non-dimensionalization for temperature which normally is done as a ratio of differences of temperature. So there is some characteristic temperature could be that free stream temperature could be that bulk mean temperature for the internal flows and then we define some sort of a characteristic difference of temperature which will be placed in the denominator and then there is a t-tc as a standard difference or a general difference on the numerator. So this is one way of introducing the non-dimensionalization. So let us see just what happens with this. If you look at the continuity equation all that we need to do is a bunch of chain rule applications. So I have du dx for example which I simply write as d dx star and dx star dx through the use of chain rule. So this first d dx star dx star dx takes care of your original d dx partial and u if you want to go back u is simply u star times vc. So that is what I have simply substituted here. Same thing you do for the y term which is d dy star dy star dy since both x star and y star are defined in terms of the x and y coordinates and a constant length we are basically going to put a total derivative here. So looking at the way we have defined these dx star dx is going to be simply then 1 over lc. Similarly dy star dy is going to be 1 over lc. So this will be 1 over lc. This will be 1 over lc. Here I have a typo. It should be both vc. I will correct that and upload the slide. So it should have been vc. So this all comes out as vc over lc times the same form written now in terms of the start quantities. Since vc over lc is essentially a constant you will end up generating the same form but now in terms of the non-dimensional velocities and non-dimensional distance. So nothing new seems to be coming out of the non-dimensionalization of the continuity equation. Remember that this is something that we are doing only for an incompressible. When it comes to the momentum equation alright. So it looks like what I seem to have done later is I am using uc as the characteristic value of the velocity here. So perhaps I should go back here and put uc to make my life simple. I will do that and here as well. Fine. So x momentum equation we are assuming that the body force component in the x direction is 0 which is fine. We have chosen the coordinate system in such a manner. Remember that we said we will deal with a steady flow so that the time derivative is gone and you are also dealing with even though I have not written let us say constant density type situation it does not matter one way or the other but let us assume that it is a constant density. So I have divided through by the constant density situation so that mu which normally shows up here is now mu over rho which is the kinematic viscosity so it is the mu rather than mu. And that is it. All I do is the same exercise as what was done for the continuity equation. ddx is written as ddx star times dx star dx. There are two u's so accordingly there will be a u star uc, u star uc same thing for the y term. On the right hand side there will be constant density so nothing for that. ddx is again written as partial x star and x star x star x. Pressure is simply written as p star times pc. For the viscous terms now let me go back for a minute. Viscous terms normally we write in this Laplacian form or the second derivative of u and v with respect to x and y. For the purpose of illustration and simplicity in some sense I am writing this d2 u dx squared and d2 u dy squared as ddx of dudx. This is just for simplicity in the sense that then the non-dimensionalization you can just do fairly mechanically. So then if you see here mu as it is this first ddx goes as ddx star dx star dx and keep on doing it. If you end up simplifying this altogether finally what seems to end up with is the left hand side comes out as it was with everything replaced by starred quantities, the non-dimensional quantities. The right hand side shows up with a coefficient for the pressure term as the characteristic value of the pressure over rho uc squared plus this mu over uc lc as the coefficient for the viscous term. So now you can easily recognize what is happening here. You can see that this mu over uc lc is something that we know. This is the inverse of the Reynolds number. So this is 1 over Reynolds number. This fellow is not in this particular form let us say but a p minus pc if you introduce some sort of a delta p sort of a expression this is normally called as an Euler number pc over rho uc squared but that is fine. So let us just go ahead and see what happens to the y momentum. So since the x momentum procedure was more or less worked out completely here I am not sorry I am not rewriting the entire thing here. What ends up happening is for the y momentum the so called Euler number and of course this is 1 over Reynolds number will show up the way they are and additionally because we assume here that the y component of the body force is equal to minus g again choosing the coordinate system in a particular manner you end up introducing a so called Froude number which I think the civil engineer people will know. So what the idea here is that if you knew for example that these were my governing equations for constant density situation steady flow with the gravity force such that it is along a minus y direction you end up generating through this non-dimensionalization process the relevant non-dimensional parameters of interest automatically without really going through that Buckingham's pi theorem etc. So that is the only idea behind in some sense doing this non-dimensionalization from the point of view of CFD also some people can point out some advantages. So you will see that you know some people will actually prefer solving the CFD equations in a non-dimensional form such as what has been worked out let us say here. So some of you are perhaps already familiar with this idea so any input on that any comment on that why would you want to do that perhaps in a non-dimensional form? Sir the problem is generalized so that the solution can be used for any problem. So that is absolutely right the problem is in some sense generalized the way I would explain this generalization is think about a very simple situation of flow inside a channel. So if you non-dimensionalize this problem that entire channel problem is generalized so I do not want to necessarily solve channel of this width channel of that width and channel of this width separately. I can basically combine all solutions in terms of a relevant Reynolds number and as long as I am able to match the Reynolds number for whatever channel I am talking about I am in a position to immediately utilize those results. So there is some utility in that sense as well the issue with those generalized solutions though is that they will work only for simplified situations. So what I mean by simplified situations is where these characteristic lengths and the characteristic velocities are easily identifiable you can actually employ the non-dimensional approach of solving the problem. If you are trying to solve a very complicated shape let us say geometrically for example it is perhaps not that easy to figure out what could be a characteristic length in that problem. So in that case it is not really utilized I mean utility why it is not there for doing a non-dimensionalize. Similarly many times the boundary conditions can be time dependent there are problems where boundary conditions can be time dependent and that time dependency is not necessarily exactly the same from one problem to the next. So they are also utilizing a non-dimensionalization perhaps may not work. Usually such complexities are where the non-dimensional approach will not work and people will actually prefer solving the problems using the so called dimensional variable directly without really going through this. But for a simplified situation like what I just talked about channel flow or there is that classic benchmarking case that we all will use in our CFD class that so called driven cavity flow. I am very sure most of you have seen this driven cavity flow. So the driven cavity flow is another candidate for solving it with a non-dimensional form because then all cavities are mapped onto the one solution. So you do not have to do this cavity or that cavity it is the same thing. So such simplified situations are fine but the moment you are dealing with some complex geometry or for example temperature dependent thermo physical properties. You really cannot come up with a good way of non-dimensionalizing in that case because there is hardly any characteristic value that you can think about in which case you cannot really do a non-dimensionalization. But for many situations you can and that is why we are looking through this. So the specific energy equation now that we want to look at is written in terms of some specific heat could be for liquids it is just one for gases it is at constant pressure or constant volume but the way it is written is some sort of a general expression. So rho Ct times u is so Ct is basically your I let us say and for a constant density situation that p times del dot v is not there and for the purpose of non-dimensionalization I again write this d2 tdx squared as d dx of d tdx and so on. If I introduce my non-dimensionalization as was expressed here same process and what you will end up is with a product which looks like 1 over Reynolds number times. So any idea what this is called? So in the fundamental sense actually I have already written some things in a manner to bring about this Reynolds number and Prandtl number explicitly but fundamentally it will come out in the form of a Peclet number 1 over Peclet number perhaps which is simply a product of the Reynolds number and the Prandtl number. So the energy equation then is mapped in terms of the Peclet number or the product of Reynolds number and Prandtl number and you do not have to solve the dimensional form of the energy equation all the time if you are dealing with a simple situation. So that is more or less what I wanted to point out this was very fast and very quick but the idea of non-dimensionalization is to express the equations governing equations in terms of the relevant non-dimensional parameters which will essentially automatically pop out as you are doing the non-dimensionalization itself. Then when it comes to presenting some of these CFD results having solved these equations then you know exactly what are the parameters of interest and if you want to perhaps then figure out how if this parameter changes what happens to the solution then you know exactly what to do. Then here for example in the case of energy equation you can think about changing the Peclet number in a suitable manner you can change the Peclet number whichever way you want it does not matter by changing either the u or the l or any of these thermo physical properties it does not really matter and then you can study the effect of change of any of these relevant non-dimensional parameters on how the solution behave. Since people mentioned Peclet number let us clearly obviously there is a physical interpretation associated with each of these numbers correct more or less everyone knows how we want to interpret the Reynolds number how do we normally end up interpreting the Reynolds number ratio of inertia forces to viscous forces right fine. In that sense anyone wants to comment on how we want to interpret the Peclet number although it is not written that Peclet form but so in general the strength of convection to strength of diffusion is what we would like to that is absolutely right. So in this case we are talking about the diffusion and diffusion of temperature and convection of temperature or energy you can see in that sense Reynolds number is same as Peclet number is that acceptable it is the same it basically is talking about the convection of momentum versus diffusion of momentum that is all. So in principle there is no difference in my opinion between the Reynolds number from a physical point of view and the Peclet number so that is fine. In fact those who perhaps follow Professor Suhas Patankar's book is anyone following that? So there you will see that it is written essentially in terms of Peclet number and you appropriately interpret it whether it is a fluid flow problem then you say that Peclet number is essentially Reynolds number because it is a generalized diffusion coefficient in terms of which that Peclet number is written and the diffusion coefficient can be the dynamic viscosity or it can be the thermal diffusivity if you are dealing with a energy equation. So good that is all that really is to be pointed out in this non-dimensionalization form that the governing parameters of the non-dimensional parameters of interest are automatically brought about so that it helps in representing the CFD solutions later when you bring about I think that is more or less I have for the differential analysis. Again I think we have talked more about where to find this material as we were going but on the whole if one of these three needs to be followed or read I will say chapters 5 and 6 in Fox, chapter 6 in Gupta and Gupta and chapter 5 in the Potter's book deal with this material. The energy equation the way I have outlined is available exactly in the same for in Potter's book that is the only book that I see which has outlined the energy equation the way I have done it in the sense that without really including that viscous dissipation term but at least giving some idea as to what energy equation on a differential basis is. The other two books do not even talk about the differential forms of the energy equation so if you want to get some idea of how to go about it in addition to what we have discussed you can please refer to one of those.