 When you go to the derivation of transport equation, you may be saying that I am calling it as a transport. If you see books on fluid, actually in chemical engineering study, not fluid may not mostly not fluid mechanics is a separate course, but they study a course which is called as transport phenomena because this law of conservation of momentum and law of conservation of energy basically represent what we call as transport processes. So when you talk of transport phenomena in simple language, when something is transported, what do you need? There should be a driver and there should be a passenger. So here also I will emphasize later on that there is a driver and there is a passenger. So basically what is happening is transport equation is momentum and energy are transported and they are quite analog then the transport mechanisms are quite analogous. In fact, I would start this by what we call as the heated momentum transport mechanism, where basically two transport mechanism first which is called as the diffusion transport and second which is called as the adaptive transport. Now what is this diffusion transport mechanism? This transport mechanism what it involves? It involves how it occurs, what is the driver in this case in the diffusion transport? Random motion of the molecules. Random motion of the molecule is the driver for this transport mechanism. What is the driver in the adaptive transport when a flow occurs other than the random motion of the molecule? There is a bulk motion or what you call as the macroscopic motion of the fluid. The individual motion of the molecules results in a bulk motion and in fluid mechanics when we use a continuum concepts, we are not capturing this random motion of the molecule. What we are defining as point by point information is this bulk or macroscopic motion of the fluid, note that. So when the molecules in aggregate, so this random motion is always there and it gives rise to the bulk motion. And what is bulk motion? A large number of molecules are moving collectively or as aggregates. So these are the two drivers, but what it effectively results? What is the passenger? What it transfers during this process? Transport process. What is transferred? When you look into the energy transport, the molecular heat flux transports the conduction heat flux. Random motion of the molecules transfers what we call as the conduction heat transfer. Whereas if you go to the adaptive transport, then there is a bulk motion of the molecule. What it transports in heat transfer is what we call as the enthalpy transport. So there are two transports in heat transfer. Conduction heat transfer which is by the random motion of the molecules and then enthalpy transport which constitutes the advective term by the bulk motion. And convective heat transfer is a combination of this conduction heat transfer and the enthalpy transport. If you look into a good book on heat transfer such as in Cropper and David, this is the way they define convective heat transfer. There is a combination of diffusion transport, conduction heat transfer and advection transport which is enthalpy transport. When you go to the momentum transport, the random motion of the molecules results in a pressure force. You know from kinetic theory how do we define pressure by the random motion of the molecules. It also constitutes the viscous forces for the momentum transport. The advective transport as I said that in heat transfer it contributes to the enthalpy transport. In momentum transport they result into momentum influx, momentum outflux. So these are the two transport mechanisms which are involved in heat and momentum transport. So let us try to understand those concepts through some example. Let us suppose you are standing in between ice and fire and you are standing exactly in the middle. And let us assume that the heat transfer is one-dimensional in the x direction, horizontal direction. And let us assume that there is no flow. Then what is the temperature which you experience? You experience 0 plus 100 divided by 2. Why divided by 2? Because you are standing exactly in the middle. In this case you are having equal effect of ice as well as fire at 100 degree centigrade. Note that here I am assuming that the heat transfer is one dimensional and there is no bulk flow. This is what is diffusion transport. This is what is pure conduction phenomena. Now let me take an example of pure advection. Now when we talk of advection there is a bulk motion of the molecule. So there is in previous case in conduction there is no flow. When I want to have pure advection not only there is some flow but pure advection means that the flow velocities are extremely high. Now in this case when you say flow then you may think that there are two options. Flow is from the ice side or flow is from the fire side. And note that flow velocities are extremely high. So if there is a very large velocity from the ice side whether you will still experience 50 degree centigrade. If it is extremely high although it is hypothetical situation you will experience a temperature close to 0 degree centigrade. If the flow is from the fire side you experience a temperature close to 100 degree centigrade. Note that this pure advection is a hypothesis or is an assumption because hardly we get such large velocities. What happens in most of the real world cases is shown in the next slide that you have combination of what we call as advection and diffusion. The velocities are of certain magnitude. So if the flow is from the ice side let us suppose you will experience let us say I am taking 4 cases. Let us take a first case. If the flow from the ice side is 1 meter per second then you will experience a temperature less than 50 degree centigrade which is from note that conduction is always there whether there is a flow or whether there is no flow. But in previous slide in pure advection we had considered conduction heat transfer as negligible as compared to advection. Why? Because the flow rates were very very high. So random motion of the molecule is anyway there, conduction is anyway there, if there is no flow you experience a temperature of 50 degree centigrade. If the flow is from the magnitude 1 meter per second you experience a temperature less than 50 degree centigrade let us assume that you experience a temperature 40 degree centigrade. If the flow velocity increases from 1 meter to 100 meter per second you experience a temperature as low as 10 degree centigrade. This is the flow in the positive x direction if the flow is in the negative x direction that is from the fire side. Similarly, you can experience a temperature 60 degree which is when it is 1 meter per second 90 degree when it is 100 meter per second. So, this pictures may give you feel about what is molecular transport and what is advective transport. Now, let us come back to a more detailed conservation laws momentum conservation energy conservation. And finally, I am writing a single expression corresponding to those two. So, as what is law of conservation of momentum rate of change of momentum inside the control volume which constitutes the unsteady component and across the control volume where you do the balance of the momentum fluxes which constitutes the advective component. And this is equals to the momentum source and the momentum source is generated by the pressure force, viscous force as well as if there is a body force like gravitational force. In case of energy analogously you have rate of change term for internal energy or enthalpy of the fluid inside as well as across the control volume. And this is equal to heat gained by conduction and volumetric heat generation. This two can be combined and we can write a single expression where we can say rate of change of momentum or energy of the fluid inside and across the control volume is equal to momentum or heat gained by molecular transport and volumetric source term. Now, let us try to write a what I am doing here. I am showing you the derivation of x momentum equation y momentum equation and energy equation component by component. So, what is the first component of the conservation law rate of change of x momentum or for x momentum equation y momentum for y momentum and energy for energy equation. So, what is this term density here we are note that here mostly discussing on a 2D control volume because when you want to take it is easy to draw a 2D figure as compared to a 3D figure. So, what is in when you take a 2D control volume the volume of the control volume is delta x into delta y. So, this is the volume of the control volume density multiplied by volume is the mass and the mass multiplied by u velocity is the x momentum. Similarly, this is the y momentum and this is the enthalpy and when you take the derivative it is rate of change time rate of change of x momentum y momentum and internal energy or enthalpy. So, with this I am showing you the derivation of the unsteady component of this transport equation. I would like to point out that in the lecture slides which had given you there had changed the direction of this arrow because specially after discussion with professor Puranik as well as I know you had lot of confusion into this how this x momentum will be going inside the control volume from the in the vertical direction. It is easy to understand and appreciate that x momentum is going in the horizontal direction but how x momentum is transported in the vertical direction is I know it is little confusing but I will emphasize it here. So, what is small m all through this in my course will be the mass flux mass flow rate per unit area. So, there is a and this delta y is the surface area of the vertical surfaces of the control volume this the delta x is the surface area of the horizontal surface of the control volume. So, the mass flux multiplied by surface area is the mass flow rate mass flow rate for vertical surfaces it is multiplied by the u velocity as well as for the horizontal surface. So, mass flow rate multiplied by u velocity is the rate at which x momentum is entering or leaving the control volume. Note that here we want to know what is the net momentum x momentum inflow and x momentum outflow. Note that here x momentum is going in by the mass flow rate in the x direction which constitute the velocity in the x direction. In this case the velocity in the x direction which is u velocity is going inside the control volume by a mass flow rate which involves velocity in the normal direction that is a v ok. So, the u velocity is taken inside this control volume by the mass flow rate in the y direction note that ok now earlier I had shown you that we get del f by del x what is that small f here is the product of mass flux and u velocity product of mass flux into u velocity gives you x momentum flux ok. So, when you do a balance of this x momentum inflow and outflow and divide by the volume of the control volume you end up with a derivative. Note that how we are getting the derivative if you do not divide by the volume you will not get delta x delta y you cannot apply limit. So, note that all the differential equation which you get in fluid matrix and heat transfer are law of conservation laws per unit volume similarly we do for y direction. Now here you see that the v velocity is taken inside by the mass flow rate which involves the velocity in the x direction u velocity ok. So, here there is a net y momentum inflow from the left and bottom and net y momentum outflow from the top and the right face. And when you balance you get derivative of y momentum flux now when you do a balance in case of energy equation now energy equation analogously you see what is the flux analogous to momentum flux here you have what we call as mass flux multiplied by specific it multiplied by temperature this is called as enthalpy flux. Note that x momentum y momentum and enthalpy all these are scalar they are being just they are basically there is as I said that there in this transport equation there is a passenger and there is a driver. So, in all this what is the driver where does this transport going is goes based on the direction of the driver wherever driver moves that is the direction in which this passenger moves. So, who is the driver in all this three note that mass flux is the driver mass flow rate is the driver. And what is the passenger u velocity in the x momentum equation v velocity in the y momentum equation and temperature in the energy equation. And there is quite analogy there is a same transport mechanisms in this adaptive transport this occurs due to the bulk motion of the molecule. So, there is a commonality and we will use this commonality in CFD. So, in CFD what we say is that there are fluxes x momentum flux y momentum flux and enthalpy flux. Let us denote let us use one symbol for all these fluxes let us use a symbol small a. So, we have used small m small f is a general flux. So, what I am doing is that that small f I am showing in different ways small m is one of the flux mass flow rate per unit area small a is what we call as the advection flux. So, here you get a gradient of advection flux in continuity equation you get gradient of mass flux and what is this advection flux in the momentum transport is it is x momentum flux and y momentum flux and energy equation it is enthalpy flux. So, this completes the left hand side of the conservation law rate of change inside the control volume plus rate of change across this across note that in involves the fluxes inside is basically represent in terms of del by del theta this across is represented by del dot flux term del dot x momentum flux del dot y momentum flux del dot enthalpy flux when you want to write a general expression of this advection flux we can use one constant c which is one in case of momentum equation and which is specific heat in case of energy equation. Now, this when you go to the momentum and energy transport there are two transport mechanism advective which I had already talked of I had already shown you the derivation of from the advective transport now there is a diffusive transport. So, in momentum what is the diffusive transport the forces and in case of heat transfer what is the diffusive transport the conduction heat transfer. Let us start with the forces in fluid mechanics there are two types of forces there are two types of forces are the body force there are forces in fluid control volume which is directly proportional to the volume of the control volume such as gravitational force centrifugal force force due to magnetic and electric field and there are certain forces like fluxes which are proportional to surface area as I already mentioned that this viscous stresses and pressure are per unit area they are directly proportional to area. So, they are basically flux term their surface forces so this is that left hand side of this is the conversion of the left hand side of the conservation laws into the differential equation where this is what we call as unsteady term this is this represents rate of change inside the control volume this second term is the advective transport it represent rate of change across the control volume and the right hand side is surface force and body force in case of momentum equation what is the surface force how do we represent surface force the force which acts on the control volume I will separately show you the viscous force and the pressure forces and the body forces the sigma which I am using here is the viscous force pressure is Puranic I had already mentioned how do we represent the viscous stresses there are two subscript first subscript represents the plane in which this stress is acting and the second subscript represent the direction in which it is acting. So, note that in this figure second subscript is always x so it represents the direction of the stress. So, the stress is acting in the x direction in this figure and in this figure it is acting in the y direction note that if you take a fluid control volume this stresses viscous stresses will act in any arbitrary direction it is just that we are resolving into components while we are doing that because we are applying the momentum conservation laws component wise. So, from this figure we will obtain the net viscous force in the x direction this is like a free body diagram which you have studied in your let us say first class first course first year course in engineering mechanics. So, from here you can calculate net viscous force in the x direction from here you can calculate net viscous force in the y direction and divide by volume because if you want a differential then you divide by volume surface area divided by volume gives you a length scale in the denominator and you take the limit of that length scale and you get a differential term this is the key to getting the differential term. So, if you do that you get so earlier we got del dot m mass flow rate del dot a which was adduction flux here it is del dot sigma which is viscous stresses for x momentum you get del dot viscous stresses in the x direction for y momentum you get del dot sigma which is viscous stress in the y direction I will not go into the detail of the how do we obtain convert this stresses into strain rates and how we convert those strain rates into the velocity gradient this all this was taught by professor Puranic in chapter on flow kinematics and so I will just show you that in fluid mechanics stress is related with the strain rates and the strain rates using flow kinematics are converted into velocity gradients and finally I will show you the direct relationship of the viscous stresses in terms of the velocity gradients. So, what I am showing you in this slide is expression for viscous stresses in the x direction in terms of velocity gradients here note that that is CFD which we will be taking here is mostly for in fact all for incompressible flow we are not talking of CFD for compressible flow because as a mechanical engineer mostly we encounter the incompressible flow situation and most of you are a mechanical engineer. So, for a incompressible flow this relationship or the viscous forces acting on the control volume in the x direction can be simplified further what we will do is just to show you the simplification in the next slide we will take out of this two we will take 1 mu del u by del x and out of this two term we will take del v by del x and we will apply the continuity equation and we will simplify this further how I will simplify I will I will show you next to next slide figure where instead of 2 there will be 1 and instead of this two term I will show you this term only. So, what I am trying to say is that one of this two mu del u by del x and this if you do a balance and I apply the continuity equation it comes out to be 0 for incompressible flow which I will show you in the next slide. So, out of the two mu del u by del x I take 1 mu del u by del x and calculate net viscous force in the acting in the x direction. So, this 1 mu del u by del x x plus delta x minus mu del u by del x at x multiplied by if you take the if you divide by the this is the total viscous force acting I have not divided by the volume you can divide it by the volume this then this delta x delta y can cancel down. So, out of the two mu delta u by del x I am taking 1 del u by del x and out of this two term I taken one of this term. So, this del v by del x if you divide by the volume surface area divided by volume in the denominator here in this case note this mu del u by del x is in the vertical surface. So, if you divide by volume this delta y cancel down and in this case you get in the denominator delta x in this case when you do a balance and divide by volume you will get this delta x will cancel down and you will get delta y in the denominator and finally, you get del square u by del x square plus mu del square v by del y del x when you apply the limits. Now, in this you can take del y del x as common and you get del u by del x plus del v by del y and by applying continuity equation for incompressible flow this comes out to be 0. So, the previous slide can be simplified as this. So, I am not showing you two gradients I am showing you one gradient why I am doing this is that there is a particular relationship as far as this gradient is concerned this gradient is in which direction if you compare with the surface area tangential or normal this stress is acting on which surface and the surface finish on which it is acting what is the tangential direction and what is the normal direction. So, this is acting on this surface vertical surface tangential direction is y normal direction is x and what is the gradient you are having here it is a tangential gradient or a normal gradient it is a normal gradient. If you look into this stress this is a horizontal surface what is the normal direction y and here the gradient is in the y direction. So, note that viscous stress in the x torsion for an incompressible flow is expressed in terms of normal gradient of u velocity. Note this word normal because we will use this characteristics in our discretization finite volume method. Similarly, you can do for viscous forces in the y direction out of this 2 mu del v by del y if you take one of them and you take del u by del y from here and you simplify and apply the continuity equation this can be converted into that this. So, here again note that viscous stresses in the y direction are expressed in terms of normal gradient of v velocity and what is the constant of proportionality dynamic viscosity and in case of conduction. So, analogous to stresses here you have a conduction heat fluxes and how do you express conduction heat fluxes in terms of normal gradient of temperature and what is the constant of proportionality here minus k. So, normal gradient of u velocity gives viscous stresses in the x direction normal gradient of v velocity gives viscous stresses in y direction and normal gradient of temperature gives conduction heat flux. Note that all this term constitute the what we call as the diffusion phenomena which is occurring by random motion of the molecule what is the transport mechanism molecular transport. So, I can write a general expression. So, like for advection we have used one symbol small a which represented momentum flux for momentum and enthalpy flux for energy here we will use small d as a diffusion flux. And we will use a symbol gamma phi which is called as a diffusion coefficient which is dynamic viscosity for momentum and minus k for energy equation. And phi is a general variable which is u for x momentum it is like a passenger in this transport process it is u velocity for x momentum v velocity for y momentum and temperature for energy equation. And note this symbol eta what is this eta it is the coordinate direction coordinate in the normal direction surface normal note that this fluxes are acting on a surface and this is a normal direction of the coordinate system note that I am not showing you arrow here why because when if I want to draw arrow for the viscous forces acting in the x direction the arrows are in horizontal direction for viscous forces acting in the y direction the arrows all arrows are in vertical direction y direction for conduction it is in the surface normal direction. And what is this diffusion flux I would again like to correlate with small m sorry small f that small f was mass flux for mass balance advection flux for advection transport. And what is this diffusion flux small d it is viscous stresses in case of momentum equation and conduction it fluxes in case of energy equation. Note that this viscous stress consist of normal component as well as shear component the figures which are shown you on to surface the stresses where viscous normal and viscous another two phases it where shear viscous shear stresses. And this is the differential form of those diffusion terms and we do a balance of this. What I am showing you is the capital D this is the small d term multiplied by surface area this is the total d this is capital D y plus y this is capital D at y this is the capital D at x this is the capital D at x plus delta x it is a product of small d small d is this flux term multiplied by surface area. When you do a balance and divide by the volume surface area divided by volume gives you elemental length denominator and you end up with a differential term. Note that here you get double derivative for mass flow rate and adduction adduction transport you are getting first derivative why and here you are getting second derivative why because the mass flux and the momentum flux and the enthalpy flux they are related with velocities and temperature by the value not the gradient whereas the viscous stresses and the conduction heat fluxes are not related with the they are related with velocities or temperature in terms of gradients normal gradients okay. So, when the fluxes are having gradients then the differential term is a double derivative if the fluxes are having the value then the gradient is a first derivative which you get for so you get del dot m del dot a but here you get del square this phi again is u velocity in x momentum v velocity in the y momentum and temperature in energy equation. So, let us we had talked about viscous forces now let us talk about pressure force for x momentum we want pressure force in the x direction note that pressure is always compressive in nature. So, note that pressure and here I want to show you the pressure force in the x direction pressure is always compressive. So, on the two vertical faces it will act in the horizontal direction on the two horizontal surfaces it will act in the vertical direction but in this slide I want you to show you the pressure force the differential term representing the pressure force acting in the x direction per unit volume. So, I am showing you the arrows only in the x direction and note that pressure force always acts in the normal direction it is always a normal stress and it is always compressive in nature. So, there it is shown inward into the control volume but when you do a balance you want net pressure force in the x direction and when you divide by the volume as pressure is compressive but you are calculating pressure force acting in the positive x direction you end up with minus del p by del x I hope you understand why this minus sign is coming because pressure is compressive it is not tensile if it would be in tensile then you will get plus here and we are finally calculating the pressure force in the positive x direction. So, this is the reason you get negative in the differential for pressure terms similarly the pressure force in the y direction per unit volume is expressed in derivative as minus del p by del y and finally you get x momentum as this different component this is the rate of change of x momentum per unit volume when there is a x momentum inflow and outflow this is the net x momentum change across the control volume which is expressed by this this is the net viscous force acting in the x direction which whose differential term is this this is the net pressure force acting in the x direction which is expressed by this similarly this slide is for y momentum equation four terms I am showing you the fluxes also here similarly for energy equation rate of change of enthalpy or internal energy inside the control volume this is across the control volume when you do a balance this is the total heat gained by conduction this is the total heat gained by volumetric generation. So, with this I have shown you the derivation of the differential equation in fluid mechanics which with the advent of CFD is called as a name stroke equation which constitute not only momentum equation, but it also constitute continuity and energy equation earlier definition in fluid mechanics momentum equation is only called as name stroke equation, but in CFD we call this system of equation as a name stroke equation I will here again point out that this equation right now I am showing you for any incompressible flow in a conservative form because the derivation which I had shown you here is a conservative type of derivation now here this is the continuity equation this is the x momentum equation this is the y momentum equation and in the next slide there is an energy equation as I mentioned that this x momentum y momentum and energy are basically transport equations. So, there is a driver I mentioned which is the mass flow rate there is a passenger which is u velocity next momentum u velocity in the y momentum and temperature in energy equation. So, this equation is very easy to remember if you just draw your attention to this boxes I am showing you two boxes small box big box in the energy equation also there are these two boxes are there is certain commonality and we use this commonality when we want to develop programs in CFD like if you are doing a programming and if some calculation have to be performed repeatedly some set of calculation then you create functions and subroutines. So, that is the idea that here we draw commonality we understand what are the commonalities and then appropriate develop functions or subroutines and use them. So, we develop subroutines for let us say unsteady term advection term diffusion term and send appropriate general variables. So, here we are trying to draw our attention to what is the general variables. So, this small box whatever I am showing you here it is u velocity and here it is v velocity and in the next slide it is temperature and what is this is like a passenger in the transport process and what is the big box minus del p by del x in x momentum minus del p by del y in the y momentum and q bar which is the volumetric heat generation this is called as source term. So, in computational fluid dynamic this small box we write a small general variable phi which is u in case of x momentum v in y momentum and temperature in energy equation and this big box I am defining a general source term capital S phi which is minus del p by del x in x momentum minus del p by del y momentum and q bar in energy equation. This constant C is 1 in case of momentum transport and specific it in case of energy transport. This diffusion coefficient is dynamic viscosity in case of momentum transport and conductivity in case of energy transport. So, with this I had not only shown you the derivation but I also shown you the general transport equation which consist of and study term which represent rate of change inside the control volume, advection term which represent rate of change across the control volume, diffusion term which represents the net viscous forces or heat gain back conduction, source term which represents pressure force or volumetric heat generation. So, this is just what I had discussed in the previous slide. I will again emphasize the fluxes because this is it is very important for you to appreciate and understand get a feel of this fluxes because we use this fluxes very commonly in later on in finite volume method. So, I would again like to draw your attention in a different way that capital F here is a capital M, small f here is small m and here the fluxes mass flux mass flow rate per unit area and this mass flow rate is consist of velocity normal to the control volume and this is not only used in the mass conservation but it is also used in the x momentum equation y momentum equation and energy equation as driver to evoke the passenger which is u velocity v velocity and temperature in transport equation. And there is an advective flux this advective flux again constitute of the driver which is the mass flux and the passenger which is the velocity in case of momentum equation and temperature in case of energy equation and the diffusion flux. So, capital F here is a capital D small f here is a small d and the small d constitute the viscous stresses in momentum and conduction it fluxes in the energy. So, if you understand feel and have a feel about this you can do the derivation you do not need to remember the derivation. If you have got a feel of this phenomena transport process transport phenomena it constitute of two transport mechanism molecular transport advective transport and the terms are directly coming out from those transport processes ok. So, after derivation and emphasizing about the fluxes whenever you do CFD simulation whenever you do a research if I want to be little broader you want to do a research which is applicable for different types of people or let us say different types of fluids different size of the domains different let us say in a flow there is a some inlet velocity. So, here there is a concept what we call as non dimensionalization. So, we try to do our study whether it is experimental or computational in a non dimensional form. So, in the next slide I will show you the non dimensional form of this differential equations whenever you take up a problem like flow across a car or flow across a aircraft or flow in a pipe there is some characteristic velocity there is some characteristic length scale. So, because using those characteristic velocities and length scale we non dimensionalized the dimensional form and obtain the non dimensional form of the given equations and we get certain non dimensional variables like Reynolds number and Prandtl number shown here capital U here is the non dimensional velocity. In this lectures I will use small letters for the non dimensional form capital letters for the non dimensional form when we talk of properties. I talk about the property when I talk about the moving fluid mechanics the moving the fluid mechanics is expressed in terms of the fluid property in case of comparable flow density varies sorry the fluid property when we create movie we create movie for what we call as flow property velocities pressures temperature and vorticity. Now, what is fluid property this is a thermo physical property like density two types of viscosity specific thermal conductivity and thermal diffusivity. So, note that there are two types of properties and when you talk of parameters there are two types of parameters what is what you call as input parameter and second which you call as the output parameter. This non dimensional parameters like Reynolds number Weber number which comes when surface tension is there this Froude number which comes when gravitational force is there this Grashev number comes when it is a buoyancy force this are what is called as the input parameter or non dimensional governing parameter. These are what we can call as output parameter or output engineering parameters which engineers are interested this is a non dimensional wall shear stress non dimensional pressure drop non dimensional drag force non dimensional lift force non dimensional representation of the frequency of unsteady flow non dimensional rate of heat transfer and so on. So, I will stop here.