 of momentum essentially is force balance and just as we wrote energy in, energy out, energy generated equal to energy stored. Similarly, we are going to write in this way also rate at which momentum is coming in, rate at which momentum is going out, rate at which momentum is generated is equal to rate of change of momentum associated with that control volume. But Newton's law tells me we will call it by a different name it is balanced by summation of all the forces. So, instead of me saying rate of change of momentum of the control volume Newton's law says F is equal to ma that is nothing but summation of all forces. So, a force balance is basically what we are doing we are doing a force balance and equating it to momentum coming in, momentum coming out and momentum generated. So, capital M in minus capital M out plus momentum generated equal to whatever is the balance of the forces. So, we will first do the left hand side this integral that you see is nothing but can anybody interpret this for me rho there is a v in front deliberately I have not told that v rho v dot n cap dA what is that? What is so rho v dot n cap dA and there is a v here and this is integrated over the CS what is this integral when you look at this integral what do you understand is this a surface integral surface how do we know by this symbol and dA we know that it is a surface integral what is I will look at it this way rho v dot n cap dA what is rho v dot n cap dA? So, essentially what this is telling me is m mass flow what is v dot n vector. So, in fact the governing equation for mass conservation was just integral rho v dot n cap dA and that when it is evaluated over the surface becomes a negative sign for mass going in and positive sign for mass coming out because the velocity vector and the outward normal are in the same physical same direction outward for mass flow coming out and it is in the opposite direction for the dot product when mass is going in that is why this summation becomes a positive and a negative term which we write as m dot n minus m dot or whatever way you want to write equal to 0. So, same thing here the v times the associated mass flow rate. So, this quantity is the vector quantity. So, v times the mass flow rate associated with that direction. So, we are going to write this as what are the momentum which is going to come in go out for each of the control surfaces. So, this is the control volume control surfaces are these shaded I mean unshaded rectangles which are there. So, rho d y d z is the area of the surface u is the x component of the velocity v bar through the surface that is the mass flowing through that surface that mass carries a momentum in the x direction which is given by product with u. So, that is the momentum entering through the surface momentum leaving through the corresponding phase at a distance dx from it is nothing but what is given by our Taylor series expansion. So, u rho u delta x delta y delta z plus d by dx of this whole bracket which I am just going to write as rho u square for the derivative because delta x delta y delta z is just the volume. I will write the same terms for the y direction same terms for the z direction what am I doing here. How did I write this y perhaps you two are thinking same thing how did I write this term u into rho v delta x delta z is that right. What am I doing there momentum in the x direction. So, what am I multiplying with what mass flow rate in the in the y direction is multiplied with the velocity in the x direction because why are we doing this because we are deriving what are we deriving x momentum. So, we are book keeping the x momentum in all directions that is the reason why we are writing it like that these arrows are indicative of the mass flow rates direction the u which is multiplying each of these terms. So, there is a u here u here u here everything that will signify that it is the x momentum equation x component of the momentum. So, what this means I am going to ask himself ask. So, this is the mass flow through this control surface multiplied by the x component of the velocity which is contributing to the x momentum equation corresponding term of going out same thing for the z direction rho dx dy times w which represents the mass flow rate in the z direction multiplied by the u that is the x component of the velocity which is the momentum in the corresponding term for the upward. So, this when I write it is going to this become next slide. So, when I do this budget I would get the first term here here here will cancel off with these three terms and I would be left with d by dx of rho u square plus d by dy of rho u v plus d by dz of rho u w whole thing multiplied by delta x delta y delta z. So, what we have done is essentially we have opened that up this is what it is. So, this can now be expanded as u times d rho by dt plus d by dx of rho u I am keeping the product rho u in the second bracket in the third term I am taking u as one term and rho v as the second term and keeping that together. So, u times d by dy of rho v plus u times dy d z it should be of rho w this should be a dz here plus now I will take the other term of the first term here that is rho times du by dt plus rho u was here. So, rho u comes out. So, rho times u du by dx plus rho times v du by dy plus rho times w du by dz this essentially if you look at this is your continuity equation. So, this whole thing goes to 0 the first brick bracket goes to 0 and I am left with rho du by dt plus u du by dx plus v du by dy plus w du by dz everything has this delta x delta y delta z. So, we are divided through therefore, it is per unit volume basis. Now, one question I have what is this term sorry what is this term physically mean what is this term and what is this term I have four terms one is rho du by dt other is u du by dx v du by dy w du by dz what are these terms who said convective acceleration somebody said. So, u times du by dx plus v times du by dy plus w times du by dz if you just go back to your conservation of mass equation the concept of material derivative was introduced here if you look at the d rho terms. So, transport of density because of the x component of the velocity y component and the z component of the velocity. So, these first four terms became capital D by dt of rho same thing this will be your material derivative of the x component of the velocity du by dt is this clear any questions this is clear then y and z are the same we are not going now left hand side is taken care of right hand side summation of the forces what is summation of the forces what are the forces. So, for that we have to write the force balance properly we have to do with the appropriate side. So, this convention again is can get confusing to the students. So, we have to emphasize this again this sigma xx sigma xy and sigma xz first subscript denotes the direction of the normal to the plane on which the stress first subscript xxx here that denotes the direction of the normal. So, for this left phase the normal is in this direction which is the x x coordinate direction that is why the first subscript x and this is corresponding to the second subscript direction of the stress therefore this is the stress in the x direction this is the sigma in the y direction this is the sigma in the z direction acting on a surface whose normal is in the x direction. So, every phase has this. So, we have 18 such arrows which will have to be drawn all we are saying is each surface is having three stress components just to make life easy we are saying the first subscript represents the surface for first subscript represents the normal direction to the plane on which the stress acts the second subscript denotes the direction of that particular stress and the same convention will be followed for the others also. So, what we are saying is if this is the cuboid again we have sigma xx sigma yx and sigma zx it is we are not put it on anyway yeah it becomes messy. So, which of these is the normal stress which is the shear stress color color what is p delta y delta z how many say that is normal force it is a pressure force. So, pressure is compressive. So, pressure acting on the left surface is p times area and the corresponding term on the right face will be the Taylor series expansion of that normal stress acting on the left surface sigma xx times the area associated with that surface on the right hand side again it will be in the outward normal direction of sigma xx dA plus the first order term what is this violet color arrow here sigma yx why am I doing in the previous diagram I have shown sigma xz sigma xy sigma yz yy xz xy what happened to all those terms x momentum equation. So, we are concerned with the balance of forces only in the x direction. So, of these nine which are here this one this one and this one are the ones which are acting in the x direction. So, we are taking that only. So, this purple arrow is representing the shear stress in the direction like this. So, sigma yx dx dz why is the direction opposing we are writing for a velocity in the positive x direction this is going to be in the opposite direction. So, sigma yx that is the shear stress on this surface times the area associated with that surface and the corresponding Taylor series expansion of that. Why is it not in this not in this direction on top why is this why is the arrow direction reversed in all these this is outward this is outward this is inward pressure is inward whereas, here this is left direction this is right direction all this I have had a student terrible time what is that. You go to the name what is shear this motion it is when it is shear right scissor what does it do it is the opposite motion correct. So, shearing is essentially tearing apart like this so that is what acting on that solid surface on that fluid surface sorry. So, when I do the balance of these forces I will get dp by dx will come with a negative sign because it is in the negative x direction plus sigma xx which corresponds to the normal stress sigma yx which corresponds to the shear stress on this plane there will be a sigma zx which will correspond to the shear stress on the front and the back surface and then there will be a body force which would have to be multiplied by delta x delta y delta z because body force per unit volume times the total volume of the qubit. So, we take everything divide through by delta x delta y delta z yeah a function f f of x plus h is equal to what is what does calculus tell me that that is definition of limit we can write here it is what he told. So, that it is the first term of the Taylor series expansion that we are using right from mass conservation we have used this even in differential equation for heat diffusion. So, when I do the balance I am left with the left hand side which is rho du by dt which is the material derivative and the right hand side which is nothing but these things. So, everything has dx dy dz which gets cancelled off this is called as the Cauchy's equation. How do I explain body forces pressure force normal now these forces we understood reasonably well because we can feel the stresses because we know that they are gradients but how do I understand body forces one is gravity okay that is because of which centrifugal force you have you get yes electromagnetic force Coriolis force what is Coriolis force for example, if I have a pipe in which fluid is taking place like that at the same time this pipe is rotating. So, there is a linear motion and the same time rotary motion. So, the fluid particle whichever is there here it is going to have what is called from right hand from rule. So, this is the flow direction this is the Coriolis this is the direction of rotation this is the Coriolis acceleration Coriolis force would be in the opposite direction okay. So, when this body force is there all our things are going to go everywhere whatever future factor everything what we have derived they are all for no body force you take one of the body forces it will be one phd it will be one phd for any configuration because everything will change if you just take Coriolis force friction factor will change heat transfer coefficient if you take centrifugal force everything will change if you take electromagnetic force the way he said that is a phd by itself okay you just take one and its influence is significantly different I mean its influence is significantly high which is what makes my velocity distribution pressure distribution and my integral parameters friction factor and if I take temperature energy everything subsequently will change. So, that is what we need to understand. So, body force generally we are not taking when we are deriving in our regular thing but if we take every complete nature of the problem will change that is true body force by definition is per unit to volume per unit to volume this one there is a error there both are enters exit exit one is one is one is that the last one is exit rate of increase minus rate at which it enters plus rate at which it exists I will correct and actually I do not personally I do not think it matters because it is a budget see we are getting back to our if I put E out and E in still the equation is valid that should preserve a say it is a convention q coming in in thermodynamics the people have broken their heads over its q coming in is positive q going out is positive how does it matter if you follow the same convention throughout in a given problem you will always be right within the problem do not change the convention that is all okay. So, now what are we doing complete thing is over we have to just substitute for sigma xx sigma xy sigma xz we have to substitute the terms they are all in terms of velocity gradient yes here I am stating without deriving especially normal stresses shear stresses at least I think I can feel what it is but this derivation is again a lengthy one which we do not intend to do it at all here but basically one thing I can tell how we arrive at these equations what we do here is we assume that stresses are proportional to strains stresses are proportional to strains and we get set of equations if we solve those equations I will get stress in terms of what is del u by del x that is a strain that is a normal strain what is a del u by del y that is a shear strain. So, if I take stresses as proportional to strains then I will be able to relate my stresses that is sigma xx sigma xy sigma xz in terms of in terms of gradients in terms of gradients okay I am not deriving consciously I am well aware of that that will be again 3 to 4 transparencies I do not intend to do that but this I need to keep in the back of my mind that stresses are proportional to strain I am taking and taking and going ahead okay fine is that is that the right way of doing that is how I introduce viscosity you know how is viscosity introduced that is how I introduce you know so that is that is basically so this I can easily do for Newtonian fluid so stresses are proportional to strain so I get these values for normal stress shear stress and again shear stress I substitute that what was this sigma xx sigma xy sigma xz I substitute that if I substitute that I will be if I expand that I will be getting rho du dt minus del p del x plus mu del squared u okay plus del by del x of mu by 3 del dot v plus fx what is this equation called or all put together what are these 3 equations called Navier Stokes equations okay so we can write very nicely vectorial form although I do not like vectorial or tensorial I would like to expand it so that I can feel each term easily okay so but then people try to write that in vectorial form this is vectorial form but all that I want to emphasize here is this these were the equations how many equations I have and how many unknowns I have these were found by Navier and Stokes independently Navier from France and Stokes from English England so they were they found independently these equations that is why we tend to call them as Navier Stokes equations how many unknowns are there and how many equations are but then only problem is that I cannot get closed form solutions all the time why because why why can I not get the closed form solutions that easily for few restricted cases for flow through a pipe fully developed flow flow between two parallel plates all that we can get closed form solutions why not generally I solve once and keep it and use it for any situation because my equation is nonlinear my equations are nonlinear so they are not mathematically tractable they are not mathematically tractable that is that is that is what makes my life difficult okay so with this I think we are through with momentum equation but now what I would do is instead of starting energy equation I will spend time on morning what professor broached up that is turbulence we left one question if you remember rho u prime v prime now that we have Navier Stokes equation in our hand I am going to go to this is my fluid mechanics notes so I have not put this yet now that in the morning discussion it was logically leading to that so I thought it is okay that we thought that we should introduce you this so now how did we handle turbulence someone said random the moment the word random comes which mathematics comes into picture what is the recourse of maths which maths I take into account yes statistics so I take recourse of statistics and that is what I am doing so I take an average value u bar and u prime is the velocity at any instant minus u bar okay so now what is u bar what is u bar I am integrating it for for some time t here in this case I am starting from time t0 to t0 plus t that is all the difference morning professor derived it for 0 to t here I am showing it between t0 to t0 plus t but now how do I decide this capital T how do I decide this capital T can I take half a second now let us say I have an imaginary sensor hot wire anemometer is the typical sensor what we use for measuring velocity as a function of time mungage is used with this you can feel it you can discuss with him left and right in the guest house what is hot wire and all you bug him and get to know what is hot wire is a small wire which is 5 times thinner than my hair diameter 15 micrometer I will try to keep that with a wheat try to keep that wire at a constant temperature okay and that wire is one arm of a wheat stone bridge now when the flow takes place what will happen to my temperature of the wire it will dip when it dips again I will apply certain amount of power to bring it back to the same constant temperature which I had said earlier so that measure of change of power is the measure of velocity that is very simplistic very simplistic way of telling okay so the point is if I have such a thin wire whose thermal inertia is so less you can understand what I mean by thermal inertia what I mean by thermal inertia here rho vcp is very less so time constant it can respond very fast so then only I can get this velocity distribution no no no no no no no no no I am not going to take a question I have not answered my question yet so how long should I integrate this I will that hot wire will give me the velocity as a function of time how will I decide this capital T one minute because I can integrate for long time half a minute one minute two minutes three minutes four minutes five minutes six minutes how do I decide while doing experiments I will not allow mangas to answer others anyone can answer should professor said in the morning should average velocity be a function of time should average velocity be a function of time no then that means whether I take five minutes six minutes seven minutes it should not matter but if I take two shorted time two shorted time my average velocity might be erroneous I should take five minutes six minutes seven minutes fifth minute average velocity should be same as sixth minute that should be same as seventh minute that should be same as eighth minute if they are different I should integrate still longer duration what is your question okay that is not in the purview of this but still no if you have a single wire you cannot if I have a wire now this is my wire let us say this is my wire and flow is in this direction it is going to measure velocity in this direction now if I orient this in this direction it will give me it will give me what why velocity similarly if I orient normal to z direction I will get the z velocity but I need to orient all the time with single wire but there are cross wires there are triple wires all sorts of things are there let us not get into that telling is easier than doing okay fine so but in principle yes we can do now coming back average velocity is this now u is equal to u bar plus u prime what is u prime now u prime also u prime bar I am saying u prime is u minus u bar u prime bar what will happen if I integrate this fluctuating component without doing statistics let us see this and see let us not have to do integration we will do that integration anyways but if I average only the fluctuating components for long time what will happen sometimes it is plus sometimes it is minus sometimes it is plus sometimes it is minus what should happen it should get cancelled out so ultimately I should be ending up with zero that is what happens that is what happens if I integrate that you can go through this integration u bar minus u bar so u bar so if you integrate this you get again u bar and you get yeah one analogy which I just was thinking how to tell it you think of any of these prices of of course everything is increasing now price but you think of a stock price yeah base price and then it fluctuates ok so if you are a smart fellow you will not sell it immediately or you know you are lazy fellow you might just keep it for several years or months on end at the end of the day net value will be the same as what you have purchased by and large talking of normal thing of course if you look at gold or something right now it is always increasing we are not taking such a case but by and large a stable thing which minor fluctuations up and down integrated over a long period of time it will average out to be the base value so these fluctuations essentially die out with time they are not dying out the magnitudes of the fluctuations are such that the positive and negative almost cancel off to give you almost zero value that is what you before I flash the next transparency I am now going to ask you a question now will my u prime v prime bar is it going to be zero u prime v prime bar we now saw that little while ago u prime bar is it is not visible no problem this is a different okay sorry so u prime bar we saw that this is so now will u prime v prime bar be zero what does u prime means fluctuating component in the x velocity now similarly as he asked hot wire I will orient and measure the v velocity and get v prime so at every instant I am going to get u prime and also v prime if I multiply that and then average over a period of time will it become zero need not be it need not get zero okay so that is what is the what is this what was this what is this this was my what did we tell in the morning this is turbulent shears see physically also we can think like this u prime is what random motion in the x direction v prime is what random motion in the y direction if I have both of them only I can have some mixing happening some mixing happening otherwise mixing cannot happen is that right is that right that is how physically I have to understand this minus rho u prime v prime but you can always ask how how on earth I should know that I should be defining this normal stresses and all that it is like a religion professor that is terms we have one professor by name aw that a with whom I have learned all this in fact what y plus and all I taught you in the morning it is so ingrained in my mind because of professor that because he has learnt it from folding that k epsilon model itself has come from launder and spalding okay the one which I was answering in the morning so he taught us so that is why it is so universal velocity profile I do not have to remember at all because he has taught us it so well so well because he has learnt it from the originator laundering spalding himself okay so point is this shear stress is because of u prime this is shear stress now normal stresses if I take normal stresses again u prime squared bar u prime whole squared bar can it become 0 u prime whole squared bar u prime can be plus or minus why am I squaring it to make it nonzero okay so it cannot become 0 if I average so this is minus rho u prime squared bar is my normal stress so how do I understand this I think this is all fine I will leave it there so we understood no we understood what is the normal stress I mean what will happen to my u prime if I average u prime for long time what will happen u prime if I average for long time it becomes 0 but u prime v prime bar if I average u prime v prime if I average it is nonzero so similarly if I average u prime squared for long time it is nonzero okay now let me take the neighbor stokes equation okay and now what are we going to do is that what did we do by telling what what did we do in turbulent flow we divided the flow into two domains what was that one was no no no no no no no no no that is why I wanted to tell I have I have I have I have divided into two domains one is mean motion another one is fluctuating motion it is like mean motion on mean motion I have superimposed the fluctuating motion so there are two motions so one is mean motion another one is fluctuating motion okay whatever I measure with the pitot probe what is it going to measure mean motion u bar is it is going to capture it cannot capture u prime okay so now now let us take the momentum equation instead of taking continuity equation we will just take the momentum equation now why do not you put the mean component and the fluctuating component for each term come on I want all of you to derive because many of you are sleeping I want all of you to derive because I cannot derive it but I want you to derive come on let us take the first term that is let us say del u what is that yeah before doing that I have to do the small trick I will take the x momentum equation left hand side okay please write this in this form that is you have rho del u by del t plus u del u by del x plus v del u by del y plus w del u by del z let me rewrite this as del u squared by del x is that okay how how is this equal to this can anyone tell me how is the left hand side of the second equation same as the left hand side of the first equation you expand this now what do I get continuity equation you will get if you expand that and that will that will become okay please I have not put that here because it becomes too lengthy if I go on putting so that I have not put that so you get this now you average this you average the first term now you just take del u squared bar u squared upon del x now you do the Reynolds averaging that is what is this I am going to put u squared I have to replace u by u bar and u prime for you I am replacing that is my yeah for you I just said a little while ago that I am having two components one is the mean motion another one is the fluctuating so if I put u bar and u prime and I have to square it know I have to square it so if I square it and expand it what do I get del u prime squared bar by del x plus del u bar squared by del x what happened to my u prime u bar bar that is I get two three terms I have put two terms only what happens to my u prime u bar bar it is 0 it is 0 can you see that I think I have done that yeah I have done that here u prime u bar if I expand that blindly if I expand that see u prime that is u bar I can pull out pull out of the integration and I know that this integration is already 0 so u prime u bar bar has to be 0 agreed so now that is the reason I have only two terms two terms so that is del u prime squared bar by del x plus 2 u bar del u bar by del x now this is only for the first term what am I doing what am I doing I am doing what is called as famously Reynolds averaging apparently Reynolds did it so that is why it is called as Reynolds averaging okay so similarly let me just go ahead and do it for me u v the previous term was next term is del u v by del y if I do that what will I get u has to be replaced by u bar plus u prime next one is v bar plus v prime if I take the product of that u bar u bar u prime u prime bar u bar v prime bar what should happen to that 0 similarly v prime v bar u prime bar also has to become 0 so I end up with only two terms you see here I had one term extra and here I am getting one term extra because u bar v bar anyway is there so u v bar I get similarly I can do it for del u v bar so you get u bar del u bar by del z plus w bar del u bar by del z plus del u prime w prime by del z is that okay are you all with me okay now let me do it for pressure same thing for pressure pressure is I have to replace it by p bar plus p prime p prime bar has to p prime bar has to vanish so I get only del p bar by del x so if I write complete equation complete equation what I wrote earlier I have u prime bar squared by del x 2 u bar del u bar by del x plus u bar del v bar by del y plus v bar del u bar by del y this is the additional term and this is the additional term is that okay yeah which one so p prime bar is getting no problem no problem no problem first time when you are doing it will happen so when I rearrange that I am coming back to my momentum equation what were we doing to remind you this was my momentum equation I was trying to replace each term with mean motion component and the fluctuating component so that is what is called Reynolds averaging when I do that I get you see what is that I get if you just see only this this this what are what are these three terms they are what they are what first three terms left hand side del p bar by del x and this they look similar to my old momentum equation but this is representing mean motion mean motion you see every term is having what what is called bar now you see on the right hand side with minus rho what are the other additional terms which are there which were not there earlier fluctuating those are what are called as Reynolds stresses okay so the first term this is contributing for the normal stresses this is contributing for the shear stress and this is also contributing for the shear stress this was done for which equation x momentum equation I can do similarly for y and z now you see when I do that here in u and v let us take x as this y as this z as this so u v means it is going to be u prime in this direction and v prime in this direction so mixing in this plane okay so u w means this plane so u prime is in this plane in this direction and w prime is in this so that is mixing in see it is so physically it is very easy to see if I imagine this picture u prime v prime means mixing has to occur in this plane u prime w prime means mixing has to occur in this plane is that okay so that is how you can imagine now v prime w prime v prime u prime all sorts of combinations you can yes now now if I put these in complete x y and z completely if I do that what are the additional stresses I get these are what first three terms in all of these equations or what name is tox equations applied for mean flow additional terms which are coming is r sigma xx prime tau xy prime that is minus rho u prime squared minus rho u prime v prime minus rho u prime v prime w prime bar okay I am not going to answer you one question now but I am going to ask you none of you asked me which I was waiting that you will ask me but none of you asked why I write always minus rho u prime v prime does that mean that it is negative sign tau wall I wrote lamp today morning tau laminar plus tau turbulent and I wrote for tau laminar it is mu del u by del y plus tau turbulent is minus rho u prime v prime does that make my shear stress negative compared to laminar I am not going to answer this question please cook this in your minds overnight and give me the answer tomorrow morning okay you can check wikipedia whatever whatever no problems but if you as long as you can come up with the answer you come up it does not matter okay so these are these three are what are these three normal stresses in x y and z direction and these are additional shear stresses in x y z direction is that okay yeah so these are called Reynolds stresses so now you see already in laminar flow my life was so difficult for laminar flow I had four equations four unknowns at least number of unknowns were equal to number of equations so but then we said okay we cannot solve it because it is non-linear but now I have same number of equations but number of unknowns have gone terribly high so rho u I mean rho u prime squared v prime square w prime they are all very high how do I solve no solution you have to measure them morning professor was saying how do I get them these shear stresses these fluctuating components how do I get them no way I will have to measure them measure them and use in the form of models that is what has been done in k epsilon turbulence model and that is why it does not work for all class of problems some reason for flat plate and internal flows these stresses have a definite pattern so if I have to come up with a turbulence model properly for a impinging flows I have to take proper these measured stresses is that okay that is the only way I can get closure in my problem what do I mean by closure number of equations should be equal to number of unknowns the unknowns are additional how many I have got into how many I have got into 6 unknown stresses I have to only way out is I have to take hot wire animometer or laser Doppler velocimetry or particle image velocimetry and measure so that is why I keep saying experiments and numerical work go together they are not separate they are complementary to each other so they have to complement one has to feed to other okay fine so this was done what we are studying today and trying to understand today was done more than a century back okay we are studying nothing nothing nothing absolutely nothing new there is a saying also if you want a new idea read an old book it only means that there is nothing new in this world everything someone has thought it is just that we are trying to understand again okay anyway so this is Reynolds stresses in 1895 Reynolds has done all this okay I think that is what we went up to laminar shear laminar shear stress turbulent shear stress and that is how we got into this viscous layer overlap layer and all that is that okay okay I think I think I think that is about turbulent stresses so coming back now okay so now okay before doing the non-dimensionalization now I have written can I if I take a flow over a flat plate flow over a flat plate what is my conservation of mass equation let us write I want all of you to write along with me so what do I what is my momentum equation yes you can flip your pages we have derived in the morning just before lunch d rho by dt plus rho of del dot v that is del u by del x plus del v by del y plus del w by del z is equal to 0 what does this mean what does this fellow mean to us what does this physically mean stretching in x direction stretching in y direction stretching in z direction this is dilation volumetric dilation it is getting swollen or maybe compression but people call it as volumetric dilation if it is incompressible if it is incompressible it cannot it cannot it cannot dilate it cannot expand okay so only it can dilate when it is compressible that is what it means so nice isn't it so now let us take momentum equation what is my momentum x momentum equation rho rho come on one of you should tell I will have to catch someone to yes du by dt this is total derivative remember equal to minus del p by del x plus mu del squared u by del x squared plus del squared u by del y squared plus del squared u by del z squared plus mu by 3 is that right mu by 3 del of let me go back I cannot also remember yeah del by del x of it is there del by del x of if I assume property is a constant I can pull this mu by 3 out but let us not do that del by del x of mu by 3 into del dot v del dot v I don't think I need to expand that because I can write this as del dot v so that you know what is what it is del dot v plus fx equal to 0 what is this x momentum equation so similarly let me write for y momentum equation no problem d by dt is equal to minus del p by del y plus mu into del squared u by del squared sorry del squared v by del x squared plus del squared v by del y squared plus del squared v by del z squared plus del by del y of mu by 3 del dot v plus f y is equal to 0 now is that okay no I have not told sorry 0 is not right oh sorry you see I am also sleeping okay so so it is it is it is 0 has to be it has no meaning if I write it as 0 absolute nonsense okay so now if I take my fluid as incompressible if I take my fluid as incompressible what will happen what all terms will vanish what will this equation reduce to del dot v is equal to 0 is that right now what will this equation reduce to let us say there are no body forces this term is 0 because del dot v is 0 and this term is 0 and I am neglecting this I am neglecting this is that okay and if it is steady flow if it is steady flow what will happen to this term I will have to with u del u by del x plus v del u by del y plus w del u by del z I will ask you a question let us say I say that viscosity is 0 when do I say that viscosity is 0 inviscid flow okay ideal flow where there is no viscosity so what will happen to my equation what is that equation what is that equation I do not know whether I have kept it here or not but anyway what is that equation yeah what is that equation what is much used abused whatever yeah yeah yes Euler's equation you will get because that is how historically hydrodynamics came into picture where viscosity was not considered that is why D L M birds paradox what is D L M birds paradox D L M birds suffered from this paradox that is why it is called D L M birds paradox D L M birds tried to calculate force around a cylinder what is the net pressure force around a cylinder pressure net force around a cylinder he got 0 all the time he calculates he gets 0 but he knows that there is some resistance because viscous some if not viscosity was not defined that time he knew that there is a resistance for the flow but he is getting 0 so that is why it is called as D L M birds paradox because in his derivation always he neglected viscosity okay so now this equation if I neglect viscosity I will get Euler's equation which can be reduced to Bernoulli's equation okay so this is in fact we can introduce back we can relate in the class we can relate if you are teaching fluid mechanics you have you have you can relate back to Euler's and Bernoulli's equation from this okay so I think we will stop here it is time that we stop here I think I will not even go to non-dimensionalization because it was good that we took a recap so but one thing as a teacher I would believe that as a teacher I should be able to write continuity equation momentum equation energy equation in the middle of the night if I am open up also yeah yeah now let us ask ourselves now before we wind up we have three more minutes that is a good question how do I feel each of these here I did d rho by d rho by dt means what density variation with time and space because by virtue of velocity by virtue of velocity now similarly what is this this is coming from pressure force you go to its mother who is the originator this is the pressure force what about this you can call diffusion term but this is actually viscous forces this is actually viscous forces when I take viscid flow in viscid flow I make it as 0 so what are these terms yeah but if I have to write Reynolds number in Reynolds number I always use one force upon another force inertia force by viscous force inertia upon viscous force so what is this inertia force mass into acceleration no so it is inertia force what was this this was mass into acceleration no it is mass into acceleration so we should be able to see the picture what the equation is in fact if I have to recollect surely I am joking in Feynman you know Feynman Feynman was the scientist who got Nobel Prize on something I do not know on what I got but he was involved in diagnosing why challenger Shetl failed okay he what he tells in that is each term I should be imagining as a picture he tells so that is what everyone says you know if I can think through pictures I have understood that yeah actually fluid mechanics is a wonderfully visual subject unfortunately it is taught in such a boring way that the whole beauty of the subject is lost in fact I had the opportunity to take compressible flow that is where you have density changes that gas dynamics under one very famous person who had so many pictures in his lectures every lecture would have some 5 to 6 pictures and the it was probably the most enjoyable course and doing compressible fluid mechanics through the pictures lot of this incompressible fluid mechanics became clear actually I mean at that point oh this is like this how foolish we are we could not understand because we could not understand because there was no feel for those terms so we should try to make it as visual as possible and unfortunately we we are behind completing the syllabus getting the exam in time this that that but it is just teaching we are not educating you had some question how do I feel it total acceleration all that I can see it is having both convictive term and the transient I do not think I can tell anything other than that inertia I understand always everyone has inertia now we have inertia to get up to go to coffee so we have inertia in the morning to get up so that is how fluid particle is also having inertia to move so that is what is this inertia I think every term as I mean it is having a meaning okay so I think here we will stop and get started tomorrow morning but I want you fresh in the morning because we are going to start on a fresh note the energy equation okay which is going to be involved in what we have derived so far.