 So, here again we realize that while discussing so much the fluid mechanics is, I think now you will appreciate my word indispensable. So, fluid mechanics is indispensable, not my word, incorporeal. So, fluid mechanics is indispensable. So, that is the reason why we are harping so much on fluid mechanics. So, on a fast track what we will do is in the next one half an hour we will try to touch upon whatever basics we need for differential analysis before we really go to the order in which we are planning to go in the next couple of hours because now it is seeming like very invincible force to we are trying to enter. So, it is not so invincible as it appears on the face of it. So, what we are trying to do is the big picture plan is what we will do is we will give some basic definitions and interpretations of various terms and then we will go for conservation of mass, momentum and energy in the differential form. Once we derive them what we will do is we will then non-dimensionalize them. Once we non-dimensionalize them which is called as dimensional similarity. So, if we non-dimensionalize them we are going to get the non-dimensional numbers right Nusselt number Reynolds number Prandtl number we are going to see. So, what is the significance of each one and how do we handle them for laminar flows that is what we are going to see. So, this is on a generic note this is nothing to do with whether we are handling internal flow or external flow. These basic principles or fundamentals are valid for both internal and external flows. So, to that extent these are quite generic. So, that is the reason why we are going to spend time on. We could have as well gone to energy equation directly, but it is quite logical to go step by step and understand mass, momentum and then go to energy. So, now as usual there is finite control volume approach and the differential approach. So, I will not have too much time on what is what. So, if it is control volume approach we are going to take the whole control volume what happens within the control volume is a black box for us. We do not care about the velocity distributions we only get the integral parameters that is drag force and things like that in one go. So, but in a differential approach we get the whole lot of information that is complete velocity distribution, temperature distribution. Once we have all the fundamental velocity temperature distribution you can always compute the integral parameter. So, that is the reason why we are going to go to differential approach. Before we go to the differential approach what happens typically to a fluid particle while it is moving from one location to another location. What happens it undergoes we all know that it undergoes all sorts of deformations and things like that, but we have to go from simple to complex I cannot just say that it will change. So, I will have to say it can undergo translation, it can undergo linear deformation it can only stretch linearly or it can undergo rotation or it can undergo angular deformation. Whether they will occur all together or only few sometime it depends on situation to situation. So, how do we put this mathematically? If I say translation, how do I see that? This translation this has translated that is it has moved from one location to another location. How much it has moved in the x direction u delta? How much it has moved in the y direction p delta? So, what about linear deformation? I am just trying to put this deformations and translations and rotation in mathematical form. Why because what is the emphasis here is that tomorrow I mean later on if we see in the differential equation various terms I should be feeling what it is. When I see del u by del x, this fellow is stretching. When I see del u by del y, I should be feeling that it is shear stress. Who is causing the shear stress? I should be knowing that. That is the reason why we are approaching it in this form. So, linear deformation what is happening? u only when there is a gradient of u in the x direction. So, then it can stretch itself in the x direction. So, what do I understand from this 4 point slide? Del u by del x represents stretching in x direction because of the velocity del u by del x for me should not like a mathematics u varying with x. I should not think that way. I should not feel it that way. It should be stretching. It should be stretching that is how I should feel. For a mathematician del u by del x is velocity gradient of u in x direction, but not for a fluid mechanism or a heat transfer expert. It is stretching. Similarly del v by del y should be stretching in y direction. Del w by del z should be stretching in z direction. Now, I think del u by del x plus del v by del y plus del w by del z equal to 0 means what? Net stretching is that is exactly what we write for. What is that equation equal to? Continuity equation. Continuity equation for what? Is that continuity equation general continuity equation? For incompressible. So, that is how I should start feeling that is del dot v equal to 0 means net stretching is 0 that is how I should feel. Del dot v is not simply del u by del x plus del v by del y plus del w by del z. It is not so. Net stretching is 0 that is what I should feel. So, similarly you can just go ahead that is what volumetric dilation rate that is how much it is getting dilated. If it is getting stretched means what? It is getting dilated. It is getting swollen. The net swelling should be 0. See the words also are so nicely put dilation. Of course, it is a medical term mostly dilation is used for medical. Fine. So, what about linear deformation? So, what about angular deformation? That was about linear deformation. Stretching was all about linear deformation. What about angular deformation? What is this? There is a velocity gradient of v in the x direction. So, net velocity gradient in the x direction is del v by del x. Similarly, there is a velocity gradient of v in the y direction or sorry u in the y direction. So, that is del u by del y del u by del y. Now that this deformation from a to a prime is del v by del x and from b to b prime is del u by del y. It depends. These two fellows can create either angular motion or angular deformation depending on the direction. Depending on the direction. Now, can you tell me when can this become angular motion? When can this rotate? When can this rotate? When the direction is same. When the direction is same. So, that is what I do not have shown. I have not shown the figure, but if the direction is same, if the direction is same that is del v by del x and del u by del y. They are same direction. Then I am going to get the rotation. If they are opposite then I am going to get the angular deformation. So, the point is when I get the gradient del v by del x v gradient in x and u in y not in the same direction. U is in the x direction, but the gradient not in the same direction always represents either angular deformation or rotation. For us usually it is stress that is shear stress. You wrote tell tau equal to mu into del u by del y that is although velocity gradient in y that is fine but it is actually the deformation. It is actually my fluid. What is happening to my fluid particle? It is getting angularly deformed. That is the understanding I should get. Of course, I can derive vorticity by rotation, take limits and say that omega o a is del v by del x, omega o b is del u by del y and then show the rotations. Clockwise is positive. We have to take some notation. So, I am taking clockwise as positive. So, that is why the other one is getting when I take when I average it, the other one del u by del y becomes negative because it is in counter clockwise. Is that okay? Are you with me? You see del v by del x del v by del x is counter clockwise and del u by del y is. So, one is positive another one is negative based on the notation what you choose that is all it is. A few text books use clockwise as positive. Other guys use counter clockwise as positive. It does not matter. Fine. So, net rotation you can do rotation in x direction, y direction, z direction all that is fine and you go ahead and show that vorticity equal to 2 omega and del cross v all that. So, let us not get into that because you can derive this from del cross v if you expand 1 meter. Yeah, that is it. So, do you do all this in your fluid mechanics when you teach fluid mechanics? So, that way students should be knowing. Curl divergence. But still I think it is a good idea to prime them again before you take them along with you because they would have studied for exam and as usual forgotten or they will complain that the earlier teacher did not teach us. This is the normal thing. I always tell oh no I my your fluid mechanics teacher is my friend. I know I know what he has got. So, that way they cannot deceive me, but in university I do not know maybe the teacher has left then you have no option. So, it is it is a good idea to go on a faster note, give them the notes earlier the way I am going faster with you. You can give them the notes in advance ask them to read and come and you teach it you know on a faster track. So, that at least they know that they need to know that before they study they have a small quiz. And to ensure that they study it have a small quiz. Before you start convective. So, that way they are forced to read. Otherwise whatever you talk for energy equation is just going to go above the head and we will end up teaching convection as matching of the correct correlation for the correct problem. So, all this effort that you will put for energy equation derivation will be a waste. So, I think a good idea is to test them a little bit about the fluid mechanics part. Because as I told compartmentalization is the main reason for all this. They are compartmentalizing courses and students are sticking to that compartmentalization too much that linkage is not happening. In fact, anyway talk goes on even in IITs that we should teach these courses as transport phenomena. It is transport of momentum heat transfer and mass. So, we have to teach everything together because ultimately everything is same. So, but anyway we never teach that. So, why I said that because it is all compartmentalization fine. So, now shear stress means what? Shear stress means this is rotation this is rotation. Now, if it is shear stress means shear stress is what? What is shear stress? It is undergoing the same way I had shown the angular deformation. Rotating means I will have the deformations in opposite direction. The figure which I showed earlier this is what? Is this rotation? This is undergoing angular deformation. In fact, if I have to draw the particle I should extend this up like this and extend this up like this. So, then it would have undergone angular deformation. So, this angular deformation is the shear stress if I add them del u by del y plus del v by del x. So, that is my so shear stress when I am telling all the time to my student. What does that mean to me? It is actually it means that my fluid particle is deforming angularly. It is undergoing angular deformation. If whenever I come across a gradient of that velocity not in the same direction not in the same direction it means that it is shear stress. See we were taking till tau equal to del u by del y. Why we were taking tau equal to del u by del y in flow over a flat plate? Because v is was 0. So, there was no question of del v by del x. So, point is I just want to summarize that whenever a velocity gradient is there in the same direction it means to me that it is stretching. Velocity gradient in the other direction any other direction for example, u del u by del y or del u by del z it means that it is undergoing angular deformation. My fluid particle is undergoing angular deformation. That is how I need to feel my terms in my equations whenever I see not the gradients. Gradients are there. That is for maths not for a fluid machination. So, this insight we need to draw home to the students. This that is the reason I put these colorful pictures. In fact, I first learned this in U1 textbook U1. In fact, as a teacher all of us we should be reading fluid mechanics textbook by U1. Fine, he is anyway student of Tumoshenko, fine. So, now coming back to that is about the gradients. So, I will not spend time I would request you to read this in great detail when you come back. So, I think on a quick note we can do conservation of mass. I think all of us can just like that do conservation of mass. So, how do I do conservation of mass? Let us not worry about these transparencies. Let us derive all of us. I know we are all hungry and we are not able to focus much, but still let us force ourselves to utilize these 15 minutes. Let us get back to that E dot business. What was the equation we used there? Instead of E I am just going to use m dot. What do I do? m dot in, m dot in minus m dot out plus m dot g is equal to m dot s t. Is that right? So, now let us write a cuboid. I want all of you to derive along with me. This is delta x delta y delta z and as usual x y z with velocities u v w. Is that okay? So, now what is the mass flow rate which is getting in here on this surface? Rho into velocity into area. Rho into area is here. Yes delta y delta z into velocity which velocity u. Now what is the velocity which is getting out all u same m dot plus if I have to tell m dot x plus del m dot x by del x plus del into del x plus del squared. So, I always remove the higher order terms. I do not think I need to write the Taylor's expansion. So, I write rho u delta y delta z plus of plus of del by del x of rho u delta x delta y. Is that right? Come on, write for me in the y direction and z direction. What will be in the y direction? Yes, we will go from left hand side. Yes, Mr. Samath, why do not you help me? Rho v it will be. Rho v delta z. Very good. Then what is getting out? The getting out is rho v delta x delta z plus del of del of rho v by del y into delta y delta x delta z. My figure is going to become clumsy, but let me make it clumsy no problem. Let me get it in the z direction. Let me catch youngsters. Who is teaching for the first time? Someone said who is teaching for the first time. I forget his name. I think you are teaching. Your name Nagraj, right? Yeah, okay Mr. Arun. Come on, help me. I am hungry. I am not able to think. Yeah, rho rho delta x delta y into w that is getting in. Getting out rho delta x delta y w plus del by del z of rho w delta z delta x delta y. So if I just apply m dot in and m dot out, can you tell me what are the terms which we will get out and what am I left out with? All in terms and in the out terms the first these three terms are going to get cancelled. So what am I left out with? Minus. I am not going to, okay, we will write delta x delta y delta z as common minus sign minus minus also let me get it out. What do I get? Del del rho u by del x plus del rho v by del y plus del rho w by del z. Let me for a minute, let me keep m dot g as m dot g only, okay or for a minute I can neglect that. Let me remove that m dot g, okay. So there is no generation term let us say. What is m dot s t? dm by dt, right? So that is dm by dt means what? Yes, del rho by del t delta x delta y delta z. So delta x delta y delta z is going to cancel out. What am I left out with? What am I left out with? Let me throw out delta x delta y delta z. So I get del rho by del t plus I am pushing left hand side terms to right hand side. What am I left out with? Del rho u by del x plus del rho v by del y plus del rho w by del z is equal to 0 is equal to 0. Now let me expand this. Let me expand this. So del rho by del t what will be the expansion of del rho u by del x? Let me write this as u del rho by del x and the next term as rho del u by del x little further because I just want to say one step, okay. So plus plus v del rho by del y next rho is common. So del v by del y plus w del rho by del z plus del w by del z is equal to 0. So what is this first term? First four terms what are what is that called? This sounds quite familiar. Total derivative have you heard about? It is total derivative that is yes this is the local term varied right. This is the local term and this is the convective term because this is existing because of what? Why is it called convective? Presence of u v w otherwise it could not have convected, okay. So this is d rho by d t plus rho of del dot v is equal to 0. But then we write for continuity equation del u by del x plus del v by del y plus del w by del z equal to 0. Is there any anomaly there? Yes this equation what we have derived is a generic one. This is the generic one, okay. For incompressible flows rho is neither going to change with time nor with space. So that means that my d rho by d t is going to be 0 that implies del u by del x plus del v by del y plus del w by del z is equal to 0, okay. Total derivative all of you are familiar with total derivative material derivative. Anybody who is unfamiliar with that? Looks like everyone is unfamiliar. So any parameter if I take let me take because we can feel with temperature very easily. Let me expand d t d t can anyone help me in expanding d t d t? It is changing any parameter it can be density in the above case it was density. Now I have just changed it to temperature. Why because I can explain it a little easier and we can feel it a little easier. Del t now temperature has to change with time del t by del t plus u del t by del x plus v del t by del y plus w del t by del z. How do I explain this to my student? I will take the recourse of Anderson's example del t by del t we will wait for a minute del t by del x. I am entering an ac room from outside. I am entering means what is my u that is velocity. I am moving I am a fluid particle. I am entering the ac room that is I am carrying a velocity I am moving with velocity u. The moment I enter I undergo a temperature gradient with respect to space that is x del t by del x is actually occurring because of my u because of my convection I am getting convicted from outside world to ac room. Now what is this del t del t while I entered some naughty guy threw on to me a nice ball. So all of a sudden I am going to experience a temperature dip because of heating some unsteady transient thing has occurred. It can be a flame it can be a fire or it can be just some ball of high temperature or low temperature as heat. So that this only represents the total derivative only represents that things can change not only with time but also can change with space. But who is the initiator for space velocity because you see at least for me when I came across with this material derivative I was always in plus 2 we always study acceleration means velocity change with time. So the notion that I can have an acceleration because of space variation also is little difficult to absorb. So that is the reason this material derivative is having so much importance. Actually just to interrupt in this textbook by Fox and McDonald there is a very nice beautifully explained section on fluid particle acceleration that was the way I understood for the first time that acceleration of a fluid particle basically they will show the derivation says that you know a fluid particle is different from a solid primarily because of the change in the relative position between the two particles because of the bulk motion. So I have two particles it not only moves because of change in because of the time it also moves because of this change in the relative position because of x, y, z component of the velocities. Very nicely explained one page section I think all of us should read that I understood it like that and one classic example which all of us give is nozzle steady flow in a nozzle fluid is accelerating nozzle aim is to accelerate a fluid but thermodynamics we have studied steady flow energy equation dE by dt equal to 0 q dot 0 w dot 0 and we just get h plus v square is a constant. So what we are saying even though the flow is steady velocity change is there why because we are talking of a fluid not a solid and that whole concept comes into this material derivative business. So it is because of the change in the variable with respect to time plus the contribution of the changes in that variable because of the flow because of the velocity components that is why you have u times it is like the I do not know how to say this you know it is coupled it is always integral with the velocity the change of that variable with respect to that position is married with that corresponding velocity u times d variable by dx plus v times d variable by dy plus w times d variable by dz plus of course the contribution with respect to time. So in case of steady flow nozzle example dv by dt is 0 but these terms first three terms convective terms which we have called that is non-zero therefore there is an acceleration. So what is the big deal about this momentum equation you know as students I think it is very very scary for a student when you tell Navier-Stokes equation I was very scared I did not even know what it was. So momentum equation Navier-Stokes equation all these are from a student point of view unless it is the physical insight is given it becomes a mere exercise in mathematics for the student okay some differential equation you put something in something is coming out Taylor series cancel the terms left hand side equal to right hand side and you get something but what is the whole idea of this equation. So essentially momentum equation or Navier-Stokes equation is nothing but a balance of forces okay so balance of forces means what unlike a solid fluid is having a little bit slightly different kind of forces that we are going to deal with. So we have pressure force we have a normal force we have shear stress which is something which is different for a fluid which is what defines a fluid in fact what is the definition of a fluid substance which is capable of flowing under it cannot resist deformation under application of which force shear force so that is what is the difference between a fluid and a solid if I have a solid this is a solid it was something nothing is going to happen on the face of it but if it is a fluid the top layer is going to deform from its initial position to a new position and it will come back to its original position once the force is removed okay so that distinguishes fluid from a solid and because of this distinction lot of things are different from a solid when we deal with fluid mechanics. So once that concept is ingrained we say when a fluid element what is a fluid element you know there are two ways of doing this one is this Eulerian frame and one is Lagrangian frame everybody is familiar with Eulerian Lagrangian frame of references anybody who is hearing it for the first time please feel free it is Eulerian what is it Eulerian any other student asks sir what is Eulerian Lagrangian can you explain fixed in space which one Eulerian frame of reference what is Lagrangian okay how would you explain this with an example to a student so always our examples are engineering right smoke emitted yeah that is it you sit on the top of the train go around Lagrangian you stay on the platform see the number of trains crossing by that is finally so you fix the region in space smoke example is also perfectly fine but to a layman to a non-engineering non-technical person if you want to introduce Eulerian Lagrangian issues then I think it is easy it is not very difficult people understand it still you know such everyday examples are always useful so when a fluid particle is subjected to a set of forces what could the type of forces be pressure shear normal which is acting perpendicular to the surface one question which I have I always had got it cleared very late in my life what is the difference between a pressure force and a normal force we both of them are acting normal to the surface what is the definition of a pressure what is definition of pressure force per unit area and what is the unit area normal so what is the difference between the pressure force and a normal force all of you should know it because people will ask you then what is the body force what is body force pressure force normal force I think we have all of us use this very loosely what is pressure force is due to momentum transfer why I keep I keep a glass of water it is exerting a pressure for the bottom of the glass where is momentum transfer occurring here I am I am talking of a fluid element what is the difference between a pressure force and normal I have a dam dam concept you know pressure is at the bottom bottom of the dam it is going to be higher so the base is brought we all we all know this is the reason question in physics so pressure increases with depth all those things physically as much as we understand pressure that much we do not understand normal force what is the difference we will leave this discussion a little bit let the producer come we will have it a little bit more interactive so we have shear force pressure force body force due to the weight of the element normal force okay and gravity force of course and if it is rotating fluid element then you have centrifugal and coriolis for there is what do you understand by pressure before going to normal when does pressure occur how do I understand pressure or let us go to extreme when is pressure zero why no problem atmospheric pressure when does it become zero vacuum in vacuum what is not there molecules are not so what is creating because of the presence of molecules the pressure the intermolecular collisions so one molecule is hitting the other molecule because of which there is a pressure that's how I understand that's how I understand so if no molecule is there then naturally pressure has to become there is no collision that is how I understand pressure normal force normal force let's now take normal force here in this context pressure we have understood now that's the thermodynamic pressure what we feel here in this context normal force we are talking about normal force so we are talking about normal force this normal force who is creating a couple of people had used the word it's a reaction force who is creating this force yes stress that's that's their stress but who is creating this stress material yes yes molecules again right but then in the morning we spent quite a bit of time on gradients someone should be creating this normal force among see basically fluid flow is all because of what pressure gradients number one and because of which it is going to create my velocity gradients so these velocity gradients some of those velocity gradients are going to create normal stresses and some are going to create so those stresses which create normal force are normal stresses basically we need to or I appreciate always like this it's the gradients you are not gotten into turbulent flow luckily okay but of course there is nothing specific to laminar only these can be used for turbulent as well but stresses are all created because of gradients other than pressure is that right agreed or not is that right what are those we will understand little later on I mean we are going to derive but first we need to physically feel them you are not with me 2 o'clock slumber is still 2 o'clock slumber yeah yes yes pressure force is also one kind of normal force there is no doubt about that but in fact in few textbooks they cause the cause of the pressure force is very different from the cause of normal force so when there is no deformation or when there is no stretching yeah velocity gradients even then there will be a pressure force as long as there is a fluid that is very rightly put right yeah so tumbler with water there is no motion perfectly stand still there will be a pressure force okay now when there is a motion in addition to the pressure force then we have normal force that is why you know when we look back now as I am talking to you I am realizing that is a pressure was introduced under the head fluid statics yes and then what kind of force will act on the fine fine no problem no problem are there velocity gradients there no so they are all static forces that static force is essentially because of no that is pressure force atmospheric pressure force see what Roger is trying to emphasis is correct yeah yeah yeah yeah but is it causing a velocity gradient at the surface is there a du by dx du by dy du by dz dv by dx etcetera is that there no no if it is not there if there is no motion and associated deformation of the fluid then I cannot call it a normal force it is still a pressure all other forces which are created other than pressure are essentially because of one of these flow deformation my fluid particle is getting default so that is creating gradients few gradients which are cross that is u in y and z u in x itself will create v in y will create normal w in z will create normal basically what is trying to emphasize is when flow is there only there can be velocity gradients when flow is there only my fluid can undergo deformation it can be whichever type of deformation no problem but there will be deformation right in the absence of velocity all together the only normal force you can imagine is only force you can imagine is pressure force is that ok did we feel ok see what did we say what what are we saying there are forces on the right hand side of this equation we are having forces so we classify these forces as pressure forces normal forces and forces because of shear stresses that is shear forces we can call and these are all all these forces whatever I I listed out all these three what are this what is common in these three are surface forces which are acting on surface that is per unit area I am always worried about it is not body force which is per unit volume I am not getting into body force yet so in surface forces we have pressure forces forces due to normal stresses I should be telling that way and forces due to shear stresses pressure we all understand pressure can exist even when there is no motion p equal to rho g h v i routinely use it because that is fluid statics ok now what is normal force I mean which are the forces which are causing normal force because of the normal stresses which what is the normal stress who is creating my normal stress velocity gradient right can you physically feel me I told again in mathematical terms let me rephrase it normal stresses are created by what angular deformation linear deformations or what linear deformation I am just stretching I am pulling you in x direction v in y direction z in z direction it is basically normal stresses because of stretching it can be stretching or it can be compression also ok but the shear stress is because of angular deformation is that ok all these stresses are coming by by virtue of their flow as professor emphasized before we went for lunch it is all because of flow if you recollect in solid mechanics we never worried about rates here only we are worried about rates strain rates there we only said strain here also we use the word strain but in the back of the mind we know that that is strain rate strain rate so that is the basic difference between solid mechanics and fluid mechanics I hope I have reached it