 I begin with the third lecture called laws of convection. In a way, this is the most fundamental lecture along with the next one, because I will first be stating the fundamental laws that govern our subject of convective heat and mass transfer, but then go on essentially to enunciate the laws governing fluid motion, which are then also the tractable forms of governing equations are also called Navier-Stokes equation. So, I will end with complete derivation of the Navier-Stokes equation. In a way, this lecture is very very fundamental, because fluid motion is so very important in convective heat transfer. So, what are the fundamental laws? The first is the law of conservation of mass, which is responsible for transport of mass. This Newton's second law of motion force is equal to mass into acceleration is essentially responsible for transport of momentum and the first law of thermodynamics is really responsible for transport of energy. The first two laws namely the law of conservation of mass and Newton's second law of motion define fluid motion completely and the third one defines the transport of energy. So, today I am going to consider only the first two laws that define fluid motion completely by applying these laws to infinity similarly small control volume located in a moving fluid. The first question that arises is that how should we look at fluid and its motion? The fluid itself is viewed in two ways. The first one is called the particle approach and the second one is the called the continuum approach. In the particle approach, the fluid is assumed to consist of particles, molecules, atoms and the laws are applied to study motion of each particle. The fluid motion that we see and feel is then described by statistically average motion of a group of particles, very, very important. In this approach, the motion of each particle is studied, but the fluid motion that we speak of is described by statistically average motion of a group of particles. For most engineering applications and also environmental applications, this approach is too cumbersome simply because the significant dimensions of our flow say radius of a pipe could be anywhere from say 5 millimeters to anything up to 2 meters diameter is conceivable or a boundary layer thickness. So, these are the significant dimensions of the flow and these tend to be considerably bigger than the mean free path length between molecules, which means in a given pipe there will simply billions and billions and trillions of particles to track even in a simple 1 centimeter diameter pipe. To appreciate this, just consider something that you already know. For example, the Avogadro's number specifies that at normal temperature of 25 degree centigrade and pressure of 1 atmosphere, a gas will contain 6 into 10 raise to 26 molecules per kilo mole. So, for example, air, which has a molecular weight of say about 29 and density of about 1, it can easily be deduced that there will be 2 into 10 raise to 16 molecules per millimeter cube. You can very well imagine therefore that the mean free path length between molecules must be very very small and there will be simply far too many molecules to track even in a simple case of a flow in 1 centimeter diameter pipe. Continuum approach was devised essentially to overcome this difficulty. In this approach, we assume that the statistical averaging is already performed and we consider elements of fluids to which fundamental laws are applied. The elements of fluid are also sometimes called control volumes and these control volumes although infinitesimally small actually contain large number of particles. They are simply very very tiny very very tiny or infinitesimally small and since we already assumed that the fluid is viewed such that averaging has been taken place, obviously some information is always lost in any averaging and therefore that information must be recovered and it is this information recovery is done by invoking some additional auxiliary laws along with the fundamental laws and the three fundamental the three auxiliary laws that we that are normally invoked are firstly the stokes stress and rate of strain law which defines the fluid viscosity mu. The Fourier's law of heat conduction which is familiar to you defines the thermal conductivity k and Fick's law of mass diffusion which defines the mass diffusivity and you will recall in my first lecture I said all these are molecular phenomena. Essentially, they are molecular simply because they represent the lost information as a result of averaging at the molecular levels. The transfer properties mu k and diffusivity are typically determined from experiments although theories such as kinetic theory of gases etcetera are available to be able to determine their magnitudes. As we move towards applications involving very small dimensions called micro scale heat transfer or nano scale heat transfer a quantity called Knudsen number is a very important one. Knudsen number is simply the ratio of mean free path length between molecules divided by the characteristic flow dimension l small l divided by capital L and we say that the continuum approach is valid and is also experimentally verified when Knudsen number is less than 10 raise to minus 4. Today, we have micro channels whose dimensions could be of the order of microns or 10th or 100th of microns and clearly the dimension l itself would be comparable to the mean free path length l and therefore in such a case the particle approach would become necessary. In this course of course, we are always going to assume continuum approach by saying that we would be considering situations in which capital L is much greater than the mean free path length. I said in the continuum approach the fundamental laws are applied to a fluid element we call control volume. So, let us just define the control volume. The control volume is a region in space across the boundaries of which matter energy and momentum may flow and it is a region within which source or sink of the same quantities may prevail further it is a region on which external forces may act. So, imagine a fluid inside a flowing flow very tiny fluid element then we are saying that in this fluid element energy will flow in and flow out mass will flow in flow out momentum will flow in and flow out and it is an element on which external forces may act. In general, a C v can be large or infinitesimally small however consistent with the idea of a differential in a continuum and infinitesimally small C v is considered. Remember the idea of continuum is also invoked in mathematics. In fact, when we define a derivative we say difference between difference in the values of two variables or value of a variable at two distinct points is written as d y by d x when d x goes to 0. In other words when d x becomes very very small or the two points are brought very very close to each other a derivative is defined. The notion here is very similar. We want to represent our laws in terms of differential equations and so in order to be consistent with the idea of a differential in a continuum we shall consider infinitesimally small control volume. The C v is located within the moving fluid again there are two approaches first of them is called the Lagrangian approach and the second one is called the Eulerian approach. In the Lagrangian approach the C v is considered to be moving with the fluid as a whole. In the Eulerian approach on the other hand if I have a moving fluid the element that I consider is fixed in space. In a way the material of the element is changing continuously because some fluid is coming in and going out and the fluid element contains different materials at different times but the location of the element is fixed in space. In the Lagrangian approach on the other hand the material of the element does not change but the element itself moves with the fluid. In the Eulerian approach the C v is assumed to be fixed in space and the fluid is assumed to flow through and past the C v. When is Lagrangian approach invoked? It is invoked in only study of certain types of unsteady flows such as free surface waves for example but this is not something of interest to us and therefore we shall be preferring Eulerian approach to application of fundamental laws to control volumes. There is also one more advantage when we consider Eulerian approach. As I said in the Eulerian approach the fluid element is fixed in space although located inside the moving fluid it is fixed in space. Now recall of course that whenever we do experimental measurements using a pitot tube or hot wire for velocity or laser Doppler anometer and so on so forth they are all fixed instruments in space and the fluid moves past them and the fluid moves past them. As a result there is a great advantage in the sense that what is measured by the instruments can now be compared directly with the solution of the differential equations that the Eulerian approach defines and as a result we shall always prefer throughout this course to look at the fluid as a continuum and apply fundamental laws to the control volume within the Eulerian approach. Now there is still one more matter to be settled. The fundamental laws by themselves are pretty useless in many ways. Firstly the fundamental laws apply only to total flows of mass momentum energy. They define total flows both in magnitude as well as in direction. So there are two properties involved magnitude of the total flow and the direction of the total flow. To the extent direction is involved all these flows are vectors like this say total vector. Now the trouble is that when the fluid flows through a duct or flows over a turbine blade or something like that at every point in the flow I simply do not know the direction of the total vector a priori. I am sure you are able to imagine this for the in the entrance region of a duct where the boundary layers grow close to the wall the direction of the flow will be pointing towards the axis of the tube inside the boundary layer. But as I move away from the wall it will be more or less aligned to the and parallel to the axis of the flow and in fact at the axis symmetry the flow will be absolutely parallel to the axis of symmetry. But I simply do not know a priori the direction that the total vector will make with the fixed coordinate system x, y, z. Now do I know its magnitude? So what is done then that in the general problem of convection since we do not know magnitude and direction a priori we settle the matter in this way. We say the problem of ignorance of direction is circumvented by resolving the total vector in three directions x 1, x 2 and x 3 as I have shown here. So the total velocity vector u would be resolved in terms of velocity vector u 1 in the direction x 1, u 2 in the direction x 2 and u 3 in the direction x 3. Same thing would apply to forces, same thing would apply to all other fluxes like heat flux and mesh flux. So we shall always resolve all the vector quantities in three directions that define the space. Of course we have increased our work but at least we have made the problem tractable in the sense that now we need to be worry only about how to determine their magnitudes in three directions so that the total vector could always be constructed knowing the three vector sub vectors in three different directions. Let us consider the very first law of conservation of mass and here is a control volume. The verbal statement of the law is very simple. It says the rate of accumulation of mass within a control volume would equal the rate of mass in minus rate of mass out. Very simple to understand there is no difficulty at all. So how do we represent mathematically the accumulation of mass, the rate of accumulation of mass m dot ac? Remember the mass of the element would be simply rho m multiplied by its volume, rho m is the bulk fluid or mixture density multiplied by the volume which is delta x 1 delta x 2 delta x 3 divided by dt. So partial d by dt rho m delta v would be the rate of accumulation. Well at surface x equal to x 1 the mass rate of mass flow in will be rho m into u 1 which is the mass flux multiplied by the area delta a 1 which is delta x 2 multiplied by delta x 3. So that is what I have written here. That would be the rate of mass in at surface x equal to x 1 likewise there will be rho m u 2 delta a 2 which is the mass coming in from this side and there will be rho m u 3 delta a 3 which is the mass coming in from the back side at x 3 equal to constant surface. m dot out would likewise be rho m u 1 delta a 1 going out at x 1 plus delta x 1 surface, rho m u 2 delta a 2 at x 2 plus delta x 2 would be the mass going out of the x 2 plus delta x 2 surface and likewise mass will come out at the front surface in the x 3 direction. Incidentally notice that the directions of x 1 x 2 x 3 satisfy the right hand screw rule which means if I start with x 1 and move towards x 2 a right hand screw would make take me forward in x 3 direction. So that is what is implied here if I turn x 1 to x 2 I will move in x 3 direction. This is very very important to remember in all our subsequent slides that the coordinate directions are so chosen that they obey the right hand screw rule in cyclic manner. So having written out the replace the verbal statement by in its mathematical form we divide each term here by volume delta v which is nothing but delta x 1 delta x 2 and delta x 3 and remember that delta a 1 is delta x 2 delta x 3 delta a 2 is equal to delta x 1 delta x 3 and delta a 3 is delta x 2 delta x 1. Then the law of conservation of mass would read something like this rho m u 1 x 1 which is the mass flux in minus rho m u 1 x 1 x 1 mass flux out from in the x 1 direction divided by delta x 1 similarly in y direction and similarly in z direction and if I now let each of these delta x 1 delta x 2 delta x 3 go to 0 that means making the control volume extremely small or infinitesimally small and then each of these expressions would simply get converted to a partial derivative. This expression for example would be minus d rho m u 1 by dx 1 minus d rho m u 2 by dx 2 minus d rho m u 3 by dx 3 which I have transformed on the left hand side to read is false. So, essentially then d rho m by dt equals plus d rho m by u 1 dx 1 plus d rho m u 2 by dx 2 plus d rho m by d u 3 dx 3 equal to 0 is the statement of the law of conservation of mass in differential form. It is also called a conservative form of equation simply because this is how it is derived. It has conserved all the fluxes in all directions and non conservative form can be derived by mathematical manipulation. For example, I can treat this as differentiation of a product then you will see this will become u 1 d rho m by dx 1 and u 1 d rho m by dx 1 which I have written sorry rho m d u 1 by dx 1 which is written here on the right hand side. So, you will see I will get d rho m by dt plus u 1 d rho m by dx 1 plus u 2 d rho m by dx 2 plus u 3 d rho m by dx 3 equal to minus d rho m now minus rho m into d u 1 by dx 1 d u 2 by dx 2 d u 3 by dx 3. This way of writing is called the non conservative form of writing the law of conservation of mass and you will readily recognize that this left hand side is nothing but what we call the total derivative d rho m by dt equal to minus rho m and what is this is simply divergence of velocity vector v and therefore, written as del dot v. So, d rho m by dt equal to minus rho m del dot v is a non conservative form of the law of conservation of mass. I now turn to the second law of motion which as I said is concerned with the transport of momentum the statement goes something like this for a given direction x 1, x 2 or x 3 the rate of accumulation of momentum equals rate of momentum in minus rate of momentum out plus some of the forces acting on the c v in the same direction. Remember Newton's second law of motion says that the force equal to mass into acceleration, but implies that the force and acceleration are in the same direction. What are the forces that act on a control volume firstly there are shearing stresses shearing forces tau 2 1 is a shearing force acting in x 1 direction and likewise tau 2 1 would be at x 2 equal to constant surface would act in this direction tau 3 1 likewise acts at x 3 plus delta x 3 surface and tau 3 1 also acts in the other direction. The difference in this traces provide the shear of the control model sigma is the normal stress it is tensile and therefore, points outwards from all surfaces sigma 1 here sigma 2 here and sigma 3 on the back and likewise sigma 1 here sigma 3 here and sigma 2 there at the other surfaces. In addition there could well be body forces due to buoyancy or coriolis force or a centrifugal force or an electromagnetic force a fluid can experience variety of forces body forces in particular. So, these are all the forces, but the important thing is we can consider the Newton's law of motion in one direction at a time. So, therefore, since there are three direction we shall have three equations as I mentioned here. Let us write each term in mathematical form momentum accumulation well simply what is the mass of the control volume is rho m into delta v and since I am considering direction 1 I must multiply that by u 1 to get mass into velocities momentum and the rate of change of that is the accumulation. So, d by d t of rho m delta v u 1 is the accumulation of momentum what about momentum in rate of momentum in would be rho m u 1 into delta a 1 is the mass coming in from surface x 1 x 2 constant that must be multiplied by velocity u 1 to get momentum in x direction. There is also mass coming in from x 2 equal to constant surface that also must be multiplied by velocity u 1 to get momentum in direction 1 and likewise there is also coming mass is coming from the back at x 3 equal to constant surface which must also contribute to momentum in direction 1. So, I have term three terms mass coming in at surface x 1 multiplied by u 1 mass coming in at surface x 2 multiplied by u 1 mass coming in at surface x 3 multiplied by u 1 going out fluxes would be likewise mass coming in at u 1 multiplied by u 1 at x 1 plus d x 1 similarly x 2 plus d x 2 and x 3. So, these are the way mass momentum in and momentum out terms what about the forces remember as I said sigmas are tensile forces sigma 1 acts on the area delta a 1 sigma 2 acts on the area delta a 2 and sigma 3 acts on delta a. So, the first term is in direction x 1 positive direction I shall have minus sigma 1 at x 1 minus sigma x 1 at delta x 1 multiplied by delta a 1 this would be the force in the positive direction 1 likewise tau 2 1 at x 2 plus delta x 2 surface is acting in positive direction where the tau 2 1 at x 2 surface is acting in the negative direction and therefore, I have tau 2 1 x 2 plus delta x 2 minus tau 2 1 x at x 2 multiplied by delta a 2 and likewise in the surfaces in the z direction tau 3 1 at x 3 plus delta x 3 minus tau 3 1 at x 3 multiplied by delta x 3 plus there if there is a body force b 1 in extend direction which acts on the control volume as a whole is written as rho m into delta v which is the mass of the control volume multiplied by b 1 meaning thereby that b 1 has units of Newton's per kilogram and therefore, it has been multiplied by rho m into delta v stresses on the other hand have units of Newton's per meter square and therefore, are multiplied by area a 1 a 2. So, in other words the units of each term here is simply Newton's the force. So, if you substitute these mathematical terms into the statement of the Newton's second law and we divide each term by volume delta v and let delta x 1 delta x 2 delta x 3 go to 0 then it is not very difficult to show that d rho m u 1 by d t plus d rho m u 1 u 1 by d x 1 plus d rho m u 2 u 1 by d x 2 plus d rho m u 3 u 1 by d x 3 would simply equal d sigma 1 by d x 1 plus tau 2 1 by d x 2 plus d sigma tau 3 3 1 by d x 3 rho m b 1. This would be the momentum equation in direction x 1. The left hand side as you will readily appreciate is the net rate of change of momentum in x 1 direction and the right hand side is the net forces in x 1 direction by fluid stresses and body forces. Now, clearly an exercise similar to this can be carried out in direction 2 and direction 3. I leave that as an exercise for you it is not very difficult at all. You will notice if you write down 3 equations together then they can be represented in a tensor by tensor notation. So, for example, law of conservation of mass can be written as d rho m by d t d rho m u j by d x j equal to 0 where j goes from 1 2 3 in direction in cyclic order. Momentum equations in direction x i can be written as d rho m u i by d t plus d by d x j of rho m u j u i plus d by d x i of sigma i delta i j where delta i j is the chronicle delta. It equals 0 when i is not equal to j and equals 1 when i is equal to j. So, these terms will survive only when i is equal to j d by d x j tau j i 1 minus delta i j i these terms will survive only when i is not equal to j plus rho m v i and in tensor notation we write it as in this fashion for i equal to 1 2 3 and j equal to 1 2 3 cyclic, but now we have a little closure problems. As you can see these represent three equations these represents one. So, the fluid motion has been described by four equations, but we have many more unknowns. First of all we have three velocity components which are not known in fact that is what we wish to determine sigma i 3 tensile stresses which we do not know and six shear stresses tau i j or tau j i. So, we have essentially 6 plus 3 9 and plus 12 unknowns and only four equations. So, essentially this is a not a solvable set. In the early days say around 1960 Euler simply assumed that the tensile stresses and the tau j i the shear stresses would be extremely small and simply ignored them that was the situation is 1960. Around 1825 a man called Navier found that now those terms cannot be neglected this the fluid is indeed stressed as it flows through the tube through when it flows and therefore fluid stresses could be as big as the body forces that it experiences and therefore decided to retain them which gave rise to these nine unknown stresses. But the way forward was found by Stokes in England and he said that the shear stresses would be related to velocity gradients or the rates of strain through viscosity tau i j is equal to mu equal to d u i d x j plus d u j d x i. Remember this is simply a definition or a model of a stress related to rate of strain introducing an entirely new quantity mu into our set of equations. From the form of the stress strain law you can readily appreciate that tau i j will indeed be equal to tau j i because when you change the indices the expression does not change when i is not equal to j and this is precisely what we call complementary stresses. So, the tau 2 1 is a stress in that direction the stress complementary to that is tau 1 2 acting on another phase. Likewise tau 2 1 at x 2 has a complementary stress tau 1 2 which acts on x 1 surface. So, the 6 stresses which were unknown are now reduced to 3 unknowns due to complementarity and they are now reduced to the velocity components if you know the value of viscosity. Normal stresses which are tensile are written as minus p plus 2 mu d u d x i minus p because pressure is always compressive and using Stokes notation you will see 2 mu d u i d x i is nothing but tau i i where i is equal to j. We had 4 equations and 12 unknowns but sigma i and tau i j are now replaced by velocity gradient and pressure. So, now we have 4 equations and 4 unknowns 3 velocity components u i and pressure p and also one more additional constant called fluid viscosity mu. Now, this viscosity was simply a constant of proportionality between stress and strain. It is our great fortune that viscosity has turned out to be the property of a fluid rather than the flow. Imagine if I had if I had if the stress and strain were connected in such a way that viscosity of water a when it flows in a circular tube is different from when it flows in a square section tube we would have a much bigger problem on our hand. We are very lucky it is an accident of history if you like our accident of nature if you like that viscosity has turned out to be a property of the fluid and not surprisingly because it is essentially trying to capture the information loss during statistical averaging of molecular motions. So, we must supply to the equations now the value of viscosity. As you will see our equations will now read d rho m by d t plus d rho m u j by d x j and the three momentum equations would read in this fashion d rho m by d u i d t d rho m u j u i d x j equal to d p d x i plus d by d a mu d u i d x j plus rho m b i which are the body forces and this is the remaining part of the stress stokes stress. So, these equations written in this form are known as Navier stokes equations. Navier was a French scientist engineer and Stokes was a English scientist engineer both are credited with formulating these set of equations by including the stress terms which were ignored by Euler in 1760. When of course, these three equations are ignored essentially we are saying viscosity is assumed to be 0 and therefore, when these terms are 0 we say the momentum equations applied to inviscid fluid, inviscid meaning fluid having zero viscosity or really an ideal fluid. Such an ideal fluid can explain quite a few things in fluid mechanics, but not others principally it cannot explain the drag offered by a body when fluid flows past it and as I said this drag is of paramount importance to a convective heat transfer engineer because he must design his surfaces such that the drag is reduced or the pressure drop caused by the drag is reduced and therefore, these terms are very important to a convective heat transfer heat and mass transfer analyst. These equations then describe the fluid motion completely. Incidentally, I may mention that when mu happens to be an absolute constant which is only a property of the fluid then we say the fluid is Newtonian because stress and rate of strain are then linearly related but there are fluids like blood or polymers and so on and so forth in which the viscosity or the magnitude of viscosity itself depends on the rate of strain and sometimes in the manner in which the fluid was strain through time and therefore, viscosity also happens to be function of time, function of the flow in which the fluid is situated and so on and so forth. So, my remarks about viscosity apply only to Newtonian fluids such as air and water at pressures that we are interested, pressures and temperatures in which we are interested in mechanical engineering. We are leaving out exceptional applications like blood flows and other thing where the flows are non-Newtonian. In the next class I will take up the fundamental law of energy which is the first law of thermodynamics.