 Moving ahead, let us talk about quickly the mathematical background that is required for this course. Usually you will see that fluid mechanics as well or even later the CFD part in many textbooks will be treated from a mathematical point of view. And we believe that both the physical aspects as well as the mathematical aspects should be known to a teacher to a good extent because you should be able to connect the two in some sense. So what I have listed here is the absolute minimal mathematical background that we will be utilizing for our course. And to the best of my knowledge typically this is the background that you will require even when you go and teach a fluid mechanics course or even a CFD course. So I have listed here the Taylor series expansion for a function of one variable. So if I am talking about the function value at a point x plus delta x, I express the function value at x plus delta x in terms of its value at the point x. And then these additional terms as you keep adding if you want only the first term in the Taylor series, we call it a first order Taylor expansion which will then be then f at x plus delta x equal to f of x plus the derivative of x with respect to x times the distance delta x between these two points. And that is what we used in the previous couple of slides when we are coming up with the expression for the rate of shear strain. Additionally if you want to keep on adding higher order terms as they are called, you can add a second order term, a third order term and so on. Keep in mind that in fluid mechanics because of the assumption of continuum and the resulting smooth distribution of fluid properties, we will actually always be restricting ourselves to a first order Taylor expansion. The reason is because when we are going to come up with differential equations of motion etc., we will be dealing with two points which are separated from each other by an elemental distance such as this delta x, delta y let us say delta z. These are all elemental distances and for elemental distances in continuum situations a first order Taylor series expansion is found to be sufficient. What I mean by that is it is found to be sufficient in terms of the accuracy that is required. In principle you can always keep on adding terms as you wish but for continuum situations it is found to be sufficient that a first order Taylor series expansion is what we can deal with. Thus immediately I have tried to list the Taylor series expansion for a function of two variables. So, now we are dealing with a function which has a value at the point x, y and then the function value at the point x plus delta x and y plus delta y. So, now we are moving both in x direction and y direction by the distances delta x and delta y respectively. When we are dealing with a two dimensional situation such as this the Taylor series expansion will involve partial derivatives with respect to x and y. So, the first three terms will form a first order Taylor series expansion and I have placed in brackets the second order terms. The second order terms as you can see will involve the second order differentials again partial with respect to x as well as y and there will be what is called as a cross derivative as well right here the third term. You can keep on adding terms here again the third order and the fourth order and so on. But again coming back to the kind of situation that we are dealing with we will deal with a first order Taylor series expansion. Then there is something called a Leibniz rule which we will utilize time and again you will see that many of the textbooks will routinely use this Leibniz rule. Some of you may be familiar with the Leibniz rule as the differentiation under an integral sign. So, what I have listed here is a one dimensional version of the Leibniz rule. So, what we have is a function of two variables x and t with variable limits on the integral u of x and v of x the lower limit is u of x and the upper limit is v of x and this entire integral is differentiated with respect to the variable x. So, what we can do is that using the Leibniz rule this differentiation with respect to x which is sitting outside the integral can be brought inside the integral in the form of a partial differentiation now which will operate on this function f. But then you need to add a few more terms and those terms are essentially evaluation of the function at the upper limit times the differentiation of the upper limit which is v of x with respect to the x minus the evaluation of the function at the lower limit which is u of x. So, the t here that you see inside the argument for the function is essentially a parameter and depending on the value of the parameter you will use either the lower limit or the upper limit. So, here we have used the upper limit and we are subtracting the lower limit multiplied by the differentiation of the lower limit with respect to the variable x. So, this is what we call a Leibniz rule in one dimension if you are dealing with a situation where both limits u of x and v of x are actually constants meaning that they are not really a function of x at all what you realize is that this dv dx and du dx these two differentiations will actually turn out to be 0 and what you have then is only the first term here which is inside the integral then a partial differentiation with respect to x of the function which was the integrand here. The reason I point out this special situation of constant limits is because we will utilize this routinely when we go to the integral analysis what we will be actually dealing there with is a fixed control volume. So, that the limit of the control volume is actually fixed and in which case these third second and third terms will not really be there because the differentiation of the limits upper or lower will actually be exactly equal to 0 and we will deal with only the first term. So, this is what formally is called as again the Leibniz rule in one dimension and this is what we will use when we go to the integral analysis later. It is good to know some basic vector algebra and vector calculus as well. In fact, we will be utilizing the vector algebra and vector calculus routinely through this course. Many of these are actually familiar to you. So, I am simply listing some of the common vector products and differentiation rules that we should be familiar with. So, at the top we have the dot product then we have the cross product in terms of their magnitudes. In terms of the vector representation I am listing the cross product as our standard determinant expression. When it comes to the differentiation I am dealing with in the first instance here a scalar which is small a multiplying a vector and the whole product then is differentiated with respect to another variable t. Then I have a dot product which is differentiated with respect to t and then we have a cross product which is differentiated with the variable t. When it comes to the differentiation of the cross product we make sure that the order of the cross product is maintained when we carry out the differentiation. So, you see how it is we have d dt of a cross b where both a and b are vectors. So, we first differentiate a with respect to t and then cross that with b. So, the order is maintained and then plus a crossed with d b dt. So, these are standard vector products and differentiation rules which we have been routinely using right from perhaps basic engineering mathematics and these are something that will be required often on in the course. So, I am trying to list those right at the beginning. Let us talk about the notion of a field. So, as a result of our continuum assumption we will necessarily be dealing with what we call a field and field is nothing but you can say a continuous distribution of a physical quantity such as pressure, density, temperature, etc. In space it can be a function of time as well if you are dealing with unsteady flows. But typically we will talk about fields as continuous distribution of distributions of physical quantities such as temperature, velocity, etc. So, if you are dealing with a scalar such as temperature we have a scalar field if you are dealing with a vector such as velocity we will have a vector field. And in general the vector field v say can be a function of x coordinate, y coordinate, z coordinate and time as well. Once we have defined this field we are in a position to talk about the directional derivative and what is called as a gradient. So, directional derivative as the name implies is simply the rate of change of a field variable with respect to distance in a particular direction that is simply called as the directional derivative. And the directional derivative for example in the x direction will be the derivative of whatever field variable that we are talking about with respect to the x direction. Similarly, the derivative of whatever field variable that we are talking about with respect to y will give you a y directional derivative. So, here I am talking about a scalar field actually phi and I am talking about phi as a function of only x, y, z right now. So, from our vector calculus we recall that the directional derivative you can obtain as a dot product of what is called as the gradient of phi with a unit vector in the direction of interest. So, since our direction of interest is given by the letter s which can be x, y, z whatever the unit vector in that direction is denoted here by a hat over the s. So, s hat is the unit vector in the direction s we are familiar with the Cartesian unit vectors. For example, i hat is the unit vector in the x direction j hat in the y direction and k hat in the z direction as I have written in here actually. So, s hat is simply the unit vector in the given direction s and if we form a dot product of the so called gradient of phi with the unit vector in the s direction we obtain what is called as the directional derivative in the direction of interest which is s. So, d phi d s is simply gradient of phi dotted with s. So, what is this gradient of phi then in Cartesian coordinates gradient of phi will be represented as the derivative partial derivative of phi with respect to x in the x direction. So, the i hat is going with the partial derivative of phi with respect to x plus j hat times d phi d y partial derivative plus k hat times d phi d z which is in the z direction. So, note that if this phi is a scalar function gradient of phi is actually a vector. So, you can imagine that this d phi d x here d phi d y and the d phi d z which are the three partial derivatives with respect to x y and z can be considered as the components in x y and z directions for this gradient of phi vector. So, continuing to talk about the gradient if you recall from your vector analysis for the scalar function phi as a function of x y z we can always devise surfaces of constant phi. So, phi equal to phi 1 will be a constant surface, phi equal to phi 2 is another constant surface and so on. So, this gradient phi turns out to be normal to the surface phi equal to constant and in fact, it represents the largest spatial rate of change of phi at a given point on the phi equal to constant surface. So, physically speaking this gradient of phi turns out to be the largest spatial rate of change of phi at any point on phi equal to constant surface. We normally talk about the vector operator which is given by this inverted triangle which we call as the del operator and the vector operator del is simply given as the unit vector i times the partial derivative with respect to x plus unit vector in j times the partial derivative with y plus unit vector k times the partial derivative with respect to the z direction. So, that this vector operator del operates on functions such as phi. So, if I can form del phi or in principle I should call it as the gradient of phi all I will do is I will add a phi here, here and here to come up with let me go back in a second the gradient of phi as we had seen on the previous slide. So, this del operator is a very important and frequently encountered operator in vector calculus. We will be using the methods of vector calculus sufficiently in this class. So, we will come across this del operator all the time and that is the reason I would like to point it out here as part of the mathematical background. Now, note that again I had talked about this few minutes back if the field variable phi is a scalar function then gradient of phi when this del operator operates on this scalar function it returns a vector as we saw earlier. Then we will deal with what is called as a divergence of vector field and it turns out to be a scalar. So, what is meant by divergence of a vector field? If you want to denote this through a symbol we use this as del dot a where a is the vector field that we are talking about. So, in general a will have three components a suffix x in the x direction, a suffix y in the y direction and a suffix z in the z direction. So, this del dot a will return partial derivative of a suffix x with respect to x plus d dy that is partial derivative of y with respect of a y and partial derivative of z with respect to a z and this turns out to be a scalar. So, this divergence of a vector field is a quantity that we will encounter often again and when we go to some of these governing equations we will specifically try to point out the physical meanings of some of the specific divergences that we are going to talk about. So, for example, if this a vector turns out to be the velocity field divergence of velocity has a certain specific physical meaning which we will try to point out when we come to it in a few lectures from now. The next operator that or the next relations I should say that we have listed here is that we first form a gradient of a scalar function phi. So, this grad phi is a vector and then we take the divergence of this gradient. So, the divergence is the outer operator you can say gradient of phi gradient that is is the inner operator and if you work out the algebra in the Cartesian coordinates you will get del dot del which is something called as del square which is what we call as a Laplacian operator and that Laplacian operator in Cartesian coordinates is given by the second derivative of x partial operating on phi plus second derivative of y partial operating on phi plus second derivative of z operating on or second derivative with respect to z I should say operating on phi adding together is what we will call a Laplacian of phi. So, this del dot del operating on phi will result into del square phi and the explicit expression for del squared in Cartesian coordinates is simply partial square divided by partial x squared partial squared divided by partial y squared partial squared divided by partial z squared. So, these are second derivatives all of them partial with respect to x, y and z added together. Moving on with few more vector operators there is something called a curl of vector field which is simply given by the cross product of the del operator with respect with a vector field. So, in this case I have chosen b as the vector field and this del cross b is simply using our standard cross product rule of the determinant expression I can evaluate this using the determinant expression. So, when it comes to kinematics we will see this curl of a velocity field which will result into what we will call a vorticity field and the vorticity field then will be simply del cross v where v will be the velocity field and it will be utilized or sorry calculated utilizing this determinant expression. What I have listed down here is a set of vector identities which we can come across during the study of our fluid mechanics. I am not going to talk about these these are for your reference whenever we use one of these I will simply remind you that we had discussed or we had pointed out some of these vector identities at the beginning. So, for example, if you look at the very first one what we are dealing with here is a curl of a gradient of phi. So, the curl is the outside operation gradient of phi is the inside operation phi is a scalar here. So, gradient of phi is giving you a vector and then that vector is you are finding actually the curl of that vector in the outer operation and it turns out that whatever be the phi the curl of a gradient of phi will always be 0. It is very very easy to show this using the Cartesian coordinates all that you need to do is you first find the gradient of phi and then take the curl of it using an expression such as this and when you work out the algebra you will see that this turns out to be always the case as long as phi is a scalar. And similarly there are a few more useful vector identities which I do not want to discuss, but you can easily prove many of these using the standard Cartesian coordinates. So, the second one here is what we will call it divergence of a curl of a vector is equal to 0 always and so on. So, at this point what I will point out to you is that you simply note down that these are some of the mathematical requirements that we will be utilizing from time to time. At this point we are not discussing those from a proof point of view clearly we have not discussed any formal proof for any of these, but for the purpose of our course we are not really required to get into those kinds of proof. We are going to utilize these as mathematical tools and at this point we want to note down these whenever we are going to use one of these specifically we will point out again that this was the identity that was pointed out to you long time back. The one last point here that needs to be talked about is the utilization of couple of integral theorems from vector calculus which we will routinely use especially in the integral analysis and converting the integral equations of motion into a differential equation of motion. You will always require some of these some of the textbooks on CFD that deal with the finite volume method. We will actually utilize some of these vector integral theorems as well in their derivations. So, you should be familiar with these integral theorems when it comes to the use for CFD type material as well. So, what we have here is an enclosed volume which is shown by this roughly ellipsoidal shape and the volume itself is shown by this symbol V and it is an enclosed volume. The surface area that is enclosing the volume is shown by this letter A here. At any point on the surface that is enclosing the volume we can talk about a unit normal which is pointing outward from the surface and when it is pointing outward this n hat for example, at this point which is pointing outward at this point will be taken as positive if it is pointing outward from the enclosed volume. The two integral theorems from vector calculus that we will utilize regularly are what are called as the Gauss theorem which will deal with converting a volume integral into a surface integral and a divergence theorem which will also convert a volume integral into a surface integral. You can always think about these as converting an area integral into or a surface integral into a volume integral as well going in the reverse manner. Similarly, for the divergence theorem going from surface integral to volume integral. So, either way these two theorems of integral calculus vector calculus I am sorry relate volume integrals to surface integrals. So the Gauss's theorem here deals with the gradient of a scalar function phi integrated over the enclosed volume V can be represented equivalently as a surface integral where you take the field variable phi and integrate it over the surface. Only thing is that the integral is actually a surface integral in the vector sense because we are utilizing the vector element dA using this n hat as the qualifier for the area element as a vector element. The divergence theorem deals with taking the divergence of a vector field integrated over the entire volume V and expressing that as an equivalent surface integral where you form a dot product of the vector B with the unit outward normal n hat and integrate that dot product over the entire surface. So, again here we are not talking about the formal proofs of the Gauss's theorem and the divergence theorem. You will find the formal proofs in one of the two applied mathematics books that I listed at the beginning of this session. It is quite possible that during your engineering mathematics courses you have undergone the formal proof of these theorems. For our point of view we are going to utilize these theorems more as a couple of tools rather than a mathematical entity. So, three things treat them as such that these are going to be the tools which will be utilized as we move on in our course material. So, this is more or less the first set of slides that I wanted to talk about as part of the introduction. So, to summarize this introduction session what I have done is that we have just introduced the basic ideas that we will deal with when it comes to this fluid mechanics analysis. We will deal with a continuum description of the fluid medium. As a result of the continuous or continuum description, we will make sure that our field variables like temperature, pressure, velocity, density, etcetera are all continuously varying, smoothly varying throughout the field. As a result of this continuum assumption, we will employ methods of differential calculus and the vector integral theorems. What we have listed toward the end is a list of useful vector identities and integral theorems from the vector calculus which will be routinely using as we move on in this course. Typically this will be used in the fluid mechanics part. Some of it may also be used during the CFD part. As I mentioned some of the finite volume techniques can be derived through the utilization of these integral theorems and many of the textbooks are actually utilizing these integral theorems as well. So, it is useful to know all this information. Again keep in mind that at this point we are utilizing this as a set of tools from applied mathematics. You can always go back to those nicely written couple of mathematics books that I have listed to get more information on this vector calculus and the integral theorems. So, with this we have essentially come to the end of the first session which we will call the introduction session. What I will do is that since there is about half an hour left for this session let me try to take a few questions just to sort of get going on this. Mufakumjha of Hyderabad you have a question. So, please go ahead. Thank you Mr. Paranik for giving us overall idea of CFD and its importance. Some limitations with respect to the approach to draw these profiles etcetera because even the solutions that we adopt for solving these equations because we will rather energy equation or other way another two equations. Whereas, the limitations could be just as slightly zeroes a proof idea about the limitations of CFD. Though the benefits are appreciated they also find that there are some limitations with respect to the study of CFD. Yeah, so the question seems to be if we can point out some of the limitations of the CFD analysis. If you go back to the set of slides that I just showed the basic limitation think about is the limitation of the model that you begin with. For example, if you cannot handle a full set of Navier-Stokes equations for whatever reason that will be a limitation from the CFD point of view. However, let us say that you have a model that you have chosen and then you are going to solve it numerically you will generate some numerical solution. How accurate that is is going to depend upon how accurate your numerical schemes are. So, what happens is that when you execute a numerical solution you evaluate various terms in the governing equation through some approximations. So, how good are those approximations are going to decide how good is your solution. So, here we many times come up with terminologies such as a first order accurate solution a second order accurate solution and so on. So, especially when it comes to things like compressible flows for example, there are entities such as shocks which you want to capture through your CFD simulation. Many times depending on the accuracy of the method that you are employing you are not in a position to actually capture the shocks correctly. So, that turns out to be one limitation of a CFD analysis. Usually if I want to point out in one line what can be the main limitation of a CFD analysis is that you are trying to approximate each term in your governing equation somehow. So, depending on how good is that approximation that you are implementing your solution is going to be decided on that. So, technically that is what I will say is that given all other let us say computer power and all other resources available to you how good is your approximation for a given term within your governing equation will decide how good is your CFD result. So, that to me is one of the main limitations. Of course, you know some problems are very, very complicated to be simulated. So, for that you will need enormous amount of computation power. So, that is an inherent limitation as well that you just do not have that sort of a computational power available to analyze a very complicated real life problem. So, what you end up doing many times is that you simplify that problem in terms of either the governing equations or in terms of the geometry that you are going to handle. So, again there is a limitation from the point of view of a CFD result that you are not really solving the entire real life situation because of the lack of computational power you are actually simulating something which is a subset of that and that also turns out to be another limitation. But to me the primary limitation is how good are your approximations for the terms in the governing equations and you will actually notice as we proceed especially the next week when we talk about finite volume method as one of the techniques for the CFD analysis. You will see that we will employ several approximations for each of the terms that comes in the governing equation and there you will come to know how the approximations are actually resulting into let us say a poor CFD solution. So, that is what the answer would be at this point. Thank you. So, R C Patel, Sherpur you have a question please go ahead. My question is related with the continuum model before considering the continuum model how we can define the continuum means what does the continuum exactly and what is the significance of the continuum? Yes. So, the question is about continuum model and specifically the question is what is really a continuum? So, very simply if you want to talk about physical terms a fluid continuum you can think about as a continuous distribution of fluid material where we are not bothered about the molecular nature or molecular structure of the fluid material at all. We are going to only focus on the overall behavior of the fluid medium and to get to that we do not really need to get into a molecular level. What we say is that the fluid material is continuously distributed and distributed in the domain of interest and corresponding to the material distribution there will be a distribution of the fluid properties as well like the velocity and the density and the pressure which will be also continuously distributed through the domain. Now the point really is that as you said how do we know whether a situation is a continuum situation or not correct before we get on with it. So, that is when I had pointed out that governing parameter called the Knudsen number and in many situations that you start analyzing you can actually calculate many times the value of the Knudsen number based on the mean free path of a molecule that in that particular situation to the characteristic length of that problem you form that ratio and if that ratio turns out to be much smaller than 0.001 or there about say rule of thumb you are fairly sure that the continuum situation is valid and then you do not have to bother about anything. Quite roughly most of the engineering applications that we normally deal with in mechanical engineering are perfectly within the continuum application only if you are dealing with a highly rarefied gas flow or a micro or a nano scale flow type situation you are in a situation where the continuum model is not applicable. But roughly speaking continuum is more or less always applicable what we mean by continuum is that sufficiently large number of molecules are always present in a very very small volume that you can think of within the domain so that some sort of a point value of density or pressure can be assigned meaningfully to that particular volume and then you can get a distribution of these points in the domain. Thank you. Shivaji University please go ahead you had a question. Is the scalar quantity or the vector quantity? If it is vector quantity is it important to specify its direction? Shear strain is actually a completely scalar quantity it is essentially a ratio of two lengths as if you go back to one of the slides one of the beginning slides we had calculated a rate of shear strain. So first we calculated the shear strain and then we calculated the rate of shear strain. So if you see the expression for the shear strain which is by the way very similar from what you get in solid mechanics also strength of materials is exactly the same it is simply the ratio of two lengths and as such it is a scalar quantity. Thank you. NIT Trichy you have a question please go ahead. Yeah I have a question I am not a mechanical engineer I am a metallurgist I would like to discuss about the main pre-parked of the atoms or molecules whatever you call it. Okay suppose the density of the metal is very high the main pre-park decreases is it correct? Yeah the question is about mean free path of the molecules and specifically the question is that if the density of material increases whether the mean free path goes down and that is absolutely correct. We are actually talking about the mean free path in a fluid medium though please keep that in mind. Typically we are talking about gases really if we are talking about mean free path one can talk about liquids also but mean free path is nothing but the distance that a molecule fluid molecule will travel between successive collisions. So as the density of a material increases the molecule is actually start colliding more and more in a given time and therefore the mean free path will actually go down as the density of the material increases that is correct. So that is why I said that if we are dealing with a rarefied situation a rarefied gas dynamic situation for example we are dealing with a very low density fluid flow such as what will happen in a vacuum situation for example or very high altitude from the earth surface where the density would be very low and therefore the mean free path will be very large. So what ends up happening is that if you form the ratio of the mean free path to the characteristic length of the problem in a situation where the density is very very low the Knudsen number turns out to be very high and therefore the continuum model in such situations is normally not applicable. So if you are dealing with a rarefied gas dynamic problem or a vacuum situation you will see that typically continuum models are not applicable. However we are going to restrict ourselves completely to continuum situations as far as this course is concerned but I hope that answers your question. Thank you. J. N. T. Hyderabad you have a question please go ahead. At what conditions the gas theorem and divergence theorem is to be used first question and the next question is shear stress tau directly proportional to rate of shear strain whole power n. What is the physical significance of n? What do you say? Yeah, so there are two questions. One is with respect to the use of the Gauss theorem and the divergence theorem when they are they are supposed to be used. Actually what I will say on that use is that we will utilize them very soon in the next couple of lectures. So I will request that if you can just have some patience maybe for two more sessions we will actually end up using those for situations where we are going to convert volume integrals into surface integrals and vice versa. So just wait for a couple of sessions and you will know how to use them. The second question was tau directly proportional to du by dy. What is the physical significance of that? Actually the physical significance is that there is a whole lot of fluids which we will call Newtonian fluids which are found to obey that relation experimentally. So if you were to do an experiment with with a Newtonian fluid you will actually find that the shear stress in case of a Newtonian fluid behaves or is directly proportional to the rate of shear strain. So physical significance is that experimentally a large number of fluids have been found to be conforming to that law and those are what are called as Newtonian fluids. So in this course we are going to be dealing with only Newtonian fluids. Non-Newtonian fluids if you want to deal with typically you will deal with those in chemical engineering situations. So slurries for example are non-Newtonian many times. Even our blood the human blood is also non-Newtonian. So it does not follow this tau equal to mu times du dy type situation. It is actually mu root times du dy raised to something. However, we are going to follow only Newtonian. So that is that is the answer to that. Thank you. Nitte Minakshi you have a question please go ahead. Can you verify the flows and where exactly can we find the application of this highly rarely failed verified flows in real world applications that is in real time applications and also can you share some more knowledge on this close form solutions. Yeah so there are two questions. One is dealing with the applications of rarefied gas flows where those are encountered. So as I was mentioning earlier if you look at situations such as reentry flows which are very very high altitude aerospace applications. You will typically deal with rarefied situations because at very high altitudes let us say about 90 kilometers from the surface of the earth or 100 kilometers from the surface of the earth. The density is very very low and there you are actually dealing necessarily with a rarefied gas flow. So that is one application. The other application is somewhere where you can think of a vacuum technology application. So if you are dealing with a high vacuum equipment and if you are analyzing flows in high vacuum equipment again because the density is very low you will necessarily deal with a rarefied gas flow. So that is about the rarefied part. The second question was about close form solutions. So what I will answer is that we are actually going to work out in detail 5 or 6 close form solutions maybe in 3 more days as part of the course. So I will request that you just wait till about Thursday or at the most Friday of this week and we will actually work out several of these exact solutions close form solutions. So you will know what we are talking about there. Thank you. Ajay Somaia you have a question please go ahead. Yes of the printout page number 5. The question is on the last line when the fluid is at rest it cannot resist a shear force no matter how small. But there is an internal resistance. So how do we justify this statement? Yeah so that is how actually we define the fluid. So the question is about whether the fluid can resist a shear force no matter how small when it is at rest. In fact that is the definition of a fluid material. What we say is that something which is fluid which behaves like it cannot stay in static equilibrium when it is in a static equilibrium and if some sort of a shear force is applied it will start flowing it just cannot stay in the static equilibrium. However we will see little later when we are working out some of these exact solutions is that when the fluid is in motion it can actually keep moving under what we will call a dynamic equilibrium situation where it will not really experience any acceleration it will keep moving with a constant velocity but it will be moving with a couple of different types of forces acting. So for example there will be a pressure force acting and there will be a viscous force acting but at the same time the fluid will keep on moving without an acceleration in some sort of a dynamic equilibrium situation. So there it is not continuously accelerating mind you. So that way it is able to resist a applied shear force when it is in motion but if it is stationary and if you apply a shear force it just cannot remain stationary it will start flowing. So that is the way we actually define a fluid. Thank you. This is a question regarding application of computational fluid dynamics in manufacturing process such as electrochemical machining or electrolysis. Can we apply computational fluid dynamics for analysis of variation of conductivity of electrolyte between parallel plates? Yeah the question is about application of CFD in a specific manufacturing process. To be honest with you I am not really familiar with those kinds of manufacturing processes but I have come across some mention of the use of CFD in the kinds of processes that you are mentioning and in principle I would think that it should be applicable. There should not be any question of that. I am not particularly in knowledge of any of those but if you read some of the manufacturing journals especially the latest journals I am very sure that you will come across the application of CFD for the kinds of manufacturing processes that you are referring to. Unfortunately I am not really an expert in any of those. So I cannot really specifically answer that question. Thank you. Amrita Kolam you have a question please go ahead. Could you explain how a Newtonian fluid differ from a non-Newtonian fluid with an example? Yeah the question is how do we differentiate a Newtonian fluid from a non-Newtonian fluid. So the best way to really look at these two types of fluids is if you go to a rheological diagram where you plot the shear stress on the y axis and the rate of shear strain on the x axis. So if you see the Newtonian fluid since tau is directly proportional to mu times du dy there will be a straight line describing this situation passing through the origin. However if you are talking about a non-Newtonian fluid because tau is proportional to du dy raised to some power which is not equal to 1 the plot that will describe this situation will not be a linear one passing through the origin. So it will be a non-linear curve passing through the origin in case of a non-Newtonian fluid. So the typical non-Newtonian fluid that you can think of is a toothpaste for example. So if you start squeezing the toothpaste harder and harder it will start flowing more and more. But you will need some sort of a certain shear stress before which it will actually start flowing. So it is not like a Newtonian fluid where you employ a very small shear force and it start flowing but it requires certain finite force which will make it move and then as you start increasing the shear force it will start flowing better and better or more and more. So that way it is a classic example of physical example of a non-Newtonian fluid. Thank you. Manipal, please go ahead you have a question. Yeah I am. Let me come back to the concept of continuum. Hello. Yes, please go ahead. Concept of continuum. See when I say the distribution of matter is continuous and it is homogeneous. Distribution of matter is continuous or homogeneous whether those statements are analogous or else is there any significant factor when we do the statistical averaging? So the question is about continuum assumption and whether if the material is continuous and homogeneous will essentially mean the same thing. Roughly speaking it is correct. The material does not have to be necessarily homogeneous. However there will be a continuous distribution of properties like the density and the pressure throughout the fluid medium. See what we mean by this continuous distribution is that if there are sufficient number of molecules present in a very small volume that you can think about you can assign a value of properties such as density for that volume and then you can think about several such volume sitting right next to each other forming the entire fluid continuum. So that way you can imagine a continuous distribution of a property that can be formed as a result of the continuum. So the continuum assumption will lead into the formation of a continuous distribution of the fluid properties and for the continuum assumption to be valid you need to have a very large number of molecules present in a fairly small volume that you can think of in the medium. Thank you. So what we will do now is that we will break for lunch. So thank you very much.