 find something called as Knudsen number which is the ratio of the molecular mean free path and the characteristic length scale of the system. So this L we give it a symbolic note for a pipe it may be the diameter of the pipe for something else it may be some other dimension but we call it as a characteristic length scale of the system. So if this ratio is small what does it indicate? It indicates that the mean free path is much smaller than the characteristic length scale of the system which implicitly tells that it is a sufficiently densely packed system. On the other hand if that is not the case that is if the Knudsen number is large that means the mean free path is much it may be even larger than the characteristic length scale of the system it is if it is very very rarefied. In such cases what happens? In such cases you have very very few number of molecules and then there are lots of uncertainties with respect to presence of molecules in individual elemental volumes. So in those cases the macroscopic point of view is not expected to work so efficiently or the macroscopic way of defining the characteristics or properties of the fluid might not work so efficiently. So whenever the macroscopic way works we call it we call the fluid as a continuum and the hypothesis concerned is known as continuum hypothesis. So what is the continuum hypothesis? Continuum hypothesis tells that we treat the fluid medium as a continuous matter disregarding the discontinuities in the system. There are discontinuities if you look into the molecular level. There are molecules, there are gaps and so on but if those are sufficiently compact then you may treat it as a continuous matter. Once you can treat it as a continuous matter then you may use the well known rules of differential calculus to talk about the changes in properties from one point to the other. So you can talk about simple gradients, second order derivatives and so on. So continuum hypothesis works if there are sufficiently large number of molecules so that the Knudsen number that we are talking about is very very small. If the Knudsen number is greater than 0.1 so to say then it comes to a state where your mean free path threatens to be 10% of the characteristic system length scale and when it goes on larger and larger there is a stronger and stronger deviation from the continuum hypothesis. So if there are situations when we cannot use continuum hypothesis and one needs to have a different treatment altogether. Throughout this course we will be bothered mostly about situations when continuum hypothesis works. So that means there are sufficiently large number of molecules in the system so that uncertainties with regard to individual molecules do not influence the prediction regarding the fluid flow to a significant extent because we are not looking from a molecular viewpoint but treating the fluid as a continuous matter. Keeping that in view what we will do is we will next see that no matter whether we are treating it as in a macroscopic viewpoint or in a macroscopic viewpoint how should we describe the fluid as a system. So for that we will introduce two important concepts system and control volume. We have identified an approach now we have to identify that what should be that fluid over which we apply that concept or approach. So when we talk about a system by definition system is something of fixed mass and identity. So something which must have its mass fixed it must have its identity fixed that means those are identified. For fluids sometimes this is not such a simple concept to implement. Let us again take an example of flow through a pipe. So we have a pipe there are many molecules or even particles whatever are entering the pipe and leaving. So at a particular instance of time so these are the molecules which are present. Now at a different instant of time you may have different entities different molecules which are present. The reason is quite clear something is entering and something is leaving. So it is continuously being replenished. In such a situation if you want to track the motion of these particles or individual molecules as the something of fixed mass and identity it becomes difficult and tedious because then you have to put a tag on individual entities and follow it as it is moving. This kind of approach or the so called particle tracking approach in mechanics is known as Lagrangian approach. In fluid mechanics it is not many times convenient to follow that approach. So what we do instead is like we focus our attention on a fixed region in space. So let us say that we have focused attention on this identified region. So as if we are sitting with a camera focusing the camera on this zone what we are observing we are observing whatever is coming into this zone and leaving that we are only keeping track up to that much. We are ignorant about where from it has come and where it is going. So rather than focusing attention on individual particles we are focusing attention on a specified region in space across which matter can flow. So that region we call as control volume. So control volume approach is more convenient for fluid flow because you do not have to track individual particles and fluid is a continuously deforming medium. So it is very difficult to track individual particles. It is much easier to focus your attention on a specified region and see what happens across that. And this particular approach where you use a control volume and analyze what happens across that is also known as Eulerian approach. Just as the Lagrangian approach is following the name of the famous mathematician Lagrang Eulerian is according to the name of Euler. So whatever we will be discussing in this context of fluid flow no matter whether it is a system approach no matter whether it is a microscopic approach or a macroscopic approach we will be mostly using the control volume concept for analyzing the flow behavior. With this basic understanding we will now go into the concept of a fluid. So till now we have loosely talked about fluid we have not seriously defined what is a fluid. Now common sense wise if we are asked that what is a fluid obviously we will say something what flows is a fluid. And loosely speaking this is not a bad definition but there are many things which flow but those are not so called classical fluids those may be in a borderline between fluids and solids so to say. A very formal definition of fluids is like this that fluids are substances which undergo continuous deformation even under the action of very small shear force. So if you are applying a shear force or there is some shear force acting on the fluid then even if it is very small it will continuously deform the fluid for a solid. Obviously if you are applying a shear force it will not spontaneously deform it till may be it comes to a threshold limit when it will it will it will appear to be seriously deformable. Obviously there are substances which are fluids but which require a threshold shear to be deformed and therefore the borderline between the fluid and the solid is sometimes not so strict. But for most of the practical cases this definition is something which we will be keeping in mind and will be classifying fluids or solids according to this behavior. So one of the important consequences is that if there is a fluid which is non-deforming so non-deforming fluid may be fluid at rest. So if you have a container within the container you have put some water and water is at rest. What is the implication of that? The implication is very straight forward there is no shear which is acting on it. So if there is some shear which is acting on a fluid the fluid is deforming. Converse is also true that means if the fluid is deforming there must be some shear which is acting on it. So when there is a fluid which is there at rest that means there is no shear component of force that is acting on it that means there is only normal component of force acting on it and that normal component of force per unit area which is acting inwards is called as pressure. So obviously whenever there is a fluid at rest the entire situation of the forcing which is there on a fluid element may be expressed in terms of pressure. When the fluid is moving it does not mean that there is no pressure. Obviously pressure is very much there which would have been there if the fluid is at rest but there are additional forcing components which come into the picture which are directly related to the deformation of the fluid and this situation all together may be tackled in continuum mechanics that is mechanics of a continuous medium through the concept of stress. So we will now introduce the concept of stress which we will be introducing in the context of continuum mechanics. So it is valid for both solids and fluids. Now let us say that we have an element when we say we have an element it may be an element of a solid fluid whatever we are not very particular about it and we are identifying a small chunk from that. On that small chunk we are taking a small area say dA. What we are interested is to see that whenever we are taking out this small chunk the other part of the body will exert some force on this just by Newton's third law. So we are interested to identify that force and let us say that that force is directed like this it is absolutely arbitrary. So it depends on many situations. Let us say that the force is dF. So if we want to define some force per unit area then we are just giving it a name. We are calling it T or traction. We will call it a vector because it is having the nature of a force. It will be implicitly determined by this one where F is a force. But we have to remember that it is not unique until and unless we specify the area. So what it means let us say that centered around the same point we take a different elemental area same dA but differently oriented. So if we take that differently oriented area now if we find out the resultant force on that it is likely to be different. That means given the point around which we take that differentially small area as fixed that is the location fixed given the magnitude of the so called elemental area as fixed still this ratio is going to be different. So this strongly depends on not just that dA but how that dA is oriented. So it is important to give a kind of superscript or subscript to this. So let us give a superscript n. So this n denotes that we are talking about an area which is having its outward normal in the direction of the n vector. So whenever we denote the orientation of area it is constant to denote it by the unit vector in the outward normal direction. Let us say that n is such a vector even if it is not a unit vector there is no problem because we can always normalize it in the form of a unit vector that direction is what is important. If n is changed that means the orientation of the area is changed obviously this traction vector will change. Now this traction vector therefore is not denoting something which is ordinarily like any other vector. So if you are talking about say a force so whenever you are writing the component of a force say f is a force you are using an index i to denote the component of the force. So this index i equal to 1 will mean the x component, i equal to 2 will mean the y component and i equal to 3 will mean the z component. Therefore by using one index i and varying it from 1 to 3 we may denote the components of a vector but when we are trying to denote this traction vector yes it has a component because it is it is like a force per unit area. So based on the direction of the force it has its own direction it has its own components but its specification depends on also another sort of index n which denotes the choice of the area orientation that has been employed to calculate this t. So it is something more general than an ordinary vector. How general it is to understand that we will take some special examples. What are the special examples? Let us say that we take an element of a fluid which is of a rectangular shape like this. We may orient axis like x, y, z in terms of the index we call this as x 1, x 2, x 3. Index using the index is a very convenient way because just by varying the index you can vary the directions. Now let us try to see that what are the special cases of traction vectors on the faces of these q y. So this has 6 faces and these faces are special surfaces. Why these are special surfaces? These have their normal directions either along x or y or z. This is absolutely an arbitrary area and what we are interested to do is to figure out what happens for an arbitrary area by referring to special areas which are either oriented along x, y or z. So that is the motivation of taking such an element. So once we have taken this element what we are going to see is we are going to write such expressions for different faces. So when we come along this face we are interested to write the components of the traction vector. So let us write that. So components of the traction vector there is a component along x. In terms of this notation what should be the subscript 1? What should be the superscript? See what does this represent? It represents the unit vector outwards normal of the surface chosen. So here the surface chosen has unit vector along x 1. So this should be 1. Alternatively we use a notation equivalent to this is tau 1 1. So 2 indices are there. What are these indices representing? The first index is representing the direction normal of the surface and the second index is representing the direction of action of the force component. So in general it is like tau i j where i represents the direction normal of the surface which we have chosen and j represents the direction of action of the force component itself. So let us take a second example. Let us say we write the y component on this surface. So this one. So tau what should be the subscripts? First one is 1 and second one is 2. So this is very simple and you can do it for all the surfaces. So what I will advise you to do is that you repeat the same thing for all the 6 surfaces to get a feel that you understand the notation. This is like notation grammar. If you want to learn say classical music you require to know the notation. So we are starting with the notation. The notation is important because that will help us in developing the basic equations in a very elegant manner. Interestingly let us look into the opposite phase of this one. So for this phase the outward normal is along negative x. So we will develop a sign convention that if the outward normal of the surface is along negative i, we will have the sign convention such that positive tau i j is along negative j. That means here thus so this one we will call as positive tau 1 1 for this surface. Why? Because the first one is actually along negative x the first index therefore the positive sense of the j which is the second one is along negative x. So on this surface if you want to draw positive sense of tau 1 2 so that should be downwards. These are sign conventions. So if it actually is the other way it will come as minus of this number. Just like in free body diagram you draw a force. The force might have come in the negative. That means it is actually in the opposite sense than what you have drawn in the figure. So these are also like that. So as if you are drawing the free body diagram of a chunk or an element. So we are establishing sign conventions for that. So what we have learnt as the sign convention is if the direction i is along the negative of one of the coordinate axis then the tau i j positive sign will be oriented along the negative j direction and if it is positive it is the other way. So this tau i j so when you write tau i j this i may vary from 1 to 3 and j may vary from 1 to 3. So these are certain quantities. In general you may have 3 into 3. 9 tau i j components. We will later on see that actually out of these 9 you have 6 which are independent and utilizing those independent 6 components which are called as components of a stress tensor. We can actually find out the state of stress for any arbitrary plane which is neither oriented along x nor y nor z. So these particular quantities which are like called as components of a stress tensor we may understand that these are not like vectors. So what are the differences between these and the vectors? So very logically you can see a vector requires a single index for its specification i. This requires 2 indices for its specification. What are the special indices? One index is just like making it act like a vector but the other index is specifying the direction normal chosen to calculate that quantity. So it is something more general than a vector. This actually is called as a second order tensor. We will not be defining in general what is a tensor because it is an involved mathematical concept and there is not enough scope here that we discuss about that but at least from common sense you can appreciate that the order of tensor in this Cartesian notation is like the number of indices that you are requiring to specify it. So vector is also a tensor. It is a tensor of order 1. Scalar does not require any index for its specification. So it is like a tensor of order 0. So we have very easily come across 3 different orders of tensors. Tensor of order 0 which is a scalar, tensor of order 1 which is a vector and tensor of order 2 example is a stress. And we will see examples where we will be having tensor of order 4 as of course there may be nth order tensor in general but we will see that there are certain important tensors in the context of mechanics in a continuum or fluid. So fourth order tensor is one such example which we will come across later on in this course. So we will now go to our next objective that is given this components of the stress tensor. How we may utilize these concepts or these components to designate the state of stress on any arbitrary surface which is neither oriented along x, y or z. For that what we do is consider an elemental volume like a tetrahedron. We give the point certain names for convenience. So there are surfaces like this the surface A, O, B let us call it as S1, surface A, O, C we call it as S3 why such 1 and 3 because this index we are trying to preserve for the direction normal of those surfaces. So this one is for the fact that the direction normal of this A, O, B is along x. Similarly this one is S2 and A, B, C let us go to our next objective which is A, B, C let us give it a name S. Can you tell what is the motivation of this taking such a volume? See whenever we are deriving something in the class it is like it will appear to you that yes it has to be done like that. Remember this is not a ritual. Do not accept anything whatever we are learning in the class as a ritual. Always try to ask yourself a question why have we taken such a volume? What is the motivation behind taking such an element? So if you see this element has 4 faces out of these 4 faces 3 are the special ones which have their direction normals either along x, y or z. The fourth one is not a special one it is arbitrarily oriented. So now by considering the equilibrium of this element by considering the forces which are acting on it we will be able to express what is there on that odd surface in terms of what is there on the special surfaces. So that is what is the motivation behind taking this one. Now whenever we are coming to such an element our objective will be say to write the Newton's second law of motion for this. So just resultant force equal to mass into acceleration. Question is what forces are acting on this element? So when we say what forces are acting on the element we will be classifying the forces in continuum mechanics into categories. One is a surface force another is a body force. These names are almost self-explanatory. When you say surface force it means that these are forces which are acting on the surface or surfaces which are comprising the volume element chosen. And body force is a force which is acting over the volume of the body. Example is body force. One of the examples of body force is the gravity force which acts throughout the body volume of the body. Surface force pressure is an example. Force due to pressure is a surface force. So whenever we are having forces we will categorize in terms of surface force and body force. So whenever we have surface force the surface force may be expressed in terms of the traction vector because the traction vector we have defined in such a way that on a surface it represents the resultant force per unit area. So it represents a cumulative effect of all forces which are acting at that point on the surface. So this particular element has four surfaces. Let us write the forces which are acting on these four surfaces and write the Newton's second law of motion along say x direction. So what we are going to write is resultant force along x is equal to the mass of the fluid element times the acceleration along x. So when we write the resultant force we will write it in terms of the surface force and body force. So first let us come to the surface say aob. So on the surface which component of the stress tensor will give a force along x in terms of tau ij, tau 11 right. So what would be the positive sign convention direction of tau 11 along this. So tau 11 this is the force per unit area. So what is the area on which it is acting into s1. Similarly for s2 what is the force which is acting along 1 tau 21 right. What would be its direction just like this because its outward normal is along negative 2 direction. Therefore the positive sense of tau 21 on this surface is along negative 1. So and this multiplied by s2. Similarly this will be tau 31 s3. There is a force surface which is really the back surface here which has its normal neither along x, y or z. Let us say the normal to this is n cap the normal vector outward normal vector of s. So this n cap say it has its components like this n1 i cap plus n2 j cap plus n3 k cap where n1 n2 n3 are the components of this along x, y and z. We are now going to write the force on s. So the force on s let us say that it is t with superscript n. So now we cannot use the tau notation for that because it is not a special surface. Tau notation you can write for a special surface where the normals are along x, y or z. So we use the t notation this is for the a, b, c. This component wise it is along 1 and the area on which it is acting is s and obviously by default we are taking it as along positive x. These are the surface forces. What is the body force? Say we call that b is the body force per unit mass. If we find out what is the mass of the fluid element, so what is the mass of the fluid element? Let us say that we find out first what is the volume of the fluid element. So for this type of element we can say that it is 1 third into the area of this a, b, c times the perpendicular distance from o to a, b, c. Let us say that perpendicular distance is h. So we first find what is b, 1? So what is b, 1? First the volume 1 third into s into h where h is the perpendicular distance from o to a, b, c. This is the volume mass of this you multiply it by the density say rho is the density. So this is the mass and this multiplied by the body force per unit mass along x will give the total body force along the direction 1 or x. So this is the mass. This is the body force per unit mass along x. So the product is the total body force along x. So we write the Newton's second law of motion. This particular expression now we write the forces. So minus tau 1 1 s 1 minus tau 2 1 s 2 minus tau 2 1 s 2 minus tau 2 1 s 2 tau 3 1 s 3 plus t 1 with superscript n into s that is the total surface force plus the total body force that is the net force which is acting on it and that net force is equal to its mass into acceleration. So that is equal to what is its mass? One third s h rho that is the mass. Let us say a is the acceleration. So a 1 is the acceleration along x 1. The physical definition and the mathematical definition is identical here. Surface force is a force which is distributed over the surface which is the envelope of the volume that is being considered and body force is something which is acting within the volume of the body. Yes. Here force is the surface force as an example. Also normal force that is like pressure is a normal component. It is not a shear component. Yes. But it is a force which is acting on the surface. I mean whether normally to the body or not it is it is a matter of direction but the force may act on the surface or the force may be acting throughout the volume of the body that is how this is classified. So obviously when we are considering this this is something which is acting on the surface. Obviously it will have a direction right. I mean there is no contradiction with that with the surface force and body force. Surface force will have a direction. Body force will also have a direction. So when you have this expression next is you can write S1, S2 and S3 in terms of S. How can you write it? See S1 is like the projection of S on the YZ plane right. So how do you find out the projection? You find out basically the component of this so called vectorial representation of ABC on the YZ plane. So that means when you want to find out the component you basically find the dot product of the corresponding unit vectors. So this has a unit vector in the direction of ABC has unit vector in the direction of N. AOB has unit vector in the direction of minus i. So but obviously here the plus minus you are already taking care of through this sign convention. So you are not duplicating it once more. So the dot product of those 2 directions will be N1 is this vector and another is i. So the dot product will be N1. So in terms so these are all magnitudes. Their senses have already been taken care of with plus minus. So S1 is nothing but SN1 by taking the component of so called S in the direction of the so called S1. Similarly S2 is SN2 and S3 is SN3. So in that way if you substitute in this equation you will see that S gets cancelled out. So what you will get? Minus tau 11 N1 minus tau 21 N2 minus tau 31 N3 plus T1 plus one third row plus T1 plus T1 plus T1 plus T1 plus T1 into h into B1 is equal to one third row into h into A1. When you have these equations the next consideration that we have to make is something which is subtle but important to understand. We will shrink this volume to a point such that this entire volume as if converges to the point O because our end objective is to find out the state of stress at a point in terms of an area chosen around that point. So we will be considering a vanishing area or a vanishing volume so to say not a vanishing area so that everything converges to O that means we will taking the limit as h tends to 0. So when you take h tends to 0 the entire volume will converge to the point O then whatever we describe basically is the description of state of stress at a point O. So when you take that limit as h tends to 0 you will see very beautifully these terms will tend to 0. So in that case you are left with a very simple expression for T1 that is tau 11 N1 plus tau 21 N2 plus tau 31 N3. You can see that this is a very excellent expression because it relates the traction vector on an arbitrary surface with the components of stress tensor. What are the inputs? The inputs are that you must know the state of stress on those specified planes with specified orientations and the components of the unit vector direction of the arbitrary plane that you are considering everything at a particular location. The other thing is that there is a way of writing this symbolically in a more compact manner. You can see that here you are having two indices. So the first index is what is varying? The second index is something which corresponds to this one. So you can just generalize it and say so you have one index i which is fixed the other index j which is a variable which varies from 1 to 3. Now because this type of notation is very common the general rule is again the general notation is that this summation is omitted. So this becomes invisible. So this is also written as just tau ji nj. How you will know that there is a summation? Whenever there is a repeating index you have to keep in mind that there is an invisible summation in it. So from now onwards many times we will be using this notation without using the summation symbol but you have to keep in mind that whenever there is a repeating index there should be an invisible summation over that. This is known as Einstein index notation. This was first introduced by Einstein. So this type of notation gives a very compact way of writing these terms of writing this traction vector in terms of the components of the stress tensor. So this is known as Cauchy's theorem this expression. So what it does is it expresses the traction vector on any arbitrary plane in terms of the corresponding stress tensor components. It is also possible to write it in a matrix form. So what you can do? You are having components of the traction vector on a given plane say with orientation n 3 components you have. So this if you just follow these expressions you will see that so you can see that this is nothing but a matrix way of writing the 3 components of the traction vector. So whatever equations that we have written this is not something new. Just put i equal to 1 it will correspond to the first row of this i equal to 2 and i equal to 2 n 3. So what you can see is that here you get those so called 9 components of the stress tensor and these all these 9 are not independent we will see but this is something what it is mathematically doing you can see here look into these quantities. What is this? This is a vector it has its 3 components n 1 n 2 n 3 this is also a vector. So this is acting like a transformation which maps a vector onto a vector. So it is a very important characteristic of a second order tensor that a second order tensor maps a vector onto a vector. Similarly like a fourth order tensor maps a second order tensor onto a second order tensor like that as an example. So tensor is also like a transformation tool or a transformation which tries to transform one vector into another vector if it is a second order tensor. The next thing that follows from this is that are all these independent or all these tau ij is independent or we should be in a position to express this tau ij's some of these in terms of the other. So for that we will quickly do one exercise we will consider now a two dimensional element. Two dimensional element is something where everything is occurring in a plane just for simplicity. So we are assuming that the third direction is like unity or whatever. So it has its length like say this is delta x this is delta y just imagine that it has faces perpendicular to whatever has been drawn in the figure having all width as one. So let us write the components of the stress tensor on these surfaces. So very quickly we will write it because we have learnt by this time how to write it. So this is tau 11 this is what is this tau 21 here this is tau 11 this is tau 21 here this is what will be here tau 12 it is along the positive 1 because the outward normal is along positive 2 yes. So these two will be reversed. So I would expect that you always correct it. So the first index is what direction normal. So direction normal is what 1. Second index is the direction on which it is acting. So this is tau 12 and this is tau 21 here same. Now we are interested about the equilibrium of this element. So when we are interested about the equilibrium of this element we will consider the rotational equilibrium as an example. So rotational equilibrium let us consider that as if it is an element where we will be writing an equivalent form of Newton's second law for rotation. As if like we are writing rotation of a rigid body with respect to a fixed axis something like that but the axis is this center. So as if we are writing a rotational equilibrium equation with respect to an axis which passes through this center O and is perpendicular to this plane of the board. So that is why it is a two dimensional thing we are considering a rotation in this plane basically. So what we can write resultant moment of all forces which are acting on this is what say we are writing the moment with respect to which axis z axis here is equal to what tells that it is 0 it might be having an angular acceleration. So it is I with respect to the same axis which is passing through this one and perpendicular to the plane of the board say z times the angular acceleration say alpha. So the resultant moment of all these forces what will be that. So you will see that tau 11 and minus tau 11, tau 22 and minus tau 22 they cancel. So moment contributors will be tau 12 and tau 21. So tau 12 and this tau 12 they form like a couple. So it is tau 12 what is the area on which it is acting into delta y into 1 which is the width times the arm of the couple moment delta x for the other one it is clockwise. So minus tau 21 delta x delta y delta x delta y we are assuming there is no body couple which is acting on it just like body force there could be body couple fluids usually do not sustain body couple. So there is no body couple which is acting on it is equal to the moment of inertia is like it is having a dimension of m into length square. So m is like delta x delta y into 1 into rho that is the m if you write it properly it is delta x square plus delta y square by 12 with respect to this axis times the alpha. Keep in mind that delta x delta y are all small and tending to 0. So if you cancel by considering that these are tending to 0 but not equal to 0. So you are left with what in the right hand side you have terms because delta x and delta y are tending to 0 you have the right hand side tending to 0 and that will give you a very interesting result tau 12 is equal to tau 21. So in general tau i j is equal to tau ji that means tau 21 and tau 12 are same tau 31 and tau 13 are same and tau 32 and tau 23 are same. So you are left with 6 independent components in this stress tensor and you see that tau i j equal to tau ji what are the assumptions under which it is valid? There is no body couple that is the only assumption it does not depend on whether it is at rest or in motion. This is a very common misconception that people have that this is valid only for static systems. No I mean it might be accelerating but it does not matter because in the limit as delta x delta y tends to 0 the acceleration angular acceleration term becomes insignificant and that is how you get tau i j equal to tau ji. We stop our discussion today we will continue in the next lecture. Thank you.