 I'll share my screen. All right, kinetic theory is going on. All right, thermal properties, exams next week. Okay, revision for exams. When is your exam, next one? Where is the next exam? 16th is physics. Is there any exam tomorrow? Tomorrow there is nothing, right? Starts on 1st of October, very good. Mute yourself. Who is Shri Devi Guttu? Is there anyone of that name in our class? Pradyun, okay, it's Pradyun. Is it that, that's your mom's name, is it? All right, I think you can't change it now. All right, so guys, we'll start, all right? Today, we have to complete this chapter and very little of this chapter is left and once this chapter gets over, we can get down to probably thermal properties of matter and you have your physics on 15th of October, right? 15th of October and till thermal properties is coming, right? In your exam, till thermal properties is coming. Okay, so I guess, till thermodynamics, till thermodynamics is coming. That's little too much. So what we can do at max that we could just finish up the thermal properties of matter, till that. So thermodynamics, you can do it your own this time round, right? Otherwise, we'll not be able to do anything properly. Whatever we do, we have to do it systematically so that it is useful for next two years, okay? Not just for your UTs, all right? So I hope all of you are agreeing to whatever I'm saying here and do you guys do the homework assignment? Like I receive not more than 50, 60% of students assignment. You can see every week we put it on the group, all right? So those who are not doing it, do it, all right? Start doing it, otherwise it will be a problem, right? So if you're not doing the assignment, please do not expect any marks, any good marks in the monthly test, right? It'll be an unfair expectation that you do not do what is required and then you are expecting good marks in the monthly test. That never will happen, okay? So if you want good marks in the exams, in the monthly exams, you need to do the assignments regularly, fine? And the chemon test is the coming Sunday. You guys are all aware of it, right? Coming Sunday is your main level test. I hope all of you are aware of it, okay, great. Anyways, so let us start this topic of the mechanical properties of fluids, all right? This topic may get over probably in two hours or a little bit more than two hours and once we complete this chapter, we will then look at the probably, we'll solve questions from this chapter rather than starting a new chapter today itself. Once this gets over, we'll be solving questions on it, okay? All right, guys, so this is the section. I will have to reshare the screen. So write down this, the first topic that we are going to discuss is viscosity, right, Tom? When we talk about viscosity, what comes in your mind? Honestly, what comes in our mind? The first thing that will come in your mind, very good. So this is the first thing that come in everybody's mind, honey, right? I'm sure that many of you would have thought about it. All right, so we associate viscosity with honey somehow, all right? Now what it is basically, it is a viscosity is a property, right Tom? It is a property in which we are talking about force between the layers, layers of it. So basically if we assume, if we assume a laminar flow, what is a laminar flow? In the laminar flow, the flow you can think the flow like this. I guess I might have discussed it earlier, but again, it's like when the liquid is flowing, it is moving in the form of layers, okay? One layer is moving relative to the other, okay? It's not that the entire liquid has the same velocity, same together they're moving, it is not like that, okay? Probably this layer has higher velocity than the lower layer. And the lower most layer could be at rest, okay? So there is some sort of shear deformation is happening. You discussed it last time, okay, great. So anyways, so basically, we are talking about this phenomena only for the laminar flow, in which we can imagine the flow as if it is happening in layers, layer after layers, okay? So in this scenario, the every layer, where down every layer will try to oppose, oppose the relative motion of the adjacent layer. This is the phenomena of the viscosity, all right? Now, this is just a phenomena and it is a property of a fluid. This property can be, you know, this property, viscosity is a property that varies from fluid to fluid. So the honey which comes in our mind the first time you hear the viscosity is a very viscous fluid. How can you say it is a very viscous fluid? You can see the way it flows. It flows very, very slowly, isn't it? Suppose you have a honey like this, there's a piece of honey and it starts moving like that. The layer below this top layer will try to resist its motion a lot. So it flows, ultimately entire honey will be flattened out on the surface, but it does very, very slowly, okay? So same thing is not observed with the water or with the oil, okay? It does vary quickly, yeah, previous slide. This one, copy it quickly. All right, so suppose I take this example, the top most surface is moving with velocity V, let's say, okay, and bottom most surface is at rest, okay? It is at rest. The reason why it is at rest because the contact surface, this is the surface which is in contact with the fluid, the surface is at rest, okay? So if the liquid has to make a contact with the surface, then at a point of contact, the motion between liquid and surface should not be there. So the point which is in touch with the surface will have the same velocity of that top surface, which is zero. So the bottom most part is at rest. The upper part is going with velocity V and let's say the thickness, okay? This thickness is, let's say L, okay? Now I have to quantify something related to viscosity, right? Now when you look at this kind of thing in solid, you will have shear modulus coming in your mind or modulus of rigidity, right? Which is what? Modulus of rigidity, you had defined it as F by A divided by what? Divide by X by L. You remember this? X is the displacement of the top most surface. Do you all agree? Okay, now here, when it comes to fluid, you know, when it comes to fluid and suppose this is a fluid body, you are applying force at the top most surface. Can you tell me how much X can be? Does X depends on the force which you are applying on top most surface or X will be almost the same every time, no matter what force you are applying? Everyone, X will always be same no matter what force you are applying or it will be different for different force you are applying because in case of solid, X was different for different forces you are applying at top most surface, talking about a shear deformation, right? In case of liquid, liquid will always, if you wait long enough, if you wait long enough, liquid will always take the shape of the container in which it is in. Or suppose you are putting it on the floor, if you come after a long time, you'll see that entire honey which was a square shape initially has flattened out and spread evenly on the surface. Same thing will happen with the water, same thing will happen with the oil. Only difference is that the spreading of water will happen lot faster than the spreading of the honey. Okay, but ultimately, the spreading if you calculate the, what is the displacement X? It will be as much as possible. So X will be same in all the liquids, okay? So the amount of displacement of top most surface, X, is same for all the liquids, right? So what we do is that, since we know that the rate of deformation is different, not the amount of deformation, rate of deformation is more relevant in case of liquid. All of you are able to understand whatever I'm telling here, a highly viscous fluid will not deform as quickly as a non-viscous fluid will be, okay? So how can you quantify the rate of deformation? This deformation I do not understand. They've got deformation is X, understood? I don't know, deformation is X or you can say no. Strain is deformation, strain is X by L. You can say this is my deformation. Amount of deformation in solid or in liquid doesn't matter. This is what the deformation is. I'm saying X by L, if you wait long enough, will be same for every fluid. The only difference is somebody will get deformed quickly, some liquid will get deformed later on. So rather than calculating the deformation, which is same for every liquid, I will worry about the amount of deformation. I don't know, that is a property of liquid. Liquid will ultimately take the shape of the container, amount of deformation will be same. But the rate of deformation, which is strain rate, if you differentiate it, it will become equal to one by L, DX by dt. Now DX by dt is what? Everyone, velocity, right? So it'll be V by L, okay? So this is the strain rate. So strain rate is more meaningful in the case of liquid, not the strain itself. Strain is meaningful for solids, not for liquids. Strain rate for liquids. So rather than defining the coefficient of rigidity for liquid, we are defining here coefficient of viscosity. Okay, which is what? It is very simple. It is stress, which is F by A divided by, rather than epsilon, I'll have D epsilon by dt. I'll have V by L. This is my coefficient of viscosity. It will be different for different flutes. Everybody understood how coefficient of viscosity is defined? What will be the units of coefficient of viscosity? Units of coefficient of viscosity. Tell me, unit of coefficient of viscosity. You can tell in terms of Pascal's also. F by A is Pascal's, F by A is Pascal. Velocity is meter per second. So it'll be Pascal's second. Numerator has L also. You can do a little bit of dimensional analysis. You'll directly get it, okay? Everyone understood? All of you understood, right? Okay, those who are joining, please mute. Now, what is F? F is what? What is F? Which I've written F here, what it is. F represents restoring force. It tells you the resistance. It is about the resistance. How much the top layer is getting resisted by the lower layer, okay? So this is what that force is, okay? Now, what if I take, let's say this much, if I take only L by two thickness, rather than taking L, if I take L by two thickness, okay? Then will I get the same coefficient of viscosity? Will I get this only or not? What would be my answer then? Everyone, rather than taking L, I take L by two, okay? I am comparing with this layer. Will I get different coefficient of viscosity? Should I get different or not? First answer me that. Should I get the same one or different one? Should it depend on the amount of thickness I am taking in a liquid? It should not, right? It should be same. It should be same because it is a property of liquid, right? It should not depend on how much liquid I'm taking. So, but then if you look at this formula of coefficient of viscosity, the way we have derived it here, instead of L, you will write L by two, because you're taking L by two length. So, the coefficient of viscosity will become half. I got a new formula. What is a catch here? What is a catch? Now, area remains same. The area remains same. It is a square thing. Area is on the top surface area. When you take the shear stress, you take the top most surface, right? L becomes L by two. Force is, see, what you got? Force is between the two layers. Any two layers, this is the force, okay? This F is the force between any two layers, which is same. F doesn't change. Then what is changing here in the equation? Yes, that is correct, Rubau. That is correct. V is what? V is basically relative velocity between two layers that you have considered, two layers that are considered. When you're taking the L, when you're taking the L length, the bottom layer has zero velocity. Top most layer has velocity of V. So, relative velocity is V only, okay? When you're taking L by two, when you're taking L by two length, the top layer has velocity V. Middle velocity is V by two. So, relative velocity is V by two. Are you getting it? So, L becomes L by two, V also becomes V by two. So, coefficient of viscosity remains same. Do you understand all of you? Please type in quickly. So, this remains unchanged, okay? All right, so I hope coefficient of viscosity is very clear. If I tell you coefficient of viscosity of a liquid, you can find out the force between the two layers, okay? And there are certain numericals based on it. So, let us see numerical on that. There is one numerical from your textbook only, which I'm sure you might have done it in your school also, but let's do it again. Anyone has any doubt, quickly type in. It's not opening up. Okay, anyways, here is the question, guys. All of you, I'll draw the figure here. I thought I could directly project it. So, you have a situation in which there is a mass over here, okay? And there is a pulley. Draw it with me, everyone. Do you do the school questions in the classroom itself, in the school classroom? Some of them, okay. See, sometimes now you have to prioritize what you have to do. You may have to wonder, should I do the school work? Should I do the symptom work? Should I prepare for the UT? Should I prepare for the monthly test? So, you know, you have to be mindful about it, okay? If you're spending X amount of time in your school, it requires three times more time than for the community exam preparation. That's how the wide it is. Make sure to do that. All right, so there is a metal block of area. There's a metal block of area 0.1 meter square, okay? It is connected to, this is a metal block, okay? It is connected to 0.010 kg, okay? Mass wire string that passes through the pulley as shown in the diagram, okay? There is a liquid, liquid of thickness 0.3 mm. 0.3 mm is a thickness. It is placed between the metal block and the surface. So, this red one is the liquid. The red color thing is the liquid, all right? Now, when it is released, everything is released, this mass is appeared to be moving with velocity of 0.085 meter per second, okay? You need to find out coefficient of viscosity of the liquid, okay? This moves with constant velocity. Yeah, you can say that this is Newton laws of motion. The entire chapter is a modified form of Newton laws of motion, okay? Metal block has, the mass is not given. For the metal block, the mass is not given, okay? Anyone close to the answer? All right, you know, there is a basic premise here. You guys should start, should continue solving question. I'll write here, every time we will not argue this, I'll write here that the assumption is, okay, Arabi got something, all right? Pascal seconds, Arabi. The liquid in contact with a surface moves with the surface's velocity. Okay, Rubau got something, Parvati got something. Yeah, something like that. Radhiyan also got it. Nobody else? No one else? Okay, so coefficient of viscosity is F by A divided by velocity divided by L. This is the formula, right? So, FL by AV, so F is what? F is the force between the two layers, any two layers, so that has to be equal to the T. Okay, the reason, what is the reason? If you draw the free by diagram of the metal block, okay? It will be getting pulled with tension T. It is getting pulled with by tension T. There is this viscous force. Why there's a viscous force? Because the upper layer is trying to move relative to the lower layer. Upper layer where it is, it is in contact with the metal. So, it tries to move with the metal. So, the layer just below it applies a force F and expression is zero. So, T should be equal to F, okay? And over here also expression is zero. So, MG should be equal to T. So, basically F is equal to MG over here. So, I can write over here MG into L. L is what? L is the thickness of the liquid. So, MGT divided by area of cross section into velocity. I can directly take the velocity of the movement that is the velocity of the topmost surface relative to the ground. Because thickness I am taking from the ground. So, A into V, which is given to us. So, MGT by AV, all right? This comes out to be 3.45 into 10 raise to power minus three Pascal seconds. Okay, everybody is clear about it. Nothing will happen, Siddharth. It is independent of it. All right, so now, you know, when we talk about the viscous flow or the viscosity, another very important thing that we will notice is the movement of object, right? The movement of object inside a viscous fluid. See, we know very well that if it is a non-viscous fluid, what will happen? We have continuity equation. We have the Bernoulli's theorem and we have many other things to take care of. There will be buoyant force, there will be pressure force and all that. But because now we have to consider viscosity also, it is a viscous fluid. There will be another force, additional to the buoyant force, gravitation force, pressure force, additional to all these forces. There will be one more force, which will be viscous force. Fine, and this is what we are going to include now. All right, and in the case of Bernoulli's theorem, do you guys all remember that we had assumed that the viscous or the viscosity is zero? All of you remember that, right? So, viscous and non-viscous, what exactly is the difference between viscous and non-viscous fluid? Rubber viscous fluid has some coefficient of viscosity, non-viscous fluid will have coefficient of viscosity zero. And if coefficient of viscosity is zero, look at the formula for coefficient of viscosity. If coefficient of viscosity is zero, F will be zero, right? So the force between the two layers will be zero. Or you can say that viscous force is like friction force between the two layers, okay? You know, you can treat it exactly like that. So if there is a relative velocity between the two layers, friction will be there in case of solid, right? Similar thing for the fluid also, okay? And when you are applying the viscosity, you are assuming that laminar flow is happening, okay? You are not taking a turbulent flow here, okay? Yeah, in viscous is like frictionless surface. You can say like that, okay? So now, focus here, everyone. Suppose you have a bucket, okay? A deep bucket like this. It is filled with a fluid, okay? Fluid is at rest. The fluid is at rest. Fluid is not flowing. But then suppose I drop an object inside it, okay? Suppose I drop some random object inside it, and it starts to move inside with certain velocity, okay? Because it's density is a lot higher than the density of the, density of the liquid and density of solid. Density of solid is very large compared to density of the liquid. So it starts sinking in, okay? Now, why it is sinking in? There will be a gravity force. There will be a buoyant force. And suppose it is a viscous fluid. Will there be a viscous force, everyone? Will there be a viscous force on this object which is sinking down? If yes, why? And how? How it get developed? Why viscous force is there? Don't tell him because it is a viscous fluid. Of course it is. It moves the layer as it follows. Good, right? Near the object, surface liquid has velocity, good. So yes, now you got the point. So the layers which are in contact with the solid, the layers which are in contact with the solid, these layers are moving with the solid, all right? So there is a relative velocity between the two layers. The layer adjacent to the liquid, layer adjacent to the layer which is in contact with the solid, okay? There is a relative velocity. What I'm trying to say is this, things are very simple, okay? This layer which is in contact with this surface is moving down. This layer is also moving down. But the layer next to it, this one, was not moving. This one is not moving, right? So there's a relative velocity between these two layers, okay? And because of that, there will be viscous force, all right? But if the coefficient of viscosity is zero, viscous force will be anyway zero, no matter there is a relative velocity between the layers or not. If coefficient of viscosity is zero, then it is zero. Anyways, all right. So now the thing is that there will be a viscous force. Can you tell me the viscous force on it, on the object that is moving inside depends on what? Depends on everyone? What should it depends upon? Of course, coefficient of viscosity, it should depend on coefficient of viscosity, all right? Second, what else should it depend upon? Should it depend on the mass? Should it depend on the mass? What do you think, everyone? Does liquid care about the mass when it is applying the viscous force? Does it really matter to the liquid? It doesn't matter, it doesn't depend on it, okay? So does it depend on the area of contact? Area, can I say area? So, you know, when you say area, it also depends on how the area is oriented, right? Whether you're talking about a sphere, a rectangle, a triangle, right? So I would say geometry here, geometry, right? What else? Okay, somebody said velocity. Will it depend on the velocity, everyone? Viscous force, does it depend on the velocity with which the object is moving? Everyone, should it depend on the velocity? Yes, it is, velocity of the object. Surface area and everything comes under geometry, all right? Now, over here, this eta is fine, okay? Eta is fine, coefficient of viscosity is called eta. This is a Greek letter. Eta is fine, velocity of object is fine, but this geometry, because of its dependence, because of its dependence on geometry, the viscous force will be different for different shape and sizes. Shape and sizes. So it becomes very difficult to create a formula that is valid for every shape and size. In fact, it is not possible, okay? So depending on every geometry, there will be a formula, okay? There was this guy called Stokes, all right? There was this guy called Stokes who had derived the formula for the spherical geometry. The viscous force on the spherical object. This guy has found out that, all right? So we will talk about the viscous force on a spherical object, which goes inside the liquid of coefficient of viscosity eta and with velocity V, all right? Surface area comes under geometry or shape. Everything is under geometry, all right? Can you tell me one thing? If velocity is increased, if I increase the velocity of the object, which is sinking down, will the viscous force increase, decrease or remain same? If velocity with which the object is sinking down, if that is increased, that is increased, then the viscous force will increase, decrease or remain same, everyone, it will increase, guys. Why it will decrease? Why it will decrease? Tell me the reason, tell me, tell me the reason. If velocity increases, the force also increases. You remember eta is what? What is eta? Coefficient of viscosity is fL by AV, okay? So if velocity is increased, f has to increase, right? So you can roughly say that, yes, if velocity keeps on increasing, viscous force keeps on increasing, all right? Everyone understand, right? Okay, so there was this person Stoke, who has given us Stokes law. What does it talk about? It tells us, it gives us the viscous force formula. Sorry, viscous force equation for a spherical object, a spherical shaped object, okay? Now that object could be hollow inside or whatever it is, it doesn't matter because viscosity, viscous force between the object and the liquid totally depends on what kind of surface it is, surface area, geometry and everything, okay? So Stokes law gives us the viscous force to be equal to this. All of you write down six pi eta r into v, okay? Where in your textbook, they have taken radius as r or a, let me just find out, they have taken a. So we'll write down radius as a, okay? So this is the viscous force between the solid, a spherical shape object having radius a, a is a radius, v is the velocity with which the object is going down, okay? So this is the viscous force only, okay? This is not the net force. Apart from this force, there'll be gravity and buoyant force also, all right? Now let us try to analyze what kind of motion it will have inside a liquid which is filled in a container like this, which is, you can say the height of the container is very large, infinite you can say, okay? So suppose this is a container, liquid has a density of rho, coefficient of viscosity is eta, this is the property of the liquid and you have a sphere, a metal sphere that is dropped inside, which has radius of a, okay? You can say density of this is sigma, sigma is large compared to the rho and it is a solid object so it will sink, okay? Now, while it is sinking, its velocity will increase or not? Everyone, initially while it is sinking, the net downward force is more than net upward force or not, everyone? So its velocity will increase, velocity will keep on increasing in downward direction, okay? As a by the way, viscous force is always, this is always opposite to the velocity, okay? It doesn't matter where is gravity or whatever it is, it only depends on which direction the velocity is, viscous force will be opposite to that velocity, which velocity it is, it is a velocity relative to the liquid in which it is moving, okay? Most of the time liquid will be at rest so you can say the velocity is velocity of the object only. So viscous force is opposite to the velocity. So now this velocity keeps on increasing. Now look at the expression of the viscous force. Will this viscous force keeps on increasing or not? Because velocity keeps on increasing, okay? It will keep on increasing. This is viscous force, okay? Apart from this force, apart from the viscous force, there will be two other forces. The forces are gravity, mg, and you have a buoyant force also, okay? Buoyant force. Will mg and buoyant change depending on how deep the object is sinking? Will mg and buoyant force depends on what is the velocity and how deep it is? No, it doesn't depend on it, right? But viscous force keeps on increasing. So can you describe the motion of it object when it is sinking down? Tell me, what can you tell? How does it move? Everyone, tell me about its velocity, how it will be? It goes down with constant acceleration. Somebody is saying, okay, acceleration is constant. Don't you see that viscous force? Don't you think that it will keep on increasing? Viscous force is increasing as a velocity is becoming more and more. It will fast at the start and then slow down. What is fast? Velocity is fast or acceleration is more? What do you mean by that fast? Initially, net force is in downward direction. Viscous force initially was zero almost because velocity was very less. So downward force was very large. When viscosity keeps on increasing, do you guys see that downward force keeps on decreasing because viscous force is upward? If it increases, net downward force reduces, okay? There will be a time that the velocity, velocity can become large enough, becomes so large, downward force becomes zero. Can it happen by the way? Can it happen? It will happen if the bucket you're considering is deep enough because before even reaching that velocity, suppose it hits the bottom surface, it will hit the bottom surface and stop, getting it. So if it is deep enough, there will be a time where velocity will be so high and viscous force will become so much so that net force will become zero. And if net force will become zero, what I can say about the acceleration? What is it? Say acceleration is what? Zero, it gives us constant velocity. This constant velocity which is slowly achieved in this process is called terminal velocity. After a long time, this velocity, constant velocity is reached. It is called terminal velocity referred as VT, okay? I want you to find out what is VT, okay? Viscous force is given to you and every other parameter is given. Mass M is not given, okay? You can consider mass M as density into volume. So can you find out what is a terminal velocity formula? Everyone, derive it, it is from your school, okay? Your school exam, this is one of the favorite question they ask. Let's say that velocity is VT. Just equate the forces at that moment. Net force becomes zero, so net downward force is equal to net upward force. Should I solve or should I wait? Everyone, okay. I'll give one more minute. Everyone get the answer. Very important for your school exam, if not anything else. Shethi's got something, others. You haven't done this in school? Not done. All of you have written this. Mg minus viscous force minus bow and fore should be equal to zero. M is what? The density of the solid into volume that is four by three pi a cube into G minus, huh, those who have got it, that is correct. Viscous force is six pi eta a terminal velocity is VT minus bow and force, which is rho times four by three pi a cube into G. In the bow and force you take density of the liquid, isn't it? So pi will get canceled away throughout. Then this will be two, this will be three, this is two, okay. So from here terminal velocity, you can see you'll get two by nine, two by nine a cube into G sigma minus rho divided by eta and one of the a's also gone like this, a square, okay. Oshik, we have done it. Derivation of bow and force, we have done it. Look at the first video of the fluid mechanics, okay. Everyone, so this is the terminal velocity, okay. So we had only, we have in our syllabus only a geometry of sphere to be considered, okay. For other geometries, we don't have it in our curriculum, okay. So only sphere we have learned in which we can find out the viscous force between the sphere and the liquid when the sphere is moving inside the liquid, okay. There is something called Reynolds number also that is defined with respect to the viscosity only. So write down and somebody send something. Let's write down this Reynolds number. Yes, not buoyant viscous force. Okay, do you talk about a Stokes law? Derivation of a Stokes law? No, the derivation is not there, okay. It is empirically found. You can plot a graph and you know, it's not there. You just assume that this is the viscous force, okay. All right, so Reynolds number, there was this person. Name was Osborne Reynolds, okay. So in his name, on his name this Reynolds number is given. Now what does it indicate? Reynolds number will tell us this Reynolds number is RE. Reynolds number can tell us if the flow is laminar that is streamline or turbulent. Looking at the Reynolds number, you can tell that, okay. Now I'll just tell you the, you know, thought process beyond the Reynolds number then we'll discuss what it is, okay. First of all, you tell me if viscosity of some liquid is very high. Let's say I have, I'm telling you that I have two liquid, liquid one and liquid two. Velocity is same. The flow velocity, they're flowing in a pipe, okay, velocity is same. This one has coefficient of viscosity eta one. This has coefficient of viscosity eta two. Eta one is large compared to eta two. And I have told you only one flow is laminar. Which one that can be, which has higher viscosity or which has a lower viscosity, which has more chance of being laminar. That's what I'm asking here. What do you think? There's a split, I could have taken a poll but now it's okay. But Siddish, can you explain why higher viscous has more chance of being laminar flow? What is the reason? That you're saying that high viscosity implies that it has more chance of becoming a laminar flow. See, it is, if not laminar, consider viscosity as in separation is lesser, okay. Good. So see, if well, a viscosity is high. If viscosity is high, it means that the layers, they don't like to get separated. They want to move together, okay. They will try to maintain some sort of sanity in the movement. Together they will move. They'll not start moving zigzag manner, all right. Now you can imagine the, let's say in a pipe, water is flowing in one pipe and honey is flowing in another pipe, okay. And both the pipes velocity is very high. So now you will understand that the smooth flow will be there in the pipe in which honey is there, which has higher viscosity, okay. There is a chance of having a turbulent flow with the water because it has a lesser viscosity. So viscosity actually promotes the laminar flow because the layers will try to be together, okay. They will move in a defined manner, okay. Do you understand this logic, everyone? So good. So if viscous force is more, so viscosity is higher, more chance that it is a laminar flow. Chance of laminar increases, okay. Now suppose you have two liquids, liquid one and liquid two, which has the same coefficient of viscosity, but their velocities are different. Same liquid you've taken into pipes, similar pipes, their velocities are different. V one is more than V two. Where is the chance that laminar flow will be happening at higher velocity or at a lower velocity? Lower, right, lower. Okay, so this is very clear. So if velocity is lesser, then the chance is that it become laminar will be higher, okay. So you have two things to look at. You have viscosity to look at and you have velocity to look at, all right. You cannot just look at the velocity and say that it should be a turbulent flow. What if viscosity is very high? Then there's a chance that it can be a laminar flow. And at the same time, you can't just look at the viscosity and say that viscosity is very less. So it will be a turbulent flow. What if velocity is very less? There's a chance that it can be laminar. So it becomes complicated to look both, right. So that is the reason why both of these factors are combined together. Both of the factors are combined together and Reynolds number has been put forward. When you look at the Reynolds number, you're accounting for both the variables together, okay. So it is, you know, as good as, suppose you have to look at a variable X also, a variable Y also. What you do is that you define a variable Z which is equal to X by Y. And rather than looking at X and Y separately, you just look at X divided by Y and depending on that you can say. So whatever it is, let's discuss it again here. So Reynolds number is basically a ratio of, it is a ratio of the inertial force right down by viscous force. I'll come back to it. First write it down, viscous force. It's a ratio between the two forces, okay. Inertial force represents velocity in a way. Viscous force represents coefficient of viscosity in a way. Okay, so in a way you're taking the ratio of two things and it will be dimensionless, no dimension or no units. Okay, now what is this inertial force and what is this viscous force? Inertial force is simply, you know, it is rate of change of momentum is inertial force. Simply put, ignoring the viscosity, it is rate of change of momentum. Now, how do you find out that? If you consider a pipe in which liquid is flowing, let's say if liquid is flowing with velocity V, okay. Then DM by dt is what? DM by dt we have found out earlier, last class only, rho A V. All of you remember this, DM by dt? Everyone? Okay, so if velocity is constant, V DM by dt is your rate of change of momentum. Rate at which momentum is changing is rho A V square. Clear to everyone, this is called the viscous inertial force, sorry. This is the inertial force, rate of change of momentum. Okay, see I'm telling you how it is derived. I could have directly written, but then I thought I should tell you that's the reason why. Now the viscous force can be written in terms of eta. Okay, eta is F by A divided by V by L. All right, so this is F L by A V. So the viscous force can be written as eta A V divided by L. So all you have to do is to take the ratio between the inertial force and the viscous force. You'll get the Reynolds number, okay. So Reynolds number, if you take a ratio rho A V square divided by eta A V by L. So Reynolds number will become equal to rho V L by eta. All right, and if the pipe's diameter is D, if the diameter of the pipe is D, you can write the Reynolds number as rho V D by eta. Okay, now this is a variable which you can use to find out whether the flow is laminar flow, streamline flow, or it is a turbulent flow. Anyone has any doubt in this derivation that we have done quickly? Tell me, by the way, this is there in your textbook, okay. It is there in your textbook under Reynolds number. Somewhere hidden, you just have to read it carefully. Type in quickly, have you understood this? Okay, then all right, so this is a number. Okay, why it is called number because it doesn't have any units. So it is found out, okay. It is found out that when Reynolds number, wait down, that when Reynolds number is less than 1000, it is a laminar flow. When Reynolds number is greater than 2000, it is found to be turbulent in nature. This is turbulent flow, okay. And Reynolds number between 1000 and 2000. Don't ask me exactly at 1000 what happens, exactly at 2000 what happens, okay. It is, it gives you a range, okay. It is, it's not that a very sharp line is there, that one point this side laminar, one point that side turbulent. It doesn't suddenly transition. Just give you an indication, okay. So Reynolds number between 1000 to 2000, it is a transient flow, transient or you can say unsteady flow. The flow is getting transitioned from laminar to turbulent. It is, you can say half laminar, half turbulent, something like that you can say, okay. These ranges you should remember and this is true every time. This is true, not just the flow in a pipe, this is true for any kind of pipe. Let's say you're considering a pipe which has, let's say cross-section like this, a square cross-section and water is coming out through it. All right, so you can use a Reynolds number formula, rho VD divided by eta and find out the Reynolds number, okay. What you'll take D as if you have certain irregular shape, let's say hexagon is there instead of circular cross-section. If there's circular cross-section, you will take D as diameter. But if it is not, then what do you take D as? If it is not a circular, then what do you take D as? Then you'll take D as hydraulic diameter. It is given as four times area of cross-section divided by perimeter, okay. Four times area of cross-section divided by perimeter. In case of circle, it comes out to be two R, which is diameter only, okay. In case of any of the cross-section, you can find out hydraulic diameter to be equal to four times area of cross-section divided by the perimeter, all right, and use that over here, fine. Anyway, so there is a numerical on whatever we have just done. Let us take that from your textbook. Anyone has any doubt, quickly type in, meanwhile, why we use diameter again? See, it is not that you have to use diameter, only you could have used radius. Then Reynolds number range will be different. Reynolds, if you take radius, Reynolds number range will be less than 2000, greater than 4000, okay. Just that we have assumed, let's take diameter. Then the numbers will be changed accordingly, all right. That is not required, a shortage, that discussion is not required. Just take it as it is, hydraulic diameter. You learn that in engineering, this is flutes. But any, that hydraulic diameter is anyway not part of your curriculum itself. I thought I could just tell you because you might be wondering what if the cross-section of the pipe is not circular. So just additional information. Okay, I want you to solve this, solve this question. Okay, do it completely yourself and do it. All of you should do it. You will see the kind of calculation errors probably that could creep in. You need a lot of practice in calculation itself. Forget about the concepts. It should be good with numbers. So I'm seeing that a lot of you are making silly errors in calculation that require some practice. Are there any liquid with zero viscosity? You can say near zero, okay, some of the oils. But yeah, that is an ideal scenario. That doesn't happen naturally. Basically they're asking you whether the flow is laminar or turbulent. So first tell me the Reynolds number. You can modify the equation of Reynolds number in terms of flow rate also, if you want. Anyone close to the answer? Okay, Harabi got something. Others, did you get any notification from school reopening? As in physical classes? Do you want school to get reopened? No. Except start at 10 a.m. Now it will be very difficult to wake up early and attend the classes, travel and attend the class. See the main problem you guys face is that you have not seen the hard work which is required along with school managing the symptom and the comedy exam becomes more difficult for you. Okay, because you're anyway complaining that the time is very less. Now when school starts, travel is there. You get tired also. So you're not used to that kind of rigor. So probably that may have some impact. Anyways, Reynolds number is four times. See, I have modified the Reynolds number formula in terms of flow rate. Do you all see that? Why have modified? Because flow rate is given to me. Isn't it? That's the reason why. Density is thousand. What should I write instead of q? I have to write in meter cube per second, right? So 0.48 into 10 raise to power minus three that is meter cube. This is per minute. Per second will be divided by 60 into pi into eta that is 10 raise to power minus three into dia that is 1.25 10 raise to power minus two. Anybody calculated this? What is this equal to? See, do the calculation yourself. I'm telling you again and again, I can't emphasize it enough how important it is. Do not look at the textbook and all that. It's very easy to fool yourself, okay? And make you believe that, yeah, you got it. Do it yourself. What is the answer you're getting? Even if you get the wrong answer, it is fine. It is, at least you've tried it. Calculate this quickly. All of you, let's see how well you can deal with the numbers. Everyone, all of you understood, right? How it comes, no one has any doubt in that. Understood, all of you, this is understood, right? Whatever I've written, okay, get the answer. You want me to solve? Parvati got something, Arrabbi got something. Nobody else got anything other than these two. Anyways, 10 raise to minus three is gone. One zero is gone. It's 12.8, is it? It is 12.8, that less. What can it be? 81.56, Gurman is saying, okay? So this is four into 48 into 10 is part two divided by six pi into 1.25. How much it is equal to? See how pathetic you have the calculation skills. You're not able to, everybody's getting different if I answer. This is very bad. It will impact you during the exam. All right, anyways, so this will be 815.2, roughly. If you do your calculation properly. So you can be proud of yourself, look at this. You're not able to multiply or divide, add, subtract. It'll not lead you anywhere. So make sure you are practicing a lot of calculation here and otherwise someone come from grade five will beat your hands down because of the calculations only. All right, so this is the Reynolds number. So this is laminar or turbulent? Everyone, laminar or turbulent? Laminar, it is less than 1,000, right? Now, if the flow rate is increased to three liter per minute, what will be the Reynolds number then? Do we need to calculate it again all over again? Reynolds number will be what? If the flow rate becomes three liter per minute. Do we need to calculate it again? Everyone, what should you do? See Reynolds number is proportional to q, right? Proposal to q. So three divided by 0.48 times 815.2, like this you can do. So 0.48 flow rate, Reynolds number is 815. So for three flow rate, it'll be three divided by 0.48 into 815.2, okay? So it is roughly six times of the earlier this thing. So Reynolds number will come out to be around 5060. Okay, 5175.5, okay? 5175.5. Now, this is a turbulent flow, right? So if you increase the flow rate to this much, your flow will become turbulent, okay? Anyone has any doubt on the viscosity, whatever we have done till now? Typing quickly, is it clear to everyone?