Fluid Dynamics Lecture 1 - Introduction

Added: 2026-05-12

[00:00:01]Okay. So, hi guys, welcome to my channel, the harmonic oscillator. So, this is my first video and uh so first I will give a brief introduction about the channel and you know uh what is this channel about. So yeah in this channel I am planning to start a series called the completing the book series. In this series uh we will be taking certain books and you know completely finishing that book. So typically the idea is that uh in normal post-graduate or undergraduate studies uh in most of the universities and I think in almost all the universities in India so how it works is like uh the professor or the teacher uh he or she will be following some book and some materials and they will cover let's say two chapters, three chapters maybe four or five or maybe half of the book because in one semester you can cover only that much and the rest of the thing will be you know um taken for the student to study on her own or maybe his or her own or maybe it is most often it is completely ignored.

[00:01:25]So the idea of this series is that to make a well uh you know a series which has a complete where we will complete the entire book books like for example here I'm starting with fluid dynamics okay so in fluid dynamics that's a wonderful book called by London and lipstick u so we will be covering the entirety of that book from starting to end it along with the problems in that book that also we will be discussing.

[00:02:00]So along with this uh one so I will also parally run two more courses. One is string theory for undergraduates.

[00:02:19]It's an undergraduate level course but uh you need some amount of you know higher under like a senior undergraduates or you know some yeah even postgraduates can also do that and here we will be covering a book written by student professor professor uh button back lot of you will be familiar with him because he has done some lecture series wonderful lecture series in quantum mechanics his quantum mechanics 1 2 and three are really good and his book on mastering quantum mechanics is also a really nice book if you want to study quantum mechanics anyways he will be covering this whole book string theory for undergraduates Okay. U written by Martin and then introduction to smooth marold.

[00:03:25]Introduction to smooth marles.

[00:03:34]This is a quite difficult uh uh postgraduate level mathematics course.

[00:03:40]Okay. So this is written by JM Lee. It's a very good book and it's a very big book. So I don't think I can finish that book in one year. So see this the plan.

[00:03:53]So for the fluid dynamics and study theory I will try to finish this by next year's May. Okay. Hopefully if everything goes well but the manifold part basically this is a course on geome differential geometry. So this is a course on differential geometry.

[00:04:16]So I'll try to uh finish this in two years. So maximum it might go by 2 years. So that's the main idea. So I will be parally running these three courses. Okay. So whenever I get time I will upload videos and so you'll get a good documentation of the entirety of this and this London lipstick. So speaking about this London lips so if you are a theoretical physicist or an aspiring theoretical phys or a student physics student so you might have heard of this series by Londo and it's basically a 10 book series. So this series has 10 books starting from classical mechanics, classical field theory, quantum field theory. The book title is not quantum field theory but it is called quantum electronamics. Then you have thermodynamics, statistical mechanics one, two, kinetic gases lot of so so the main idea of that is that if you are a theoretical physicist so this is the minimum amount of knowledge that you should have in the physics subject. So ideally you should be knowing all the 10 books thoroughly completely right but most universities won't cover this. So so I will try to cover as much as possible. So once this fluid dynamics part is over I will take up on some other you know some other uh one from the book. So London is here for a long time. It will be here for a long time this channel. So I'll be covering one book by book and complete the series.

[00:05:56]Okay. And the rest of this we will see what can be done.

[00:06:02]Okay. So so that was about fluid dynam sorry that was about this channel and what all I'm going to do in this channel. So now let's start with the topic. The topic this title has two things. one is fluid and other is dynamics. So as usual I'll be starting with a very basic okay introduce what is fluid dynamics but uh there is going to be math in this course a good level of math but not too hard but you can think of it a good level of math will be there. So it has two terms one is fluid and one is dynamics.

[00:06:50]Okay. Now what does these two terms mean?

[00:06:55]Dynamics. Okay. First you will look at the second term. So what is dynamics?

[00:07:01]So you might have heard this. You might have studied this electrostatics.

[00:07:08]So what what did we have in electrostatics?

[00:07:12]Statics. So in electrostatics uh the charged particle was at rest, right? So that is the main difference between static is a static thing system and a dynamic system. In a static system the things or the particles under the consideration or the system under the consideration will be in rest or the particles will be in rest. Okay? It will not be moving. Once it starts moving on the charge starts moving it will create magnetic field. It can create radiations etc etc. So a lot of other phenomenas you can see right?

[00:07:48]So that is a field of electronamics.

[00:07:51]So dynamics means there is some motion.

[00:07:55]There is some motion or you see something is moved.

[00:08:03]Okay. A fluid is some so what is a fluid? So vaguely speaking fluid is something something that flows.

[00:08:17]So what do we mean by a flow? So we will come to what do we mean by a flow in a bit. Okay. But first see in school we might have seen some fluids. So there are this there was this classification of matter right. So we had solids, we had liquids and we had gases. These were the three states of matter. Okay. Uh that we studied in school in early schools. So let's ignore solids for now and look at liquids and gases. Liquid flows. So in literal sense you know in English sense you can think of it a liquid flows a gas also flows.

[00:08:59]You can imagine right physically what do you mean by a liquid which is flowing right? And physically what do you mean by gas is flowing? So these two things flows. So liquids and gases they flow.

[00:09:14]So hence these two both of them are fluids.

[00:09:22]So but these are not the only fluids.

[00:09:24]Okay. So there are lot of fluids in our system in our nature in our universe. So what is a fluid? So I'll start with the definition of it. So definition the definition that we see is a fluid is anything that has a flow.

[00:09:55]Now you can ask the question what do you mean by flow? So then what do you mean by a flow?

[00:10:08]So what do you mean by a flow? So by flow okay I think literally you know what is a flow but I would like to make it more precise. Okay, but I won't go it as precise as a mathematical definition because that is something that we will cover in differential geometry at some point. Okay, but I'll give you an idea of what how what the definition goes like. So suppose you have some surface or you know you take any mathematically you say it's a manifold or you can think of it as some surface let's say surface of a sphere or surface of a you know rectangle sorry surface of a cuboid or something okay the surface okay and in surface I'm drawing some lines okay and these lines I call them vectors okay these are some lines and these are lines which has direction right so these are vectors and they have direction okay now let's say this is some point okay now as we move from as we move from here to here the vector is changing right as you move from here to here point to P2 the vector is changing and it could also see that you stay at one point. Let's say P is the point that you're staying. At one moment the vector could be or this line could be facing in this direction and at the next moment it could be facing in this direction. So essentially all these collection this collection all these vectors or all these lines or rays are determined by two quantities. So X denotes a vector.

[00:12:04]So I will put XAR.

[00:12:06]It depends on the point P and also some parameter T. You can think of T as a time parameter. So at t=0 as I said at t=0 the vector might be like this. Okay.

[00:12:19]And at t = to 1 it might be like this.

[00:12:22]So vector change right? So that vector or that thing is now this collection.

[00:12:28]So if you find this thing throughout the manifold or m is the surface of the manifold.

[00:12:35]M basically you can think of as this object surface of the this surface. So this thing okay so if you are able to do this then you can say that there is some flow there.

[00:12:51]There is some flow on that surface.

[00:12:54]Okay. So for example uh let's say water.

[00:12:58]Okay. At every point in the water you can associate a vector right. That will be the direction in which that particle or that point is moving. Right? So that is a so a water is a flow. So it has a flow.

[00:13:13]At every point you can associate a vector. Similarly wind suppose you have this air where wind is flowing. Okay. At every point you can associate a vector.

[00:13:24]That vector is the velocity okay of the wind particle. Right. And that will give you the direction in which that wind is flowing or that point in the wind is flowing at that instant of time. Okay.

[00:13:36]So that is what we mean by a flow. So a fluid is anything that has a flow. And now I have also defined what do you mean by a flow. Right? Now the question is okay I understood what a flow is. I understood how you define a fluid by flow. Now then why are we learning this chapter like why why do we have to study fluid? So why okay and this answer to this is very tricky you will see okay okay so thing is this fluid dynamics fluid dynamics is a universal thing so that is a statement that I making and I will explain that statement in a It means uh universal means you take any object anything okay any fluid okay and that will obey the same set of equations. So basically what happens is that uh fluid dynamics in fluid dynamics you have some set of equations you have lot of equations in fluid dynamics I mean lot in the sense there are three four so these equations are governed by something known as conservation laws these conservation laws are not some unfamiliar to you have been studying this uh since school conserv conservation of mass, conservation of energy, conservation of momentum, etc., etc. So, you have this conservation laws.

[00:15:26]So, so you have few equations based on this conservation laws because you don't have infinitely many conservation laws, right? You have very few conservation laws and almost everything in the universe which has a flow. Okay, you don't have to modify the equation. You can take the exact same equation and apply it there. So that is the power of fluid dynamics. So I'll give examples to support the statement. So for example, so you take a container. Suppose you take a container and in that container you are filling with some atoms. Let's say you are filling with some hydrogen atoms or helium atoms. Okay. And now what you're doing is that you are putting that on fire. So let's say this is some fire and there is some fire here. Okay. So what will happen because of this fire? Okay. Uh these particles fluid particles here or these atoms or hydrogen atoms here will gain some energy and will start moving move randomly. So some random motion or maybe some periodic motion some kind of motion. So there will be a flow here.

[00:16:36]So because of heating it so some flow is happening. The particles are moving.

[00:16:42]This equation this system could be described by the equation of fluid dynamics. So instead of this atoms let's say now you filled the container with water.

[00:16:53]So this is water and now again okay water. This is H2 and then you again put fire on it.

[00:17:02]Okay. So you put fire on it. You again put fire on it. Now what happens? You have seen this phenomena I think. So there will be some small small what bubbles which comes here and it goes up it sometimes get evaporated and sometimes it get condensed. So there's a lot of motion which is happening here.

[00:17:25]Right. So there is a flow here also.

[00:17:29]Okay. This whole system again can be described by the set of equations which are nothing but the equations of fluid dynamics.

[00:17:40]Okay. So we have seen examples of atoms which are very small and we have also seenam an example of water which is a very observable quantity right now I will go to some other extreme example which is okay. So let's say you take a neutron or a proton it can be anything you can take neutron or proton. What is inside this? Inside this there is something which is called quarks.

[00:18:14]So there is something known as quarks.

[00:18:16]The quarks are very tiny particles. You can think of it as tiny particles. The point is that in the present theory so in the theory of the standard model quarks are you know one of the fundamental particles that we have in nature as of now. So in previous in school you might have seen that atoms are fundamental most fundamental and then you saw that atoms are divisible electrons protons neutrons and now you see inside neutrons and protons also there are this quarks so presently according to the theory that we have now quarks are fundamental electrons are fundamental and there are other bones also which are fundamental etc but I'm not going into that discussion that is for some other Okay.

[00:19:06]Again, you put these quarks inside a container. So, let's say these are quarks. These are very very tiny. These are smaller. These are even smaller than the nucleus, right? It's very very small thing and then you put fire on it or you it. Okay. Now, what will happen again?

[00:19:27]It will have some kind of flows, right?

[00:19:31]So it will start moving and this flow the flow here is also governed the the equation that governs the flow in this case is also the equation of fluid dynamics. Exactly same equation. The exact equation that governs the uh motion of water governs the flow in this case also. In case of quarks also in case of atoms also you see these are completely different entities right you have qu uh this electrons or atoms and water and it's a very interesting you know observation you have there is another very exotic and very very interesting example and that is that of a black hole okay Now then if uh since most of you might not be familiar with it, I will explain what a black hole is briefly but if you if you know u general activity then you might be able to uh appreciate it much better.

[00:20:46]You see there is something known as space time.

[00:20:52]Okay. So the idea is that space and time are not distinct. So for example suppose you something happened. Let's say let's call it as an event. Okay event A. Now how do you say that that event A happened. So you say that this event A happened? By specifying the place at which it happened and also the time at which it happened.

[00:21:20]So if something happens in the universe, you need two things.

[00:21:27]One is the position in which it happens and the time in which it happens. Right?

[00:21:31]That is how you describe that system.

[00:21:34]Only position will not describe it. So another point is the space and time.

[00:21:41]This place and time are not different objects. They come in the same they are they are part of the same object. Okay.

[00:21:53]Is it fine? So they are not listening.

[00:21:54]They are connected to each other. They are connected to each other. for example.

[00:22:00]Uh so yes, you have some equation which uh you know corresponds to space and time and time and space something like that and there is something known as a singularity in that equation.

[00:22:18]Now what do you mean by a singularity?

[00:22:20]So in a layman term not not a layman terms but in a very simplistic term. So in school you have learned this right?

[00:22:30]So suppose this is some function f of x and now you look at this point at this point a function is not differentiable right because you know whenever there is a sharp corner it is not differentiable.

[00:22:45]So that is what here we call it as non-s singular or not non-s singular that is the point of singularity.

[00:22:55]So singularity is non- differentiability or it can also be discontinuous. You can also have situation like this. So yeah situations like this. Okay. So here there is no continuity. You have to lift the pen to draw this graph completely.

[00:23:15]Right? So there is no continuity. So this is discontinuity.

[00:23:22]This also you can consider as a singularity. Okay. So there is something called a singularity and there is something called a space time. Okay. A singularity is basically this. So you have some you know discontinuity or non- differentiability in in the space or the function. And then there's something known as a spacetime. And what is a black hole? A black hole is a singularity.

[00:23:49]a singularity singularity in space time a black hole is a singularity in space time so I will explain what does that statement mean so suppose a universe is a circle mind you a universe is described by both space and time not only space and let's say it is a circle and it obeys this particular equation so that means the universe looks like this right so now so this is the universe x² + square= to 1 is universe so this is the x coordinate and this is the a coordinate now suppose you are moving in the universe so you are moving like this okay so there is no problem so you move you it's a very smooth graph right a circle is very smooth you can move throughout the circle okay in this arrow in this purple arrow okay without any trouble Right? There is no issue that happens there.

[00:24:51]Now suppose in the same circle I am putting a hole a tiny hole here.

[00:25:01]Now what is now let's say you start from here you're traveling you're traveling you're traveling you're traveling you're traveling. Okay you're moving like this and at this point you don't know what is there. Okay you can't go further. Okay.

[00:25:16]But yeah, you can't go further. So, and you don't know what is there. There is nothing there or there is something there. You don't know. Okay. So, this part, okay, this here is known as the singularity in space or in in our case.

[00:25:39]Okay. So, that is a blackboard. So this is what you call here a black hole.

[00:25:47]Okay. So that is a black hole.

[00:25:50]Now okay. So that is what a black hole is. But why did I say this? It's because there is something which is there is one fantastic result which is discovered by Hawking is known as the Hawking radiation.

[00:26:08]So what it does is it associates black hole radi. So it is basically black hole radiation. So black hole radiation means there is some sense of temperature. So you can give a sense of temperature to the system.

[00:26:25]Okay. So once you give a sense of temperature to the system what happens is that so you have some set of black holes. Okay. and then you are in a way heating or you are giving temperature because of this these black holes will flow. Okay, mind you these black holes are not particles.

[00:26:49]Black holes are not particles. The black holes are this kind of pores in space time and but still when you heat it okay it will have some flow and because of this this will obey the same equations of dynamics.

[00:27:17]Okay. So it does not matter which system. So in previous cases at least you can argue oh these are particles and there are when when you give heat to particles it will gain some energy and because of that it is moving and therefore it is you know oasis but what about this black hole? So there is no particle here technically. Okay. So you don't have a particle here. It's just a space time and there some force in it.

[00:27:47]Okay. And then you have some temperature there and because of that it is moving and that is also described by the same set of equation.

[00:27:56]Okay. So the conclusion that we have here is that a lot of phenomena phenomenon in the universe could be described by the same set of equations.

[00:28:24]And this here is the idea it's a very beautiful idea of universality.

[00:28:31]So this is what we call very very important idea of universality.

[00:28:43]So it basically means that the same set of equation okay described a lot of phenomena in the universe. You can describe thousands of phenomena using the same set of equations.

[00:28:56]This might seem trivial to some people.

[00:29:00]This might seem uh you know very out of the box for some people but this is the fact and this is a very very beautiful fact in physics at least theoretical physics that we have.

[00:29:14]Okay. Now you can ask the question why does this happen?

[00:29:20]Why?

[00:29:21]Why does you know the universe or lot of the phenomena in the universe? Yeah. Why does this idea of universality come into being?

[00:29:32]And the answer to that question lies in the way in which we make the theory. So that is the idea of average.

[00:29:50]Okay. What does it mean that? So we don't care we don't care about the microscopic topic details.

[00:30:07]How do we make a theory in general? So as a physist physicist or theoretical physicist you are being asked to make some theory. Okay, theory of something.

[00:30:18]You are given a system and then someone asks you to make a theory for that particular system.

[00:30:24]So how do you make that theory? Okay. So the first thing that you'll go about is that you look at the observations that you have and think about how can you describe or you know how can you make that observations how can you make some theory which will describe that observation.

[00:30:42]So the theory that we construct has a constraint that it has to obey the observations or it has to the predictions that you get from the theory could be observed. So if you are not able to observe it or even if you are not able to put make an experiment to observe it then you can't be sure whether that theory itself is correct.

[00:31:07]So, so you care, so what we care about is the observations observations.

[00:31:18]So in theory, so for example, here we have general activity or quantum. So these two aspects so this black hole things these are these are mathematical things which arises from the theory of from the way we have constructed general relativity and these things quarks and electrons and atoms they are the fundamental things in quantum field theory and these two are very you know disjointed in nature. So no one has actually connected them. But still even though these two theories are very different in its own right.

[00:32:01]They both obey the same rules. That is because we have constructed both the theories. Keeping in mind the observations that we get and the observations that we get uh are kind of like an average phenomena like we don't care how these individual atoms interact. The ultimate goal is how that system as a whole how does system as a whole behave. So when so there will be some interaction between the atoms some interaction between the atoms and the molecules different kind of interaction and some of them will cancel some of them will add up lot of things will happen and finally you'll get something okay which is what you observe and based on that observation you make the theory since every theory is made like this okay so these theories should obey okay you can expect that to have some kind of universality because there is something which is common in all of them which are observations.

[00:33:08]Okay. So that is the motivation for studying this uh one that is basically fluid dynamics.

[00:33:18]It will help you to describe the universe. A lot of thing in the universe could be described by the same set of equations of fluid dynamics. So it's a very good theory. Okay. So in its own right it's a very strong and powerful theory. So that is why it's always good to learn this.

[00:33:38]Okay. So let's start with the subject.

[00:33:44]Yeah. So so we'll begin with some basics where you will see what are the underlying assumptions that we have in the theory that we are going to construct. So to construct any theory so you need some assumption and we will give that assumptions here.

[00:34:06]So the first assumption that we get is the most important one that a fluid is considered as a continuous Okay. So what do you mean by this? So that is so we don't this is what I said exactly now we don't look at the microscopic details.

[00:34:47]Okay. So we consider it has some continuum that means if you are looking at it microscopically then you have to consider the fluid as infinite set of point particles there are lot of point particles which are interacting to each other as a discrete system but here the fluid that we are considering is a continuous medium that means such system like you are not considering any microscopic interactions etc. So the phenomenas so this means that the phenomenas that we describe are called macroscopic macroscopic.

[00:35:30]So we describe only macroscopic phenomenas. Okay.

[00:35:35]So yeah. So microscopic phenomena we'll see what are the macroscopic phenomena that we described. So this has some implication.

[00:35:48]Okay. So what are the implications of this statement that if you take any quote unquote small fluid element?

[00:35:59]So what do we mean by small fluid element? So what is small here? It is small small with respect to the entire the total volume.

[00:36:21]Okay. So it's small with respect to the total volume of the fluid. Okay. Mean you have a large fluid and you take a small part of it. Okay. So if you take a small fluid element.

[00:36:33]Okay. So it has infinitely many fluid particles.

[00:36:45]Okay. So if you take some element like this, let's say this is a fluid element inside this. There is lot of fluid particles, right? So there are a lot of infinitely many particles are there.

[00:37:04]Okay, this is fine. So again you have a big container which has a volume B. You take a small element which is V 0. So V 0 will be much much much less than V. V 0 is the volume of the element and this is the total volume of the fluid under concentration.

[00:37:28]And if you look at V 0, if you zoom in V 0, let's say you zoom in V 0, V 0 will look like this. Okay? And inside V 0, there will be this fluid particles.

[00:37:44]Let's say D is the distance between them. This is V prime, which is the distance of this particle. So D will be much much much less than D prime.

[00:37:55]So the fluid element that you're considering okay is much much larger than the inter intermolecular or inter particular distance between two fluid particles but is much much smaller when you look at the total volume of the fluid. So that is how uh you are going to describe uh that is how that's the first uh and the most important assumption that you have that a fluid is a continuous medium.

[00:38:31]Uh okay.

[00:38:36]Secondly, what do you mean by uh secondly?

[00:38:49]So we'll use this term of a point in a point in a or fluid particle.

[00:39:07]For example, at sometimes sometimes we may use a term like for example displacement displacement of the fluid particle.

[00:39:25]So what do you mean by displacement of a fluid particle? As I said before, if you had a container, big container with volume B, okay, and you take a small fluid element is something like this.

[00:39:40]Okay, it's V 0 and V 0 is much much much less than B and again it is much much larger than the inter particular distance.

[00:39:49]Okay, so that is basically the fluid particle. So when you say the displacement of a fluid particle, it means the displacement of this thing. It does not mean that you it is it is not a displacement of individual fluid particles. It is a displacement of this fluid elements. This whole element this collection of particle it has infinitely many particles in it.

[00:40:14]Thirdly, as I said, this fluid uh so we are learning fluid dynamics that means uh there is some dynamics or there is some motion to this theory or the system that we are studying. So whenever there is a motion so the question that one should ask is that what are the dynamical variables of the system? So the dynamical variables in the system dynamical variables of the system. So what do I mean by this? So so by dynamical variables of the system I mean that what are the variables okay with which we can completely determine the system. So those are here. So you have velocity okay it depends on X Y Z and time density of the fluid it also depends on X Y Z and T and also the pressure of the X Y Z and T.

[00:41:27]So you have velocity, density and pressure. In total you have five dynamical variables.

[00:41:36]Dynamical variables.

[00:41:38]You have five dynamical variables. So if you have five dynamical variables to completely determine the system that means you I need to give you or we need to derive five equations. Okay. Five independent equations from which we can solve and find these five uh independent variables.

[00:41:58]Okay. So the main goal of this part or the first part will be to find the set of equations and solve some of them.

[00:42:10]Okay. But before going to that what do you mean by the velocity here?

[00:42:18]So it is the velocity of the fluid particle. As I said before, a fluid particle we mean that you have some element fluid element like this. Okay.

[00:42:30]Now it's the velocity of this thing.

[00:42:33]Okay. But it is not the velocity. So at this moment let's say at t =0 it was here.

[00:42:41]Okay. And at t = to some t prime it is at some other position because obviously it is flowing. It's a flow. So it will flow.

[00:42:51]So at t = to some t prime it is here or t= to what maybe sometime. So it has moved a bit right. It has moved a bit some distance it has moved. Okay.

[00:43:08]But this here describes the velocity of the exact same point here.

[00:43:18]Okay. Initially there was a point here which was described by x uh x y z and x y and z. If you take x y and z you get some point here. Okay.

[00:43:33]And then this velocity describes the same velocity of the same point x y z and p. Okay. Okay. So this is a very important thing to understand is that you have a fixed point XY Z fixed by XY Z and then the velocity vector or the density or the pressure.

[00:43:53]All of these all these three all these three are like that. So you have a fixed point XY Z and then you are measuring the velocity of that fixed point at different time intervals different times. Similarly you are measuring the density of that fix point at different time or the pressure at that point at different time. Okay. So that is how these three are determined. Okay. So that means you are looking in the fluid. Okay. At the fixed point at some fixed point. Okay. Uh so this this uh is known as the olarian description.

[00:44:35]Uanian description.

[00:44:38]So in ulan description what you do is that you stay at one point and analyze the velocity, density, pressure of that point and that point is arbitrary for that matter like XY Z can be anything.

[00:44:52]Okay. So there is another description which is known as a negian description.

[00:45:00]Okay. description of the fluid where you move along with the fluid element. So you will describe so you are at the fluid element and the fluid element moves and you are also moving along with the fluid element and then you will describe the motion of the entire food from that perspective from that viewpoint. But uh we will look at this udarian description oh sorry lranian description towards the end of this chapter that's the first chapter on ideal but mostly throughout this course we will be working with the ulian description. So wherein this x y z are fixed. So you're looking at the velocity pressure and density of fixed points and that fixed point is arbit.

[00:45:47]Okay. So those are some basic uh notions that one must know before you know starting to learn or dwell deeply into fluid dynamics. Okay.

[00:46:00]And with that we will uh start the discussion on dynamics with the fundamental fundamental equations of dynamics.

[00:46:21]Okay. The fundamental equations of dynamics. So as I said these equations of fluid dynamics are governed by the conservation laws.

[00:46:36]Conservation laws.

[00:46:39]Now what are the familiar conservation laws? We have few of them. One is conservation of mass or matter.

[00:46:51]Okay. So we have this conservation law and conservation of energy and conservation of moment.

[00:47:00]Okay. So these are the three conservation laws that you have.

[00:47:06]Okay. So we have to describe or we have to find the equation or equation that describes [clears throat] these conservation laws for the fl.

[00:47:16]Okay. So that is the goal that we have and the first one that we have is the conservation or equation of continuity. I will write that in its own term that is equation of 38.

[00:47:42]Okay. So that is the thing that we are going to look at equation of continuity. Now what is this uh equation of continuity? So equation of continuity describes the conservation of matter.

[00:48:05]Okay. So it tells us about the conservation of matter. So what it basically means is that if you take some fluid elements. Okay, this is a fluid elements and then there is something which is flowing. The fluid is flowing in and some fluid will go out of it.

[00:48:28]So consideration of matter says that if you take a fluid element okay the amount of matter inside that fluid element must be same. Okay, that would mean that whatever matter is there inside the fluid elements. Okay, or the change whatever matter should be same as it will be same as the amount of matter which is entering.

[00:49:00]So the difference in the amount of matter which is entering and leaving.

[00:49:04]Okay. So both of them should be same. So that is what we are going to look at.

[00:49:11]Okay. So let's uh let's see how we will describe or find out that.

[00:49:19]So consider the fluid. Let's say the entire volume of the fluid is V 0. Okay.

[00:49:27]Now let's say the fluid is flowing with some velocity V bar. Okay.

[00:49:36]And uh there is a normal vector here which is mhat okay which is in the opposite direction.

[00:49:46]Okay. Now in some small time dt okay some amount of fluid would have entered this. So this will be the amount of fluid which has entered and let's say the density of the fluid. So we also have the density which is zero.

[00:50:07]Okay. Now what is the total amount of mass or the matter which is inside this volume V 0? So that will be so total total matter within this fluid will be equals to you have to integrate over whole v 0 row d basically density into volume will give you the total mass which is inside it is a very simple idea.

[00:50:44]Now that should be same as okay. So the amount which is entered. Okay. So how you find the amount which is entering into that element.

[00:50:58]Okay. So that you have to find out.

[00:51:03]So see so the amount entering that surface.

[00:51:20]So here I am considering only the surface number. Let me label it surface number one. Okay. So the amount uh of fluid entering surface one will be one. So if you look at it okay so you have density row. Okay. So and it is flowing with a velocity V right. So this volume what will be this volume? This volume will be dt into area. Right? Area is a vector. So you have to know that a and the direction of that area vector is n here. Okay? So a is the area of that surface. This surface okay surface surface number one.

[00:52:20]So this will be the volume.

[00:52:23]Okay. In a DJ time interval the fluid will have traveled this much volume.

[00:52:28]Right? So what will be the amount that has entered? So that will be volume into density. Density into volume uh sorry so it will be vt right because uh dt is the time interval. So the distance traveled will be dt. So this distance will be vt into a into n. So that will be the volume. Okay, that into density will give you that that will give you the amount of fluid that has entered through surface one. So that will be density into distance that is traveled okay into area.

[00:53:18]Okay. So uh this I can write as row v bar dot df. So this whole thing I'm calling df.

[00:53:34]Okay so where magnitude of df is basically area right? It will be the area of that surface.

[00:53:46]area of surface infinite decimal area basically.

[00:53:51]So now you have two quantities one is this total volume and one is the amount entering.

[00:53:59]Okay. Now if you see this uh cube as a whole. So there will be fluid entering through surface one, surface 2, surface 3, surface 4. So there are six surface right? If you think of it as a cube, there are six surface. So the total flow difference in the amount amount entering the surface will be integral over the closed surface. So you have to integrate over the whole surface right in the closed surface row var dot df vector. Okay, is it fine?

[00:54:45]So this quantity should be equal to now mind you that area vector and df uh velocity vector are in the opposite direction. So if the fluid is entering okay if the fluid is entering what will happen there will be an increase.

[00:55:03]Okay. So, so what will happen to integral row dv if the fluid is entering that will be positive. Okay. But the direction of df is opposite to the direction of v.

[00:55:19]Right? So there is you have to put a negative sign here.

[00:55:26]Okay.

[00:55:28]That means if I am calculating the rate of change. So this will be the total amount of fluid which is inside. Now I am calculating the rate of change. How much does this change with time? Okay.

[00:55:41]So if this is positive, if d / dt of integral row d is positive, that means in total.

[00:55:50]Okay. Uh there is more matter which is coming in.

[00:55:59]Okay. But then that should be equal to this quantity right the uh whatever is a net difference in the amount which is flowing and going out. Okay. And that should be the opposite side. Okay. So this must be equals to negative. So these two should be negative for each other.

[00:56:29]Okay, is it fine? So now, so the equation that we have is minus d by d t of row dv is close integral row v df.

[00:56:51]The df is the area. Now this you can convert this into gradient.

[00:56:59]row v TV right now uh del dot divergence not gradient divergence so you can bring both to the same side so you'll get integral over v 0 do row by d t plus divergence of row v bar dv =0 now this is true for all volume this equation is true for any volume.

[00:57:34]So this is this will be this will happen this will happen if and only if do by do row by dt plus divergence of row v is zero. So that is the first equation and that is known as equation of continuity and this quantity row v row v bar we redefine it as some j bar.

[00:58:03]Okay and that is known as the mass flux density is just a technical term. So in this notation the equation that we get store over do t is negative gradient of j.

[00:58:23]Okay. So that is the equation that we have.

[00:58:28]Okay.

[00:58:30]Okay. So this equation tells us about the conservation of uh mass or matter in the system. Okay. So and this is known as the equation of continuity.

[00:58:55]Okay. So yeah. So with this I think I will yeah okay so I have to explain one more thing. So this negative sign okay why is this negative sign? So you see you look at this sign of row v dot df. Okay look at the sign of this row v.tf.

[00:59:26]Now by convention DF is taken outside outward normal. Okay. So V dot DF. So suppose if the fluid fluid is going into the volume. Okay. If the fluid is going inside what will be V dot DF. So V is in this direction. DF is in this direction.

[00:59:48]So if the fluid is going inside then V dot DF is negative.

[00:59:53]But if the fluid is going outside then v dod df is positive. So see this is the surface here. So this is the surface.

[01:00:03]Fluid is going inside v bar df is in the opposite direction. So if fluid is going inside then v dot df is less than zero. If v dot df is greater than zero that means fluid is going outside because d v bar and df should be in the same sign.

[01:00:24]Now rate of change okay if the fluid is going inside that means what will happen this whole quantity should be increasing. So derivative of that should have a positive sign. But because of our convention, row v df is having a negative sign right. So therefore we have to compensate it with a extra minus sign. So minus d by dt of this is equals to this thing. And then uh by doing that further simplifications we obtain this result and this is the equation of continuity.

[01:01:09]Okay fine. So with this I think uh we'll stop today's uh lecture. It's the first lecture on this. In the next uh lecture we will look at Ula's equation.

[01:01:26]Okay. So then I can