Streamline

Okay, in the previous segment we talked about steady, we talked about unsteady, and I'll introduce something. This will be quite important. It's called streamline, pathline, and streaklines. Okay, so over here I will discuss these three different definitions that I can call. You can see there's some confusion over here. One's called stream line, path line, streak line. Okay, some people say that, hey, they're the same thing, you don't have to really cover this. Yes and no is the answer. Okay, yes and no is the answer. The answer depends on the previous segment I just covered, and the previous segment asks this question. Is this steady or unsteady? If this is steady, then I can tell you something. These three will be the same. Okay, streamline will be equal to path line, will be equal to streak line. So I know have to differentiate them, but if my flow is unsteady, unfortunately, I have to differentiate them. There will be three different definitons. One is called the streamline. So let's actually visit this right here, right now. Let's stick to what I was showing before. So let's say I have a bend over here like that. Okay, if I'm interested in streamline for this case, what I do is I look at the points, not over time, but I look at the cases and I look at the velocity. Where's the velocity? Like this velocity vector. Obviously, velocity is a vector, not a scalar, right? So it has direction. Combining like this, it's not gonna be a perfect drawing, you know, I'm doing my best over here, but this close to red is what I have that combines all these velocity vectors at different points. This is called the streamline. Velocity is tangent to streamlines. This part is important. Okay, and another point that I want to make is no flow can flow across a streamline. Let me explain. Actually, these two are kind of related to each other. As the velocity is tangent, think about it, if velocity is tangent, how can it cross it? It's tangent to it. Okay, in real life I have infinite number of streamlines, and what I'm saying is they're not gonna cross each other. Some other thing that I want to discuss about the streamline is stagnation point. Okay, this is used to, you know, something called pitot tubes. We'll talk about those things down the road as well. Let's give an example of a sphere over here, and if I have some velocity coming in here, what will happen is if I plot my streamlines, the velocity tangent to it, it's gonna be like that. It's gonna be like this, etc., right? But that will be the very first streamline in here where it's gonna kind of hit over here. The velocity over here will be zero, right? Will hit and will stop. So this is called stagnation point. So now I want to talk about the second thing that I have over here. It’s called the streakline. Okay, you maybe look at the first half of this terminology, it's called the streak. Okay, so basically what it is, is I'm gonna write it this way. Succession of, let's call it, marked particles that originated from a particular point in the flow. Succession of marked particles that originated from a particular point in the flow. Let me give the same example that I have over there. Obviously, again, I'm gonna repeat one more time. But if this is a steady flow, then streakline is streamline. I don't have to talk about this. It's the same thing. But basically what I can do in here is, let's say that this is top view. Let’s say that there's some velocity over here and the velocity over there. So things go into in here and leave here. I will go to the grocery store and I will get myself food coloring. Okay, and I'm gonna sit over here and I'm gonna drop every one second like tip, tip, tip, tip, right? What will happen to this, you know, like drops is over time I'll get some progress. Okay, these, obviously, you know, arbitrarily drawing these, but then if I combine them, this will be the streakline. Okay, like this. This is the streak line. Originally the exact this all this point, this point, this point this .., every point was originated from right over there. Everything originated from here. I'm just looking at how it changes over time. It just has this. Another good example for streakline is you can look at the sprinkler system that you have in your backyard. So what happens is if I look at it at a given snapshot, it's a time. If you think about it, you literally see it's coming out from the sprinkler at an instant. So those are the streaklines that you're seeing. And how about the pathline? That's the last one that I need to explain. So this is basically, as the name suggests, it is the path of the particle traveled over time. Okay? And so basically, it's the path of particle travel over time. So if I have a steady flow, it would be this red one in this particular case as well. And if you want to obtain your pathline experimentally, there's something called neutrally buoyant. So basically, we talked about buoyancy in Module Number Three, right right before this module. And in that one, we talked about this buoyant means it's just staying at its particular point, and I can release this and I can do a time-exposed photography of it. If you think about it, what is happening over time, because there's the path of that particular neutrally buoyant particle that is in my flow. So before I close this off segment, I want to talk about equation of streamline. Okay, over here, you can see this is my streamline, right? And the streamline, you can guess, is the coordinate is typically defined as s. So I'm going to replicate it. I'll make a point. So we have this over here and this is s, right? So let's just take this point. Okay, so what will happen over here is the velocity will be tangent, right? The velocity is tangent to this point. I'm gonna look at it. Let's say that this is x and this is y directions. Displacement that I have will be defined by this. This displacement vector will be x i plus y j plus z k. Okay, so in a three-dimensional x is, well, basically it's from here to here, right? In the x direction. This. y is again over here. This is the y direction for this particular point. And z is zero for this particular case. But this is the displacement that I have. If I write the velocities, and I will derive this in detail, but what you will get is it will be u i plus v j plus w k. So these just repeat, you know this, but i, j, k, those are the unit vectors in the x, y, and z directions, respectively. u is the velocity in the x, v is the velocity in the y, w is the velocity in the z. Now I know this. Let me derive you the equation of streamline. Okay, so here's what's gonna happen. Let's look at these two components of this. Okay, so I have this over here. The component of this will be, this will be u, and this will be v. And I have some angle over here that's called theta. Okay, and you can see over here, tangent of this theta, that angle, will be v over u. Okay, but take a look at in here, I'm actually doing a differential analysis over here. This distance can be dx. This can be dy. And if I want to write the same tangent of the angle, tangent of theta, it will be dy over dx, right? From this blue font. So, actually what I need to do is I write this. These two are the same. So I will just simply equate them to each other. And then what will happen is it v over u will be equal to dy dx. So then this will be definition, or rather, the equation of the streamline. So you can use the inverse. Sometimes I see the inverse being used as well. u over v, so the dx dy. That's fine. It's just the inverse of it, right?