Fluid flow and vector fields
A neat way to interpret a vector field is to imagine that it represents some kind of fluid flow. Created by Grant Sanderson.
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- So am I right in saying that the size of a vector is representative of the speed of the particle as it moves through that point?(6 votes)
- This is my speculation and reasoning:
This is what the fluid flow model is meant to represent - the output indicates velocity (and so the magnitude/size of the vector of course indicates speed), while the input vectors are the location. By having the output mean velocity, which is derived from location, you can nicely map the velocities on the location map, which happens to be the input map.
I think that for any vector field to make sense, you would have to be mapping the derivative (the "output" as described in the video) onto the "input" (from which the derivative would be derived) - since there would be no point in mapping the output arrows into the input, if the output vectors/the vector field does not directly affect or model the changes in the input (at least it would not be a very useful or intuitive representation). The easiest way to make sense of the vector field model is using velocity (first derivative, "output") and location, with the model of the fluid flow. The vector field can be used to represent other cases as well, that don't involve time.
This also would mean that a vector field can't be used to model all types of multivariable/multidimensional equations. (Of course, like he said in the video, there must be the same number of inputs as outputs, so these ideas about #ofinput=#ofoutput and derivative/input are probably related.)
Correct me if anything sounds funny about this explanation.(7 votes)
- Can anyone provide an online source that does something akin to the animations that are shown in the video?(4 votes)
- I know the question is pretty old, but if anyone reads it and has the same doubt, there are two free software we use in class: WxMaxima and Octave(5 votes)
- Which software has been used to show the fluid flow?(2 votes)
- What software is he using to simulate this fluid flow? I have the Grapher app but couldn't find any option for this simulation.(2 votes)
- How does a vector field differ from a slope field?(1 vote)
- They look similar, but it is my understanding that a slope field uses line segments to show the slope at a given (input, output) pair, while vector fields show the vectors that are the output at an (input, input) pair.
Hope this helps!(2 votes)
- Starting at about0:18, the diagonal particles along the path y=x can be seen compressing inwards at the origin and not moving thence, I know this is mathematics, but if this kind of equation was used to describe actual movement of water, as in a simulation, what could be done to negate this effect?(1 vote)
- Even if the vectors aren't drawn to scale, are all their tails at their true starting positions?(1 vote)
- Based on how vector fields can be correlated to fluid flow:
*)In the stream line flow of fluids(Which is the only flow that can be analysed in contrast to transient and turbulent flow) each fluid particle passing through a particular point has the same velocity when it passes through that point (i.e. velocity vector) and hence the vectors in the vector field should in this case start from their true starting points,as the velocity varies with he point being analysed.
*)This may also be generalised for most cases.Thus,the tails of the vectors are at their true starting points.(1 vote)
- [Voiceover] So in the last video, I talked about Vector fields, and here I want to talk about a special circumstance where they come up. So imagine that we're sitting in the coordinate plane, and that I draw for you a whole bunch of little droplets, droplets of water, and then these are going to start flowing in some way. How would you describe this flow mathematically? So at every given point, the particles are moving in some different way. Over here, they're kind of moving down and to the left. Here, they're moving kind of quickly up. Over here, they're moving more slowly down. So what you might want to do is assign a vector to every single point in space, and a common attribute of the way that fluids flow. This isn't necessarily obvious, but if you look at a given point in space, let's say like right here, every time that a particle passes through it, it's with roughly the same velocity, so you might think over time that velocity would change, and sometimes it does. A lot of times there's some fluid flow where it depends on time, but for many cases you can just say, at this point in space, whatever particle is going through it, it'll have this velocity vector. So over here, they might be pretty high upwards, whereas here, it's kind of a smaller vector downwards, even though, I'll play the animation a little bit more here, and if you imagine doing this at all of the different points in space, and assigning a vector to describe the motion of each fluid particle at each different point, what you end up getting is a vector field. So this here is a little bit of a cleaner drawing than what I have, and as I mentioned in the last video, it's common for these vectors not to be drawn to scale, but to all have the same length, just to get a sense of direction, and here you can see each particle is flowing roughly along that vector, so whatever one it's closest to, it's moving in that direction. And this is not just a really good way of understanding fluid flow, but it goes the other way around. It's a really good way of understanding vector fields themselves, so sometimes you might just be given some new vector field, and to get a feel for what it's all about, how to interpret it, what special properties it might have, it's actually helpful, even if it's not meant to represent a fluid, to imagine that it does, and think of all the particles, and think of how they would move along in. For example, this particular one, as you play the animation, as you let the particles move along the vectors, there's no change in the density. At no point do a punch of particles go inward, or a bunch of particles go outward, it stays kind of constant, and that turns out to have a certain mathematical significance down the road. You'll see this later on as we study a certain concept called divergence. And over here, you see this vector field, and you might want to understand what it's all about, and it's kind of helpful to think of a fluid that pushes outward from everywhere, and is kind of decreasing in density around the center, and that also has a certain mathematical significance, and it might also lead you to ask certain other questions. Like if you look at the fluid flow that we started with in this video, you might ask a couple questions about it like it seems to rotating around some points, in this case counter clockwise, but it's rotating clockwise around others still. Does that have any kind of mathematical significance? Does the fact that there seem to be the same number of particles roughly in this area, but they're slowly spilling out there. What does that imply for the function that represents this whole vector field, and you'll see a lot of this later on, especially when I talk about divergence and curl, but here I just wanted to give a little warmup to that as we're just visualizing multivariable functions.