By the end of this video you should have a better idea of velocity fields and how we
can sketch them.
We are going to start by looking at a conceptual question involving velocity fields.
Specifically we are given a velocity equal to 2*y only in the x direction and we are
given in units of m/s.
We are asked to determine which plot best describes the field.
Right away you should see that because we are only working in the x direction we can
cross out option B. Now we have to determine if it is option A,C or neither.
What we are told here is that as y changes our velocity component is also going to change.
It is not going to stay constant with y so we can get rid of A. Then we look at the last
choice and we see that as we increase in the y direction our x component of velocity is
increasing and we can see that as we increase y our x direction is doubling.
Option C is a plot that best describes the velocity field given above.
Let's take a look at another one.
In this question we are given a velocity field equaling 2*y plus 2.
You can see that the 2*y component is in the x direction and the 2 is in the y direction.
We are asked to determine which plot below best describes this field.
We can rewrite this as the x component of velocity is equal to 2*y and the y component
is equal to 2.
Also we can say that the z component is 0.
Based on these two components we can see that as we increase y we are going to to change
the magnitude in the x direction but as we increase in either x or y directions the y
component of the velocity is going to remain constant.
Because we have a change in the x direction and no change in the y direction we can cross
out option A since both the x and y components of the velocity look pretty constant.
Looking at option B we see as we increase along the x axis initially the velocity starts
to increase in both the x and y components and then eventually it looks like the x component
decreases and the y component increases.
Because the y component is increasing in the x direction we can cross out option B. Now
option C we see that the y component of the velocity is about the same regardless of position
so that would verify that part of our velocity.
As we look as we increase along the y axis our x component is increasing.
This plot option C best describes this velocity field.
In this problem we see that the x component of the velocity is equal to xy and the y component
is equal to 2.
The first thing we should do is write this out in vector form.
We have xy times the i component plus 2 times the y component of velocity.
In this problem we want to determine the equation for the stream lines and plot on a positive
y axis only.
Instead of just writing the velocities and representing those with arrows we are going
to draw stream lines.
Remember that a stream line is tangent every to this velocity field.
We went to determine what our stream line looks like and since it is tangent every where
to the velocity field we can determine what that tangent function looks like.
This is just going to be the v component over the u component.
This is 2 over xy.
We separate this and integrate to get y*dy is equal to 2/x*dx.
After integration we get the following.
Our stream lines will have the function 1/2 *y^2 minus 2*ln(x).
This will be equal to some constant.
In the problem statement we are asked to plot the stream lines on a positive y axis only.
I have drawn that here.
To do this we basically pick any constant that we want and plot that stream line.
Let's pick the first constant as equaling 0.
We see that we are going to have an equation of the form y=(4*ln(x))^1/2.
If we picked a constant of 1 we would have something very similar but we would have it
plus 2 inside of the parenthesis.
This is for 1.
This is for a constant of 0.
If we did 2 we would have 4 natural log of x plus 4 square rooted and so forth.
You can use a graphing calculator or any program at this point and you can graph these lines.
They would roughly look like this where this would be for constant 0.
This would be for constant 1, constant of 2 and so forth.
What you will notice is that both the x component of the velocity and the y component of the
velocity are positive with positive x and y.
I am drawing these arrows to show what that velocity field would look like if we picked
a point and determined the velocities at that point.
Hopefully this gives you a good idea on velocity fields and how we would determine the stream
lines based on those coordinates.