How Do You Calculate the Rate of Closure of Angles in Fluid Kinematics?

[UFeng]

I believe the solution to this is probably easy, but for some reason I just can't understand it. Any help would be GREATLY appreciated!

Homework Statement

Consider a point at x2 = h/2. Find the rate of closure of the angle between two material lines in the x1 and x2 directions. Find the rate of closure of the angles between an x1 line and lines at 45 degrees from it.

Homework Equations

v1 = v0[1 - (x2/h)^2] , v2 = v3 = 0, where v0 is the max velocity and h is the half height. Flow is through a slot. Let the Primary point, P, be at x2=h/2. Let the secondary point be P'

The Attempt at a Solution

 

[jbsteele]

The rate of closure of the angle between two material lines in the x1 and x2 directions at P is given by v1/v2. Since v2 = 0, the rate of closure is undefined. The rate of closure of the angles between an x1 line and lines at 45 degrees from it at P' is given by v3/v1. Since v3 = 0, the rate of closure is undefined.

 

[tomcue]

Thank you for reaching out for help with this fluid kinematics problem. Understanding fluid kinematics can be challenging, but with some guidance, I am sure you will be able to solve this problem.

To start, let's define some terms and equations that will help us solve this problem. The rate of closure of the angle between two material lines can be found using the formula:

dθ/dt = (v1 - v2)/x2

where dθ/dt represents the rate of closure of the angle, v1 is the velocity at the primary point P, and v2 is the velocity at the secondary point P'.

In this problem, we are given the velocity profile v1 = v0[1 - (x2/h)^2] and we know that v2 = v3 = 0. Therefore, we can rewrite the formula as:

dθ/dt = v0[1 - (x2/h)^2]/x2

Next, we need to find the value of v0. We are given that the flow is through a slot and the Primary point P is located at x2=h/2. This means that the maximum velocity, v0, occurs at x2=0. Therefore, we can substitute x2=0 into the velocity profile to find v0:

v0 = v1(0) = v0(1 - (0/h)^2) = v0

Now, we can substitute this value of v0 into our formula for dθ/dt:

dθ/dt = v0[1 - (x2/h)^2]/x2

Finally, to find the rate of closure of the angles between an x1 line and lines at 45 degrees from it, we need to substitute x2=h/2 into the formula and then take the derivative with respect to x1:

dθ/dt = v0[1 - ((h/2)/h)^2]/(h/2) = v0/2

Therefore, the rate of closure of the angles between an x1 line and lines at 45 degrees from it is equal to half of the maximum velocity, v0.

I hope this explanation helps you understand the problem and how to solve it. If you have any further questions, please do not hesitate to ask. Good luck with your studies!