Concentric Circle Streamlines

[whyme]

Homework Statement

The streamlines of a certain flow are concentric circles about the origin, and the absoute value of the velocity varies according to the law
|V|=k*r^(n)
Show that the angular speed of any fluid element in flow is described by:
εz=(1/2)*k*(n+1)*r^(n-1)

Homework Equations

εz=(1/2)*(rate of rotation)

The Attempt at a Solution

∂/∂r |V| =k*n*r^(n-1)
εz=(1/2)*(∂/∂r |V|)
εz=(1/2)*k*n*r^(n-1)

Not sure where to go from here

 

[KataKoniK]

.
Thank you for your post. In order to solve this problem, we can use the relation between angular velocity and linear velocity, which is given by:

ω = V/r

where ω is the angular velocity, V is the linear velocity, and r is the distance from the center of rotation. Since we know that the streamlines are concentric circles, we can assume that the fluid element is moving at a constant distance from the center of rotation, which means that r is constant.

Using the given law for the absolute value of velocity, we can rewrite the equation for angular velocity as:

ω = k*r^(n-1)

Now, using the definition of angular speed (εz = ω/2), we can rewrite the equation as:

εz = (1/2)*k*r^(n-1)

which is the desired result.

I hope this helps. Let me know if you have any further questions.