Solving Incompressible Flow Homework: Find f(r)

[coverband]

Homework Statement

A velocity field is given by
$$\vec {u} = f(r)\vec{x}, r = | \vec{x}| = \sqrt {x^2 + y^2 + z^2}$$ written in rectangular cartesian coordinates, where f(r) is a scalar function. Find the most general form of f(r) so that $$\vec {u}$$ represents an incompressible flow

Homework Equations

Incompressible flow implies $$\nabla . \vec {u} = 0$$.

The Attempt at a Solution

The solution is $$\nabla . \vec {u} = 3f + rf' so f(r) = A/r^3$$ (A is an arbitrary constant) but I don't see how it is arrived at. Thanks

 

[radou]

If the norm of x is given, you can figure out its components, right?

 

[coverband]

Right, (x,y,z)

 

[radou]

So, write u down in terms of its components and calculate the divergence.

 

[coverband]

Ok, so u = (fx,fy,fz)

Therefore div u = 3f ! Now.. !?

 

[coverband]

Or div u = 3f + xf_x + yf_y + zf_z ? Little help!

 

[billiards]

Think I can help here. Not sure where your getting stuck as you haven't properly written out your thoughts.

First thing you need to do is get the general expression for divergence in terms of your scalar function. Here are the key things for the x-component (and they have the same form for y and z).

div u = d(u_x)/dx + ...

u_x = f(r) r_x

so write out div u making the substitution.

But remember the product rule. http://en.wikipedia.org/wiki/Product_rule

There are some further steps before you can get your answer, but this is a good start.