Need help with fourier transformation to derive oseen tensor.

[Hoeni]

Homework Statement

Derive the Oseen tensor from the Navier-Stokes equation, Consider an incompressible fluid where the velocity field $$\vec{v}$$($$\vec{r}$$,t) is given by
$$\nabla$$.$$\vec{v}$$=0 (1)
Assume inertia forces are negligible.

Homework Equations

The Navier-Stokes eq becomes:
-$$\nabla$$P+$$\eta$$$$\Delta$$$$\vec{v}$$+$$\vec{f}$$=0 (2)
With pressure P, $$\eta$$ the fluid viscosity and f the force acting on a unit volume.

Now we define the Fourier transform as $$\vec{v}$$k=$$\int$$$$\vec{v}$$($$\vec{r}$$) exp[i $$\vec{k}$$.$$\vec{r}$$] d$$\vec{r}$$ and so on.
Show that eq (1) and (2) can be rewritten as:
$$\vec{k}$$.$$\vec{v}$$k=0 and -$$\eta$$ $$\vec{k}$$$$^{2}$$ $$\vec{v}$$k - i$$\vec{k}$$ Pk = -$$\vec{f}$$k (3)

The Attempt at a Solution

In every book it is just said that Fourier transforming eq (1) and (2) just leads to (3) but it is not explained. I've tried it but maybe someone can help me. Thanks in advance.

 

[kai_sikorski]

There is a nice derivation in the attached file.

 

[Hoeni]

Haha, thank you!
but I posted this question in 2009 and as I remember correctly was able to figure it out in the end and hand in my exercises.
To save you some time, next time check the data of the post!
Cheers

 

[kai_sikorski]

I saw the date actually; just figured it might be useful to someone else at some point.

 

[Hoeni]

You are right, didn't think about it. I never posted my answer, doubt I still have it.
Good job!

 

[jchrisuk]

thanks it was useful to me!

 

[kai_sikorski]

Good to hear!