Proving Fluid Flow Velocity & Vorticity Equation

[Fairy111]

Homework Statement

For a fluid flow of velocity u and vorticity w=∆ x u, show that:

(u. ∆ )u=-u x w + ∆(1/2|u|²)

Sorry the triangles should be the other way up!

Homework Equations

∆(u.v)=(u.∆)v + (v.∆)u +u x (∆ x v) + v x (∆ x u )

The Attempt at a Solution

I need to show this using subscipt notation, but am really stuck, any help?

 

[lanedance]

here's how you write it correctly in tex - click it on the tex to see the expression
$$(u \cdot \nabla )u=-u \times w + \nabla (\frac{1}{2}|u|^2)$$

i would start by trying to expanding one of the expressions in your equation, use the equations and product rule expansions

by subscript notation do you mean like:
$$\textbf{u} \cdot \textbf{w} = u_i v_i$$

$$\textbf{u} \times \textbf{w} = u_i v_j \epsilon_{ijk}$$

 

[Fairy111]

I don't know how to expand the expression...Sorry I am really not very good at this area of maths.

but yes that is what i mean by subscript notation.

 

[lanedance]

have a crack, I'm not just going to do it for you - how about starting with u x w?