I am trying to derive the Poisson equation for pressure, specifically that $$\bigtriangledown^2(p + \frac{1}{2}|\vec{u}|^2) = \bigtriangledown \cdot(\vec{u}\times \vec{w}) + \varepsilon \kappa^2\psi\cdot\bigtriangledown\phi $$ where the vorticity field is given by $$w = \bigtriangledown\times \vec{u}$$ I am unsure how to manipulate the left-hand side to reach the right-hand side.I know that I can write $$|\vec{u}|^2 = |\bigtriangledown\phi|^2 = |\bigtriangledown\psi|^2$$ But I am unsure what else I can do.This problem is in reference to a particular paper I am reading,so some of the notation on the RHS may not be known without context,however,I just need help with the manipulation of the LHS.
Thanks, any help is greatly appreciated.