User:Loren36/kelvin

Theory

The dispersion relation for the Kelvin-Helmholtz instability can be derived by assuming that the fluids are incompressible flow and irrotational such that the fluid velocities $u_{1}$ and $u_{2}$ may be defined by a scaler potential $\phi _{1}$ and $\phi _{2}$ corresponding to the top and bottom fluid respectively, such that:

$u_{i}=\nabla \phi _{i}$

Using Reynolds decomposition, the scaler potential is decomposed into an ensemble of time varying perturbations superimposed upon a time independent mean state:

$\phi _{i}(x,y,z,t)=\Phi _{i}+\phi _{i}'(x,y,z,t)$

$\phi _{i}'(x,y,z,t)={\hat {\phi _{i}}}(z)e^{i(kx+ly)+st}$

The interface between the two fluids $\zeta$ is described in a similar manner, such that

$\zeta (x,y,z,t)={\hat {\zeta }}(z)e^{i(kx+ly)+st}$