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:


 * $$\phi_i(x,y,z,t) = \Phi_i + \phi_i'(x,y,z,t)$$
 * $$\phi_i'(x,y,z,t) = \hat{\phi_i}(z) e^{i(k x + l y) + s t}$$

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


 * $$\zeta(x,y,z,t) = \hat{\zeta}(z) e^{i(k x + l y) + s t}$$