The Kelvin-Helmholtz instability arises at the interface of two fluid layers of different densities ρg and ρl flowing horizontally with velocities ug and ul By assuming that the flow is incompressible and inviscid, and applying a small perturbation it can be shown [Ishii (1982)] that the solution for the wave velocity is given by:
and k is the wave number, i.e., 2 π/wave length.
The displacement of the interface from the equilibrium configuration is proportional to exp[ik(x – Ct)] and can therefore grow exponentially if the imaginary part of the wave velocity is nonzero. This will occur when:
where σ is the interfacial surface tension.
When rearranged this gives:
For a system with finite depths hl and hg, modified densities of ρr coth khg and ρl coth khg should be used, leading to:
For large wavelengths k→0, the gravity term dominates and the stability criterion becomes:
Ishii, R. M. Handbook of Multiphase Systems. Ch 2.4.1, Hemisphere Publications, New York.