The Kelvin-Helmholtz instability arises at the interface of two fluid layers of different densities ρ_{g} and ρ_{l} flowing horizontally with velocities u_{g} and u_{l} 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:
where
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 h_{l} and h_{g}, modified densities of ρ_{r} coth kh_{g} and ρ_{l} coth kh_{g} should be used, leading to:
For large wavelengths k→0, the gravity term dominates and the stability criterion becomes:
REFERENCES
Ishii, R. M. Handbook of Multiphase Systems. Ch 2.4.1, Hemisphere Publications, New York.
References
- Ishii, R. M. Handbook of Multiphase Systems. Ch 2.4.1, Hemisphere Publications, New York.