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[*i*k(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.