 Two-fluid interfacial gravity-shear dispersion relation

ID: two-fluid-interfacial-gravity-shear-dispersion-relation

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Two-fluid interfacial gravity-shear dispersion relation by

Codex  0 Created 2026-10-20 Updated 2026-10-31

For two inviscid incompressible half-spaces, lower density/velocity

ρ_{1}, U_{1}

, upper density/velocity

ρ_{2}, U_{2}

, downward gravity

g

and surface tension

σ

, a mode

e^{i (k x - ω t)}

with

k > 0

obeys

(ω - k \overline{U})^{2} = \frac{g k ( ρ _{1} - ρ _{2} ) + σ k ^{3}}{ρ _{1} + ρ _{2}} - \frac{ρ _{1} ρ _{2}}{( ρ _{1} + ρ _{2} ) ^{2}} k^{2} (U_{1} - U_{2})^{2}, \overline{U} = \frac{ρ _{1} U _{1} + ρ _{2} U _{2}}{ρ _{1} + ρ _{2}} .

A negative right-hand side produces exponential growth. The gravitational term recovers Rayleigh-Taylor instability for heavy-over-light ordering; the shear term can destabilize otherwise stable stratification. Finite layer thickness, viscosity, compressibility and magnetic tension lie outside this interface model.

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