Two-fluid interfacial gravity-shear dispersion relation

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.

 Ancestors (7)

 Incoming links (1)

Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2016 / Paper 1 / 2 / a / Solution

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version