AST242 LECTURE NOTES PART 5 Contents 1. Waves and ...

AST242 LECTURE NOTES PART 5 7Figure 1. Two fluids, one lying above the other in a gravitationalfield. The top fluid has horizontal velocity U ′ and the bottom one hasvelocity U. When the top fluid is denser than the bottom fluid ρ ′ > ρthen the boundary is unstable to the Rayleigh-Taylor instability. Whenρ ′ < ρ gravity waves can propagate on the boundary. When U ′ ≠ Uthen the boundary is unstable to the Kelvin-Helmholtz instability.This problem is somewhat more complicated than our previous example of soundwaves as we must consider an interface.Recall Euler’s equation(44)∂u∂t + (u · ∇u)u = −1 ∇p − ∇ΦρWe use the vector identity( ) u2(45) (u · ∇)u = ∇ − u × ∇ × u2If the flow is irrotational we can drop the second term on the right. For an irrotationalflow Euler’s equation becomes( )∂u u2(46)∂t + ∇ = − 1 ∇p − ∇Φ2 ρIf the flow is irrotational then we can use a potential function for the velocity usuch that(47) ∇ψ = −u