10.1 The Rayleigh-Taylor instability of hydrodynamics 301The perturbation is assumed to have the form
v 1 =v 1 (y)eγt+ik·x (10.9)whereklies in thex−zplane and positiveγimplies instability. The incompressibility
condition, Eq.(10.5), can thus be written as
∂v 1 y
∂y+ik·v 1 ⊥=0 (10.10)where⊥means perpendicular to theydirection. Theyand⊥components of Eq.(10.7)
become respectively
γρ 0 v 1 y=−∂P 1
∂y
−ρ 1 g (10.11)γρ 0 v 1 ⊥=−ikP 1. (10.12)
The system is solved by dotting Eq.(10.12) withikand then using Eq. (10.10) to eliminate
k·v 1 ⊥and so obtain
−γρ 0∂v 1 y
∂y=k^2 P 1. (10.13)The perturbed density, as given by Eq.(10.6), is
γρ 1 =−v 1 y∂ρ 0
∂y. (10.14)
Next,ρ 1 andP 1 are substituted for in Eq. (10.11) to obtain the eigenvalue equation
∂
∂y[
γ^2 ρ 0∂v 1 y
∂y]
=
[
γ^2 ρ 0 −g∂ρ 0
∂y]
k^2 v 1 y. (10.15)This equation is solved by considering the interior and the interface separately:
- Interior: Here∂ρ 0 /∂y=0andρ 0 =const.in which case Eq.(10.15) becomes
∂^2 v 1 y
∂y^2=k^2 v 1 y. (10.16)The solution satisfying the boundary condition given by Eq.(10.8) isv 1 y=Asinh(k(y−h)). (10.17)- Interface: To find the properties of this region, Eq.(10.15) is integrated across the
interface fromy=0−toy=0+to obtain
[
γ^2 ρ 0
∂v 1 y
∂y] (^0) +
(^0) −
=−
[
gρ 0 k^2 v 1 y]0+
(^0) − (10.18)
or
γ^2
∂v 1 y
∂y
=−gk^2 v 1 y (10.19)
where all quantities refer to the upper (water) side of the interface, since by assumption
ρ 0 (y=0−)≃ 0.