Instability

The mechanism of the instability is kinematic: pressure gradient variations, leading to velocity variations, develop along the interface as a result of the perturbation to the interface.

We consider flow through a porous medium and choose a co-ordinate system in which x is directed vertically upwards in the direction of the base state fluid flow at velocity V, and y and z lie in a horizontal plane.

On length scales larger than the pores of the medium, flow through a porous medium is described by Darcy's law:

(where is velocity, k is permeability of the porous medium, is dynamic viscosity, p is pressure, is density, g is gravitational constant, and is the velocity potential). Hence the pressure p is harmonic.

Laplace's equation must be solved for the pressure p in two media, each with different permeability, viscosity and density, subject to two boundary conditions:

1. continuity of normal velocity along the interface;
2. a prescribed pressure difference across the interface.

Neglecting surface tension effects, this pressure difference is zero; otherwise, assuming that the system is two-dimensional it is equal to where T is the surface tension constant and is the curvature of the interface.

To examine stability, the interface in the base state is taken to be x=0 (note that this is in a co-ordinate frame moving upwards at velocity V) and a small time-dependent perturbation is added, where is an arbitrary small constant, n is an arbitrary integer and the sign of denotes stability (if negative) or instability (if positive).

The velocity potential gradients on x=0 equal the velocity of the interface and are given by

 

The velocity potential functions are harmonic functions subject to the above boundary condition () at the interface and such the the effects of the interface perturbation decay at large distances from the interface.

Neglecting surface tension effects, Saffman and Taylor found the solutions ( ) and equating the pressures on the interface showed that the stability condition is as follows: when fluid 2 is forced upwards towards fluid 1 such that the undisturbed interface at x = 0 is traveling at velocity V, the interface is stable if

and unstable if the reverse inequality holds (for all wavelengths n). Hence the effect of the more viscous fluid lying above the less viscous fluid when the lower fluid is forced upwards is destabilizing and vice versa; similarly, the effect of the more dense fluid lying above the less dense fluid is destabilizing and vice versa (as in the Rayleigh-Taylor instability). Note that when the fluid is forced at a large velocity V, gravitational effects are negligible.