Saffman–Taylor instability

The simplest case of the instability arises at a planar interface within a porous medium or Hele-Shaw cell, and was treated by Saffman and Taylor[1] but also earlier by other authors.[6] A fluid of viscosity ${\displaystyle \mu _{1}}$ is driven in the ${\displaystyle x}$-direction into another fluid of viscosity ${\displaystyle \mu _{2}}$ at some velocity ${\displaystyle V}$. Denoting the permeability of the porous medium as a constant, isotropic, ${\displaystyle \Pi }$, Darcy's law gives the unperturbed pressure fields in the two fluids ${\displaystyle i=1,\,2}$ to be

$${\displaystyle p_{i}^{(0)}=p_{\text{int}}-{\frac {V\mu _{i}}{\Pi }}x,}$$

where ${\displaystyle p_{\text{int}}}$ is the pressure at the planar interface, working in a frame where this interface is instantaneously given by ${\displaystyle x=0}$. Perturbing this interface to ${\displaystyle x=\eta _{0}\exp {\left(\mathrm {i} ky+\sigma t\right)}}$ (decomposing into normal modes in the ${\displaystyle x-y}$ plane, and taking ${\displaystyle \left|\eta _{0}\right|\ll 1}$), the pressure fields become

$${\displaystyle p_{i}=p_{i}^{(0)}+{\tilde {p}}_{i}\left(x\right)\exp {\left(\mathrm {i} ky+\sigma t\right)}.}$$

As a consequence of the incompressibility of the flow and Darcy's law, the pressure fields must be harmonic, which, coupled with the requirement that the perturbation decay as ${\displaystyle x\to \pm \infty }$, fixes ${\displaystyle {\tilde {p}}_{1}={\tilde {p}}_{1}e^{kx}}$ and ${\displaystyle {\tilde {p}}_{2}={\tilde {p}}_{2}e^{-kx}}$, with the constants ${\displaystyle {\tilde {p}}}$ to be determined by continuity of pressure. Upon linearization, the kinematic boundary condition at the interface (that fluid velocity in the ${\displaystyle x}$ direction must match the velocity of the fluid interface), coupled with Darcy's law, gives

$${\displaystyle -{\frac {\Pi }{\mu _{i}}}\left.{\frac {\partial {\tilde {p}}_{i}}{\partial x}}\right|_{x=0}=\sigma \eta _{0},}$$

and thus that ${\displaystyle {\tilde {p}}_{1}=-{\frac {\sigma \eta _{0}\mu _{1}}{\Pi k}}}$ and ${\displaystyle {\tilde {p}}_{2}={\frac {\sigma \eta _{0}\mu _{2}}{\Pi k}}}$. Matching the pressure fields at the interface gives

$${\displaystyle -V\mu _{1}-{\frac {\sigma \mu _{1}}{k}}=-V\mu _{2}+{\frac {\sigma \mu _{2}}{k}},}$$

and so ${\displaystyle \sigma =kV\left(\mu _{2}-\mu _{1}\right)/\left(\mu _{1}+\mu _{2}\right)}$, leading to growth of the perturbation when ${\displaystyle \mu _{2}>\mu _{1}}$ - i.e. when the injected fluid is less viscous than the ambient fluid. There are problems with this basic case: namely that the most unstable mode has infinite wavenumber ${\displaystyle k}$ and grows at an infinitely fast rate, which can be rectified by the introduction of surface tension[7] (which provides a jump condition in pressures across the fluid interface through the Young–Laplace equation), which has the effect of modifying the growth rate to

$${\displaystyle \sigma ={\frac {kV\left(\mu _{2}-\mu _{1}\right)-\gamma H_{f}k^{3}}{\mu _{1}+\mu _{2}}},}$$

with surface tension ${\displaystyle \gamma }$ and ${\displaystyle H_{f}}$ the mean curvature. This suppresses small-wavelength (high-wavenumber) disturbances, and we would expect to see instabilities with wavenumber ${\displaystyle k}$ close to the value of ${\displaystyle k}$ which results in the maximal value of ${\displaystyle \sigma }$; in this case with surface tension, there is a unique maximal value.

The Saffman–Taylor instability is usually seen in an axisymmetric context as opposed to the simple planar case derived above.[8][9] The mechanisms for the instability remain the same in this case, and the selection of the most unstable wavenumber in this case corresponds to a given number of fingers.

• Kelvin–Helmholtz instability
• Darcy's law