Take $x$ in the injection direction and $y$ along the flat interface. The specified $U$ must be a <Darcy velocity>, so the pore-fluid velocity and unperturbed interface speed are $U/\phi$. Write the perturbed interface as $x=X(t)+\eta(y,t)$, with $\dot X=U/\phi$, and choose the normal from fluid 1 into fluid 2. The background pressure gradients from <Darcy law> are $\partial_xp_i^{(0)}=-\mu_iU/k$.

By <incompressibility> and uniform <permeability of a porous medium>, pressure perturbations are <harmonic functions>. A transverse <Fourier mode> with positive <wavenumber> $\alpha$ has the decaying forms
$$
p_1'=A_1(t)e^{\alpha(x-X)}e^{i\alpha y},\qquad
p_2'=A_2(t)e^{-\alpha(x-X)}e^{i\alpha y},\qquad
\eta=a(t)e^{i\alpha y}.
$$
The <kinematic boundary condition> and equal normal <Darcy flux> imply
$$
\phi\dot a=-\frac{k\alpha}{\mu_1}A_1=\frac{k\alpha}{\mu_2}A_2,
\qquad A_1-A_2=-\frac{\phi(\mu_1+\mu_2)}{k\alpha}\dot a.
$$
Without capillarity the pressure is continuous at the displaced interface. Its linear perturbation is therefore
$$
A_1-A_2+\frac{(\mu_2-\mu_1)U}{k}a=0.
$$
Consequently the <planar viscous-fingering dispersion relation> reduces to
$$
\boxed{\dot a=\sigma a,\qquad\sigma=\frac{\alpha U}{\phi}\frac{\mu_2-\mu_1}{\mu_2+\mu_1}=\frac{\alpha U}{\phi}M>0}.
$$
This is the <Saffman–Taylor instability>: less viscous fluid advances preferentially through protrusions. Without a short-wave regularizing mechanism the idealized rate has no finite maximum.

For the prescribed apparent <capillary pressure>, the chosen normal is $\mathbf n=(1,-\eta_y)/\sqrt{1+\eta_y^2}$, so $\nabla\cdot\mathbf n=-\eta_{yy}+O(\eta^2)=\alpha^2\eta+O(\eta^2)$. Subtract the flat-front pressure jump $\gamma(1+\beta X)\kappa_0$. Linearization of the jump at $x=X+\eta$ gives
$$
A_1-A_2+\frac{(\mu_2-\mu_1)U}{k}a
=\gamma\bigl[\beta\kappa_0+(1+\beta X)\alpha^2\bigr]a.
$$
There is no first-order product of the surface-tension perturbation and the perturbed macroscopic <curvature>. Thus
$$
\boxed{\sigma(\alpha;X)=\frac{\alpha}{\phi(\mu_1+\mu_2)}
\left[(\mu_2-\mu_1)U-k\gamma\beta\kappa_0-k\gamma(1+\beta X)\alpha^2\right]}.
$$
Using the <capillary number> convention $\mathrm{Ca}=\mu_2U/\gamma$ and $\mu_2/(\mu_1+\mu_2)=(1+M)/2$, this is
$$
\boxed{\sigma(\alpha;X)=\frac{\alpha U}{\phi}\left[M-\frac{k(1+M)}{2\mathrm{Ca}}\bigl(\beta\kappa_0+(1+\beta X)\alpha^2\bigr)\right]}.
$$
If $x$ is measured from the unperturbed front and $\gamma$ denotes its local apparent tension, set $X=0$ to obtain the usual fixed-coefficient expression. For a physically fixed spatial gradient, $X=Ut/\phi$ and the displayed rate is an instantaneous, generally time-dependent <growth rate>; the amplitude solves $\dot a=\sigma(\alpha;X(t))a$, rather than one constant exponential law for the entire displacement. The parameter $\beta$ has dimensions of inverse length; the absolute tension gradient is $\gamma\beta$.

Assume $1+\beta X>0$, as required for positive apparent <surface tension>, and define $A=(\mu_2-\mu_1)U-k\gamma\beta\kappa_0$. If $A>0$, unstable modes satisfy $0<\alpha^2<A/[k\gamma(1+\beta X)]$. Differentiating the cubic <dispersion relation> gives the fastest-growing mode
$$
\boxed{\alpha_m^2=\frac{A}{3k\gamma(1+\beta X)}
=\frac{1}{3(1+\beta X)}\left[\frac{2M\mathrm{Ca}}{k(1+M)}-\beta\kappa_0\right]}.
$$
When $A\le0$, every nonzero mode decays, with a neutral zero-<wavenumber> translation in the infinite-domain limit. For $\beta\kappa_0>0$, <surface-tension-gradient stabilization of viscous fingering> suppresses the instability entirely at positive injection speeds satisfying
$$
\boxed{0<U\le U_c=\frac{k\gamma\beta\kappa_0}{\mu_2-\mu_1},\qquad
\mathrm{Ca}\le\frac{k(1+M)\beta\kappa_0}{2M}}.
$$
If $\beta\kappa_0\le0$, no positive injection speed eliminates every long-wave unstable mode in an unbounded interface. The prescribed wetting variation is treated as an effective normal pressure law for <porous-media flow>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 332 / 1 / Solution

{title2=$\sigma(\alpha)$}

For two fluids with <dynamic viscosities> $\mu_1<\mu_2$, uniform <permeability of a porous medium> $k$, <porosity> $\phi$, and imposed <Darcy velocity> $U$, the <Saffman–Taylor instability> of a planar interface has growth rate $\sigma=\alpha U(\mu_2-\mu_1)/[\phi(\mu_2+\mu_1)]$ without capillarity. With constant interfacial tension $\gamma$ and the stabilizing pressure-jump convention, a perturbation of transverse <wavenumber> $\alpha>0$ instead has
$$
\sigma=\frac{\alpha}{\phi(\mu_1+\mu_2)}[(\mu_2-\mu_1)U-k\gamma\alpha^2].
$$
The fastest-growing mode satisfies $\alpha_m^2=(\mu_2-\mu_1)U/(3k\gamma)$.

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 Incoming links: Planar viscous-fingering dispersion relation