Rayleigh–Taylor density instability

Place an upper fluid of density $\rho_h$ above a lower fluid of density $\rho_l$, with $\rho_h>\rho_l$ and gravity $-g\mathbf e_y$. Begin with no base shear. A bump of lower-density material extending upward displaces heavier fluid downward. Their combined gravitational potential energy decreases to second order in interface amplitude. The resting state can therefore release energy into disturbance motion. Swap the densities and the same displacement raises potential energy, giving a restoring response instead.

This argument identifies the source of instability without claiming that every possible flow shares one dispersion relation. We now choose deep, immiscible, incompressible, inviscid layers and an initially sharp material interface. Surface tension is constant. Small disturbances permit linear kinematic and pressure-jump conditions. Finite depth, viscosity, a diffuse miscible interface, mass transfer, and acceleration varying in time require modified equations or boundary conditions. Stating the ideal model keeps its insight useful without turning it into a universal mixing prescription.

For zero surface tension, the growing branch has $\gamma=\sqrt{Agk}$. Gravity-driven growth is stronger at larger wavenumber in this sharp inviscid model, though less steeply than the vortex-sheet shear result. Surface tension penalizes curvature and contributes a stabilizing term proportional to $k^3$. For positive T, unstable modes exist only below a cutoff wavenumber. The equality at the cutoff is marginal in the linear model; it is not a statement about a general finite disturbance or experimental noise.

If the heavy fluid is below, the gravity term reverses sign. The linear interface then supports gravity–capillary waves with a positive squared oscillation frequency. Avoid calling an imaginary oscillation frequency a negative physical frequency: the equation has changed from a restoring oscillator to a growing and decaying pair. Similarly, retain both branches of an unstable linear problem. Generic small initial data excite the growing branch, but a specially selected pure decaying solution remains mathematically possible.

Initially, a chosen Fourier component grows approximately exponentially while its slope stays small. Later it produces rising light-fluid structures and descending heavy-fluid structures, with evolving velocities, shapes, and interactions. A mixing layer may develop, but its thickness cannot be inferred by extending the infinitesimal exponential indefinitely. Initial spectral content, dimensionality, acceleration history, viscosity, and density contrast all matter. Empirical nonlinear growth coefficients belong to a specified regime and measurement definition rather than to the linear dispersion relation alone.

For a container accelerating bodily, the relevant body force in the container frame includes the inertial contribution. Determine its direction and then test the density arrangement against that effective acceleration. A careless substitution of an acceleration magnitude can reverse the physical conclusion. In an experiment, report the frame, time history, and interface preparation. When comparing observed growth rates, isolate the linear interval and examine whether imaging resolution removes the shortest structures before interpreting a missing mode as physical stabilization.