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Abstract This article examines the Rayleigh–Taylor instability in an rotating inhomogeneous, incompressible fluid with partial viscosity. First, using the modified variational method, we demonstrate the existence of an exponentially growing normal mode in $H^k,k\text{⩾}0$ and establish the instability of the linearised problem. Then, we obtain a nonlinear energy estimate for the problem with small initial data. In this process, we employ an innovative method to derive energy estimates for both density and velocity, effectively addressing the challenges posed by partial viscosity. Third, we prove the existence of a classical solution forH³initial data, provided it satisfies a compatibility condition. Finally, by integrating the results of the previous steps, we establish the nonlinear instability of the system in the Hadamard sense. 

Abstract We develop two totally decoupled, linear and second‐order accurate numerical methods that are unconditionally energy stable for solving the Cahn–Hilliard–Darcy equations for two phase flows in porous media or in a Hele‐Shaw cell. The implicit‐explicit Crank–Nicolson leapfrog method is employed for the discretization of the Cahn–Hiliard equation to obtain linear schemes. Furthermore the artificial compression technique and pressure correction methods are utilized, respectively, so that the Cahn–Hiliard equation and the update of the Darcy pressure can be solved independently. We establish unconditionally long time stability of the schemes. Ample numerical experiments are performed to demonstrate the accuracy and robustness of the numerical methods, including simulations of the Rayleigh–Taylor instability, the Saffman–Taylor instability (fingering phenomenon). 

ABSTRACT We analyze a quasi‐incompressible Cahn–Hilliard–Navier–Stokes system (qCHNS) for two‐phase flows with unmatched densities. The order parameter is the volume fraction difference of the two fluids, while mass‐averaged velocity is adopted. This leads to a quasi‐incompressible model where the pressure also enters the equation of the chemical potential. We establish local existence and uniqueness of strong solutions by the Banach fixed point theorem and the maximal regularity theory.