The classical Rayleigh–Taylor instability for an interface of finite extent is modified by two independently controlled perturbation effects. A component of gravitational acceleration tangent to the interface is imposed and the interface is subjected to a flow‐induced pressure field. The stability of the flat horizontal interface when surface tension holds heavier liquid above ambient gas is considered in a 2D model problem. The two perturbations are realized by tilting the interface to the horizontal and by inducing a flow with shear. It is found that the effect of tilt angle or shear on its own is destabilizing, while together, in the right combination, they can stabilize. It is thereby shown how to extend the stability limit over the classical Rayleigh–Taylor result. The framework for the analysis is the classical unperturbed pitchfork bifurcation (codimension 2). The coefficients in the unfolding are calculated by applying the Lyapunov–Schmidt technique to a pinned deformable interface that holds a shear‐induced lubrication flow. ©1996 American Institute of Physics.