Convection patterns near a lateral boundary

In Rayleigh–Bénard convection between rigid conducting planes, a roll pattern parallel to a lateral boundary is unstable to cross‐rolls in the vicinity of the boundary. Normal modes of the instability are determined and a stable finite amplitude structure which consists of a combination of rolls parallel and perpendicular to the boundary is identified in the weakly nonlinear regime. The theory incorporates the presence of a small boundary imperfection λ and assumes that the roll pattern in the bulk of the fluid is parallel to the lateral boundary. The extent of the perpendicular cross‐rolls is determined as a function of λ and a nonlinear interaction parameter μ which depends on the Prandtl number of the fluid. In the standard Rayleigh–Bénard system, where μ > 1, rolls are the preferred weakly nonlinear equilibrium state in the bulk of the fluid. However, solutions are also examined here for μ < 1, where the roll pattern is replaced by a pattern of square cells as time evolves; such patterns are relevant in a variety of related physical systems. For μ > 1, solutions are also found for the case where the roll pattern in the bulk of the fluid is perpendicular to the lateral boundary.