Pattern formation near onset of a convecting fluid in an annulus.
Numerical simulations of the time-dependent Swift-Hohenberg equation are used to test the predictions of Cross [Phys. Rev. A 25, 1065 (1982)] that Rayleigh-Bénard convection in the form of straight rolls or of an array of dislocations may be observed in an annular domain, depending on the values of inner radius r(1), outer radius r(2), reduced Rayleigh number epsilon, and initial states. As r(1) is decreased for a fixed r(2) and for different choices of epsilon and of symmetric and random initial state, we find that there are indeed ranges of these parameters for which the predictions of Cross are qualitatively correct. However, when the radius difference r(2)-r(1) becomes larger than a few roll diameters, a new pattern is observed consisting of stripe domains separated by radially oriented grain boundaries. The relative stabilities of the various patterns are compared by evaluating their Lypunov functional densities.