Double-diffusive convection andλbifurcation

Abstract

We analyze convection in a rectangular box where two ‘‘substances,’’ such as temperature and a solute, are diffusing. The solutions of the Boussinesq theory depend on the thermal and solute Rayleigh numbers $R_T$ and $R_s$, respectively, in addition to other geometrical and fluid parameters. As $R_T$ is increased, the conduction state becomes linearly unstable with respect to steady (periodic) convection states if $R_s$¯s (>R${¯}_s$). The critical value R${¯}_s$ is characterized by the frequency ω=0 appearing as a root of algebraic multiplicity two and geometrical multiplicity one of the linearized stability theory. Asymptotic approximations of the solutions of the nonlinear theory are obtained for $R_s$ near R${¯}_s$ by the Poincaré-Lindstedt method. It is found that a periodic (steady-state) solution bifurcates supercritically (subcritically) from the conduction state at $R_T$=$R_c^p$ ($R_T^s$), where $R_c^p$<$R_T^s$. The periodic branch joins the steady-state branch with an ‘‘infinite-period bifurcation’’ at $R_T$=$R_b$, where $R_c^p$<$R_b$<$R_T^s$. The shape of the resulting bifurcation diagram suggests the term, λ bifurcation. The infinite-period bifurcation corresponds to a heteroclinic orbit in the appropriate amplitude-phase plane. The stabilities of the bifurcation states are determined by solving the convection initial-value problem using the multiscale method.