Convective heat conduction and diffusion in one-dimensional hydrodynamics

Abstract

The impurity concentration in localized structures is described on the basis of analytic solutions of model equations for convective diffusion in the one-dimensional hydrodynamic approximation without pressure. The simplicity of the derivation of the analytic results depends on the ratio of the kinetic coefficients of the liquid (the Prandtl numbers). For the same kinetic coefficients, any time-dependent problem can be reduced to problems for the conventional heat conduction equation. For integer Prandtl numbers the problem of time-dependent convective diffusion in the flow field of a uniformly moving shock wave likewise reduces to problems for the heat conduction equation. Relations are established between problems whose Prandtl numbers differ by an integer. Various representations of the Green’s functions for the equations of convective diffusion are analyzed. For integer Prandtl numbers they can be expressed in terms of error functions. The asymptotic character of the solutions depends strongly on the satisfaction of global conservation laws. For global conservation of the impurity mass, coalescence of shock waves corresponds to merging of impurity solitons, i.e., clustering.