The effect of a local increase in impurity concentration in one-dimensional hydrodynamics

Abstract

The system of non-linear equations of one-dimensional convective diffusion of a passive impurity in the case of a flow of fluid described by the model Burgers equation is solved using Cole-Hopf and Darboux transformations. For equal kinetic coefficients of the fluid its solution is reduced to solving linear heat-conduction equations. For integer Prandtl numbers, differing from unity, this reduction is successful for flows of uniformly moving shock waves, smoothed by the viscosity effect. Its possibility is closely related to the factorizability of second-order differential operators in this case, their decomposability into the product of first-order operators, and the additional internal symmetry (supersymmetry) of the problem. The interaction of shock waves and the impurity solitons they transfer has an absolutely inelastic character. A local increase in the impurity concentration is a result of the merging of impurity solitons. It is pointed out that, for certain equations of state, the reduction of the solutions of the non-linear convective diffusion equations to the solution of linear heat-conduction equations is also possible for an active impurity (when the kinetic coefficients of the fluid are equal).