Kink solutions of the generalized, super-diffusive Burgers equation with memory

Abstract

Analytical and numerical solutions of the shockwave type to a one–dimensional, generalized Burgers equation with a nonlinear constitutive law that accounts for stress relaxation via a material derivative are reported as functions of the parameters that appear in the equation and the boundary conditions. It is shown that the existence of kink solutions depends on the nonlinear convection and diffusion terms, the viscosity, the relaxation time, and the upstream and downstream boundary conditions. It is also shown that the slope of the kink’s profile decreases as the viscosity coefficient, the upstream–to–downstream velocity ratio and the relaxation time are increased, and the coefficient of the nonlinear convection term is decreased. The slope of the kink’s profile is not an odd function of the coordinate in a reference system moving at the kink’s speed, but it is nearly independent of the relaxation times for small values of this parameter. The kink’s width is found to decrease and increase as the exponent of the nonlinear convection term is decreased and increased from unity, respectively. The kink’s downstream and upstream widths increase and decrease, respectively, as the exponent of the diffusion term is increased, for exponents of the nonlinear convection term greater than unity. It is also shown that the differences between the shock wave profiles corresponding to a material frame–indifferent constitutive equation and those corresponding to Maxwell’s stress relaxation model are at most six percent, but depend strongly on the relaxation time and the exponents of the nonlinear convection and diffusion terms. The local dissipation rate is also determined as a function of the coefficients and powers of the nonlinearities and the relaxation time.