The Time Periodic Solution of the Burgers Equation on the Half-Line and an Application to Steady Streaming

The phenomenon of steady streaming, or acoustic streaming, is an important physical phenomenon studied extensively in the literature. Its mathematical formulation involves the Navier-Stokes equations, thus due to the complexity of these equations is usually studied heuristically using formal perturbation expansions. It turns out that the Burgers equation formulated on the half line provides a simple model of the above phenomenon. The physical situation corresponds to the solution of the Dirichlet problem on the half-line, which decays as $x\to \mathrm{\infty }$ and which is {\it time periodic}. We show that the Dirichlet problem, where the usual prescription of the initial condition is now replaced by the requirement of the time periodicity, yields a well posed problem. Furthermore, we show that the solution of this problem tends to the inner'' and outer’’ solutions obtained by the perturbation expansions.