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High order asymptotic expansion for the acoustics in viscous gases close to rigid walls

Schmidt, Kersten; Thöns-Zueva, Anastasia

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

We derive a complete asymptotic expansion for the singularly perturbed problem of the acoustic wave propagation inside gases with small viscosity, this for the non-resonant case in smooth bounded domains in two dimensions. Close to rigid walls the tangential velocity shows a boundary layer of size $O(\sqrt{\eta})$ where $\eta$ is the dynamic viscosity. The asymptotic expansion based on the technique of multiscale expansion is in powers of $\sqrt{\eta}$ and takes into account curvature effects. The terms of the velocity and pressure expansion are defined independently by partial differential equations, where the normal component of velocities or the normal derivative of the pressure, respectively, are prescribed on the boundary. The asymptotic expansion is rigorously justified with optimal error estimates.