Asymptotic Analysis of Acoustic Waves in a Porous Medium: Microincompressible Flow
Jose Diaz-Alban, Nader Masmoudi · Communications in Partial Differential Equations · 2014
This is the second in a series of three papers that studies acoustic waves governed by the linearized compressible Stokes equations in a porous medium. In particular, we want to analyze the simultaneous inviscid and high frequency limits of fluid flows in a porous medium. The presence of time-space boundary layers decouples the flow into an incompressible (that we call microincompressible) and an acoustic part (that we call micro-acoustic) on the microscopic scale. While this paper employs the two-scale methods used in our first paper [10 Diaz-Alban , J. , Masmoudi , N. ( 2012 ). Asymptotic analysis of acoustic waves in a porous medium: Initial layers in time . Comm. Math. Sciences 10 ( 1 ): 239 – 265 .[Crossref] , [Google Scholar]], the present boundary layer phenomenon requires additional weak convergence tools. Using the Bloch decomposition, we introduce modified Helmholtz operators, enabling us to split the flow into its microincompressible and microacoustic parts. Closed equations for the microincompressible flow are obtained using two-scale convergence, while closed equations for the microacoustic flow are given in our forthcoming paper.