Spatial decay estimates for plane flow in Brinkman-Forchheimer model

In comparison to the large amount of work that has been carried out on the analysis of Saint-Venant's principle for solutions of various elliptic boundary problems in elasticity (see [6,8], and the references therein), the related studies to investigate the spatial decay of solutions for flow problems in fluid dynamics are still considerably fewer in the literature.Horgan and Wheeler [11] presented Saint-Venant type results within the general framework of the Navier-Stokes equations governing the steady laminar flow of an incompressible viscous fluid in a cylindrical pipe of arbitrary cross section.These authors studied the classical entry flow problem of viscous laminar flow theory and showed that the decay rate, which depended upon the Reynolds number, the prescribed entry profile of the base flow, and the cross section of the pipe, was exponential.A two-dimensional version of this problem with more explicit results was also investigated by Horgan [9].Ames and Payne [2] readdressed the problem studied in [11] with a somewhat different approach in formulation of the problem and methodology.They relaxed the assumption that the flow is fully developed at the exit section and obtained more explicit estimates for a weighted energy integral associated with the flow.It is well known that the stream function in two-dimensional Stokes flow satisfies a biharmonic equation; hence the study of the spatial evolution of stationary Stokes flows in a semi-infinite parallel plate channel has direct relevance to the results of Saint-Venant's principle in plane elastostatics, where the latter problems have been well studied by many authors [7,13,15,17,19].On the other hand, it was shown by a number of investigators [3,4,10,14] that for some initial boundary value problems, the solutions of parabolic equations enjoy the spatial decay behavior of their counterparts for related elliptic equations.A natural question then arises as to whether or not the same exponential spatial decay behavior is present for the solution of transient Stokes flow.The investigations of this question have only been carried out recently.