Stagnation-point flows in a porous medium

In this paper, we present non-linear exact and asymptotic solutions to a Navier-Stokes equation of Brinkman type proposed by Joseph et al. (Water Resour. Res. 18(4) (1982) 1049) for the flow in the stagnation-point laminar boundary layer on a cylinder or sphere if fibers of increasing concentration are uniformly added to a porous medium surrounding these blunt bodies. Although one cannot perform a rigorous averaging of the (u·∇)u term, one is able to gain useful insight into the transition in behavior that occurs between the classical solutions of Hiemenz (Dinglers Polytech. J. 326 (1911) 321) and Homann (Z. Angew. Math. Mech. 16 (1936) ; Forsch. Geb. Ingenieurwes. 7 (1936) 1) for the two-dimensional and axisymmetric stagnation-point boundary layers and the local expansion of the Brinkman solution for the flow past a cylinder or sphere in the stagnation regions as the Darcy permeability is decreased. In this analysis, a new fundamental dimensionless parameter emerges, β=υ/KA, where A is the characteristic velocity gradient 4U /D imposed by the external flow, υ is the kinematic viscosity and K, the Darcy permeability. β denotes the ratio of the square of two lengths, the classical boundary layer thickness for a high Reynolds number flow D/(2ReD1/2) and the fiber-interaction layer thickness K1/2. The exact solutions of the non-linear Brinkman equation for the stagnation-point flow presented herein show the structure of a new type of boundary layer that evolves as β varies from zero, the classical limit of the Hiemenz and Homann solutions, to β≫1, the classical Brinkman limit where inertial effects are negligible. Using asymptotic analysis we shall show that when β≫1 the classical boundary layer thickness decreases as β-1/2. Because of the introduction of the Darcy term, the pressure field differs greatly from the classical stagnation-point flow. The pressure does not increase monotonically along the stagnation streamline, and for β≫1 there is a pressure minimum that approaches the origin as β-1/2.