Chemical Engineering Science 60 (2005) 123 – 134 www.elsevier.com/locate/ces Stagnation-point flows in a porous medium Q. Wu, S. Weinbaum ∗ ,Y. Andreopoulos Departments of Biomedical and Mechanical Engineering & New York Center for Biomedical Engineering, The City College of the City University of New York, Convent Avenue at 140th Street, New York, NY 10031, USA Received 27 October 2003; received in revised form 12 May 2004; accepted 19 July 2004 Abstract 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,  = /KA, where A is the characteristic velocity gradient 4U /D imposed by the external flow,  is the kinematic viscosity and K, the Darcy permeability.  denotes the ratio of the square of two lengths, the classical boundary layer thickness for a high Reynolds number flow D/(2Re 1/2 D ) and the fiber-interaction layer thickness K 1/2 . 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  varies from zero, the classical limit of the Hiemenz and Homann solutions, to  ≫ 1, the classical Brinkman limit where inertial effects are negligible. Using asymptotic analysis we shall show that when  ≫ 1 the classical boundary layer thickness decreases as  -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  ≫ 1 there is a pressure minimum that approaches the origin as  -1/2 .  2004 Elsevier Ltd. All rights reserved. Keywords: Porous medium; Stagnation-point flow; Brinkman equation; Inertia; Exact solutions; Asymptotic solutions 1. Introduction In this paper, we explore the transition in flow from the classical solutions of Hiemenz (1911) and Homann (1936) for the two-dimensional and axisymmetric stagnation-point boundary layers to the local solution of the Brinkman equa- tion (1947) in the stagnation region of a cylinder or sphere as fibers of increasing concentration are uniformly added to the medium surrounding these blunt bodies and the inertia of the fluid is dissipated by the viscous resistance of the ∗ Corresponding author. Tel.: +1-212-650-5202; fax: +1-212-650-6727. E-mail address: weinbaum@ccny.cuny.edu (S. Weinbaum). 0009-2509/$ - see front matter  2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2004.07.062 fibers. This study has been motivated by a host of biological problems that have arisen in the past decade in which the Brinkman equation has been applied to describe transport problems and problems in mechano-transduction where flow is either impinging on cells or flowing past cells that are either imbedded in an interstitial matrix or whose sur- face is covered by a matrix like layer at their apical surface. These applications include such diverse flow phenomena as the flow through the endothelial glycocalyx in the motion of red cells through capillaries (Wang and Parker, 1995; Damiano, 1998; Secomb et al., 1998; Feng and Weinbaum, 2000), the flow past muscle cells in the artery wall (Tada and Tarbell, 2002), the flow through brush border microvilli in the proximal tubule (Guo et al., 2000), the flow through