Stokes flow singularity at a corner joining solid and porous walls at arbitrary angle

Abstract

Motivated by the internal flow geometry of spiral-wound membrane modules with ladder-type spacers, we consider the Stokes flow singularity at a corner that joins porous and solid walls at arbitrary wedge angle \(\varTheta \) . Seepage flux through the porous wall is coupled to the pressure field by Darcy's law; slip is described by a variant of the Beavers–Joseph boundary condition. On a macroscopic, outer length scale, the singularity appears like a jump discontinuity in the normal velocity, characterized by a non-integrable 1 / r divergence of the pressure. For arbitrary \(\varTheta \) , we develop an algebraic criterion to determine the admissible radial exponent(s) in a leading, inner similarity solution—which represents a weaker, integrable singularity in the pressure. A complete map of exponent versus \(\varTheta \) is provided for \(0< \varTheta < 2 \pi \) : this has an intricate structure with infinitely many solution branches clustering around \(\varTheta = \pi \) and \(\varTheta = 2 \pi \) . By generalizing the similarity form with a ( \(\ln r\) ) term and iterating on the slip and seepage conditions, we can carry the outer and inner power series to arbitrarily high order. Nevertheless, a numerical splice is required in between. For this purpose, we apply an iterative, numerical-asymptotic patching scheme described by Nitsche and Parthasarathi (J Fluid Mech 713:183–215, 2012). Detailed velocity and pressure profiles are calculated for three wedge angles ( \(\varTheta = 3\pi /4, \pi /2, \pi /4\) ) and two dimensionless slip lengths ( \(\sigma = 20, 40\) ). The general trends for decreasing wedge angle are (i) weakening of the pressure singularity, (ii) increasing magnitude of the radial component of velocity, and (iii) movement of the inner–outer transition farther from the corner. Wedges with \(\varTheta < \pi \) are seen to differ fundamentally from the flat wedge ( \(\varTheta = \pi \) ) previously considered by Nitsche and Parthasarathi (2012).

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