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 Θ. 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 Θ, 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 Θ is provided for 0 < Θ < 2π : this has an intricate structure with infinitely many solution branches clustering around Θ = π and Θ = 2π . 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 (Θ = 3π/4, π/2, π/4) and two dimensionless slip lengths (σ = 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 Θ < π are seen to differ fundamentally from the flat wedge (Θ = π ) previously considered by Nitsche and Parthasarathi (2012).Keywords Low-Reynolds-number flow · Porous walls · Generalized similarity solutions · Boundary integral method Electronic Supplementary material The online version of this article

Inertial effects on the flow near a moving contact line

“…Similar modified expansions of the streamfunction for other corner flow problems, at their respective critical corner angles, have also been determined previously (Moffatt & Duffy 1980; Hancock et al. 1981; Sinclair 2010; Nitsche & Bernal 2018). Some of these authors have used the power-logarithmic series as the general solution of the biharmonic equation, and have obtained the conditions under which the log terms have non-zero coefficients viz.…”

Stokes flow solutions in infinite and semi‐infinite porous channels

“…A quantitative description of laminar flow in pipes with permeable boundaries enables mathematical modeling of heat and mass transfer in numerous engineering science applications. Examples include colloidal fouling during crossflow (or tangential flow) micro-and ultrafiltration, [1][2][3] dissolved solute transport leading to concentration polarization in ultrafiltration and forward-and reverse-osmosis including spiral-wound modules, [4][5][6] combined heat and mass transfer in heat pumps 7 and fuel cells, 8 lubrication theory, 9 kinetics of membrane reactor performance, 10 and multiphase microfluidics applications such as the dynamic growth of membranes 11 as well as tumor growth. 12 The problem of steady laminar flow in porous tubes has previously been solved exactly and asymptotically for uniform wall suction (or injection) over the entire length of the duct for various geometries, [13][14][15][16][17][18][19][20][21][22][23][24][25] as summarized in Ref.…”