Stokes flow singularity at the junction between impermeable and porous walls

Abstract

For two-dimensional, creeping flow in a half-plane, we consider the singularity that arises at an abrupt transition in permeability from zero to a finite value along the wall, where the pressure is coupled to the seepage flux by Darcy’s law. This problem represents the junction between the impermeable wall of the inflow section and the porous membrane further downstream in a spiral-wound desalination module. On a macroscopic, outer length scale the singularity appears like a jump discontinuity in normal velocity, characterized by a non-integrable $1/ r$ divergence of the pressure. This far-field solution is imposed as the boundary condition along a semicircular arc of dimensionless radius 30 (referred to the microscopic, inner length scale). A preliminary numerical solution (using a least-squares variant of the method of fundamental solutions) indicates a continuous normal velocity along the wall coupled with a weaker $1/ \sqrt{r} $ singularity in the pressure. However, inconsistencies in the numerically imposed outer boundary condition indicate a very slow radial decay. We undertake asymptotic analysis to: (i) understand the radial decay behaviour; and (ii) find a more accurate far-field solution to impose as the outer boundary condition. Similarity solutions (involving a stream function that varies like some power of $r$) are insufficient to satisfy all boundary conditions along the wall, so we generalize these by introducing linear and quadratic terms in $\log r$. By iterating on the wall boundary conditions (analogous to the method of reflections), the outer asymptotic series is developed through second order. We then use a hybrid computational scheme in which the numerics are iteratively patched to the outer asymptotics, thereby determining two free coefficients in the latter. We also derive an inner asymptotic series and fit its free coefficient to the numerics at $r= 0. 01$. This enables evaluation of the singular flow field in the limit as $r\ensuremath{\rightarrow} 0$. Finally, a uniformly valid fit is obtained with analytical formulas. The singular flow field for a solid–porous abutment and the general Stokes flow solutions obtained in the asymptotic analysis are programmed in Fortran for future use as local basis functions in computational schemes. Numerics are required for the intermediate-$r$ regime because the inner and outer asymptotic expansions do not extend far enough toward each other to enable rigorous asymptotic matching. The logarithmic correction terms explain why the leading far-field solution (used in the preliminary numerics) was insufficient even at very large distances.

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

“…Based upon the following characteristic scales, Characteristic length: The case Θ = π/2 applies to a solid spacer adjoining the membrane, and represents one of the problems treated in this paper. The abrupt end of the sealed, entry section of the lower wall corresponds to Θ = π , and was treated by Nitsche and Parthasarathi [8]. The singular points are indicated with large, closed circles…”

“…Satisfaction of both no-slip and no-shear means that the slip condition (9) is then also satisfied exactly for any value of the slip length σ . Nitsche and Parthasarathi [8] found this degeneracy to propagate beyond the leading outer solution through second order in both inner and outer asymptotic expansions, and concluded that slip does not play a role for the case of a flat abutment of porous and solid walls. The case Θ = π is now seen to be fluke: for any other wedge angle Θ, the shear stress (19) is inconsistent with the radial velocity component (16) in Eq.…”

“…These two cases are distinguished by the respective wedge angles Θ = π versus Θ = π/2. For the former angle, our previous paper [8] uncovered a subtle singularity in the pressure, which would be expected to couple with osmotic effects and thereby affect the salt concentration profile in desalination by reverse osmosis. Here we shall apply and extend those asymptotic solutions and computational techniques to elucidate the progression of angles Θ = 3π/4, π/2, π/4, emphasizing important ways in which the new cases differ from the old.…”

“…As before, the classic similarity form for Stokes flow solutions in angular wedges [9][10][11][12][13][14][15][16][17] will need to be generalized with a logarithmic dependence on the radial coordinate r [13,[18][19][20][21][22][23][24]. The extensive literature review in Nitsche and Parthasarathi [8] fleshed out the context of these papers and also provided a survey of theory and computation relevant to flow in channels with porous walls.…”

“…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 …”