(368a) Asymptotic Theory and Numerical Analysis for Unraveling the Stokes Flow Singularity At the Junction Between Solid and Porous Walls With Arbitrary Wedge Angle

We consider the two-dimensional Stokes flow singularity that arises at the vertex of an angular wedge with one solid wall and one porous wall.  In the practical context of spiral-wound membrane modules, such discontinuous transitions in wall permeability occur between the membrane and either (i) the initial and final impermeable sections of the channel^1,2 (wedge angle 180 degrees),or (ii) transverse filaments of a ladder-type spacer^3,4 (wedge angle 90 degrees).  On a macroscopic, outer length scale the well-known similarity solution^5-7 appears like a jump discontinuity in the normal velocity, characterized by a non-integrable 1/r divergence of the pressure.  Our goal is to resolve the fine structure, particularly because reverse osmosis (e.g., desalination of seawater) is sensitive the local transmembrane pressure drop.  Previous results for a 180 degree wedge⁸ are here extended to arbitrary wedge angle Q.

The porous wall is characterized by a Darcy permeability for the seepage velocity and a Beavers-Joseph condition⁹ for the slip velocity.  Preliminary numerical solutions with a least-squares variant of the method of fundamental solutions indicate a continuous velocity coupled to a weaker, integrable singularity in the pressure field: p ~ r^a with a > -1.  However, inconsistencies in the numerically imposed outer boundary condition indicate a very slow radial decay, which becomes worse with decreasing angle Q.  Thus, we undertake asymptotic analysis to: (i) understand the radial decay behavior; 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) must be augmented with logarithmic terms to facilitate iterative satisfaction of the boundary conditions along the walls.  We then use a hybrid computational scheme in which the numerics are iteratively patched to the outer asymptotics, thereby determining free coefficients in the latter.  We also derive an inner asymptotic series and fit its free coefficient to the numerics at small r.

The flat wall (Q = 180 degrees) emerges as a very special, degenerate case — in which the slip condition plays no role. For any smaller wedge angle, the slip length appears as an important, intermediate length scale in addition to the inner Darcy length scale. Based upon a determinant criterion for the leading inner similarity solution, we devise a way to determine the power-law exponent a(Q) independent of the numerics. An unexpectedly rich mathematical structure emerges, with multiple solution branches arising at both small and large Q.

Our results highlight the fruitful interplay between asymptotic theory and numerical analysis, both of which were needed to make progress on this problem.

REFERENCES

¹ Miranda, J. M. & Campos, J. B. L. M. 2001 An improved numerical scheme to study mass transfer over a separation membrane. J. Membr. Sci. 188, 49–59.

² Geraldes, V., Semiao, V. & De Pinho, M. N. 2002 The effect on mass transfer of momentum and concentration boundary layers at the entrance region of a slit with a nanofiltration membrane wall. Chem. Engng Sci. 57, 735–748.

³ Geraldes, V., Semiao, V. & De Pinho, M. N. 2002 The effect of the ladder-type spacers configuration in NF spiral-wound modules on the concentration boundary layers disruption. Desalination 146, 187–194.

⁴ Geraldes, V., Semiao, V. & De Pinho, M. N. 2002 Flow management in nanofiltration spiral wound modules with ladder-type spacers. J. Membr. Sci. 203, 87–102.

⁵ Taylor, G. I. 1960 Similarity solutions of hydrodynamic problems. In Aeronaut. Astronaut.(Durand Anniv. Vol.), pp. 21–28. Pergmon.

⁶ Taylor, G. I. 1962 On scraping viscous fluid from a plane surface. Reprinted in The Scientific Papers of Sir Geoffrey Ingram Taylor, vol. IV (ed. G. K. Batchelor), pp. 410–413. Cambridge University Press.

⁷ Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1–18. 

⁸ Nitsche L. C. & Parthasarathi P. 2012 Stokes Flow singularity at the junction between impermeable and porous walls. J. Fluid Mech. 713, 183-215.

⁹ Beavers, G. S. & Joseph, D. D. 1967 Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197–207.