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Laminar flow in a two-dimensional plane channel with local pressure-dependent crossflow

Published online by Cambridge University Press:  23 November 2007

P. HALDENWANG
P. HALDENWANG*
Affiliation:
Modélisation et Simulation Numérique en Mécanique et Génie des Procédés (MSNM-GP), UMR-CNRS 6181 – Universités d'Aix-Marseille – ECM IMT/La Jetée – Technopôle de Château Gombert, 38, rue Frédéric Joliot-Curie – 13451 Marseille Cedex 20, France
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Abstract

Long ducts (or pipes) composed of transpiring (e.g. porous) walls are at the root of numerous industrial devices for species separation, as tangential filtration or membrane desalination. Similar configurations can also be involved in fluid supply systems, as irrigation or biological fluids in capillaries. A transverse leakage (or permeate flux), the strength of which is assumed to depend linearly on local pressure (as in Starling's law for capillary), takes place through permeable walls. All other dependences, as osmotic pressure or partial fouling due to polarization of species concentration, are neglected. To analyse this open problem we consider the simplest situation: the steady laminar flow in a two-dimensional channel composed of two symmetrical porous walls.

First, dimensional analysis helps us to determine the relevant parameters. We then revisit the Berman problem that considers a uniform crossflow (i.e. pressure-independent leakage). We expand the solution in a series of Rt, the transverse Reynolds number. We note this series has a rapid convergence in the considered range of Rt (i.e. Rt ≤ O(1)). A particular method of variable separation then allows us to derive from the Navier–Stokes equations two new ordinary differential equations (ODE), which correspond to first and second orders in the development in Rt, whereas the zero order recovers the Regirer linear theory. Finally, both new ODEs are used to study the occurrence of two undesirable events in the filtration process: axial flow exhaustion (AFE) and crossflow reversal (CFR). This study is compared with a numerical approach.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 593 , 25 December 2007 , pp. 463 - 473
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Bacchin, P., Aimar, P. & Field, R. W. 2006 Critical and sustainable fluxes: theory, experiments and applications. J. Membrane Sci. 281, 42–69.CrossRefGoogle Scholar
Berman, A. S. 1953 Laminar flow in channels with porous walls. J. Appl. Phys. 24, 1232–1235.CrossRefGoogle Scholar
Brady, J. F. 1984 Flow development in a porous channel and tube. Phys. Fluids 27, 1061–1067.CrossRefGoogle Scholar
Chellam, S. & Liu, M. 2006 Effect of slip on existence, uniqueness, and behavior of similarity solutions for steady incompressible laminar flow in porous tubes and channels. Phys. Fluids 18, 083601.Google Scholar
Galowin, L. S., Fletcher, L. S. & DeSantis, M. J. 1974 Investigation of laminar flow in a porous pipe with variable wall suction. AIAA J. 12, 1585–1589.CrossRefGoogle Scholar
Gupta, B. K. & Levy, E. K. 1976 Symmetrical laminar channel flow with wall suction. Trans. ASME I: J. Fluids Engng 33, 469–475.Google Scholar
Mellis, R., Gill, W. N. & Belfort, G. 1993 Fluid dynamics in tubular membrane: theory and experiments. Chem. Engng Commun. 122, 103–125.CrossRefGoogle Scholar
Regirer, S. A. 1960 On an approximate theory of viscous incompressible fluid flow in channels with permeable walls. J. Tech. Phys. (Zh. Tekh. Fiz.) 30, 639–643.Google Scholar
Robinson, W. A. 1976 Existence of multiple solutions for laminar-flow in a uniformly porous channel with suction at both walls. J. Engng Maths 10, 23–40.CrossRefGoogle Scholar
Rudraiah, N. & Dinesh, P. A. 2004 Nonlinear flow between permeable disks using computer extended series method. Stud. Appl. Maths 113, 163–182.Google Scholar
Terrill, R. M. 1964 Laminar flow in a uniformly porous channel. Aeronaut. Q. 15, 299–310.CrossRefGoogle Scholar
Terrill, R. M. & Shrestha, G. M. 1965 Laminar flow through parallel and uniformly porous walls of different permeability. Z. Angew. Math. Phys. 16, 470–482.Google Scholar
Zaturska, M. B., Drazin, P. G. & Banks, W. H. H. 1988 On the flow of a viscous fluid driven along a channel by suction at porous walls. Fluid Dyn. Res. 4, 151.Google Scholar