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Licensed Unlicensed Requires Authentication Published by De Gruyter June 17, 2015

Influence of Wall Structure on the Laminar Coupling Flow

Li Aifen, Xie Haojun, Tao Ke and Li Gangzhu

Abstract

In order to study the effect of wall structure on the pressure-driven flow with slip condition at the fluid/porous interface, three cylindrical arrangements with different internal and surface structure are built. We solve the incompressible Stokes equation in the fluid phase for the cylindrical arrangements using the finite element analysis and solver software, Comsol-Multiphysics. The most suitable interface for determining the slip coefficient is the nominal position above the top cylindrical layer with a characteristic size. Then the streamlines of the fluid flow are shown. With the increase of pressure gradient and Reynolds number, the inertial effect rises and the exchange of fluid at the interface is visible. The permeability value and its growth rate both increase with the increment of the porosity and the internal structure of the porous medium affects the permeability value in an important way. Finally, our results show that the slip coefficient depends on the permeability and the surface structure instead of the porosity.

Funding statement: Funding: This paper is supported by National Basic Research Program of China (973 program, 2011CB201004) and National Science and Technology Major Project of China (2011ZX05014-003-006HZ).

Notation

u

fluid velocity in the channel, m/s

ui

tangential fluid velocity at the interface, m/s

μ

dynamic viscosity of fluid, kg/(m · s)

μe

effective viscosity, kg/(m · s)

K

permeability of the porous medium, m2

α

empirical dimensionless coefficient

uD

Darcy velocity, Pa · s

u

velocity vector

p

pressure fields, Pa

G

negative pressure gradient, Pa/m

um

mean velocity under the Poiseuille flow assumption, m/s

uis

scale quantity of ui, m/s

Rec

channel Reynolds number

Rei

interfacial Reynolds number

Rep

pore-scale Reynolds number

r

radius of the cylinder, m

ξ

gap between two adjacent cylinders, m

h

height for the free fluid region, m

H

total height for the channel, m kinematic viscosity, m2/s

Φ

porosity

δ

deviation from the nominal interface in the y-axis positive direction, m

x

xcoordinate

y

ycoordinate

References

1. EllahiR, ShivanianE, AbbasbandyS, HayatT.Analysis of some MHD flows of third order fluid saturating porous space. J Porous Med2015;18:8998.10.1615/JPorMedia.v18.i2.10Search in Google Scholar

2. EllahiR, BhattiMM, RiazA, SheikholeslamiM.Effects of magnetohydrodynamics on peristaltic flow of Jeffrey fluid in a rectangular duct through a porous medium. J Porous Med2014;17:14357.10.1615/JPorMedia.v17.i2.50Search in Google Scholar

3. EllahiR, HassanM, Afsar KhanA, MaqboolK.Analytical solution for non-Newtonian nanofluid with heat transfer and nonlinear partial slip boundary conditions by means of optimal homotopic asymptotic method. Adv Sci Eng Med2013;5:74451.10.1166/asem.2013.1304Search in Google Scholar

4. EllahiR, WangX, HameedM.Effects of heat transfer and nonlinear slip on the steady flow of Couette fluid by means of Chebyshev spectral method. Zeitschrift für Naturforschung A2014;69:18.10.5560/zna.2013-0060Search in Google Scholar

5. KhanAA, EllahiR, UsmanM.The effects of variable viscosity on the peristaltic flow of non-Newtonian fluid through a porous medium in an inclined channel with slip boundary conditions. J Porous Med2013;16:5967.10.1615/JPorMedia.v16.i1.60Search in Google Scholar

6. ZeeshanA, EllahiR, HassanM. Magnetohydrodynamic flow of water/ethylene glycol based nanofluids with natural convection through a porous medium. Eur Phys J Plus2014;129:110.10.1140/epjp/i2014-14261-5Search in Google Scholar

7. BeaversG, JosephDD. Boundary conditions at a naturally permeable wall. J Fluid Mech1967;30:197207.10.1017/S0022112067001375Search in Google Scholar

8. BeaversGS, SparrowEM, MagnusonRA, Experiments on coupled parallel flows in a channel and a bounding porous medium. J Basic Eng1970;92:8438.10.1115/1.3425155Search in Google Scholar

9. BeaversGS, SparrowEM, MashaBA. Boundary condition at a porous surface which bounds a fluid flow. AIChE J1974;20:5967.10.1002/aic.690200323Search in Google Scholar

10. LarsonRE, HigdonJJL. Microscopic flow near the surface of two-dimensional porous-media: 1. Axial-flow. J Fluid Mech1986;166:44972.10.1017/S0022112086000228Search in Google Scholar

11. LarsonRE, HigdonJJL. Microscopic flow near the surface of two-dimensional porous-media: 2. Transverse flow. J Fluid Mech1987;178:11936.10.1017/S0022112087001149Search in Google Scholar

12. SahraouiM, KavianyM. Slip and no–slip velocity boundary conditions at interface of porous, plain media. Int J Heat Mass Transfer1992;35:92744.10.1016/0017-9310(92)90258-TSearch in Google Scholar

13. ZhangQ, ProsperettiA. Pressure-driven flow in a two-dimensional channel with porous walls. J Fluid Mech2009;631:121.10.1017/S0022112009005837Search in Google Scholar

14. RashidiS, Nouri-BorujerdiA, ValipourM. S, EllahiR, PopI. Stress-jump and continuity interface conditions for a cylinder embedded in a porous medium. Trans Porous Med2015;107:17186.10.1007/s11242-014-0431-3Search in Google Scholar

15. RashidiS, DehghanM, EllahiR, RiazM, Jamal-AbadMT. Study of stream wise transverse magnetic fluid flow with heat transfer around an obstacle embedded in a porous medium. J Magn Magn Mater2015;378:12837.10.1016/j.jmmm.2014.11.020Search in Google Scholar

16. TangHS, KalyonDM. Unsteady circular tube flow of compressible polymeric liquids subject to pressure-dependent wall slip. J Rheol2008;52:50725.10.1122/1.2837104Search in Google Scholar

17. TangHS, KalyonDM. Time-dependent tube flow of compressible suspensions subject to pressure dependent wall slip: ramifications on development of flow instabilities. J Rheol2008;52:106990.10.1122/1.2955508Search in Google Scholar

18. DurlofskyL, BradyJF. Analysis of the Brinkman equation as a model for flow in porous-media. Phys Fluids1987;30:332941.10.1063/1.866465Search in Google Scholar

19. BeckermannC, RamadhyaniS, ViskantaR. Natural convection flow and heat transfer between a fluid layer and a porous layer inside a rectangular enclosure. ASME J Heat Trans1987;109:3639.10.1115/1.3248089Search in Google Scholar

20. SongM, ViskantaR. Natural convection flow and heat transfer within a rectangular enclosure containing a vertical porous layer. Int J Heat Mass Trans1994;37:242538.10.1016/0017-9310(94)90284-4Search in Google Scholar

21. VafaiK, KimS. Fluid mechanics of the interface region between a porous medium and a fluid layer–an exact solution. Int J Heat Fluid Flow1990;11:2546.10.1016/0142-727X(90)90045-DSearch in Google Scholar

22. Valdes-ParadaFJ, Ochoa-TapiaJAAlvarez-RamirezJ. Validity of the permeability Carman–Kozeny equation: a volume averaging approach. Physica A2009;388:78998.10.1016/j.physa.2008.11.024Search in Google Scholar

23. YunJ. W, LombardoS. J. Permeability of green ceramic tapes as a function of binder loading. J Am Ceram Soc2007;90:45661.10.1111/j.1551-2916.2006.01444.xSearch in Google Scholar

24. LiuQ, ProsperettiA. Pressure-driven flow in a channel with porous walls. J Fluid Mech2011;679:77100.10.1017/jfm.2011.124Search in Google Scholar

Published Online: 2015-6-17
Published in Print: 2015-9-1

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