Abstract
Flow-through and over a Brinkman porous layer of variable permeability, immersed in a fluid-filled channel, is modeled. The model governing equation through the porous layer inevitably gives rise to an inhomogeneous Weber's differential equation, solved in this work and solutions expressed in terms of parabolic cylindrical functions. Using state-of-the-art computational techniques and a body of knowledge, the parabolic cylindrical functions are evaluated for a range of flow and medium parameters in order to illustrate intrinsic characteristics of the flow quantities. The approach followed in this work is novel and sets precedent in the study of flow through general porous media configurations and flow domains with variable permeability.
References
1.
Vafai
,
K.
, and
Thiyagaraja
,
R.
, 1987
, “
Analysis of Flow and Heat Transfer at the Interface Region of a Porous Medium
,” Int. J. Heat Mass Transfer
,
30
(7
), pp. 1391
–1405
.10.1016/0017-9310(87)90171-22.
Beavers
,
G. S.
, and
Joseph
,
D. D.
, 1967
, “
Boundary Conditions at a Naturally Permeable Wall
,” J. Fluid Mech.
,
30
(1
), pp. 197
–207
.10.1017/S00221120670013753.
Whitaker
,
S.
, 1999
, The Method of Volume Averaging
,
Kluwer Academic Publishers
,
Dordrecht, The Netherlands
.4.
Darcy
,
H.
, 1856
, Les Fontaines Publiques de la Ville de Dijon
,
Dalmont
,
Paris
.5.
Nield
,
D. A.
, and
Bejan
,
A.
, 2017
, Convection in Porous Media
, 5th ed.,
Springer
,
New York
.6.
Kaviany
,
M.
, 1995
, Principles of Heat Transfer in Porous Media
, 2nd ed.,
Springer-Verlag
,
New York
.7.
Alazmi
,
B.
, and
Vafai
,
K.
, 2001
, “
Analysis of Fluid Flow and Heat Transfer Interfacial Conditions Between a Porous Medium and a Fluid Layer
,” Int. J. Heat Mass Transfer
,
44
(9
), pp. 1735
–1749
.10.1016/S0017-9310(00)00217-98.
Rudraiah
,
N.
, 1986
, “
Flow Past Porous Layers and Their Stability
,” Encycl. Fluid Mech. Slurry Flow Technol.
,
5
(14
), pp. 567
–647
.9.
Chandesris
,
M.
, and
Jamet
,
D.
, 2007
, “
Boundary Conditions at a Fluid-Porous Interface: An a Priori Estimation of the Stress Jump Coefficients
,” Int. J. Heat Mass Transfer
,
50
(17–18
), pp. 3422
–3436
.10.1016/j.ijheatmasstransfer.2007.01.05310.
Nield
,
D. A.
, 2009
, “
The Beavers–Joseph Boundary Condition and Related Matters: A Historical and Critical Note
,” Transp. Porous Media
,
78
(3
), pp. 537
–540
.10.1007/s11242-009-9344-y11.
Sahraoui
,
M.
, and
Kaviany
,
M.
, 1992
, “
Slip and No-Slip Velocity Boundary Conditions at Interface of Porous, Plain Media
,” Int. J. Heat Mass Transfer
,
35
(4
), pp. 927
–943
.10.1016/0017-9310(92)90258-T12.
Brinkman
,
H. C.
, 1949
, “
A Calculation of the Viscous Force Exerted by a Flowing Fluid on a Dense Swarm of Particles
,” Appl. Sci. Res.
,
1
(1
), pp. 27
–34
.10.1007/BF0212031313.
Neale
,
G.
, and
Nader
,
W.
, 1974
, “
Practical Significance of Brinkman's Extension of Darcy's Law: Coupled Parallel Flows Within a Channel and a Bounding Porous Medium
,” Can. J. Chem. Eng.
,
52
(4
), pp. 475
–478
.10.1002/cjce.545052040714.
Rudraiah
,
N.
, 1985
, “
Coupled Parallel Flows in a Channel and a Bounding Porous Medium of Finite Thickness
,” ASME J. Fluids Eng.
,
107
(3
), pp. 322
–329
.10.1115/1.324248615.
Auriault
,
J. L.
, 2009
, “
On the Domain of Validity of Brinkman's Equation
,” Transp. Porous Media
,
79
(2
), pp. 215
–223
.10.1007/s11242-008-9308-716.
Chandesris
,
M.
, and
Jamet
,
D.
, 2006
, “
Boundary Conditions at a Planar Fluid-Porous Interface for a Poiseuille Flow
,” Int. J. Heat Mass Transfer
,
49
(13–14
), pp. 2137
–2150
.10.1016/j.ijheatmasstransfer.2005.12.01017.
Nield
,
D. A.
, 1991
, “
The Limitations of the Brinkman- Forchheimer Equation in Modeling Flow in a Saturated Porous Medium and at an Interface
,” Int. J. Heat Fluid Flow
,
12
(3
), pp. 269
–272
.10.1016/0142-727X(91)90062-Z18.
Nield
,
D. A.
, 1983
, “
The Boundary Correction for the Rayleigh-Darcy Problem: Limitations of the Brinkman Equation
,” J. Fluid Mech.
,
128
(1
), pp. 37
–46
.10.1017/S002211208300036119.
Hamdan
,
M. H.
, and
Kamel
,
M. T.
, 2011
, “
Polar Fluid Flow Through Variable-Porosity, Isotropic Porous Media Spec
,” Top. Rev. Porous Media
,
2
(2
), pp. 145
–155
.10.1615/SpecialTopicsRevPorousMedia.v2.i2.8020.
Málek
,
J.
, and
Rajagopal
,
K. R.
, 2007
, “
Mathematical Properties of the Solutions to the Equations Governing the Flow of Fluids With Pressure and Shear Rate Dependent Viscosities
,” Hand-Book of Mathematical Fluid Dynamics
,
Elsevier
, Amsterdam, The Netherlands.10.1016/S1874-5792(07)80011-521.
Subramanian
,
S. C.
, and
Rajagopal
,
K. R.
, 2007
, “
A Note on the Flow Through Porous Solids at High Pressures
,” Comput. Math. Appl.
,
53
(2
), pp. 260
–275
.10.1016/j.camwa.2006.02.02322.
Allan
,
F. M.
, and
Hamdan
,
M. H.
, 2006
, “
Fluid-Particle Model of Flow Through Porous Media: The Case of Uniform Particle Distribution and Parallel Velocity Fields
,” Appl. Math. Comput.
,
183
(2
), pp. 1208
–1213
.10.1016/j.amc.2006.06.04523.
Hamdan
,
M. H.
, and
Barron
,
R. M.
, 1990
, “
A Dusty Gas Flow Model in Porous Media
,” Comput. Appl. Math.
,
30
(1
), pp. 21
–37
.10.1016/0377-0427(90)90003-I24.
Haider
,
F.
,
Hayat
,
T.
, and
Alsaedi
,
A.
, 2021
, “
Flow of Hybrid Nanofluid Through Darcy-Forchheimer Porous Space With Variable Characteristics
,” Alex. Eng. J.
,
60
(3
), pp. 3047
–3056
.10.1016/j.aej.2021.01.02125.
Ochoa-Tapia
,
J. A.
, and
Whitaker
,
S.
, 1995
, “
Momentum Transfer at the Boundary Between a Porous Medium and a Homogeneous Fluid—I: Theoretical Development
,” Int. J. Heat Mass Transfer
,
38
(14
), pp. 2635
–2646
.10.1016/0017-9310(94)00346-W26.
Ochoa-Tapia
,
J. A.
, and
Whitaker
,
S.
, 1995
, “
Momentum Transfer at the Boundary Between a Porous Medium and a Homogeneous Fluid—II: Comparison With Experiment
,” Int. J. Heat Mass Transfer
,
38
(14
), pp. 2647
–2655
.10.1016/0017-9310(94)00347-X27.
Ehrhardt
,
M.
, 2010
, An Introduction to Fluid-Porous Interface Coupling
,
Weierstrass Institute for Applied Analysis and Stochastics
,
Berlin, Germany
.28.
Hamdan
,
M. H.
, and
Barron
,
R. M.
, 1991
, “
Analysis of the Darcy-Lapwood and the Darcy-Lapwood-Brinkman Models: Significance of the Laplacian
,” Appl. Math. Comput.
,
44
(2
), pp. 121
–141
.10.1016//00963003(91)90014-e29.
Vafai
,
K.
, 1986
, “
Analysis of the Channeling Effect in Variable Porosity Media
,” ASME J. Energy Resour. Technol.
,
108
(2
), pp. 131
–139
.10.1115/1.323125230.
Hamdan
,
M. H.
, and
Kamel
,
M. T.
, 2011
, “
Flow Through Variable Permeability Porous Layers
,” Adv. Theor. Appl. Mech.
,
4
(3
), pp. 135
–145
.http://mhikari.com/atam/atam2011/atam1-4-2011/hamdanATAM1-4-2011-2.pdf31.
Kaviany
,
M.
, 1985
, “
Laminar Flow Through a Porous Channel Bounded by Isothermal Parallel Plates
,” Int. J. Heat Mass Transfer
,
28
(4
), pp. 851
–858
.10.1016/0017-9310(85)90234-032.
Gil
,
A.
,
Segura
,
J.
, and
Temme
,
N. M.
, 2011
, “
Fast and Accurate Computation of the Weber Parabolic Cylinder Function
,” IMA J. Numer. Anal.
,
31
(3
), pp. 1194
–1216
.10.1093/imanum/drq01233.
Segura
,
J.
, and
Gil
,
A.
, 1998
, “
Parabolic Cylinder Functions of Integer and Half- Integer Orders of Non-Negative Arguments
,” Comput. Phys. Commun.
,
115
(1
), pp. 69
–86
.10.1016/S0010-4655(98)00097-634.
Temme
,
N. M.
, 1996
, Special Functions: An Introduction to the Classical Functions of Mathematical Physics
,
Wiley
,
New York
.35.
Temme
,
N. M.
, 2010
, “
Parabolic Cylinder Functions
,” NIST Handbook of Mathematical Functions
, U.S. Dept. Commerce, Vol.
12
,
Cambridge University Press
,
Washington, DC
, pp. 303
–319
.36.
Rees
,
D. A. S.
, and
Pop
,
I.
, 2000
, “
Vertical Free Convection in a Porous Medium With Variable Permeability Effects
,” J. Heat Mass Transfer
,
43
(14
), pp. 2565
–2571
.10.1016/S0017-9310(99)00316-637.
Nield
,
D. A.
, and
Kuznetsov
,
A. V.
, 2009
, “
The Effect of a Transition Layer Between a Fluid and a Porous Medium: Shear Flow in a Channel
,” Transp. Porous Media
,
78
(3
), pp. 477
–487
.10.1007/s11242-009-9342-038.
Siginer
,
D. A.
, and
Bakhtiyarov
,
S. I.
, 2001
, “
Flow in Porous Media of Variable Permeability and Novel Effects
,” ASME J. Appl. Mech.
,
68
(2
), pp. 312
–319
.10.1115/1.134912039.
Hamdan
,
M. H.
, and
Kamel
,
M. T.
, 2011
, “
On the Ni(x) Integral Function and Its Application to the Airy's Non-Homogeneous Equation
,” Appl. Math. Comput.
,
217
(17
), pp. 7349
–7360
.https://doi.org/10.1016/j.amc.2011.02.02540.
Alzahrani
,
S. M.
,
Gadoura
,
I.
, and
Hamdan
,
M. H.
, 2016
, “
Ascending Series Solution to Airy's Inhomogeneous Boundary Value Problem
,” Int. J. Open Probl. Compt. Math.
,
9
(1
), pp. 1
–11
.10.12816/002635041.
Abu Zaytoon
,
M. S.
,
Alderson
,
T. L.
, and
Hamdan
,
M. H.
, 2016
, “
Flow Through Layered Media With Embedded Transition Porous Layer
,” Int. J. Enhanced Res. Sci. Technol. Eng.
,
5
(4
), pp. 9
–26
.42.
Abu Zaytoon
,
M. S.
,
Alderson
,
T. L.
, and
Hamdan
,
M. H.
, 2016
, “
Flow Through Variable Permeability Composite Porous Layers
,” Gen. Math. Notes
,
33
(1
), pp. 26
–39
.43.
Abu Zaytoon
,
M. S.
,
Alderson
,
T. L.
, and
Hamdan
,
M. H.
, 2016
, “
Flow Through a Variable Permeability Brinkman Porous Core
,” J. Appl. Math. Phys.
,
04
(04
), pp. 766
–778
.10.4236/jamp.2016.4408744.
Abu Zaytoon
,
M. S.
,
Alderson
,
T. L.
, and
Hamdan
,
M. H.
, 2016
, “
Flow Through a Layered Porous Configuration With Generalized Variable Permeability
,” Int. J. Enhanced Res. Sci., Technol. Eng.
,
5
(6
), pp. 1
–21
.45.
Abu Zaytoon
,
M. S.
,
Alderson
,
T. L.
, and
Hamdan
,
M. H.
, 2016
, “
Weber's Inhomogeneous Differential Equation With Initial and Boundary Conditions
,” Int. J. Open Probl. Compt. Math.
,
9
(2
), pp. 1
–11
.10.12816/003391746.
Schulten
,
Z.
,
Gordon
,
R. G.
, and
Anderson
,
D. G. M.
, 1981
, “
A Numerical Algorithm for the Evaluation of Weber Parabolic Cylinder Functions U(a, x), V (a, x), and W(a, +/-x)
,” J. Comput. Phys.
,
42
(2
), pp. 213
–217
.10.1016/0021-9991(81)90241-247.
Temme
,
N. M.
, 2000
, “
Numerical and Asymptotic Aspects of Parabolic Cylinder Functions
,” Comput. Appl. Math.
,
121
(1–2
), pp. 221
–246
.10.1016/S0377-0427(00)00347-248.
Dunster
,
T. M.
, 2021
, “
Nield-Kuznetsov Functions and Laplace Transforms of Parabolic Cylinder Functions
,” SIAM J. Math. Anal.
,
53
(5
), pp. 5915
–5947
.10.1137/21M140159049.
Dunster
,
T. M.
, 2021
, “
Uniform Asymptotic Expansions for Solutions of the Parabolic Cylinder and Weber Equations
,” J. Classical Anal.
,
17
(1
), pp. 69
–107
.https://doi.org/10.7153/jca-2021-17-06Copyright © 2022 by ASME
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