Full length article
Natural convection heat and mass transfer in MHD
fluid flow past
a moving vertical plate with variable surface temperature
and concentration in a porous medium
K. Javaherdeh
a,*, Mehrzad Mirzaei Nejad
a, M. Moslemi
baFaculty of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran bAyandegan Institute of Higher Education, Tonekabon, Iran
a r t i c l e i n f o
Article history:
Received 25 September 2014 Received in revised form 17 February 2015 Accepted 1 March 2015 Available online 29 March 2015 Keywords:
Heat and mass transfer Moving plate MHD effect Porous medium
Variable surface temperature and concentration
a b s t r a c t
A numerical investigation of two-dimensional steady laminar free convectionflow with heat and mass transfer past a moving vertical plate in a porous medium subjected to a transverse magneticfield is carried out. The temperature and concentration level at the plate surface are assumed to follow a power-law type of distribution. The governing non-linear set of equations is solved numerically employing a fully implicitfinite difference method. Results are presented to illustrate the influence of different pa-rameters such as Grashof number (Gr), porosity parameter (Kp), magneticfield parameter (Mn) and exponents in the power law variation of the surface temperature and concentration, m and n. The dimensionless velocity, temperature and concentration profiles are analyzed and numerical data for the local Nusselt number and Sherwood number are presented. The study accentuates the significance of the relevant parameters.
© 2015 Karabuk University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
During the past several decades, convectiveflow through porous media has been a subject of considerable research interest of a large number of scholars due to its diverse engineering applications. These applications include, but are not limited to, for example heat exchangers in high heat flux applications such as electronic equipment, insulation of the heated body, thermal energy storage and sensible heat storage beds, drying process (wood and food products), air conditioning andfiltration process. During the last decades, several researchers studied free convection heat and mass transfer in a porous medium[1e3]. Mass transfer effects onflow past an accelerated vertical plate has already been well studied
[4,5]. Furthermore, the problem of heat and mass transfer of non-Newtonianfluids in porous media has been a subject of interest in many research projects[6e10].
In recent years, considerable attention has been devoted to the study of magetohydrodynamics (MHD)flow and heat. In addition to
useful features of MHDflows, such studies can be helpful in pre-diction of the effects of magnetic intrusions. Raptis and Singh[11]
studied MHD free convection flow past an accelerated vertical plate. Taza Gul et al.[12]studied heat transfer of MHD thinfilm flow of an unsteady second grade fluid past a vertical oscillating belt by analytical techniques.
In recent years there has been a growing interest in studying the combined application of MHDflow and porous media. Since the use of magnetic field can influence the heat generation/absorption process in electrically conductingfluid flows, the rate of cooling in many metallurgical processes and consequently the desired prop-erties of the end product can be controlled. Furthermore, the in-fluence of magnetic field on boundary layer flows has brought about its application in geothermal energy recovery, oil extraction and thermal insulations. Abdulhameed et al. [13]obtained exact solution for an unsteady two-dimensional MHD flow of incom-pressible viscous fluid over a flat plate with wall transpiration embedded in a porous medium. Aldoss et al. [14] investigated combined free and forced convection flow from a vertical plate embedded in a porous medium in the presence of a magneticfield. Chamkha[15]considered a plate embedded in a uniform porous medium which moves with a constant velocity in theflow direction in the presence of a transverse magneticfield.
* Corresponding author.
E-mail address:[email protected](K. Javaherdeh). Peer review under responsibility of Karabuk University.
H O S T E D BY Contents lists available atScienceDirect
Engineering Science and Technology,
an International Journal
j o u rn a l h o m e p a g e : h t t p : / / w w w . e l s e v i e r . c o m / l o c a t e / j e s t c h
http://dx.doi.org/10.1016/j.jestch.2015.03.001
2215-0986/© 2015 Karabuk University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
Reddy[16]studied radiation effects on MHD natural convection flow along a vertical cylinder embedded in a porous medium with variable surface temperature and concentration. Also, Olajuwon et al. [17] investigated the effect of thermal radiation and Hall current on MHDflow of a viscoelastic micropolar fluid through a porous medium. Furthermore, unsteady MHDflow in porous media has been studied by several researchers[18e20].
Combined buoyancy-generated heat and mass transfer are characterized by highly non-linear coupled partial differential equations which are required to be solved by numerical computations.
In recent years different researchers have used and discussed various numerical modeling techniques[21e25]. Afinite difference solution of MHD mixed convectionflow with heat generation and chemical reaction was developed by Ahmed and Mahmud[26]. Abd El-Naby et al.[27]presented afinite difference solution of radiation effects on MHD unsteady free-convection flow over a vertical porous plate. A numerical solution of the transient free convection MHDflow of an incompressible viscous fluid past a semi-infinite inclined plate with variable surface heat and massflux was dis-cussed by Ganesan and Palani [28]. Combined heat and mass transfer effects on a moving vertical cylinder was investigated by Takhar et al.[29]by using a Crank- Nicolson type implicit finite-difference scheme.
Although there are numerous practical applications in indus-trial processes, few previous published papers discussed heat and mass transfer from a moving surface with variable temperature and concentration. In this article, we attempt to investigate free convective heat and mass transfer for MHDflow past a vertical moving plate in a porous medium. It is noticed that the tem-perature and concentration do not remain constant in so many fluid flow problems of practical interests. This prompted the au-thors to consider a plate at varying, particularly linear tempera-ture and concentration distribution to approach real cases. The governing equations are solved by using an implicitfinite differ-ence method. The effect of relevant parameters like Grashof number (Gr), porosity parameter (Kp), magneticfield parameter (Mn) and exponents in the power law variation of the surface temperature and concentration, m and n, on heat and mass transfer is explored.
2. Mathematical formulation
Consider steady two-dimensional laminar free convective heat and mass transfer flow of viscous incompressible electrically conductingfluid past a vertical moving plate in a porous medium where a uniform magnetic field is applied in the direction perpendicular to thefluid flow.Fig. 1shows the physical model and coordinate system of problem. The x-axis is assumed to be taken along the plate in the vertically upward direction and the y-axis normal to the plate. The moving plate is under variable temperature boundary condition (Tw ¼ T∞ þ bxn) and variable
concentration (Cw¼ C∞þ axm). It is also assumed that the effect of viscous dissipation is negligible in the energy equation and there is no chemical reaction between thefluid and the diffusing species.
The governing boundary layer equations of mass, momentum, energy and species concentration for natural convectionflow with Boussinesq's approximation are as follow:
vu vxþ vv vy¼ 0 (1) uvu vxþ v vu vy¼ n v2u vy2þ gbTðT T∞Þ þ gbcðc c∞Þ n rK uþ F ! m r (2)
where Fmis magneto hydrodynamic force and is calculated as:
F ! m¼ F!em¼ J!c B ! 0¼ s V! B!0 B!0 ¼ su i!þ v j!B0!j B!0¼ s u B20 uvT vxþ v vT vy¼ a v2T vy2 (3) uvc vxþ v vc vy¼ D v2c vy2 (4)
The boundary conditions are as follows[30]:
Fig. 1. Physical model and coordinate system.
T¼ Tw¼ T∞þ bxn; c ¼ Cw¼ C∞þ axm; u ¼ up; v ¼ 0 on the surface : y ¼ 0
T/T∞; c/C∞; u/0 on free surface : y/∞
Introducing the following non-dimensional quantities: X¼xup n ; Y ¼ yup n ; U ¼ u up; V ¼ n up; Mn ¼ nsB2 0 ru2 p q ¼TTw T w T∞; C ¼ Cw c Cw C∞; Pr ¼ n a; Sc ¼ n D; N ¼ bcðCw C∞Þ bTðTw T∞Þ ; Gr ¼gbTðTw T∞Þn u3 p ; Kp¼ n 2 rku2 p (5)
into Equations (1 e 4) gives the following set of differential equations: vU vXþ vV vY¼ 0 (6) UvU vXþ V vU vY ¼ v2U vY2þ Grðð1 qÞ þ Nð1 CÞÞ Kpþ Mn U (7) Uvq vX n XUð1 qÞ þ V vq vY¼ 1 Pr v2q vY2 (8) UvC vX m XUð1 CÞ þ V vC vY¼ 1 Sc v2C vY2 (9)
and the corresponding boundary conditions are:
q ¼ 0; C ¼ 0; U ¼ 1; V ¼ 0 on the Y ¼ 0
q ¼ 1; C ¼ 1; U ¼ 0 on the free surface: Y/∞
q ¼ 1; C ¼ 1; U ¼ 0 on the plate attack edge
The local Nusselt number can be defined as:
Nux¼hxx
K (10)
Table 1
The comparison of Nusselt number values with Cheng[33]for n¼ 1 and 1/3. n¼ 1 n¼ 1/3
Cheng[33] Present result Cheng[33] Present result Nux=Ra1=2x 1.0000 0.99978 0.6776 0.67773
Using Newton's-law of cooling and Fourier's-law the local Nusselt number can then be expressed as:
NuX¼ XvYvq: (11)
Also, the Sherwood number is defined as:
Shx¼hDDx (12)
which can be expressed as:
ShX¼ XvCvY: (13)
3. Numerical method
The non-dimensional equations can be solved numerically using a fully implicitfinite-difference method with the relevant boundary conditions. The governing equations are considered as unsteady, untilflow state becomes steady.
Fig. 3. Axial distribution of (a) local Nusselt number (b) local Sherwood number for Pr¼ 1,Sc ¼ 0.7, Kp ¼ 0.5, Mn ¼ 1,N ¼ 1,n ¼ 1,m ¼ 2 and different Grashof number.
Fig. 4. Velocity profile for Pr ¼ 4, Sc ¼ 2, Mn ¼ 1,N ¼ 1, n ¼ 1.5, m ¼ 2.5, Gr ¼ 10.
Fig. 6. (a) Velocity profile, (b) Temperature profile, (c) Concentration profile for Pr ¼ 1,Sc ¼ 2, Kp ¼ 0.7, N ¼ .5,n ¼ 2.5,m ¼ 2,Gr ¼ 15.
To transform the momentum, energy and concentration equa-tions intofinite difference equations, the axial convection term is approximated by an upstream difference and the transverse con-vection and diffusion terms by a central difference. Also the un-steady terms are approximated by a backward difference during each transient and axial step[31,32]. The resulting system of linear algebraic equations can be written in tridiagonal form as:
Ai;jUnþ1
i;j1þ Bi;jUnþ1i;j þ Ci;jUnþ1i;jþ1¼ Di;j (14)
where
U
represent U,q
or C.Eq.(14)can be iteratively solved through Tri Diagonal Matrix Algorithm (TDMA). The TDMA is an approach based on the Gaussian elimination procedure and is only applicable to matrices that are diagonally dominant. It is worth mentioning that the
Dirichlet boundary conditions assist the dominance of matrix's main diagonal[31,32].
In this work, the same process is continued until the conver-gence criterion Unþ1 i;j Uni;j Un max < 5 105 (15) is achieved.
Because of higher velocity and temperature gradients near the moving surface and plate attack edge, smaller gridding is used here. In the study, 301 non-uniform grid points are employed in the transverse direction (Y). Some of the calculations were tested using 301 non-uniform grid points in the Y-direction, but no significant improvement over the 501 grid points was found. Additionally, there are 120 non-uniform grid points in the marching direction. Testing the program, afiner axial step size was tried and the current accuracy was found to be acceptable.
Furthermore, in order to ascertain the accuracy of the present computer code, the present study was compared with available solutions in the literature. Computations werefirst made for free convection about a verticalflat plate with ramped temperature embedded in a porous medium.
The obtained local Nusselt numbers were compared with the available solution of Cheng[33], inTable 1and are in good agree-ment with those of Cheng.
4. Results and discussion
In this research, the effects of the Grashof number, porous me-dium parameter, magnetic field parameter, power law index of temperature and concentration on local Nusselt and Sherwood number are shown graphically. Also, velocity, temperature and concentration profiles are studied and discussed.
Fig. 2aec illustrates the variation of velocity, temperature and concentration distribution of theflow field for different values of Grashof number. The thermal Grashof number characterizes the relative effect of the thermal buoyancy force to the viscous hy-drodynamic force in the boundary layerflow. Increase of Gr number leads to a rise in the values of velocity owing to the assistance of thermal buoyancy force which induces a favorable pressure
Fig. 8. (a) Velocity profile, (b) Temperature profile for Pr ¼ 2,Sc ¼ 1,Mn ¼ 1,Kp ¼ 0.8, N ¼ .7,m ¼ 1.5,Gr ¼ 20.
Fig. 9. Axial distribution of local Nusselt number for Pr¼ 2,Sc ¼ 1,Mn ¼ 1,Kp ¼ 0.8, N¼ .7,m ¼ 1.5,Gr ¼ 20.
gradient. This implies that thermal buoyancy force tends to accel-erate velocity. The fluid velocity attains a distinctive maximum value in a region near the plate surface and then decays to the free stream value.
The acceleration offluid velocity corresponds to a reduction in temperature and concentration boundary layer thicknesses. Higher temperature and concentration gradients, as displayed inFig. 2b and c, yield higher rates of heat and mass transfer and an increase in local Nusselt and Sherwood number (Fig. 3a and b).
It is evident fromFig. 4that fluid velocity decreases with an increase in the porosity parameter (Kp). Physically, this refers to the
fact that increasing the tightness of the porous medium which is represented by increase in Kpresults in increasing the resistance
against theflow.
Fig. 5a and b shows the effect of porous medium parameter (Kp)
on the local Nusselt and Sherwood number. A negligible decrease can be seen in heat and mass transfer.
Fig. 6(aec) depicts the influence of magnetic field on the ve-locity, temperature and concentration profiles. The range of mag-neticfield is taken from 0 to 2. The velocity of the flow field is found to decrease in presence of magneticfield. Physically, it is true due to the fact that the application of a transverse magneticfield to an electrically conductingfluid gives rise to a body force known as Lorentz force which tends to resist thefluid flow and slow down its motion in the boundary layer region. This, in turn, reduces the rate of heat convection in theflow and this appears in increasing the flow temperature (Fig. 6b).
It is clear inFig. 6(b) and (c) that the temperature and concen-tration gradients decrease for higher values of magnetic field parameter (Mn) which result in lower heat and mass transfer (Fig. 7a and b)
InFig. 8a and b the velocity and temperature profiles are drawn against Y for three different values of variable wall temperature index (n). Here the values of n are chosen 1, 2, and 3.
Increase in the wall temperature power law index (n) is accompanied by simultaneous reductions in thefluid velocity and temperature. Higher temperature difference between the moving plate and the fluid yields to higher Nusselt number as demon-strated inFig. 9.
Velocity and temperature profiles are shown inFig. 10a and b for three different values of variable wall concentration index (m). Here the values of m are chosen 1, 2, and 3. A comparative study of
Figs. 8 and 10reveals that the variable wall temperature and con-centration indices, n and m, have similar influences on the velocity profiles. The concentration difference between fluid and surface increases as m increases from 1.0 to 3.0 which contributes to the increase of Sherwood Number (Fig. 11).
5. Conclusion
Free convective heat and mass transfer for MHD flow past a vertical moving plate in a porous medium have been studied. The temperature and concentration level at the plate surface were assumed to vary as power law type functions in the streamwise co-ordinate. The momentum, energy and concentration equations were solved numerically using a fully implicit finite difference method. Effects of different parameters such as Grashof number (Gr), porosity parameter (Kp), magneticfield parameter (Mn) and exponents in the power law variation of the surface temperature
Fig. 10. (a) Velocity profile, (b) Concentration profile for Pr ¼ 2,Sc ¼ 1,Mn ¼ 1,Kp ¼ 0.8, N ¼ .7,n ¼ 1.5,Gr ¼ 20.
Fig. 11. Axial distribution of local Sherwood number for Pr¼ 2,Sc¼ 1,Mn ¼ 1,Kp¼ 0.8, N¼ .7,n¼ 1.5,Gr ¼ 20.
and concentration, m and n, on velocity, temperature and con-centration profiles together with the variations in heat and mass transfer were analyzed.
The present study brings out the following significant findings: 1 The Grashof number has an accelerating effect on theflow ve-locity due to the enhancement in the buoyancy force. This in-creases the temperature and concentration gradients on the surface and leads to a rise in heat and mass transfer.
2 The higher the porosity parameter, the more sharply is the reduction in velocity which is accompanied by a reduction in both Nusselt and Sherwood number.
3 Transverse magnetic field produces a type of resistive force which opposes theflow. This contributes to the thickening of the thermal and mass boundary layer which, in turn, reduces the rate of heat and mass transfer.
4 The increase in the power law index for either wall variable temperature (n) or wall variable concentration (m) is seen to decelerate theflow while increasing the temperature and con-centration gradients on the surface. This measure helps to in-crease the Nusselt and Sherwood number whereas the flow velocity is reduced.
Nomenclature
B0 magneticfield intensity
c concentration of thefluid Cp specific heat at constant pressure
D mass diffusion coefficient F magnetic force
g acceleration due to gravity Gr Grashof number
hx local heat transfer coefficient
hd local mass transfer coefficient
i grid point along the Xe direction j grid point along the Ye direction Kp permeability of the porous medium
k thermal conductivity
m exponent in power law variation of wall temperature Mn magneticfield parameter
n exponent in power law variation of wall concentration Nux local Nusselt number
Pr Prandtl number Re Reynolds number Sc Schmidt number Shx local Sherwood number
T fluid temperature
u, v fluid velocity components along x and y directions respectively
Up moving plate velocity
x, y coordinate axes along and perpendicular to the plate
a
thermal diffusivityb
coefficient of volume expansionn
kinematic viscositym
viscosity of thefluidr
density of thefluids
electrical conductivity C dimensionless concentration X,Y dimensionless x and y co-ordinatesU,V dimensionless velocity components in X,Y directions
q
dimensionless temperature Subscriptsw condition on the wall ∞ at free-stream condition
T at constant temperature condition C at constant concentration condition
References
[1] A. Bejan, K.R. Khair, Heat and mass transfer by natural convection in a porous medium, Int. J. Heat Mass Transf. 28 (1985) 909e918.
[2] F.C. Lai, Coupled heat and mass transfer by mixed convection from a vertical plate in a saturated porous medium, Int. Commun. Heat Mass Transf. 18 (1991) 93e106.
[3] A.R. Bestman, Natural convection boundary layer with suction and mass transfer in a porous medium, Int. J. Energy Res. 14 (1990) 389e396. [4] V.M. Soundalgekar, Effects of mass transfer onflow past a uniformly
accel-erated vertical plate, Lett. Heat Mass Transf. 9 (1982) 65e72.
[5] A.K. Singh, J. Singh, Mass transfer effects on theflow past an accelerated vertical plate with constant heatflux, Astrophys. Sp. Sci. 97 (1983) 57e61. [6] K.N. Mehta, K.N. Rao, Buoyancy-inducedflow of non-Newtonian fluids in a
porous medium past a vertical plate with nonuniform surface heatflux, Int. J. Eng. Sci. 32 (1994) 297e302.
[7] Yue-Tzu Yang, S. Wang, Free convection heat transfer of non-Newtonian over axisymmetric and two-dimensional bodies of arbitrary shape embedded in a fluid- saturated porous medium, Int. J. Heat Mass Transf. 39 (1996) 203. [8] H.T. Chen, C.K. Chen, Free convection of non-Newtonianfluids along a vertical
plate embedded in a porous medium, ASME J. Heat Transf. 110 (1988) 257e260.
[9] R.Y. Jumah, A.S. Mujumdar, Free convection heat and mass transfer of non-Newtonian power-lawfluids with yield stress from a vertical flat plate in a saturated porous media, Int. Commun. Heat Mass Transf. 27 (2000) 485e494.
[10] S. Abel, P.H. Veena, Visco-elasticfluid flow and heat transfer in a porous medium over a stretching sheet, Int. J. Non-Linear Mech. 33 (1998) 531e540. [11] A. Raptis, A.K. Singh, MHD free convectionflow past an accelerated vertical
plate, Int. Comm. Heat Mass Transf. 10 (1983) 313e321.
[12] T. Gul, S. Islam, R.A. Shah, I. Khan, S. Shafie, Heat transfer analysis of MHD thin film flow of an unsteady second grade fluid past a vertical oscillating belt, PLoS One 9 (11) (2014) e103843.
[13] M. Abdulhameed, I. Khan, S. Shafie, Closed form solutions for unsteady MHD flow in a porous medium with wall transpiration, J. Porous Media 16 (9) (2013) 795e809.
[14] T.K. Aldoss, M.A. Al-Nimr, M.A. Jarrah, B.J. Al-Shaer, Magnetohydrodynamic mixed convection from a vertical plate embedded in a porous medium, Numer. Heat Transf. 28 (1995) 635.
[15] A.J. Chamkha, Unsteady MHD convective heat and mass transfer past a semi-infinite vertical permeable moving plate with heat absorption, Int. J. Eng. Sci. 42 (2004) 217e230.
[16] M.G. Reddy, Radiation effects on MHD natural convectionflow along a vertical cylinder embedded in a porous medium with variable surface temperature and concentration, Front. Heat Mass Transf. (FHMT) 5 (2014) 4.
[17] B.I. Olajuwon, J.I. Oahimire, M. Ferdow, Effect of thermal radiation and Hall current on heat and mass transfer of unsteady MHDflow of a viscoelastic micropolarfluid through a porous medium, Eng. Sci. Technol. Int. J. (2014),
http://dx.doi.org/10.1016/j.jestch.2014.05.004.
[18] A. Khan, I. IKhan, F. Ali, S. Shafie, Effects of wall shear stress MHD conjugate flow over an inclined plate in a porous medium with ramped wall tempera-ture, Math. Probl. Eng. (2014) 15. Article ID 861708.
[19] A. Hussnan, Z. Ismail, I. Khan, S. Shafie, Unsteady MHD free convection flow in a porous medium with constant mass diffusion and Newtonian heating, Eur. Phys. J.e Plus 129 (2014) 46.
[20] Samiulhaq, A. Sohail, D. Vieru, I. Khan, S. Shafie, Unsteady magnetohydrody-namic free convectionflow of a second grade fluid in a porous medium with ramped wall temperature, PLoS One 9 (5) (2014) e88766.
[21] M. Mirzaei nejad, K. Javaherdeh, Numerical simulation of power-lawfluids flow and heat transfer in a parallel-plate channel with transverse rectangular cavities, Case Stud. Therm. Eng. 3 (2014) 68e78.
[22] M. Donisete de Campos, E.C. Rom~ao, L.F. Mendes de Moura, A finite-difference method of high-order accuracy for the solution of transient nonlinear dif-fusiveeconvective problem in three dimensions, Case Stud. Therm. Eng. 3 (2014) 43e50.
[23] M. Hatami, D.D. Ganji, Natural convection of sodium alginate (SA) non-Newtonian nanofluid flow between two vertical flat plates by analytical and numerical methods, Case Stud. Therm. Eng. 2 (2014) 14e22.
[24] M.A.Y. Bakier, Flow in open C-shaped cavities: how far does the change in boundaries affect nanofluid? Eng. Sci. Technol. Int. J. 17 (2014) 116e130. [25] M.H. Mohamed, A.M. Ali, A.A. Hafiz, CFD analysis for H-rotor Darrieus turbine
as a low speed wind energy converter, Eng. Sci.Technol. Int. J. (2014),http:// dx.doi.org/10.1016/j.jestch.2014.08.002.
[26] T. Ahmed, Md Mahmud Alam, Finite difference solution of MHD mixed con-vectionflow with heat generation and chemical reaction, Procedia Eng. 56 (2013) 149e156.
[27] M.A. Abd El-Naby, M.E. Elbarbary Elsayed, NaderY. AbdElazem, Finite differ-ence solution of radiation effects on MHD unsteady free-convectionflow over vertical porous plate, Int. Comm. Math. Computation 151 (2004) 327e346.
[28] P. Ganesan, G. Palani, Finite difference analysis of unsteady natural convection MHDflow past an inclined plate with variable surface heat and mass flux, Int. J. Heat Mass Transf. 47 (2004) 4449e4457.
[29] H.S. Takhar, A.J. Chamkha, G. Nath, Combined heat and mass transfer along a vertical cylinder with free stream, Heat Mass Transf. 36 (2000) 237e246.
[30] P.H. Oosthuizen, D. Naylor, An Introduction to Convective Heat Transfer Analysis, McGraw-Hill, 1999.
[31] R.H. Pletcher, J.C. Tannehill, D. Anderson, Computational Fluid Mechanics and Heat Transfer, Third ed., CRC Press, 2012.
[32] C. Hirsch, Numerical Computation of Internal and External Flows, in: Funda-mentals of Computational Fluid Dynamics, Second ed., vol. 1, Butterworth-Heinemann, 2007.
[33] P. Cheng, W.J. Minkowyez, Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike, J. Geophys. Res. 82 (1977) 2040e2044.
