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Numerical and analytical solutions for magneto-hydrodynamic 3D flow through two parallel porous plates
International Journal of Heat and Mass Transfer 108 (2017) 322–331
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Review
Numerical and analytical solutions for magneto-hydrodynamic 3D flow through two parallel porous plates Sahin Ahmed a, Joaquín Zueco b,⇑, Luis María López-González c a
Fluid Dynamics Research, Department of Mathematics, Goalpara College, Goalpara, Assam 783101, India ETS Ingeniería Industrial, Departamento de Ingeniería Térmica y Fluidos, Universidad Politécnica de Cartagena, 30202 Cartagena (Murcia), Spain c Department of Mechanic Eng., Universidad de La Rioja, C/Luís de Ulloa, 20, E-26004 Logroño (La Rioja), Spain b
a r t i c l e
i n f o
Article history: Received 3 August 2016 Received in revised form 10 October 2016 Accepted 29 November 2016
Keywords: Semi-analytical solution Perturbation technique Network model MHD Injection/suction Couette flow Porous medium
a b s t r a c t The present work analyzed the hydromagnetic effect on three-dimensional Couette flow of viscous incompressible, electrically conducting and Newtonian fluid through a porous medium bounded by two horizontal parallel porous flat plates with transverse sinusoidal injection of the fluid at the stationary plate and its corresponding removal by periodic suction through the plate in uniform motion. The flow becomes three dimensional due to this injection/suction velocity. Approximate solutions are obtained for the flow field, the pressure, the skin-friction, the temperature field, and the rate of heat transfer. It is found that the cross velocity w is reduced considerably with a rise in the magnetic body parameter (M), permeability parameter (K) and injection/suction (k) in the forward flow, while a reverse effect is observed in the backward flow. An increase in K or k is found to escalate the main flow velocity whereas an increase in the magnetic body parameter (M) is shown to exert the opposite effect. Similarly, the shear stress due to main flow is considerably increased with an increase in M and K. The results show that both methods, perturbation and electrical network schemes provides excellent approximations to the solution of this nonlinear system with high accuracy. The acquired knowledge in our study can be used by designers to control Magnetohydrodynamic (MHD) flow as suitable for a certain application. Other possible applications include materials processing, MHD propulsion thermo-fluid dynamics, boundary layer in aerodynamics, chemical engineering etc. Ó 2016 Elsevier Ltd. All rights reserved.
Contents 1. 2. 3. 4. 5. 6.
Introduction . . . . . . . . . . . . . . . . . . . . . . . Formulation of the problem . . . . . . . . . . Semi-analytical solution . . . . . . . . . . . . . Numerical method (Network Simulation Discussion and results . . . . . . . . . . . . . . . Conclusions. . . . . . . . . . . . . . . . . . . . . . . . Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction Magnetohydrodynamics deals with dynamics of an electrically conducting fluid, which interacts with a magnetic field. The study ⇑ Corresponding author. E-mail addresses: [email protected] (S. Ahmed), [email protected] (J. Zueco), [email protected] (L.M. López-González). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.11.102 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
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322 324 325 325 327 329 330 330
of heat transfer and flow, through and across porous media, is of great theoretical interest because it has been applied to a variety of geophysical and astrophysical phenomena. Practical interest of such study includes applications in electromagnetic lubrication, boundary cooling, bio-physical systems and in many branches of engineering and science. In engineering, it finds its application in MHD pumps, MHD bearings, nuclear reactors, geothermal energy extraction and in boundary layer control in the field of aerodynam-
S. Ahmed et al. / International Journal of Heat and Mass Transfer 108 (2017) 322–331
323
Nomenclature !
B B0 CP d ! E0 ! J ! ! J XB J K K M Pr p T ; v ; wÞ ðu
magnetic induction vector uniform magnetic field (Tesla) specific heat at constant pressure (Jkg1K) distance between the plates electric field electric current density Lorentz force per unit volume electrical current (A) permeability coefficient permeability of the porous medium Hartmann number/magnetic parameter Prandtl number pressure (Nm2) temperature (K) and z-directions velocity components along x, y respectively
ics. A survey of MHD studies could be found in Crammer and Pai [1]; Moreau [2]. In view of the increasing technical applications using magnetohydrodynamics (MHD) effect, it is desirable to extend many of the available viscous hydrodynamic solutions to include the effects of magnetic field for those cases when the viscous fluid is electrically conducting. Rossow [3], and Greenspan and Carrier [4] have studied extensively the hydromagnetic effects on the flow past a plate with or without injection/suction. The hydromagnetic channel flow and temperature field was investigated by Attia and Kotb [5]. An analysis of the fully-developed MHD free convective flow between two vertical, electrically conducting plates has been carried out by Soundalgekar and Haldavnekar [6]. The nonlinear integro-differential equations, governing the flow, have been solved by perturbation method. An analysis of a two-dimensional unsteady flow of a viscous, incompressible electrically-conducting fluid through a porous medium bounded by two infinite parallel plates under the action of a transverse magnetic field is presented by Hassanien and Mansour [7]. An unsteady hydromagnetic flow through a porous medium between two infinite parallel porous plates with time varying suction is studied by Hassanien [8]. Nabil et al. [9] investigated an analytical study of unsteady magneto-hydrodynamic flow of an incompressible electrically conducting fluid filling the space between two parallel plates is presented, taking into account the couple stresses and pulsation of the pressure gradient effect. Makinde and Chinyoka [10] studied the unsteady flow and heat transfer of a dusty fluid between two parallel plates with variable viscosity and electric conductivity and the fluid is driven by a constant pressure gradient and an external uniform magnetic field is applied perpendicular to the plates with a Navier slip boundary condition. The governing non-linear partial differential equations are solved numerically using a semi-implicit finite difference scheme. Channel flows through porous media have several engineering and geophysical applications, such as, in the field of chemical engineering for filtration and purification processes; in the field of agricultural engineering to study the underground water resources; in petroleum industry to study the movement of natural gas, oil and water through the oil channels and reservoirs. For example, Raptis et al. [11] analyzed the problem of hydromagnetic free convection flow through a porous medium between two parallel plates; while Kearsley [12] studied problem of steady state Couette flow with viscous heating. Makinde and Osalusi [13] considered a MHD steady flow in a channel with slip at permeable boundaries. Choudhury and Bhattacharjee [14] analyzed the unsteady two-
u; v ; w U V V
dimensionless velocity component in x; y; z-direction Uniform velocity of the upper plate (ms1) injection/section velocity (ms1) electrical voltage (V)
Greek symbols a thermal diffusivity (m2K1) e small reference parameter ðe 1Þ h dimensionless fluid temperature, j thermal conductivity (Wm1K1), c kinematic viscosity (m2s1), q density (kgm3), r electrical conductivity k injection/suction parameter s shearing stress (Nm2)
dimensional incompressible visco-elastic MHD free convective fluid flow over a radiative vertical porous plate with Dufour effects in presence of chemical reaction using the perturbation technique. Beg et al. [15] solved the numerical problem of MHD of a viscous plasma flow in rotating porous media submitted to the influence of Hall currents and an inclined magnetic field. A numerical study is made by Kim et al. [16] on the heat transfer characteristics from forced pulsating flow in a channel filled with fluid-saturated porous media where the channel walls are assumed to be at uniform temperature and the BrinkmanForchheimer-extended Darcy model is employed. Kuznetsov [17] presented an analytical solution to the flow and heat transfer in Couette flow through a rigid saturated porous medium where the fluid flow occurs due to a moving wall and it is described by the Brinkman-Forchheimer-extended Darcy equation. Sharma and Chaudhary [18] studied the effects of magnetic field on the steady flow and heat transfer in a horizontal channel. Forced convection with viscous dissipation in a parallel plate channel filled by a saturated porous medium is investigated numerically by Hooman and Gurgenci [19]. The fully developed laminar mixed convection flow in a vertical plane parallel channel filled with a porous medium and subject to isoflux-isothermal wall conditions is investigated Barletta et al. [20]. The problem of free convection heat transfer flow through a porous medium bounded by a wavy wall and a vertical wall is studied by Sahin and Hiren [21]. Recently, Fasogbon [22] studied the simultaneous buoyancy force effects of thermal and species diffusion through a vertical irregular channel by using parameter perturbation technique. Sahin and Zueco [23] investigated the effects of Hall current, magnetic field, rotation of the channel and suction-injection on the oscillatory free convective MHD flow in a rotating vertical porous channel. Bég et al. [24] obtained comprehensive network computational solutions for the two-dimensional transient hydromagnetic flow in a parallel-plate channel containing a Darcian porous material with dissipation and Hall/ionslip currents. Bég et al. [25] studied the transient problem optically-thick radiative convective double-diffusive boundary layers in a Darcian porous medium adjacent to an impulsively started surface. Turkyilmazoglu [26,27] used the homotopy analysis method to solve the equations governing the flow of a steady, laminar, incompressible, viscous, and electrically conducting fluid due to a rotating disk subjected to a uniform suction and injection through the walls in the presence of a uniform transverse magnetic field. Turkyilmazoglu [28] analyzed the MHD permeable heat and fluid flow fields induced by stretching or shrinking two-three dimensional objects, obtain-
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S. Ahmed et al. / International Journal of Heat and Mass Transfer 108 (2017) 322–331
ing exact analytical solutions is targeted either in general or in restrictive forms. In the present article we study the twin effects of the magnetic field and permeability parameter on the three-dimensional Couette flow of viscous incompressible, electrically conducting and Newtonian fluid through a porous medium bounded by two horizontal parallel porous plates. The stationary plate and the plate in uniform motion are, respectively, subjected to a transverse sinusoidal injection and uniform suction of the fluid. The dependence of solution on Hartmann number, porosity and injection/suction is investigated by the graphs and tables. The solutions are obtained using both semi-analytical and numerical methods.
The equations governing the steady motion of an incompressible viscous electrically conducting fluid through porous medium bounded by two permeable horizontal plates in presence of a magnetic field in vector form are: !
Div q ¼ 0; !
Div B ¼ 0;
Conserv ation of Mass;
ð1Þ
Gausss law of magnetism;
ð2Þ
! ! !
!
!
!
!
qðq : $Þ q ¼ $ p þ l$2 q þ J X B; Conserv ation of Momentum; ð3Þ ! !
qC P ðq : $ÞT ¼ j$2 T; Conserv ation of Energy; 2. Formulation of the problem
!
!
!
!
J ¼ r½E 0 þ q X B;
We consider the Couette flow of a viscous incompressible electrically conducting fluid between two parallel flat porous plates with transverse sinusoidal injection of the fluid at the stationary plate and its corresponding removal by periodic suction through plane lie along the the plate in uniform motion U. Let the xy -axis be taken normal to the free-stream velocplates and let the y ity. The distance d is taken between the plates. Denote the velocity ; v ; w in the ; z-directions, respectively. The components by u x; y lower and upper plates are assumed to be at constant temperature T 0 and T 1 , respectively, with T 1 > T 0 . We derive the governing equations with the assumption that the flow is steady and laminar, and is of a finitely conducting fluid. The magnetic field of uniform strength B0 is applied to the perpendicular of the free-stream velocity (see Fig. 1); at lower magnetic Reynolds number, the magnetic field is practically independent of the flow motion and the induced magnetic field is neglected. The Hall effects, electrical and polarization effects also have been neglected. All physical quantities are independent of x for this problem of fully developed laminar flow, but the flow remains three-dimensional due to the injection/suction velocity Vð zÞ ¼ Vð1 þ e cosp z=dÞ.
0
ð4Þ
0
ð5Þ
Ohm slaw;
!
^ be the fluid velocity at the point ðx; y k ; zÞ and ^i þ v^j þ w where q ¼ u
!
B ¼ B0^j be the applied magnetic field. All physical quantities are defined in the Nomenclature. Thus, with the foregoing assumptions and under the usual boundary layer approximations, the Eqs. (1)–(4) reduce to:
@ v @ w ¼0 þ @z @y
v
ð6Þ
! @2u @u @u @2u þw ¼c þ 2 @ z2 @y @ z @y
!
rB20 c ðu UÞ þ q K
@ v @ v 1 @p @ 2 v @ 2 v v þ w ¼ þ c 2 þ 2 @y @z q @y @y @z
!
ð7Þ
c
K
v
! @2w @w @w 1 @p @2w þw ¼ þc þ 2 @y @z q @z @y @ z2
v
@T @T @2T @2T þw ¼a þ 2 @z2 @y @z @y
v
ð8Þ !
rB20 c þ w q K
ð9Þ
! ð10Þ
=d; z ¼ z=d; u ¼ u =U; v ¼ v =V; w ¼ w=V; =qV 2 ; Pr ¼ where y ¼ y p¼p c=a,
1
M¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rB20 d2 =qc;
k ¼ Vd=c;
h ¼ ðT T 0 Þ=ðT 1 T 0 Þ;
2
K ¼ K=d U
( )
The non-dimensional forms of (1)–(5) are
v y þ wz ¼ 0; v uy þ wuz ¼
ð11Þ uyy þ uzz ðM 2 þ K 1 Þðu 1Þ ; k
vv y þ wv z ¼ py þ
v yy þ v zz K 1 v
0
v wy þ wwz ¼ pz þ
0
( )
v hy þ whz ¼
k
ð12Þ
;
ð13Þ
wyy þ wzz ðM 2 þ K 1 Þw ; k
ð14Þ
hyy þ hzz ; k
ð15Þ
The corresponding boundary conditions in the dimensionless form are
y ¼ 0 : u ¼ 0; Fig. 1. Model of Couette Flow through Porous Medium with MHD.
y ¼ 1 : u ¼ 1;
v ðzÞ ¼ 1 þ ecospz; v ðzÞ ¼ 1 þ ecospz;
w ¼ 0; h ¼ 0 w ¼ 0; h ¼ 1
ð16Þ
S. Ahmed et al. / International Journal of Heat and Mass Transfer 108 (2017) 322–331
h 1
325
3. Semi-analytical solution
p1 ðy; zÞ ¼ ðp2 kAÞ
Since the amplitude of the injection/suction velocity e 1 is very small, we now assume the solution of the following form:
þA2 J 4 ðJ 1 J 4 kJ 4 M2 K 1 ÞeJ4 y A3 J 5 ðJ 2 J 5 kJ 5 M 2 K 1 ÞeJ5 y i A4 J 6 ðJ 2 J 6 kJ 6 M2 K 1 ÞeJ6 y sinpz; ð29Þ
f ðy; zÞ ¼ f 0 ðyÞ þ ef 1 ðy; zÞ þ þe2 f 1 ðy; zÞ þ ;
h n 1 1 u1 ðy; zÞ ¼ Eeg1 y þ Feg2 y þ b1 A1 J 1 eðJ2 þJ1 yþJ3 yÞ b2 J 2 eðJ1 þJ2 yþJ3 yÞ b6 1 1 þA2 J 1 eðJ2 þJ1 yþJ4 yÞ b3 J 2 eðJ1 þJ2 yþJ4 yÞ b7 1 1 A3 J 1 eðJ2 þJ1 yþJ5 yÞ b8 J 2 eðJ1 þJ2 yþJ5 yÞ b4 oi 1 1 A4 J 1 eðJ2 þJ1 yþJ6 yÞ b9 J 2 eðJ1 þJ2 yþJ6 yÞ b5 ð30Þ cospz;
ð17Þ
where f stands for any of u; v ; w; p; and h. When e ¼ 0, the problem is reduced to the well-known two-dimensional flow. The solution of this two-dimensional problem is
u0 ðyÞ ¼ 1 þ
h0 ðyÞ ¼
expðJ 2 þ J 1 yÞ expðJ 1 þ J 2 yÞ ; expðJ 1 Þ expðJ 2 Þ
expðkPryÞ 1 ; expðkPrÞ 1
v 0 ¼ 1;
ð18Þ
w0 ¼ 0;
p0 ¼ constant;
ð19Þ
When e–0, substituting (17) in (11)–15) and comparing the coefficient of e, neglecting those of e2 ; e3 ; . . ., the following firstorder equations are obtained with the help of solutions (18) and (19):
v 1y þ w1z ¼ 0; v 1 u0y þ u1y ¼ v 1y ¼ p1y þ
ð20Þ
u1yy þ u1zz ðM 2 þ K 1 Þu1 ; k
v 1yy þ v 1zz K 1 v 1 k
ð21Þ
;
ð22Þ
w1yy þ w1zz ðM2 þ K 1 Þw1 ; k
ð23Þ
v 1 h0y þ h1y ¼
h1yy þ h1zz ; kPr
ð24Þ
y ¼ 1 : u1 ¼ 0;
w1 ¼ 0; h1 ¼ 0
w1 ¼ 0; h1 ¼ 1
ð25Þ
This is the set of linear partial differential equations, which describe the three-dimensional flow. To solve these equations, we assume v 1 ; w1 ; p1 ; u1 ; and h1 of the following form:
9 u1 ðy; zÞ ¼ u11 ðyÞcospz; > > > > > > v 1 ðy; zÞ ¼ v 11 ðyÞcospz; > > > = 0 v 11 ðyÞsinpz ; w1 ðy; zÞ ¼ p > > > > p1 ðy; zÞ ¼ p11 ðyÞcospz; > > > > > ; h1 ðy; zÞ ¼ h11 ðyÞcospz;
Now, after knowing the velocity field, we can calculate skinfriction components sx and sz in the main and transverse directions, respectively, as follows: ds du0 du11 sx ¼ x ¼ þe cospz; lU dy y¼0 dy y¼0 " ( ! J eJ2 J eJ1 ðJ2 þ J J ÞeJ2 ðJ2 þ J J ÞeJ1 sx ¼ 1 J1 2J2 þ e Eg 1 þ Fg 2 þ b1 A1 1 1 3 2 2 3 e e b2 b6 ! 2 2 J2 J1 ðJ1 þ J1 J4 Þe ðJ þ J J Þe 2 24 þ A2 b3 b7 ! !)# ðJ21 þ J1 J5 ÞeJ2 ðJ22 þ J2 J5 ÞeJ1 ðJ21 þ J1 J6 ÞeJ2 ðJ22 þ J2 J6 ÞeJ1 A4 cospz; A3 b8 b4 b9 b5
sz ¼
z ds dw1 ; ¼e lU dy y¼0
¼ eðpAÞ1 ½A1 J 23 þ A2 J 24 A3 J 26 sinpz
ð33Þ
dq dh0 dh1 þe cospz; ¼ jðT 0 T 1 Þ dy y¼0 dy y¼0 kPr A1 ðJ 3 þ kPrÞ A2 ðJ 4 þ kPrÞ ¼ kPr þ þ e RS1 þ SS2 þ C 1 e 1 C2 C3
A3 ðJ 5 þ kPrÞ A4 ðJ 6 þ kPrÞ cospz C4 C5
Nu ¼
ð34Þ
4. Numerical method (Network Simulation Method)
ð26Þ
where the prime denotes differentiation with respect to y. Expressions for v 1 ðy; zÞ and w1 ðy; zÞ have been chosen so that the equation of continuity (20) is satisfied. Substituting (26) in (21)–(24) and applying the corresponding boundary conditions, we get the solutions for v 1 ; w1 ; p1 ; u1 ; and h1 as follows:
v 1 ðy; zÞ ¼ A1 ½A1 eJ y þ A2 eJ y A3 eJ y A4 eJ y cospz; 3
ð31Þ
From the temperature field, we can obtain the heat transfer coefficient in terms of Nusselt number as follows:
The corresponding boundary conditions reduce to
v 1 ¼ cospz; v 1 ¼ cospz;
h n ðJ 4 þkPryÞ 1 h1 ðy; zÞ ¼ ReS1 y þ SeS2 y þ C 1 A1 eðJ3 þkPryÞ C 1 C2 2 þ A2 e oi ðJ 6 þkPryÞ 1 C5 A3 eðJ5 þkPryÞ C 1 cospz; 4 A4 e
ð32Þ
w1y ¼ p1z þ
y ¼ 0 : u1 ¼ 0;
A1 J 3 ðJ 1 J 3 kJ 3 M 2 K 1 ÞeJ3 y
4
5
6
ð27Þ
w1 ðy; zÞ ¼ ðpAÞ1 ½A1 J 3 eJ3 y þ A2 J 4 eJ4 y A3 J 5 eJ5 y A4 J 6 eJ6 y sinpz; ð28Þ
Network Simulation Method (NSM) is used in the present study to solve the partial differential equations (11)–(15) under boundary conditions (16). The starting point for this method is always the discretization of the equations that form the mathematical model of the problem under study. With a spatial discretization (after experimenting with a few sets of mesh sizes) of the dimensionless equations in Ny and Nz cells of length Dy = 1/Ny (with Ny = 100), and Dz = zmax/Nz (with Nz = 50), respectively. These partial differential equation can be transformed into a system of connected differential equations. A first-order and second-order central difference approximation is used for the first and second derivate, respectively.
uy ¼ ð@u=@yÞi;j ðui;jþ1 ui;j1 Þ=Dy þ OðDyÞ
ð35Þ
uz ¼ ð@u=@zÞi;j ðuiþ1;j ui1;j Þ=Dz þ OðDzÞ
ð36Þ
S. Ahmed et al. / International Journal of Heat and Mass Transfer 108 (2017) 322–331
uyy ¼ ð@ 2 u=@y2 Þi;j ðui;jþ1 þ ui;j1 2ui;j Þ=Dy2 ¼ 1=Dy½ðui;j1 ui;j Þ=Dy ðui;j ui;jþ1 Þ=Dy þ OðDy Þ 2
ð37Þ
¼ 1=Dz½ðui1;j ui;j Þ=Dz ðui;j uiþ1;j Þ=Dz þ OðDz2 Þ
ð38Þ
to the Kirchhoff law for currents (conservation of charge, LCK). iii) The existence of the magnitudes Vi,i-1 and Vi,i+1 denominated potential, associated to each node, so that the differences Vi,i1 = Vi Vi1 and Vi,i+1 = Vi Vi+1 are accorded to the Kirchhoff law of the voltages (is a result of the electrostatic field being conservative, LVK).
uzz ¼ ð@ 2 u=@z2 Þi;j ðuiþ1;j þ ui1;j 2ui;j Þ=Dz2
Similar expressions can be obtained for v y ; v z ; v yy ; v zz ; wy ; wz ; wyy ; wzz ; hy ; hz ; hyy andhzz . The computational domain is divided into meshes each is of dimension 2Dy and 2Dz in space-y and space-z, respectively. Based on these equations, a electrical network circuit is designed, whose equations are formally equivalent (see equivalence):
The theorems of conservation and uniqueness of flow and potential electrical variables (Kirchhoff laws) are satisfied in the circuits. Some of the equations usually contained in mathematical model, do not need to be considered in the design of the network model; for example, in the conservation of the heat flux at the boundary of two different media and the uniqueness of the temperature in the boundary. After performing the discretization of the equations, the electrical analogy is applied. A number of networks are connected in series and parallel to make up the whole medium and boundary conditions are added by means of special electrical devices (current or voltage control-sources). Once the complete network model is designed, for which few programming rules are needed since not many devices form the network, a computer code (Pspice [32]) is used to simulate it using a PC, providing the numerical solution. The finite-difference differential equations resulting from dimensionless momentum balance, Eq. (12) is,
v i;j ðui;jþ1 ui;j1 Þ=Dy þ wi;j ðuiþ1;j ui1;j Þ=Dz
In this equivalence, the variables velocity (u,v,w) and temperature (h) are equivalent to the variable electric voltage, while the variables velocity gradients (ou/oy, . . .) and temperature gradient (oh/oy, oh/oz) are equivalent to the variable electric current. In the network theory, the viability of a network model implies:
¼ ½ðui;j1 ui;j Þ=Dy2 ðui;j ui;jþ1 Þ=Dy2 =k þ ½ðui1;j ui;j Þ=Dz2 ðui;j uiþ1;j Þ=Dz2 =k ðM 2 þ K 1 Þðui;j 1Þ=k ¼ 0
The Fig. 2 shows the network models corresponds to the equation (39), where the second law of Kirchhoff is applied, therefore each term of Eq. (39) is considered a electrical current that is modelled by means a electrical dispositive (resistor, capacitor, current generator, ect). In this figure, Ju,i,j+1 and Ju,i,j1 are the currents that
i) The existence of a network independent of time. ii) The existence of the magnitudes Ji,i1 and Ji,i+1 denominated fluxes, associated to each branch that connects the nodes i with (i-1) and i with (i + 1), respectively. These fluxes follow
zi-1,j
ui-1,j
Gy,i,j
Node (i,j)
{ vi,j ,ui,j+1, ui,j-1} Ru,i-1,j
Gz,i,j ui,j-1
Jy,i,j Ru,i,j-1
{wi,j ui+1,j, ui-1,j}
Jz,i-1,j=(ui-1,j-ui,j)/Δz
Jz,i,j
Ju,i,j-1=(ui,j-1-ui,j)/Δy {ui,j}
ð39Þ
zi,j zi+1,j
326
ui,j+1
Ru,i,j+1
Ji,j ui,j
2Δz
Ju,i,j+1=(ui,j-ui,j+1)/Δy Jz,i+1,j=(ui,j-ui+1,j)/Δz
Gi,j
Ru,i+1,j yi,j-1 yi,j
ui+1,j
yi,j+1 2Δy
Momentum Equation (Eq. 39)
Fig. 2. Nomenclature and network model of the control volume. Momentum balance equation (39).
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leave and enter the cell for the friction term in the y-direction; while Ju,i+1,j and Ju,i1,j are the currents that leave and enter the cell for the friction term in the z-direction. Ji,j is the current due to magnetic and permeability effects, Jy,i,j and Jz,i,j are the currents due to the inertia terms of the velocity u in both directions (y and z), respectively. The currents Ju,i,j+1 and Ju,i,j1 are implemented by means of two resistances Ru,i,j±1 of values ‘‘kDy200 ; while the currents Ju,i+1,j and Ju,i1,j are implemented by means of two resistances Ru,i±1,j of values ‘‘kDz200 . Ji,j, Jy,i,j and Jz,i,j are implemented by means of voltage control current generators with controlactions indicate in Fig. 2. For the other equations, similar explanations can be realized to the previous case, so we will have a electrical circuit (network model) for each equation, and they will be connected and coupled together. Finally, to implement the boundary conditions are employed the following devices are used: i) a voltage source for the constant values of velocity and temperature, ii) a sinusoidal voltage-source for the sinusoidal velocity variation, v (z). More details about this numerical technique can be found in [23–24].
1.2 1 0.8
u
M = 0.0 M = 0.5 M = 1.0 M = 5.0
0.6 0.4 0.2 0
0
0.2
0.8
1
Fig. 3. Magnetic field effects on the main flow velocity.
1
The problem of steady three-dimensional Couette flow of viscous incompressible, electrically conducting and Newtonian fluid through a porous medium bounded by two horizontal parallel porous flat plates with transverse sinusoidal injection of the fluid at the stationary plate and its corresponding removal by periodic suction through the plate in uniform motion has been considered. The analytical expressions for velocities field, temperature, skin-friction and rate of heat transfer (Nusselt number) coefficients were obtained using perturbation technique. Table 1 shows the semi-analytical and numerical solutions. In this table is shown the spatial variation of shear stress sz with permeability and magnetic parameter for e ¼ 0:5; k ¼ 0:5 and z ¼ 0:5. We observe that with an increase in permeability parameter K the shear stress sz is decreased. Similarly with increasing magnetic parameter M from 1.0–5.0, there is a consistent decrease in shear stress sz . Both permeability parameter and magnetic parameter therefore have a negative influence on skin friction coefficient from the plate surface into the porous medium. An excellent agreement has been obtained between the semi-analytical solutions and NSM computations. The effects of the flow parameters such as magnetic parameter M, permeability coefficient K and injection/suction parameter k on the velocities field, temperature field and skin-friction coefficients have been studied analytically and presented with the help of Figs. 2–6. Fig. 3 illustrates the dimensionless velocity component x-direction (main flow velocity profiles) for various values of the
0.8
u
Semi-analytical solution
K 0.1 0.5 1.0 0.5 0.5 0.5
0.6 0.4
M = 5, z = 0.5, λ = 0.05
0.2 0
0
0.2
0.4
0.6
λ
0.3 0.3 0.3 1.0 1.5 2.0
0.8
1
y Fig. 4. Porosity and injection/suction effects on the main flow velocity.
Hartmann number ðM ¼ 0; 0:5; 1:0 and 5:0Þ. Profiles rise from the stationary wall to reach maximum values and then decrease to reach the velocity u ¼ 1 of the transient wall, so the velocity profiles converge in accordance with the boundary condition imposed there. With an increase in M from the non-conducting i.e. purely hydrodynamic case ðM ¼ 0Þ through 0.5, 1–5, there is a clear decrease in velocity i.e. flow is decelerated, so the Lorentz force, which opposes the flow, there is a fall in velocity maximum due to the retarding effect of the magnetic force in the region
Table 1 Comparison of the values of skin-friction component ðsz Þ for for semi-analytical and numerical schemes, with
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.6
y
5. Discussion and results
y
0.4
e ¼ 0:5; k ¼ 0:5 and z ¼ 0:5.
Numerical solution
K = 0.2, M = 1
K = 0.5, M = 1
K = 0.5, M = 5
K = 0.2, M = 1
K = 0.5, M = 1
K = 0.5, M = 5
0.47621 0.47286 0.46951 0.46318 0.46286 0.45352 0.45320 0.45272 0.44791 0.44627
0.45830 0.45504 0.43162 0.43116 0.43097 0.42790 0.42674 0.42392 0.42129 0.40737
0.38617 0.38494 0.37612 0.37460 0.37178 0.35901 0.35477 0.35172 0.34205 0.34135
0.47506 0.47309 0.46944 0.46299 0.46301 0.45303 0.45299 0.45218 0.44825 0.44705
0.46080 0.45823 0.43217 0.43559 0.43168 0.43106 0.42945 0.42763 0.42328 0.40451
0.38890 0.38656 0.37512 0.37460 0.37178 0.35989 0.35426 0.35199 0.34326 0.34155
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S. Ahmed et al. / International Journal of Heat and Mass Transfer 108 (2017) 322–331
0.05
M 1.0 1.0 0.0 1.0 5.0
0.04 0.03 0.02
w
0.01 0 -0.01
0
-0.02
0.2
0.4
0.6
0.8
1
λ 0.7 0.5 0.5 1.0 0.5
0.8
1
θ
-0.04
M = 5, z = 0.5, ε = 0.05, Pr = 0.71
0.2
y
-0.05
λ 1.5 0.5 1.0 0.5 0.5
0.4
K = 0.5, z = 0.5, λ = 0.05
-0.03
0.6
K 0.1 1.0 0.1 0.5 0.1
0
Fig. 5. Magnetic field and injection/suction effects on the cross flow velocity.
0
0.2
0.4
0.6
0.8
1
y Fig. 7. Effects of porosity (K) and suction/injection (k) on the temperature profiles.
0.04
K = 1.0
0.03
K = 0.7
w
-0.01 -0.02
M = 0.0
K = 0.5
0.01 0
1
K = 0.1
0.02
0.8 0
0.2
0.4
0.6
0.8
1
θ
M = 2, λ = 0.5, z = 0.5, λ = 0.05
M = 5.0 M = 10.0
0.6
M = 15.0
0.4
-0.03 -0.04
K = 0.1, z = 0.5, λ = 0.5, ε = 0.05, Pr = 0.71
0.2
y Fig. 6. Porosity effects on the cross flow velocity.
y 2 ½0; 1 and as a result increases the thickness of the momentum boundary layer. Fig. 4 shows the influence of the injection/suction parameter ðkÞ and permeability parameter ðKÞ on the dimensionless velocity component x-direction (main flow velocity profiles). With an increase in k from k ¼ 0:3 through 0.3–2.0, there is a clear increase in velocity with distance transverse to the walls, i.e. the flow is accelerated. Also there is clear increase in velocity values at the wall accompanying a rise in K from K ¼ 0:1 to 1.0 i.e. the flow is 2
accelerated. The parameter K ¼ K=d U is directly proportional to the actual permeability, K of the porous medium. The Darcian drag force in the momentum equations (Eqs. (21)–(23)), viz, K 1 ðu 1Þ, K 1 v and K 1 w, respectively; is therefore inversely proportional to K. An increase in K will therefore decrease the resistance of the porous medium (as the permeability physically becomes less with increasing K) which will serve to accelerate the flow and increase the velocity, u. Fig. 5 illustrates the dimensionless velocity component w-direction (transverse velocity profiles) for various values of the Hartmann number ðM ¼ 0; 0:5; 1:0 and 5:0Þ and injection/suction parameter ðk ¼ 0:5; 0:7 and 1:0Þ. It was observed that forward flow developed from y = 0 to about y = 0.4, and then onwards there was backward flow. This was due to the fact that the dragging action of the faster layer exerted on the fluid particles in the neighborhood of the stationary plate was sufficient to overcome the adverse pressure gradient and, hence, there was forward flow. The dragging action of
0
0
0.2
0.4
0.6
0.8
1
y Fig. 8. Effect of magnetic parameter (M) on the temperature profiles.
the faster layer exerted on the fluid particles reduced due to the periodic suction at the upper plate and, hence, this dragging action was insufficient to overcome the adverse pressure gradient; therefore, there was backward flow. Furthermore, it is evident from this figure that velocity w decreased with increasing k and M in the forward and back flow. Fig. 6 shows the transverse velocity profiles for various permeability parameters ðKÞ. As expected, the transverse velocity will be increased substantially with increasing permeability parameter. Fig. 7 illustrate the influence of porosity (K) and suction/injection (k) on the boundary layer variable, h. With increasing porosity the temperature in the regime is found to be decreases i.e. the boundary layer is cooled. A reduction in the volume of solid particles in the medium implies a lower contribution via thermal conduction. This will serve to decrease temperatures. Similar effect has been observed for the suction/injection parameter on the fluid temperature. Fig. 8 presents the response of temperature (h) to magnetohyqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 drodynamic body force parameter (M) where M ¼ rB20 d =qc and it signifies the ratio of Lorentz hydromagnetic body force to viscous hydrodynamic force. With increasing M, temperature, is observed to be markedly increased. This is physically explained
S. Ahmed et al. / International Journal of Heat and Mass Transfer 108 (2017) 322–331
4
K = 0.1, 0.5, 1.0
3
K = 0.5, z = 0, ε = 0.5
2 1
τx
0 0
0.2
0.4
0.6
0.8
1
-1 -2
M = 0.0, 2.0, 5.0
λ
-3
Fig. 9. Porosity and magnetic field effects on the shear stress due to main flow.
z = 0.5, ε = 0.5, Pr = 0.71
0.8
0.6
0.4
Nu
K
M
0.75 0.15 0.15 0.45 0.15 0.15
2.0 2.0 6.0 2.0 8.0 4.0
0.2
0
0
0.2
0.4
0.6
0.8
1
λ
-0.2
329
influences the temperature (h) field, only via the thermal and buoyancy terms. Porosity, injection/suction and magnetic field effects on the shear stress due to main flow sx are shown in Fig. 9. The shear stress sx increases with increasing values of M (there is a backflow entirely from the regime) and decreases with increasing value of k. Increasing K clearly increases shear stress sx values i.e. increasing the permeability serves to accelerate the flow (no flow reversal in this case) which increases shear stress function values. Fig. 10 shows the influence of porosity (K) and magnetic parameter (M) on surface heat transfer rate i.e. Nusselt number function (Nu) at the plate y = 0 plotted against suction/injection (k) parameter. An increase in porosity clearly enhances (Nu) values which are consistent with the reduction of temperatures in the boundary layer regime, computed earlier. A contrary response is computed for the effect of magnetic parameter, M, on the Nusselt number function (Nu) distributions in this figure. Increasing magnetic field enhances the Lorentzian hydromagnetic drag which serves to decelerate the flow and warm the fluid. Therefore heat will be transferred at progressively lower rates from the fluid to the plates. Moreover, an increase in suction/injection parameter from 0.0 (without suction) through 0.2 to 1.0 the Nusselt number profiles attains maximum values near the plate y = 0 and thereafter all profile decay as we continue over the plate towards the upper plate. Negative values of Nu have been observed for the maximum magnetic numbers, it means that the heat of the fluid is diffused to the plate y = 0. Finally, the pressure values p1 are reported in Table 2 for z ¼ 0 and k ¼ 0:5. We can see that the pressure decreased with the permeability parameter K. The decrease in pressure was sufficiently large for small values of Kð0:2 and 0:5Þ. Besides, increasing magnetic parameter M clearly increases pressure p1 . It was observed that positive values pressure were obtained from y ¼ 0 to about y ¼ 0:4, and then onwards negative values pressure were obtained.
Fig. 10. Effects of porosity (K) and magnetic parameter (M) on the Nusselt number.
6. Conclusions Table 2 The values of pressure (p1 ) for z ¼ 0; k ¼ 0:5. y
K = 0.2, M = 0.5
K = 0.5, M = 0.5
K = 0.5, M = 5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
11.3677 7.1748 5.4180 3.6143 1.6835 0.0259 1.3082 3.9250 5.7819 7.8307 12.3805
6.7180 4.3719 2.5303 1.4193 0.4807 0.0292 0.5371 1.8251 2.9734 4.7022 7.4509
7.3712 5.1772 2.7128 1.8617 0.6314 0.0173 0.7813 1.9033 3.1852 5.5182 7.7582
by the fact that the extra work expended in dragging the fluid against the magnetic field is dissipated as thermal energy in the boundary layer, as elucidated by Sutton and Sherman [29], Pai [30] and Hughes and Young [31]. This results in heating of the boundary layer and an ascent in temperatures, an effect which is maximized some distance away from the surface. The magnetic field influence on temperatures while noticeable is considerably less dramatic than that on the velocity field, since the Lorentz body force only arises in the momentum equations (12) and (14) and
A mathematical model has been presented for the hydromagnetic three-dimensional Couette flow of a viscous incompressible electrically conducting fluid between two parallel flat porous plates with transverse sinusoidal injection of the fluid at the stationary plate and its corresponding removal by periodic suction through the plate in uniform motion, with a magnetic field of applied to the perpendicular of the free-stream velocity. Analytical and numerical solutions for the non-dimensional momentum and energy equations subject to transformed boundary conditions have been obtained. The flow has been shown to be accelerated (main and cross velocities) with decreasing magnetic number but the main flow velocity increased with injection/suction parameter, while decreasing injection/suction parameter accelerates the cross flow velocity. However, it has been shown that increasing magnetic field generally decelerates the flow, but increases temperature values in the channel. Increasing porosity serves to accelerate the flow as well as the rate of heat transfer, but reduce temperature values. Semi-analytical solution method does not suffer from error accumulation and is able to give very accurate results accelerating the convergence to the solutions, compared with the numerical solution that involving iterative time integration. The study has important applications in electro-magnetic materials processing and nuclear heat transfer control, as well as MHD energy generators.
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Appendix A
J1 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 2
½kþ
k þ4ðM þK 2
Þ
;
J2 ¼
½k
A ¼ ½J 4 J 5 þ J 3 J 6 J 3 J 5 J 4 J 6 fe
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 2
k þ4ðM þK Þ 2 ðJ3 þJ 4 Þ
ðJ 5 þJ6 Þ
þe
ðJ 4 þJ 5 Þ
g ½J 4 J 5 þ J 3 J 6 J 3 J4 J 6 J 5 feðJ5 þJ3 Þ þ eðJ6 þJ4 Þ g ½J 6 J 5 þ J 3 J 4 J 3 J 5 J 6 J 4 feðJ6 þJ3 Þ þ eðJ5 þJ4 Þ g;
ðJ4 þJ6 Þ
A1 ¼ ½J 4 J 6 J 5 J 6 e ½J 4 J 5 J 5 J 6 e þ ½J 4 J 5 J 4 J 6 eðJ5 þJ6 Þ þ ½J4 J 5 J 4 J 6 eJ4 þ ½J 6 J 5 J 4 J 5 eJ5 þ ½J 4 J 6 J 5 J 6 eJ6 ; A2 ¼ ½J 3 J 5 J 5 J 6 eðJ3 þJ6 Þ ½J 3 J 6 J 5 J 6 eðJ3 þJ5 Þ ½J 3 J 5 J 3 J 6 eðJ5 þJ6 Þ þ ½J3 J 6 J 3 J 5 eJ3 þ ½J 3 J 5 J 6 J 5 eJ5 þ ½J 5 J 6 J 3 J 6 eJ6 ; A3 ¼ ½J 4 J 6 J 3 J 6 eðJ3 þJ4 Þ þ ½J 3 J 4 J 4 J 6 eðJ3 þJ6 Þ ½J 3 J 4 J 3 J 6 eðJ4 þJ6 Þ ½J3 J 4 J 3 J 6 eJ3 ½J 4 J 6 J 3 J 4 eJ4 ½J 3 J 6 J 4 J 6 eJ6 ; A4 ¼ ½J 3 J 5 J 3 J 6 eðJ3 þJ4 Þ þ ½J 5 J 4 J 4 J 3 eðJ3 þJ4 Þ ½J 3 J 5 J 3 J 4 eðJ4 þJ5 Þ ½J3 J 5 J 3 J 4 eJ3 ½J 4 J 3 J 5 J 4 eJ4 ½J 4 J 5 J 3 J 5 eJ5 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½J þ J 21 þ4ðp2 þK 1 Þ ½J 1 J 21 þ4ðp2 þK 1 Þ J3 ¼ 1 ; J ¼ ; 4 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ½J 2 þ J 22 þ4ðp2 þK 1 Þ ½J 2 J 22 þ4ðp2 þK 1 Þ J5 ¼ ; J6 ¼ ; 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 2 2 2 2 ½kþ k þ4ðp þM þK Þ ½k k þ4ðp þM 2 þK 1 Þ g1 ¼ ; g2 ¼ ; 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ½kPrþ k2 Pr 2 þ4p2 ½kPrþ k2 Pr 2 þ4p2 ; g2 ¼ ; S1 ¼ 2 2 b1 ¼ AðeJ1keJ2 Þ ;
b2 ¼ 3J 1 J 3 kJ 3 ;
b5 ¼ 3J 2 J 6 kJ 6 ;
b6 ¼ 2J 2 J 3 þ J 1 J 3 kJ 3 ;
b8 ¼ 2J 1 J 5 þ J 2 J 5 kJ 5 ; C 2 ¼ J 1 J 3 þ kPrJ 3 ; C6 ¼
n
C7 ¼
b4 ¼ 3J 2 J 5 kJ 5 ;
b7 ¼ 2J 2 J4 þ J 1 J 4 kJ 4 ;
b9 ¼ 2J 1 J 6 þ J 2 J 6 kJ 6 ;
C 3 ¼ J 1 J 4 þ kPrJ 4 ;
k ; AðeJ1 eJ2 Þðeg1 eg 2 Þ
h
b3 ¼ 3J 1 J 4 kJ 5 ;
2
2
C 1 ¼ AðekkPrPr1Þ ;
C 4 ¼ J 2 J 5 þ kPrJ 5 ;
C 5 ¼ J 2 J 6 þ kPrJ 6 ;
k2 Pr 2 ; AðekPr 1ÞðeS1 eS2 Þ
o n ðJ þg Þ ðJ þJ þJ Þ o n ðJ þg Þ ðJ þJ þJ Þ o ðJ 1 þg 2 Þ ðJ 1 þg 2 Þ ðJ 1 þg 2 Þ 2 2 e 1 2 4 Þ 2 2 e 1 2 5 Þ eðJ1 þJ2 þJ3 Þ Þ eðJ1 þJ2 þJ4 Þ Þ eðJ1 þJ2 þJ5 Þ Þ J2 ðe J2 ðe J2 ðe þ A2 J1 ðe A3 J1 ðe E ¼ C 6 A1 b6 b3 b7 b8 b4
J ðeðJ2 þg2 Þ eðJ1 þJ2 þJ6 Þ Þ J 2 ðeðJ1 þg2 Þ eðJ1 þJ2 þJ6 Þ Þ A4 1 ; b9 b5 J ðeðJ1 þJ2 þJ3 Þ eðJ2 þg1 Þ Þ J 2 ðeðJ1 þJ2 þJ3 Þ eðJ1 þg1 Þ Þ F ¼ C 6 A1 1 b2 b6 J 1 ðeðJ1 þJ2 þJ4 Þ eðJ2 þg1 Þ Þ J 2 ðeðJ1 þJ2 þJ4 Þ eðJ1 þg1 Þ Þ J ðeðJ1 þJ2 þJ5 Þ eðJ2 þg1 Þ Þ J 2 ðeðJ1 þJ2 þJ5 Þ eðJ1 þg1 Þ Þ A3 1 þA2 b3 b7 b8 b4
J 1 ðeðJ1 þJ2 þJ6 Þ eðJ2 þg1 Þ Þ J 2 ðeðJ1 þJ2 þJ6 Þ eðJ1 þg1 Þ Þ ; A4 b9 b5 S S S S
ðJ þkPrÞ ðJ þkPrÞ e 2 e 3 e 2 e 4 e 2 eðJ5 þkPrÞ e 2 eðJ6 þkPrÞ þ A2 A3 A4 ; R ¼ C 7 A1 C2 C3 C4 C5 ðJ þkPrÞ ðJ þkPrÞ ðJ þkPrÞ ðJ þkPrÞ
e 3 eS1 e 4 eS1 e 5 eS1 e 6 eS1 þ A2 A3 A4 : S ¼ C 7 A1 C2 C3 C4 C5 J 1 ðeðJ2 þg 2 Þ eðJ1 þJ2 þJ3 Þ Þ b2
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Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Review
Numerical and analytical solutions for magneto-hydrodynamic 3D flow through two parallel porous plates Sahin Ahmed a, Joaquín Zueco b,⇑, Luis María López-González c a
Fluid Dynamics Research, Department of Mathematics, Goalpara College, Goalpara, Assam 783101, India ETS Ingeniería Industrial, Departamento de Ingeniería Térmica y Fluidos, Universidad Politécnica de Cartagena, 30202 Cartagena (Murcia), Spain c Department of Mechanic Eng., Universidad de La Rioja, C/Luís de Ulloa, 20, E-26004 Logroño (La Rioja), Spain b
a r t i c l e
i n f o
Article history: Received 3 August 2016 Received in revised form 10 October 2016 Accepted 29 November 2016
Keywords: Semi-analytical solution Perturbation technique Network model MHD Injection/suction Couette flow Porous medium
a b s t r a c t The present work analyzed the hydromagnetic effect on three-dimensional Couette flow of viscous incompressible, electrically conducting and Newtonian fluid through a porous medium bounded by two horizontal parallel porous flat plates with transverse sinusoidal injection of the fluid at the stationary plate and its corresponding removal by periodic suction through the plate in uniform motion. The flow becomes three dimensional due to this injection/suction velocity. Approximate solutions are obtained for the flow field, the pressure, the skin-friction, the temperature field, and the rate of heat transfer. It is found that the cross velocity w is reduced considerably with a rise in the magnetic body parameter (M), permeability parameter (K) and injection/suction (k) in the forward flow, while a reverse effect is observed in the backward flow. An increase in K or k is found to escalate the main flow velocity whereas an increase in the magnetic body parameter (M) is shown to exert the opposite effect. Similarly, the shear stress due to main flow is considerably increased with an increase in M and K. The results show that both methods, perturbation and electrical network schemes provides excellent approximations to the solution of this nonlinear system with high accuracy. The acquired knowledge in our study can be used by designers to control Magnetohydrodynamic (MHD) flow as suitable for a certain application. Other possible applications include materials processing, MHD propulsion thermo-fluid dynamics, boundary layer in aerodynamics, chemical engineering etc. Ó 2016 Elsevier Ltd. All rights reserved.
Contents 1. 2. 3. 4. 5. 6.
Introduction . . . . . . . . . . . . . . . . . . . . . . . Formulation of the problem . . . . . . . . . . Semi-analytical solution . . . . . . . . . . . . . Numerical method (Network Simulation Discussion and results . . . . . . . . . . . . . . . Conclusions. . . . . . . . . . . . . . . . . . . . . . . . Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction Magnetohydrodynamics deals with dynamics of an electrically conducting fluid, which interacts with a magnetic field. The study ⇑ Corresponding author. E-mail addresses: [email protected] (S. Ahmed), [email protected] (J. Zueco), [email protected] (L.M. López-González). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.11.102 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
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322 324 325 325 327 329 330 330
of heat transfer and flow, through and across porous media, is of great theoretical interest because it has been applied to a variety of geophysical and astrophysical phenomena. Practical interest of such study includes applications in electromagnetic lubrication, boundary cooling, bio-physical systems and in many branches of engineering and science. In engineering, it finds its application in MHD pumps, MHD bearings, nuclear reactors, geothermal energy extraction and in boundary layer control in the field of aerodynam-
S. Ahmed et al. / International Journal of Heat and Mass Transfer 108 (2017) 322–331
323
Nomenclature !
B B0 CP d ! E0 ! J ! ! J XB J K K M Pr p T ; v ; wÞ ðu
magnetic induction vector uniform magnetic field (Tesla) specific heat at constant pressure (Jkg1K) distance between the plates electric field electric current density Lorentz force per unit volume electrical current (A) permeability coefficient permeability of the porous medium Hartmann number/magnetic parameter Prandtl number pressure (Nm2) temperature (K) and z-directions velocity components along x, y respectively
ics. A survey of MHD studies could be found in Crammer and Pai [1]; Moreau [2]. In view of the increasing technical applications using magnetohydrodynamics (MHD) effect, it is desirable to extend many of the available viscous hydrodynamic solutions to include the effects of magnetic field for those cases when the viscous fluid is electrically conducting. Rossow [3], and Greenspan and Carrier [4] have studied extensively the hydromagnetic effects on the flow past a plate with or without injection/suction. The hydromagnetic channel flow and temperature field was investigated by Attia and Kotb [5]. An analysis of the fully-developed MHD free convective flow between two vertical, electrically conducting plates has been carried out by Soundalgekar and Haldavnekar [6]. The nonlinear integro-differential equations, governing the flow, have been solved by perturbation method. An analysis of a two-dimensional unsteady flow of a viscous, incompressible electrically-conducting fluid through a porous medium bounded by two infinite parallel plates under the action of a transverse magnetic field is presented by Hassanien and Mansour [7]. An unsteady hydromagnetic flow through a porous medium between two infinite parallel porous plates with time varying suction is studied by Hassanien [8]. Nabil et al. [9] investigated an analytical study of unsteady magneto-hydrodynamic flow of an incompressible electrically conducting fluid filling the space between two parallel plates is presented, taking into account the couple stresses and pulsation of the pressure gradient effect. Makinde and Chinyoka [10] studied the unsteady flow and heat transfer of a dusty fluid between two parallel plates with variable viscosity and electric conductivity and the fluid is driven by a constant pressure gradient and an external uniform magnetic field is applied perpendicular to the plates with a Navier slip boundary condition. The governing non-linear partial differential equations are solved numerically using a semi-implicit finite difference scheme. Channel flows through porous media have several engineering and geophysical applications, such as, in the field of chemical engineering for filtration and purification processes; in the field of agricultural engineering to study the underground water resources; in petroleum industry to study the movement of natural gas, oil and water through the oil channels and reservoirs. For example, Raptis et al. [11] analyzed the problem of hydromagnetic free convection flow through a porous medium between two parallel plates; while Kearsley [12] studied problem of steady state Couette flow with viscous heating. Makinde and Osalusi [13] considered a MHD steady flow in a channel with slip at permeable boundaries. Choudhury and Bhattacharjee [14] analyzed the unsteady two-
u; v ; w U V V
dimensionless velocity component in x; y; z-direction Uniform velocity of the upper plate (ms1) injection/section velocity (ms1) electrical voltage (V)
Greek symbols a thermal diffusivity (m2K1) e small reference parameter ðe 1Þ h dimensionless fluid temperature, j thermal conductivity (Wm1K1), c kinematic viscosity (m2s1), q density (kgm3), r electrical conductivity k injection/suction parameter s shearing stress (Nm2)
dimensional incompressible visco-elastic MHD free convective fluid flow over a radiative vertical porous plate with Dufour effects in presence of chemical reaction using the perturbation technique. Beg et al. [15] solved the numerical problem of MHD of a viscous plasma flow in rotating porous media submitted to the influence of Hall currents and an inclined magnetic field. A numerical study is made by Kim et al. [16] on the heat transfer characteristics from forced pulsating flow in a channel filled with fluid-saturated porous media where the channel walls are assumed to be at uniform temperature and the BrinkmanForchheimer-extended Darcy model is employed. Kuznetsov [17] presented an analytical solution to the flow and heat transfer in Couette flow through a rigid saturated porous medium where the fluid flow occurs due to a moving wall and it is described by the Brinkman-Forchheimer-extended Darcy equation. Sharma and Chaudhary [18] studied the effects of magnetic field on the steady flow and heat transfer in a horizontal channel. Forced convection with viscous dissipation in a parallel plate channel filled by a saturated porous medium is investigated numerically by Hooman and Gurgenci [19]. The fully developed laminar mixed convection flow in a vertical plane parallel channel filled with a porous medium and subject to isoflux-isothermal wall conditions is investigated Barletta et al. [20]. The problem of free convection heat transfer flow through a porous medium bounded by a wavy wall and a vertical wall is studied by Sahin and Hiren [21]. Recently, Fasogbon [22] studied the simultaneous buoyancy force effects of thermal and species diffusion through a vertical irregular channel by using parameter perturbation technique. Sahin and Zueco [23] investigated the effects of Hall current, magnetic field, rotation of the channel and suction-injection on the oscillatory free convective MHD flow in a rotating vertical porous channel. Bég et al. [24] obtained comprehensive network computational solutions for the two-dimensional transient hydromagnetic flow in a parallel-plate channel containing a Darcian porous material with dissipation and Hall/ionslip currents. Bég et al. [25] studied the transient problem optically-thick radiative convective double-diffusive boundary layers in a Darcian porous medium adjacent to an impulsively started surface. Turkyilmazoglu [26,27] used the homotopy analysis method to solve the equations governing the flow of a steady, laminar, incompressible, viscous, and electrically conducting fluid due to a rotating disk subjected to a uniform suction and injection through the walls in the presence of a uniform transverse magnetic field. Turkyilmazoglu [28] analyzed the MHD permeable heat and fluid flow fields induced by stretching or shrinking two-three dimensional objects, obtain-
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S. Ahmed et al. / International Journal of Heat and Mass Transfer 108 (2017) 322–331
ing exact analytical solutions is targeted either in general or in restrictive forms. In the present article we study the twin effects of the magnetic field and permeability parameter on the three-dimensional Couette flow of viscous incompressible, electrically conducting and Newtonian fluid through a porous medium bounded by two horizontal parallel porous plates. The stationary plate and the plate in uniform motion are, respectively, subjected to a transverse sinusoidal injection and uniform suction of the fluid. The dependence of solution on Hartmann number, porosity and injection/suction is investigated by the graphs and tables. The solutions are obtained using both semi-analytical and numerical methods.
The equations governing the steady motion of an incompressible viscous electrically conducting fluid through porous medium bounded by two permeable horizontal plates in presence of a magnetic field in vector form are: !
Div q ¼ 0; !
Div B ¼ 0;
Conserv ation of Mass;
ð1Þ
Gausss law of magnetism;
ð2Þ
! ! !
!
!
!
!
qðq : $Þ q ¼ $ p þ l$2 q þ J X B; Conserv ation of Momentum; ð3Þ ! !
qC P ðq : $ÞT ¼ j$2 T; Conserv ation of Energy; 2. Formulation of the problem
!
!
!
!
J ¼ r½E 0 þ q X B;
We consider the Couette flow of a viscous incompressible electrically conducting fluid between two parallel flat porous plates with transverse sinusoidal injection of the fluid at the stationary plate and its corresponding removal by periodic suction through plane lie along the the plate in uniform motion U. Let the xy -axis be taken normal to the free-stream velocplates and let the y ity. The distance d is taken between the plates. Denote the velocity ; v ; w in the ; z-directions, respectively. The components by u x; y lower and upper plates are assumed to be at constant temperature T 0 and T 1 , respectively, with T 1 > T 0 . We derive the governing equations with the assumption that the flow is steady and laminar, and is of a finitely conducting fluid. The magnetic field of uniform strength B0 is applied to the perpendicular of the free-stream velocity (see Fig. 1); at lower magnetic Reynolds number, the magnetic field is practically independent of the flow motion and the induced magnetic field is neglected. The Hall effects, electrical and polarization effects also have been neglected. All physical quantities are independent of x for this problem of fully developed laminar flow, but the flow remains three-dimensional due to the injection/suction velocity Vð zÞ ¼ Vð1 þ e cosp z=dÞ.
0
ð4Þ
0
ð5Þ
Ohm slaw;
!
^ be the fluid velocity at the point ðx; y k ; zÞ and ^i þ v^j þ w where q ¼ u
!
B ¼ B0^j be the applied magnetic field. All physical quantities are defined in the Nomenclature. Thus, with the foregoing assumptions and under the usual boundary layer approximations, the Eqs. (1)–(4) reduce to:
@ v @ w ¼0 þ @z @y
v
ð6Þ
! @2u @u @u @2u þw ¼c þ 2 @ z2 @y @ z @y
!
rB20 c ðu UÞ þ q K
@ v @ v 1 @p @ 2 v @ 2 v v þ w ¼ þ c 2 þ 2 @y @z q @y @y @z
!
ð7Þ
c
K
v
! @2w @w @w 1 @p @2w þw ¼ þc þ 2 @y @z q @z @y @ z2
v
@T @T @2T @2T þw ¼a þ 2 @z2 @y @z @y
v
ð8Þ !
rB20 c þ w q K
ð9Þ
! ð10Þ
=d; z ¼ z=d; u ¼ u =U; v ¼ v =V; w ¼ w=V; =qV 2 ; Pr ¼ where y ¼ y p¼p c=a,
1
M¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rB20 d2 =qc;
k ¼ Vd=c;
h ¼ ðT T 0 Þ=ðT 1 T 0 Þ;
2
K ¼ K=d U
( )
The non-dimensional forms of (1)–(5) are
v y þ wz ¼ 0; v uy þ wuz ¼
ð11Þ uyy þ uzz ðM 2 þ K 1 Þðu 1Þ ; k
vv y þ wv z ¼ py þ
v yy þ v zz K 1 v
0
v wy þ wwz ¼ pz þ
0
( )
v hy þ whz ¼
k
ð12Þ
;
ð13Þ
wyy þ wzz ðM 2 þ K 1 Þw ; k
ð14Þ
hyy þ hzz ; k
ð15Þ
The corresponding boundary conditions in the dimensionless form are
y ¼ 0 : u ¼ 0; Fig. 1. Model of Couette Flow through Porous Medium with MHD.
y ¼ 1 : u ¼ 1;
v ðzÞ ¼ 1 þ ecospz; v ðzÞ ¼ 1 þ ecospz;
w ¼ 0; h ¼ 0 w ¼ 0; h ¼ 1
ð16Þ
S. Ahmed et al. / International Journal of Heat and Mass Transfer 108 (2017) 322–331
h 1
325
3. Semi-analytical solution
p1 ðy; zÞ ¼ ðp2 kAÞ
Since the amplitude of the injection/suction velocity e 1 is very small, we now assume the solution of the following form:
þA2 J 4 ðJ 1 J 4 kJ 4 M2 K 1 ÞeJ4 y A3 J 5 ðJ 2 J 5 kJ 5 M 2 K 1 ÞeJ5 y i A4 J 6 ðJ 2 J 6 kJ 6 M2 K 1 ÞeJ6 y sinpz; ð29Þ
f ðy; zÞ ¼ f 0 ðyÞ þ ef 1 ðy; zÞ þ þe2 f 1 ðy; zÞ þ ;
h n 1 1 u1 ðy; zÞ ¼ Eeg1 y þ Feg2 y þ b1 A1 J 1 eðJ2 þJ1 yþJ3 yÞ b2 J 2 eðJ1 þJ2 yþJ3 yÞ b6 1 1 þA2 J 1 eðJ2 þJ1 yþJ4 yÞ b3 J 2 eðJ1 þJ2 yþJ4 yÞ b7 1 1 A3 J 1 eðJ2 þJ1 yþJ5 yÞ b8 J 2 eðJ1 þJ2 yþJ5 yÞ b4 oi 1 1 A4 J 1 eðJ2 þJ1 yþJ6 yÞ b9 J 2 eðJ1 þJ2 yþJ6 yÞ b5 ð30Þ cospz;
ð17Þ
where f stands for any of u; v ; w; p; and h. When e ¼ 0, the problem is reduced to the well-known two-dimensional flow. The solution of this two-dimensional problem is
u0 ðyÞ ¼ 1 þ
h0 ðyÞ ¼
expðJ 2 þ J 1 yÞ expðJ 1 þ J 2 yÞ ; expðJ 1 Þ expðJ 2 Þ
expðkPryÞ 1 ; expðkPrÞ 1
v 0 ¼ 1;
ð18Þ
w0 ¼ 0;
p0 ¼ constant;
ð19Þ
When e–0, substituting (17) in (11)–15) and comparing the coefficient of e, neglecting those of e2 ; e3 ; . . ., the following firstorder equations are obtained with the help of solutions (18) and (19):
v 1y þ w1z ¼ 0; v 1 u0y þ u1y ¼ v 1y ¼ p1y þ
ð20Þ
u1yy þ u1zz ðM 2 þ K 1 Þu1 ; k
v 1yy þ v 1zz K 1 v 1 k
ð21Þ
;
ð22Þ
w1yy þ w1zz ðM2 þ K 1 Þw1 ; k
ð23Þ
v 1 h0y þ h1y ¼
h1yy þ h1zz ; kPr
ð24Þ
y ¼ 1 : u1 ¼ 0;
w1 ¼ 0; h1 ¼ 0
w1 ¼ 0; h1 ¼ 1
ð25Þ
This is the set of linear partial differential equations, which describe the three-dimensional flow. To solve these equations, we assume v 1 ; w1 ; p1 ; u1 ; and h1 of the following form:
9 u1 ðy; zÞ ¼ u11 ðyÞcospz; > > > > > > v 1 ðy; zÞ ¼ v 11 ðyÞcospz; > > > = 0 v 11 ðyÞsinpz ; w1 ðy; zÞ ¼ p > > > > p1 ðy; zÞ ¼ p11 ðyÞcospz; > > > > > ; h1 ðy; zÞ ¼ h11 ðyÞcospz;
Now, after knowing the velocity field, we can calculate skinfriction components sx and sz in the main and transverse directions, respectively, as follows: ds du0 du11 sx ¼ x ¼ þe cospz; lU dy y¼0 dy y¼0 " ( ! J eJ2 J eJ1 ðJ2 þ J J ÞeJ2 ðJ2 þ J J ÞeJ1 sx ¼ 1 J1 2J2 þ e Eg 1 þ Fg 2 þ b1 A1 1 1 3 2 2 3 e e b2 b6 ! 2 2 J2 J1 ðJ1 þ J1 J4 Þe ðJ þ J J Þe 2 24 þ A2 b3 b7 ! !)# ðJ21 þ J1 J5 ÞeJ2 ðJ22 þ J2 J5 ÞeJ1 ðJ21 þ J1 J6 ÞeJ2 ðJ22 þ J2 J6 ÞeJ1 A4 cospz; A3 b8 b4 b9 b5
sz ¼
z ds dw1 ; ¼e lU dy y¼0
¼ eðpAÞ1 ½A1 J 23 þ A2 J 24 A3 J 26 sinpz
ð33Þ
dq dh0 dh1 þe cospz; ¼ jðT 0 T 1 Þ dy y¼0 dy y¼0 kPr A1 ðJ 3 þ kPrÞ A2 ðJ 4 þ kPrÞ ¼ kPr þ þ e RS1 þ SS2 þ C 1 e 1 C2 C3
A3 ðJ 5 þ kPrÞ A4 ðJ 6 þ kPrÞ cospz C4 C5
Nu ¼
ð34Þ
4. Numerical method (Network Simulation Method)
ð26Þ
where the prime denotes differentiation with respect to y. Expressions for v 1 ðy; zÞ and w1 ðy; zÞ have been chosen so that the equation of continuity (20) is satisfied. Substituting (26) in (21)–(24) and applying the corresponding boundary conditions, we get the solutions for v 1 ; w1 ; p1 ; u1 ; and h1 as follows:
v 1 ðy; zÞ ¼ A1 ½A1 eJ y þ A2 eJ y A3 eJ y A4 eJ y cospz; 3
ð31Þ
From the temperature field, we can obtain the heat transfer coefficient in terms of Nusselt number as follows:
The corresponding boundary conditions reduce to
v 1 ¼ cospz; v 1 ¼ cospz;
h n ðJ 4 þkPryÞ 1 h1 ðy; zÞ ¼ ReS1 y þ SeS2 y þ C 1 A1 eðJ3 þkPryÞ C 1 C2 2 þ A2 e oi ðJ 6 þkPryÞ 1 C5 A3 eðJ5 þkPryÞ C 1 cospz; 4 A4 e
ð32Þ
w1y ¼ p1z þ
y ¼ 0 : u1 ¼ 0;
A1 J 3 ðJ 1 J 3 kJ 3 M 2 K 1 ÞeJ3 y
4
5
6
ð27Þ
w1 ðy; zÞ ¼ ðpAÞ1 ½A1 J 3 eJ3 y þ A2 J 4 eJ4 y A3 J 5 eJ5 y A4 J 6 eJ6 y sinpz; ð28Þ
Network Simulation Method (NSM) is used in the present study to solve the partial differential equations (11)–(15) under boundary conditions (16). The starting point for this method is always the discretization of the equations that form the mathematical model of the problem under study. With a spatial discretization (after experimenting with a few sets of mesh sizes) of the dimensionless equations in Ny and Nz cells of length Dy = 1/Ny (with Ny = 100), and Dz = zmax/Nz (with Nz = 50), respectively. These partial differential equation can be transformed into a system of connected differential equations. A first-order and second-order central difference approximation is used for the first and second derivate, respectively.
uy ¼ ð@u=@yÞi;j ðui;jþ1 ui;j1 Þ=Dy þ OðDyÞ
ð35Þ
uz ¼ ð@u=@zÞi;j ðuiþ1;j ui1;j Þ=Dz þ OðDzÞ
ð36Þ
S. Ahmed et al. / International Journal of Heat and Mass Transfer 108 (2017) 322–331
uyy ¼ ð@ 2 u=@y2 Þi;j ðui;jþ1 þ ui;j1 2ui;j Þ=Dy2 ¼ 1=Dy½ðui;j1 ui;j Þ=Dy ðui;j ui;jþ1 Þ=Dy þ OðDy Þ 2
ð37Þ
¼ 1=Dz½ðui1;j ui;j Þ=Dz ðui;j uiþ1;j Þ=Dz þ OðDz2 Þ
ð38Þ
to the Kirchhoff law for currents (conservation of charge, LCK). iii) The existence of the magnitudes Vi,i-1 and Vi,i+1 denominated potential, associated to each node, so that the differences Vi,i1 = Vi Vi1 and Vi,i+1 = Vi Vi+1 are accorded to the Kirchhoff law of the voltages (is a result of the electrostatic field being conservative, LVK).
uzz ¼ ð@ 2 u=@z2 Þi;j ðuiþ1;j þ ui1;j 2ui;j Þ=Dz2
Similar expressions can be obtained for v y ; v z ; v yy ; v zz ; wy ; wz ; wyy ; wzz ; hy ; hz ; hyy andhzz . The computational domain is divided into meshes each is of dimension 2Dy and 2Dz in space-y and space-z, respectively. Based on these equations, a electrical network circuit is designed, whose equations are formally equivalent (see equivalence):
The theorems of conservation and uniqueness of flow and potential electrical variables (Kirchhoff laws) are satisfied in the circuits. Some of the equations usually contained in mathematical model, do not need to be considered in the design of the network model; for example, in the conservation of the heat flux at the boundary of two different media and the uniqueness of the temperature in the boundary. After performing the discretization of the equations, the electrical analogy is applied. A number of networks are connected in series and parallel to make up the whole medium and boundary conditions are added by means of special electrical devices (current or voltage control-sources). Once the complete network model is designed, for which few programming rules are needed since not many devices form the network, a computer code (Pspice [32]) is used to simulate it using a PC, providing the numerical solution. The finite-difference differential equations resulting from dimensionless momentum balance, Eq. (12) is,
v i;j ðui;jþ1 ui;j1 Þ=Dy þ wi;j ðuiþ1;j ui1;j Þ=Dz
In this equivalence, the variables velocity (u,v,w) and temperature (h) are equivalent to the variable electric voltage, while the variables velocity gradients (ou/oy, . . .) and temperature gradient (oh/oy, oh/oz) are equivalent to the variable electric current. In the network theory, the viability of a network model implies:
¼ ½ðui;j1 ui;j Þ=Dy2 ðui;j ui;jþ1 Þ=Dy2 =k þ ½ðui1;j ui;j Þ=Dz2 ðui;j uiþ1;j Þ=Dz2 =k ðM 2 þ K 1 Þðui;j 1Þ=k ¼ 0
The Fig. 2 shows the network models corresponds to the equation (39), where the second law of Kirchhoff is applied, therefore each term of Eq. (39) is considered a electrical current that is modelled by means a electrical dispositive (resistor, capacitor, current generator, ect). In this figure, Ju,i,j+1 and Ju,i,j1 are the currents that
i) The existence of a network independent of time. ii) The existence of the magnitudes Ji,i1 and Ji,i+1 denominated fluxes, associated to each branch that connects the nodes i with (i-1) and i with (i + 1), respectively. These fluxes follow
zi-1,j
ui-1,j
Gy,i,j
Node (i,j)
{ vi,j ,ui,j+1, ui,j-1} Ru,i-1,j
Gz,i,j ui,j-1
Jy,i,j Ru,i,j-1
{wi,j ui+1,j, ui-1,j}
Jz,i-1,j=(ui-1,j-ui,j)/Δz
Jz,i,j
Ju,i,j-1=(ui,j-1-ui,j)/Δy {ui,j}
ð39Þ
zi,j zi+1,j
326
ui,j+1
Ru,i,j+1
Ji,j ui,j
2Δz
Ju,i,j+1=(ui,j-ui,j+1)/Δy Jz,i+1,j=(ui,j-ui+1,j)/Δz
Gi,j
Ru,i+1,j yi,j-1 yi,j
ui+1,j
yi,j+1 2Δy
Momentum Equation (Eq. 39)
Fig. 2. Nomenclature and network model of the control volume. Momentum balance equation (39).
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S. Ahmed et al. / International Journal of Heat and Mass Transfer 108 (2017) 322–331
leave and enter the cell for the friction term in the y-direction; while Ju,i+1,j and Ju,i1,j are the currents that leave and enter the cell for the friction term in the z-direction. Ji,j is the current due to magnetic and permeability effects, Jy,i,j and Jz,i,j are the currents due to the inertia terms of the velocity u in both directions (y and z), respectively. The currents Ju,i,j+1 and Ju,i,j1 are implemented by means of two resistances Ru,i,j±1 of values ‘‘kDy200 ; while the currents Ju,i+1,j and Ju,i1,j are implemented by means of two resistances Ru,i±1,j of values ‘‘kDz200 . Ji,j, Jy,i,j and Jz,i,j are implemented by means of voltage control current generators with controlactions indicate in Fig. 2. For the other equations, similar explanations can be realized to the previous case, so we will have a electrical circuit (network model) for each equation, and they will be connected and coupled together. Finally, to implement the boundary conditions are employed the following devices are used: i) a voltage source for the constant values of velocity and temperature, ii) a sinusoidal voltage-source for the sinusoidal velocity variation, v (z). More details about this numerical technique can be found in [23–24].
1.2 1 0.8
u
M = 0.0 M = 0.5 M = 1.0 M = 5.0
0.6 0.4 0.2 0
0
0.2
0.8
1
Fig. 3. Magnetic field effects on the main flow velocity.
1
The problem of steady three-dimensional Couette flow of viscous incompressible, electrically conducting and Newtonian fluid through a porous medium bounded by two horizontal parallel porous flat plates with transverse sinusoidal injection of the fluid at the stationary plate and its corresponding removal by periodic suction through the plate in uniform motion has been considered. The analytical expressions for velocities field, temperature, skin-friction and rate of heat transfer (Nusselt number) coefficients were obtained using perturbation technique. Table 1 shows the semi-analytical and numerical solutions. In this table is shown the spatial variation of shear stress sz with permeability and magnetic parameter for e ¼ 0:5; k ¼ 0:5 and z ¼ 0:5. We observe that with an increase in permeability parameter K the shear stress sz is decreased. Similarly with increasing magnetic parameter M from 1.0–5.0, there is a consistent decrease in shear stress sz . Both permeability parameter and magnetic parameter therefore have a negative influence on skin friction coefficient from the plate surface into the porous medium. An excellent agreement has been obtained between the semi-analytical solutions and NSM computations. The effects of the flow parameters such as magnetic parameter M, permeability coefficient K and injection/suction parameter k on the velocities field, temperature field and skin-friction coefficients have been studied analytically and presented with the help of Figs. 2–6. Fig. 3 illustrates the dimensionless velocity component x-direction (main flow velocity profiles) for various values of the
0.8
u
Semi-analytical solution
K 0.1 0.5 1.0 0.5 0.5 0.5
0.6 0.4
M = 5, z = 0.5, λ = 0.05
0.2 0
0
0.2
0.4
0.6
λ
0.3 0.3 0.3 1.0 1.5 2.0
0.8
1
y Fig. 4. Porosity and injection/suction effects on the main flow velocity.
Hartmann number ðM ¼ 0; 0:5; 1:0 and 5:0Þ. Profiles rise from the stationary wall to reach maximum values and then decrease to reach the velocity u ¼ 1 of the transient wall, so the velocity profiles converge in accordance with the boundary condition imposed there. With an increase in M from the non-conducting i.e. purely hydrodynamic case ðM ¼ 0Þ through 0.5, 1–5, there is a clear decrease in velocity i.e. flow is decelerated, so the Lorentz force, which opposes the flow, there is a fall in velocity maximum due to the retarding effect of the magnetic force in the region
Table 1 Comparison of the values of skin-friction component ðsz Þ for for semi-analytical and numerical schemes, with
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.6
y
5. Discussion and results
y
0.4
e ¼ 0:5; k ¼ 0:5 and z ¼ 0:5.
Numerical solution
K = 0.2, M = 1
K = 0.5, M = 1
K = 0.5, M = 5
K = 0.2, M = 1
K = 0.5, M = 1
K = 0.5, M = 5
0.47621 0.47286 0.46951 0.46318 0.46286 0.45352 0.45320 0.45272 0.44791 0.44627
0.45830 0.45504 0.43162 0.43116 0.43097 0.42790 0.42674 0.42392 0.42129 0.40737
0.38617 0.38494 0.37612 0.37460 0.37178 0.35901 0.35477 0.35172 0.34205 0.34135
0.47506 0.47309 0.46944 0.46299 0.46301 0.45303 0.45299 0.45218 0.44825 0.44705
0.46080 0.45823 0.43217 0.43559 0.43168 0.43106 0.42945 0.42763 0.42328 0.40451
0.38890 0.38656 0.37512 0.37460 0.37178 0.35989 0.35426 0.35199 0.34326 0.34155
328
S. Ahmed et al. / International Journal of Heat and Mass Transfer 108 (2017) 322–331
0.05
M 1.0 1.0 0.0 1.0 5.0
0.04 0.03 0.02
w
0.01 0 -0.01
0
-0.02
0.2
0.4
0.6
0.8
1
λ 0.7 0.5 0.5 1.0 0.5
0.8
1
θ
-0.04
M = 5, z = 0.5, ε = 0.05, Pr = 0.71
0.2
y
-0.05
λ 1.5 0.5 1.0 0.5 0.5
0.4
K = 0.5, z = 0.5, λ = 0.05
-0.03
0.6
K 0.1 1.0 0.1 0.5 0.1
0
Fig. 5. Magnetic field and injection/suction effects on the cross flow velocity.
0
0.2
0.4
0.6
0.8
1
y Fig. 7. Effects of porosity (K) and suction/injection (k) on the temperature profiles.
0.04
K = 1.0
0.03
K = 0.7
w
-0.01 -0.02
M = 0.0
K = 0.5
0.01 0
1
K = 0.1
0.02
0.8 0
0.2
0.4
0.6
0.8
1
θ
M = 2, λ = 0.5, z = 0.5, λ = 0.05
M = 5.0 M = 10.0
0.6
M = 15.0
0.4
-0.03 -0.04
K = 0.1, z = 0.5, λ = 0.5, ε = 0.05, Pr = 0.71
0.2
y Fig. 6. Porosity effects on the cross flow velocity.
y 2 ½0; 1 and as a result increases the thickness of the momentum boundary layer. Fig. 4 shows the influence of the injection/suction parameter ðkÞ and permeability parameter ðKÞ on the dimensionless velocity component x-direction (main flow velocity profiles). With an increase in k from k ¼ 0:3 through 0.3–2.0, there is a clear increase in velocity with distance transverse to the walls, i.e. the flow is accelerated. Also there is clear increase in velocity values at the wall accompanying a rise in K from K ¼ 0:1 to 1.0 i.e. the flow is 2
accelerated. The parameter K ¼ K=d U is directly proportional to the actual permeability, K of the porous medium. The Darcian drag force in the momentum equations (Eqs. (21)–(23)), viz, K 1 ðu 1Þ, K 1 v and K 1 w, respectively; is therefore inversely proportional to K. An increase in K will therefore decrease the resistance of the porous medium (as the permeability physically becomes less with increasing K) which will serve to accelerate the flow and increase the velocity, u. Fig. 5 illustrates the dimensionless velocity component w-direction (transverse velocity profiles) for various values of the Hartmann number ðM ¼ 0; 0:5; 1:0 and 5:0Þ and injection/suction parameter ðk ¼ 0:5; 0:7 and 1:0Þ. It was observed that forward flow developed from y = 0 to about y = 0.4, and then onwards there was backward flow. This was due to the fact that the dragging action of the faster layer exerted on the fluid particles in the neighborhood of the stationary plate was sufficient to overcome the adverse pressure gradient and, hence, there was forward flow. The dragging action of
0
0
0.2
0.4
0.6
0.8
1
y Fig. 8. Effect of magnetic parameter (M) on the temperature profiles.
the faster layer exerted on the fluid particles reduced due to the periodic suction at the upper plate and, hence, this dragging action was insufficient to overcome the adverse pressure gradient; therefore, there was backward flow. Furthermore, it is evident from this figure that velocity w decreased with increasing k and M in the forward and back flow. Fig. 6 shows the transverse velocity profiles for various permeability parameters ðKÞ. As expected, the transverse velocity will be increased substantially with increasing permeability parameter. Fig. 7 illustrate the influence of porosity (K) and suction/injection (k) on the boundary layer variable, h. With increasing porosity the temperature in the regime is found to be decreases i.e. the boundary layer is cooled. A reduction in the volume of solid particles in the medium implies a lower contribution via thermal conduction. This will serve to decrease temperatures. Similar effect has been observed for the suction/injection parameter on the fluid temperature. Fig. 8 presents the response of temperature (h) to magnetohyqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 drodynamic body force parameter (M) where M ¼ rB20 d =qc and it signifies the ratio of Lorentz hydromagnetic body force to viscous hydrodynamic force. With increasing M, temperature, is observed to be markedly increased. This is physically explained
S. Ahmed et al. / International Journal of Heat and Mass Transfer 108 (2017) 322–331
4
K = 0.1, 0.5, 1.0
3
K = 0.5, z = 0, ε = 0.5
2 1
τx
0 0
0.2
0.4
0.6
0.8
1
-1 -2
M = 0.0, 2.0, 5.0
λ
-3
Fig. 9. Porosity and magnetic field effects on the shear stress due to main flow.
z = 0.5, ε = 0.5, Pr = 0.71
0.8
0.6
0.4
Nu
K
M
0.75 0.15 0.15 0.45 0.15 0.15
2.0 2.0 6.0 2.0 8.0 4.0
0.2
0
0
0.2
0.4
0.6
0.8
1
λ
-0.2
329
influences the temperature (h) field, only via the thermal and buoyancy terms. Porosity, injection/suction and magnetic field effects on the shear stress due to main flow sx are shown in Fig. 9. The shear stress sx increases with increasing values of M (there is a backflow entirely from the regime) and decreases with increasing value of k. Increasing K clearly increases shear stress sx values i.e. increasing the permeability serves to accelerate the flow (no flow reversal in this case) which increases shear stress function values. Fig. 10 shows the influence of porosity (K) and magnetic parameter (M) on surface heat transfer rate i.e. Nusselt number function (Nu) at the plate y = 0 plotted against suction/injection (k) parameter. An increase in porosity clearly enhances (Nu) values which are consistent with the reduction of temperatures in the boundary layer regime, computed earlier. A contrary response is computed for the effect of magnetic parameter, M, on the Nusselt number function (Nu) distributions in this figure. Increasing magnetic field enhances the Lorentzian hydromagnetic drag which serves to decelerate the flow and warm the fluid. Therefore heat will be transferred at progressively lower rates from the fluid to the plates. Moreover, an increase in suction/injection parameter from 0.0 (without suction) through 0.2 to 1.0 the Nusselt number profiles attains maximum values near the plate y = 0 and thereafter all profile decay as we continue over the plate towards the upper plate. Negative values of Nu have been observed for the maximum magnetic numbers, it means that the heat of the fluid is diffused to the plate y = 0. Finally, the pressure values p1 are reported in Table 2 for z ¼ 0 and k ¼ 0:5. We can see that the pressure decreased with the permeability parameter K. The decrease in pressure was sufficiently large for small values of Kð0:2 and 0:5Þ. Besides, increasing magnetic parameter M clearly increases pressure p1 . It was observed that positive values pressure were obtained from y ¼ 0 to about y ¼ 0:4, and then onwards negative values pressure were obtained.
Fig. 10. Effects of porosity (K) and magnetic parameter (M) on the Nusselt number.
6. Conclusions Table 2 The values of pressure (p1 ) for z ¼ 0; k ¼ 0:5. y
K = 0.2, M = 0.5
K = 0.5, M = 0.5
K = 0.5, M = 5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
11.3677 7.1748 5.4180 3.6143 1.6835 0.0259 1.3082 3.9250 5.7819 7.8307 12.3805
6.7180 4.3719 2.5303 1.4193 0.4807 0.0292 0.5371 1.8251 2.9734 4.7022 7.4509
7.3712 5.1772 2.7128 1.8617 0.6314 0.0173 0.7813 1.9033 3.1852 5.5182 7.7582
by the fact that the extra work expended in dragging the fluid against the magnetic field is dissipated as thermal energy in the boundary layer, as elucidated by Sutton and Sherman [29], Pai [30] and Hughes and Young [31]. This results in heating of the boundary layer and an ascent in temperatures, an effect which is maximized some distance away from the surface. The magnetic field influence on temperatures while noticeable is considerably less dramatic than that on the velocity field, since the Lorentz body force only arises in the momentum equations (12) and (14) and
A mathematical model has been presented for the hydromagnetic three-dimensional Couette flow of a viscous incompressible electrically conducting fluid between two parallel flat porous plates with transverse sinusoidal injection of the fluid at the stationary plate and its corresponding removal by periodic suction through the plate in uniform motion, with a magnetic field of applied to the perpendicular of the free-stream velocity. Analytical and numerical solutions for the non-dimensional momentum and energy equations subject to transformed boundary conditions have been obtained. The flow has been shown to be accelerated (main and cross velocities) with decreasing magnetic number but the main flow velocity increased with injection/suction parameter, while decreasing injection/suction parameter accelerates the cross flow velocity. However, it has been shown that increasing magnetic field generally decelerates the flow, but increases temperature values in the channel. Increasing porosity serves to accelerate the flow as well as the rate of heat transfer, but reduce temperature values. Semi-analytical solution method does not suffer from error accumulation and is able to give very accurate results accelerating the convergence to the solutions, compared with the numerical solution that involving iterative time integration. The study has important applications in electro-magnetic materials processing and nuclear heat transfer control, as well as MHD energy generators.
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Appendix A
J1 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 2
½kþ
k þ4ðM þK 2
Þ
;
J2 ¼
½k
A ¼ ½J 4 J 5 þ J 3 J 6 J 3 J 5 J 4 J 6 fe
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 2
k þ4ðM þK Þ 2 ðJ3 þJ 4 Þ
ðJ 5 þJ6 Þ
þe
ðJ 4 þJ 5 Þ
g ½J 4 J 5 þ J 3 J 6 J 3 J4 J 6 J 5 feðJ5 þJ3 Þ þ eðJ6 þJ4 Þ g ½J 6 J 5 þ J 3 J 4 J 3 J 5 J 6 J 4 feðJ6 þJ3 Þ þ eðJ5 þJ4 Þ g;
ðJ4 þJ6 Þ
A1 ¼ ½J 4 J 6 J 5 J 6 e ½J 4 J 5 J 5 J 6 e þ ½J 4 J 5 J 4 J 6 eðJ5 þJ6 Þ þ ½J4 J 5 J 4 J 6 eJ4 þ ½J 6 J 5 J 4 J 5 eJ5 þ ½J 4 J 6 J 5 J 6 eJ6 ; A2 ¼ ½J 3 J 5 J 5 J 6 eðJ3 þJ6 Þ ½J 3 J 6 J 5 J 6 eðJ3 þJ5 Þ ½J 3 J 5 J 3 J 6 eðJ5 þJ6 Þ þ ½J3 J 6 J 3 J 5 eJ3 þ ½J 3 J 5 J 6 J 5 eJ5 þ ½J 5 J 6 J 3 J 6 eJ6 ; A3 ¼ ½J 4 J 6 J 3 J 6 eðJ3 þJ4 Þ þ ½J 3 J 4 J 4 J 6 eðJ3 þJ6 Þ ½J 3 J 4 J 3 J 6 eðJ4 þJ6 Þ ½J3 J 4 J 3 J 6 eJ3 ½J 4 J 6 J 3 J 4 eJ4 ½J 3 J 6 J 4 J 6 eJ6 ; A4 ¼ ½J 3 J 5 J 3 J 6 eðJ3 þJ4 Þ þ ½J 5 J 4 J 4 J 3 eðJ3 þJ4 Þ ½J 3 J 5 J 3 J 4 eðJ4 þJ5 Þ ½J3 J 5 J 3 J 4 eJ3 ½J 4 J 3 J 5 J 4 eJ4 ½J 4 J 5 J 3 J 5 eJ5 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½J þ J 21 þ4ðp2 þK 1 Þ ½J 1 J 21 þ4ðp2 þK 1 Þ J3 ¼ 1 ; J ¼ ; 4 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ½J 2 þ J 22 þ4ðp2 þK 1 Þ ½J 2 J 22 þ4ðp2 þK 1 Þ J5 ¼ ; J6 ¼ ; 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 2 2 2 2 ½kþ k þ4ðp þM þK Þ ½k k þ4ðp þM 2 þK 1 Þ g1 ¼ ; g2 ¼ ; 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ½kPrþ k2 Pr 2 þ4p2 ½kPrþ k2 Pr 2 þ4p2 ; g2 ¼ ; S1 ¼ 2 2 b1 ¼ AðeJ1keJ2 Þ ;
b2 ¼ 3J 1 J 3 kJ 3 ;
b5 ¼ 3J 2 J 6 kJ 6 ;
b6 ¼ 2J 2 J 3 þ J 1 J 3 kJ 3 ;
b8 ¼ 2J 1 J 5 þ J 2 J 5 kJ 5 ; C 2 ¼ J 1 J 3 þ kPrJ 3 ; C6 ¼
n
C7 ¼
b4 ¼ 3J 2 J 5 kJ 5 ;
b7 ¼ 2J 2 J4 þ J 1 J 4 kJ 4 ;
b9 ¼ 2J 1 J 6 þ J 2 J 6 kJ 6 ;
C 3 ¼ J 1 J 4 þ kPrJ 4 ;
k ; AðeJ1 eJ2 Þðeg1 eg 2 Þ
h
b3 ¼ 3J 1 J 4 kJ 5 ;
2
2
C 1 ¼ AðekkPrPr1Þ ;
C 4 ¼ J 2 J 5 þ kPrJ 5 ;
C 5 ¼ J 2 J 6 þ kPrJ 6 ;
k2 Pr 2 ; AðekPr 1ÞðeS1 eS2 Þ
o n ðJ þg Þ ðJ þJ þJ Þ o n ðJ þg Þ ðJ þJ þJ Þ o ðJ 1 þg 2 Þ ðJ 1 þg 2 Þ ðJ 1 þg 2 Þ 2 2 e 1 2 4 Þ 2 2 e 1 2 5 Þ eðJ1 þJ2 þJ3 Þ Þ eðJ1 þJ2 þJ4 Þ Þ eðJ1 þJ2 þJ5 Þ Þ J2 ðe J2 ðe J2 ðe þ A2 J1 ðe A3 J1 ðe E ¼ C 6 A1 b6 b3 b7 b8 b4
J ðeðJ2 þg2 Þ eðJ1 þJ2 þJ6 Þ Þ J 2 ðeðJ1 þg2 Þ eðJ1 þJ2 þJ6 Þ Þ A4 1 ; b9 b5 J ðeðJ1 þJ2 þJ3 Þ eðJ2 þg1 Þ Þ J 2 ðeðJ1 þJ2 þJ3 Þ eðJ1 þg1 Þ Þ F ¼ C 6 A1 1 b2 b6 J 1 ðeðJ1 þJ2 þJ4 Þ eðJ2 þg1 Þ Þ J 2 ðeðJ1 þJ2 þJ4 Þ eðJ1 þg1 Þ Þ J ðeðJ1 þJ2 þJ5 Þ eðJ2 þg1 Þ Þ J 2 ðeðJ1 þJ2 þJ5 Þ eðJ1 þg1 Þ Þ A3 1 þA2 b3 b7 b8 b4
J 1 ðeðJ1 þJ2 þJ6 Þ eðJ2 þg1 Þ Þ J 2 ðeðJ1 þJ2 þJ6 Þ eðJ1 þg1 Þ Þ ; A4 b9 b5 S S S S
ðJ þkPrÞ ðJ þkPrÞ e 2 e 3 e 2 e 4 e 2 eðJ5 þkPrÞ e 2 eðJ6 þkPrÞ þ A2 A3 A4 ; R ¼ C 7 A1 C2 C3 C4 C5 ðJ þkPrÞ ðJ þkPrÞ ðJ þkPrÞ ðJ þkPrÞ
e 3 eS1 e 4 eS1 e 5 eS1 e 6 eS1 þ A2 A3 A4 : S ¼ C 7 A1 C2 C3 C4 C5 J 1 ðeðJ2 þg 2 Þ eðJ1 þJ2 þJ3 Þ Þ b2
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