- Email: [email protected]
Effect of chemical reaction and thermal radiation on heat and mass transfer flow of MHD micropolar fluid in a rotating frame of reference
International Journal of Heat and Mass Transfer 54 (2011) 3505–3513
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Effect of chemical reaction and thermal radiation on heat and mass transfer flow of MHD micropolar fluid in a rotating frame of reference K. Das Department of Mathematics, Kalyani Government Engineering College, Kalyani, Nadia, West Bengal 74125, India
a r t i c l e
i n f o
Article history: Received 14 October 2010 Received in revised form 28 January 2011 Accepted 24 February 2011 Available online 12 April 2011 Keywords: Micropolar fluid Chemical reaction Thermal radiation Rotating frame Permeability parameter
a b s t r a c t This paper studies the effect of first order chemical reaction and thermal radiation on hydromagnetic free convection heat and mass transfer flow of a micropolar fluid via a porous medium bounded by a semiinfinite porous plate with constant heat source in a rotating frame of reference. The plate is assumed to oscillate in time with constant frequency so that the solutions of the boundary layer are the same oscillatory type. The dimensionless governing equations for this investigation are solved analytically using small perturbation approximation. The effect of the various dimensionless parameters entering into the problem on the velocity, temperature and concentration profiles across the boundary layer are investigated through graphs. Also the results of the skin friction coefficient, couple stress coefficient, the rate of heat and mass transfer at the wall are prepared with various values of the parameters. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The analysis of the dynamics of micropolar fluid has been the subject of many research papers in recent years because of its increasing importance in processing industries and elsewhere of materials whose behaviour in shear cannot be characterized by Newtonian relationships. The micropolar fluids which contain micro-constituents and can undergo rotation has been proposed by Eringen [1]. These kind of fluids are utilized in analyzing exotic lubricants, the flow of colloidal suspensions, paints, liquid crystals, animal blood, fluid flowing in brain, turbulent shear flows and body fluids both mathematically and industrially. In addition due to its practical application to boundary layer control and thermal protection in high energy flow by means of wall velocity and mass transfer, considerable attention has been paid to the thermal boundary layer flows over moving boundaries [2]. The oscillatory boundary layer flow with constant heat source in the case of MHD free convection currents and mass transfer has been considered by Rahman and Sattar [3]. Khonsari and Brewe [4] have examined the effect of viscous dissipation on the lubrication characteristics of micropolar fluids. The effect of non zero values of micro gyration vector on semi-infinite moving porous plate with constant velocity when the magnetic field is imposed transversely to the plate and the temperature of the plate is oscillating with time have been analyzed by Kim and Lee [5].
E-mail address: [email protected] 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.03.035
Combined heat and mass transfer problems with chemical reaction are of importance in many processes and have, therefore, received a considerable amount of attention in recent years. Possible applications of this type flow can be found in many industries, like design of chemical processing equipment, damage of crops due to freezing, food processing and cooling towers. Deka et al. [6] investigated the effect of first order homogeneous chemical reaction on the process of an unsteady flow past an infinite vertical plate with a constant heat and mass transfer. Muthucumaraswamy and Ganesan [7] discussed the effect of the chemical reaction and injection on flow characteristics in an unsteady upward motion of an isothermal plate. MHD flow of a uniformly stretched vertical permeable surface in the presence of heat generation/absorption and a chemical reaction has been considered by Chamkha [8]. Rahman and Satter [9] examined MHD convective flow of micropolar fluid past a continuously moving vertical porous plate in the presence of heat generation/absorption. Ibrahim et al. [10] obtained the analytical solution for unsteady MHD free convection flow past a semi-infinite vertical permeable moving plate with heat source and chemical reaction. Rahman et al. [11] studied heat transfer in micropolar fluid with temperature dependent fluid properties along a non-stretching sheet. Rahman et al. [12,13] considered heat transfer in micropolar fluid along an inclined plate with variable fluid properties under different boundary conditions. Damseh et al. [14] have investigated heat and mass transfer free convection flow adjacent to a continuous moving vertical porous plate for incompressible micropolar fluid in the presence of heat generation/absorption and a first order chemical reaction. Rahman
3506
K. Das / International Journal of Heat and Mass Transfer 54 (2011) 3505–3513
and Al-Lawatia [15] developed the problem by considering higher order chemical reaction. In all the previous investigation, the effect of thermal radiation on the flow and heat transfer have not been provided.The effect of radiation on MHD flow and heat transfer problem have become more important industrially. At high operating temperature, radiation effect can be quite significant. Many process in engineering areas occur at high temperature and a knowledge of radiation heat transfer becomes very important for design of reliable equipment, nuclear plants,gas turbines and various propulsion devices or aircraft, missiles, satelites and space vehicles. Based on these applications, Cogley et al. [16] showed that in the optically thin limit, the fluid does not absorb its own emitted radiation but the fluid does absorb radiation emitted by the boundaries. Satter and Hamid [17] investigated the unsteady free convection interaction with thermal radiation in a boundary layer flow past a vertical porous plate. Vajravelu [18] studied the flow of a steady viscous fluid and heat transfer characteristic in a porous medium by considering different heating processes. Hossain and Takhar [20] have considered the radiation effect on mixed convection boundary layer flow of an optically dense viscous incompressible fluid along a vertical plate with uniform surface temperature. Raptis [21]investigate the steady flow of a viscous fluid through a porous medium bounded by a porous plate subjected to a constant suction velocity by the presence of thermal radiation. Makinde [22] examined the transient free convection interaction with thermal radiation of an absorbing emitting fluid along moving vertical permeable plate. Ibrahim et al. [23] discussed the case of mixed convection flow of a micropolar fluid past a semi infinite steady moving porous plate with varying suction velocity normal to the plate in presence of thermal radiation and viscous dissipation. Rahman and Satter [24] studied transient convective flow of micropolar fluid past a continuous moving porous plate in the presence of radiation. The effect of the chemical reaction and radiation absorption on the unsteady MHD free convection flow past a semi-infinite vertical permeable moving plate with heat source and suction has been studied by Ibrahim et al. [25]. Rahman and Sultana [26] examined radiative heat flux with variable heat flux in a porous medium. Recently Bakr [27] presented an analysis on MHD free convection and mass transfer adjacent to moving vertical plate for micropolar fluid in a rotating frame of reference in presence of heat generation/absorption and a chemical reaction. The main purpose of the present investigation is to study the effects of thermal radiation and chemical reaction on unsteady MHD free convection heat and mass transfer flow of a micropolar fluid past a vertical porous plate in a rotating frame of reference.It is assumed that the plate is embedded in a uniform porous medium and oscillates in time with a constant frequency in the presence of a transverse magnetic field. The governing equations are solved analytically using perturbation technique. Numerical results are reported for various values of the physical parameters of interest. The paper has been arranged as follows. The Section 2 deals with the mathematical formulation of the problems. Section 3 contains the closed form solutions of velocity, temperature concentration etc. Numerical results and discussion are presented in Section 4. The conclusions have been summarized in Section 5.
Fig. 1. Physical model and coordinate system of the problem.
that there is no applied voltage which implies the absence of an electric field. The flow is assumed to be in the x-direction which is taken along the plate in the upward direction and z-axis is normal to it. Also it is assumed that the whole system is rotate with a constant frame X in a micropolar fluid about z-axis.The fluid is asumed to be gray, absorbing-emitting but not scattering medium. The radiation heat flux in x-direction is considered negligible in comparison that the z-direction. Due to semi-infinite plate surface assumption, furthermore, the flow variables are functions of z and time t only. Under these assumption, the equations that describe the physical situation are given by
@w ¼ 0; @z @u @u @2u þw 2Xv ¼ ðm þ mr Þ 2 þ gbT ðT T 1 Þ þ gbC ðC C 1 Þ @t @z @z 2 mu rB20 u @x mr ; k q @z
ð1Þ
ð2Þ
1 @v @v @ 2 v mv rB20 v @x þw þ 2Xu ¼ ðm þ mr Þ 2 ; þ mr @t @z @z k q @z
ð3Þ
1 1 1 K @2x @x @x ; þw ¼ @t @z qj @z2
ð4Þ
2 2 2 K @2x @x @x ; þw ¼ @t @z qj @z2
ð5Þ
@T @T j @2T Q 1 @qr ðT T 1 Þ þw ¼ ; @t @z qC p @z2 qC p qC p @z
ð6Þ
ð7Þ
2. Mathematical formulation of the problem
@C @C @2C þw ¼ Dm 2 Rr ðC C 1 Þ @t @z @z
Consider unsteady three dimensional flow of an electrically conducting incompressible micropolar fluid past a semi-infinite vertical permeable moving plate embeded in a uniform porous medium and subjected to a constant transverse magnetic field B0 (see Fig. 1 in the presence of thermal and concentration buoyancy effects with chemical reaction and thermal radiation. It is assumed
where u, v and w are velocity components along x, y and z-axis 1 and x 2 are micro-rotation components along x respectively, x and y-axis respectively. bT and bC are the coefficients of thermal expansion and concentration expansion, q is the density of the fluid, m is the kinematic viscosity, mr is the kinematic micro-rotation viscosity, Cp is the specific heat at constant pressure p, K is the spin gradient velocity, j is the micro-inertia density, g is the acceleration
3507
K. Das / International Journal of Heat and Mass Transfer 54 (2011) 3505–3513 B0 p rm
due to gravity, k is the permeability of porous medium, T is the temperature of the fluid in the boundary layer and T1 is the temperature of the fluid far away from the plate, C is the concentration of the solute and C1 is the concentration of the solute far away from the plate, Q is the additional heat source, Dm is the molecular diffusivity and Rr is the chemical reaction rate constant. The boundary conditions are given by 1 ¼x 2 ¼ 0; T ¼ T 1 ; C ¼ C 1 for t 6 0; u ¼ v ¼ 0; x ð8Þ 9 e int 1 ¼ 12 @@zv ; x 2 ¼ 12 @u ; u ¼ U r 1 þ 2 ðe þ eint Þ ; v ¼ 0; x > @z > > = T ¼ T w ; C ¼ C w at z ¼ 0; for t > 0; ð9Þ > and > > ; 1 ¼x 2 ¼ 0; T ¼ T 1 ; C ¼ C 1 as z ! 1; u ¼ v ¼ 0; x
where R ¼ 2UX2m is the rotational parameter, M ¼ Ur q is the magr lqC netic field parameter, Pr ¼ j p ; Sc ¼ Dmm ; Gr ¼ mgbT ðTUw3 T 1 Þ ; Gm ¼ r mgbC ðC w C 1 Þ are the Prandtl number, Schmidt number, Grashof numU 3r 4T 31 r ber, modified Grashof number respectively, F ¼ jk is the heat 2 r radiation parameter, S ¼ wU0r is the suction parameter, K ¼ kU m2 is the Rr m permeability of the porous medium, a ¼ Ur is the chemical reaction parameter, k ¼ l^j is the dimensionless material parameter, D ¼ mmr is the viscosity ratio. Also the boundary conditions become
where Ur is the uniform reference velocity and e is the small constant quantity. The oscillatory plate velocity assumed in Eq. (9) is based on the suggestion proposed by Ganapathy [19]. The continuity Eq. (1) gives
and
w ¼ w0 ;
ð10Þ
where the w0 represents the normal velocity at the plate which is positive for suction and negative for blowing. The radiative heat flux term by using the Rosseland approximation is given by
qr ¼
4r @T 4 ; 3k @z
ð11Þ
where r⁄ is the Stefan–Boltzmann constant and k⁄ is the mean absorption coefficient. Assuming that the differences in temperature within the flow are such that T4 can be expressed as a linear combination of the temperature, we expand T4 in Taylor’s series about T1 and neglecting higher order terms, we get
T 4 ¼ 4T 31 T 3T 41 :
ð12Þ
Thus we have
1 ¼x 2 ¼ 0; h ¼ 0; / ¼ 0 as z ! 1 u ¼ v ¼ 0; x
ð13Þ
Let us introduce the following dimensionless variables:
v 0 ¼ Uv ; r
z0 ¼ zUm r ;
2
t 0 ¼ tUmr ;
n0 ¼ Unm2 ; r
V ¼ u þ iv ;
2
r
r
: ;
for t > 0: ð22Þ
1 þ ix 2 x¼x
and get
@V @V @2V @x S þ iRV ¼ ð1 þ DÞ 2 N2 V þ Grh þ Gm/ þ iD ; ð23Þ @t @z @z @z 2 @x @x @ x ð24Þ S ¼k 2 ; @t @z @z 2 @h @h 1 4F @ h Q S ¼ 1þ ð25Þ h; @t @z Pr 3 @z2 Pr @/ @/ 1 @ 2 / S ¼ a/: @t @z Sc @z2
ð26Þ
The associated boundary conditions (21) and (22) are written as follows:
h ¼ 0;
V ¼ 1 þ 2e ðeint þ eint Þ; and V ¼ 0; x ¼ 0; h ¼ 0;
/ ¼ 0 for t 6 0;
v ¼ 0; x ¼ 2i @V ; @z
h ¼ 1;
ð27Þ 9 / ¼ 1 at z ¼ 0; > =
/ ¼ 0 as z ! 1:
9
01 ¼ x 12m ; = x U
1 1 02 ¼ x 22m ; h ¼ TTT x ; / ¼ CCC ; Q 0 ¼ UQ2mj w T 1 w C 1 U
> > > ;
ð21Þ
We now simplify Eqs. (15)–(20) by putting the fluid velocity and angular velocity in the complex form as
V ¼ 0; x ¼ 0;
@qr 16T 31 r @ 2 T : ¼ @z2 @z 3k
u0 ¼ Uur ;
1 ¼x 2 ¼ 0; h ¼ 0; / ¼ 0 for t 6 0; u ¼ v ¼ 0; x 9 e int int 1 ¼ 12 @@zv ; x 2 ¼ 12 @u u ¼ 1 þ 2 ðe þ e Þ; v ¼ 0; x ;> @z > > = h ¼ 1; / ¼ 1 at z ¼ 0
> ;
for t > 0:
ð28Þ 3. Analytical solutions
r
ð14Þ Then substituting Eq. (14) into Eqs. (1)–(7)yields the followingdimensionless equations (dropping primes):
@u @u @2u 1 S Rv ¼ ð1 þ DÞ 2 þ Grh þ Gm/ M 2 þ u @t @z @z K 2 @x D ; @z
To find the analytical solutions of the above system of partial differential Eqs. (23)–(26) in the neighbourhood of the plate under the above boundary conditions (27) and (28,we express V, x, h and / as [19]
Vðz; tÞ ¼ V 0 þ ð15Þ
ð16Þ
1 1 1 @x @x @2x ; S ¼k @t @z @z2
ð17Þ
2 2 2 @x @x @2x ; S ¼k @t @z @z2
ð18Þ
@h @h 1 4F @ 2 h Q S ¼ 1þ h; @t @z Pr 3 @z2 Pr
ð19Þ
eint V 1 ðzÞ þ eint V 2 ðzÞ ;
e xðz; tÞ ¼ x0 þ eint x1 ðzÞ þ eint x2 ðzÞ ; 2 e int int hðz; tÞ ¼ h0 þ
@v @v @2v 1 @x v þD 1; S þ Ru ¼ ð1 þ DÞ 2 M2 þ K @t @z @z @z
e 2
2
e h1 ðzÞ þ e
h2 ðzÞ ;
e /ðz; tÞ ¼ /0 þ eint /1 ðzÞ þ eint /2 ðzÞ : 2
ð29Þ ð30Þ ð31Þ ð32Þ
Invoking the above Eqs. (29)–(32) into the Eqs. (23)–(28) and equating the harmonic and non-harmonic terms and neglecting the higher order terms of o(e2), we obtain the following set of equations:
ð1 þ DÞV 000 þ SV 00 a1 V 0 þ Grh0 þ Gm/0 þ iDx00 ¼ 0;
ð33Þ
kx000 þ Sx00 ¼ 0;
ð34Þ
ð3 þ 4FÞh000 þ 3SPrh00 3Q h0 ¼ 0;
ð35Þ
/000 þ SSc/00 aSc/0 ¼ 0;
ð36Þ
2
@/ @/ 1 @ / S ¼ a/; @t @z Sc @z2
ð20Þ
3508
K. Das / International Journal of Heat and Mass Transfer 54 (2011) 3505–3513
ð1 þ DÞV 001 þ SV 01 a2 V 1 þ Grh1 þ Gm/1 þ iDx01 ¼ 0;
ð37Þ
kx001 þ Sx01 inx1 ¼ 0;
ð38Þ
ð3 þ 4FÞh001 þ 3SPrh01 3ðQ þ inPrÞh1 ¼ 0;
ð39Þ
/001 þ SSc/01 ðin þ aÞSc/1 ¼ 0;
ð40Þ
ð1 þ DÞV 002 þ SV 02 a3 V 2 þ Grh2 þ Gm/2 þ iDx02 ¼ 0;
ð41Þ
kx002 þ Sx02 þ inx2 ¼ 0;
ð42Þ
ð3 þ 4FÞh002 þ 3SPrh02 3ðQ inPrÞh2 ¼ 0;
ð43Þ
/002
þ
SSc/02
A8 ¼ 1 A7 ;
þ ðin aÞSc/2 ¼ 0;
ð44Þ
where the primes denote differentiation w.r.t z and a1 ¼ iR þ M2 þ 1 ; a2 ¼ iðR þ nÞ þ M2 þ K1 and a3 ¼ iðR nÞ þ M2 þ K1 . K The corresponding boundary conditions can be written as
V 0 ¼ V 1 ¼ V 2 ¼ 1; h0 ¼ 1;
x0 ¼ 2i V 00 ; x1 ¼ 2i V 01 ; x2 ¼ 2i V 02 ;
h1 ¼ h2 ¼ 0;
/0 ¼ 1;
DSkfA1 ðm1 m3 Þ þ A2 ðm2 m3 Þ þ m3 g ; ð2 þ DÞS2 2kðS2 þ a1 kÞ þ DSm3 k Dm4 m5 ; A5 ¼ ð2 þ DÞm24 2ðSm4 þ a2 Þ þ Dm4 m5 Dm6 m7 A6 ¼ 1 A5 ; A7 ¼ ; ð2 þ DÞm26 2ðSm6 þ a3 Þ þ Dm6 m7
A4 ¼
/1 ¼ /2 ¼ 0 at z ¼ 0
B1 ¼
ð2 þ DÞS2 2kðS2 þ a1 kÞ þ DSm3 k im5 fð1 þ DÞm24 Sm4 a2 g ; B2 ¼ ð2 þ DÞm24 2ðSm4 þ a2 Þ þ Dm4 m5
B3 ¼
ð45Þ Cf ¼
x0 ¼ x1 ¼ x2 ¼ 0; h0 ¼ h1 ¼ h2 ¼ 0;
/0 ¼ /1 ¼ /2 ¼ 0 as z ! 1: ð46Þ Solving Eqs. (33)–(44) under the boundary conditions (45) and (46)we obtain the expression for translational velocity,microrotation,temperature and concentration as Sz
V ¼ A1 em1 z þ A2 em2 z þ A3 em3 z þ A4 e k e þ ðA5 em4 z þ A6 em5 z Þeint þ ðA7 em6 z þ A8 em7 z Þeint ; 2 Sz
x ¼ B1 e k þ
e 2
B2 eðintm4 zÞ þ B3 eðintþm6 zÞ ;
ð47Þ
ð49Þ
m1 z
ð50Þ
;
where
p
fðSScÞ2 þ 4aScg m1 ¼ ; 2 p 2 3SPr þ fð3SPrÞ þ 12Q ð3 þ 4FÞg ; m2 ¼ 2ð3 þ 4FÞ p 2 S þ fS þ 4a1 ð1 þ DÞg ; m3 ¼ 2ð1 þ DÞ p 2 S þ fS þ 4inkg ; m4 ¼ 2k p 2 S þ fS þ 4a2 ð1 þ DÞg m5 ¼ ; 2ð1 þ DÞ p 2 S þ fS 4inkg ; m6 ¼ 2k p 2 S þ fS þ 4a3 ð1 þ DÞg ; m7 ¼ 2ð1 þ DÞ Gm ; A1 ¼ ð1 þ DÞm21 Sm1 a1 Gr ; A2 ¼ ð1 þ DÞm22 Sm2 a1 SSc þ
A3 ¼ 1 A1 A2 A4 ;
im7 fð1 þ DÞm26 Sm6 a3 g : ð2 þ DÞm26 2ðSm6 þ a3 Þ þ Dm6 m7
i V 0 ð0Þ ¼ 1 þ D 1 þ 2
SA4 e A1 m1 þ A2 m2 þ A3 m3 þ þ ðA5 m4 þ A6 m5 Þeint : k 2 þðA7 m6 þ A8 m7 Þeint :
sw jz¼0 i ¼ 1þD 1þ 2 qU 2r
ð51Þ
The couple stress coefficient at the wall Cw is given by
@ x1 @ x2 þ i ¼ x0 ð0Þ @z z¼0 @z z¼0 SB1 e ¼ þ ðm4 B2 eint þ m6 B3 eint : k 2
Cw ¼
ð52Þ
The local Nusselt Nu is given by
ð48Þ
h ¼ em2 z ; /¼e
;
The physical quantities of engineering interest are skin-friction coefficient, couple stress coefficient, Nusselt number and Sherwood number. The local skin friction coefficient Cf is given by
and
V 0 ¼ V 1 ¼ V 2 ¼ 0;
ifA1 ðm3 m1 Þ þ A2 ðm3 m2 Þ m3 gfð1 þ DÞS2 kS2 a1 k2 g
x @T Nu ¼ @z z¼0 ¼ Rex h0 ð0Þ; Tw T1
ð53Þ
where Rex ¼ Umr x is the local Reynolds number. Thus
Nu ¼ h0 ð0Þ ¼ m2 : Rex
ð54Þ
The local Sherwood number Shx is given by
x @C Shx ¼ @z z¼0 ¼ Rex /0 ð0Þ: Cw C1
ð55Þ
Thus
Shx ¼ /0 ð0Þ ¼ m1 : Rex
ð56Þ
4. Numerical results and discussion The formulation of the effect of chemical reaction and thermal radiation on MHD free convection heat and mass transfer flow of an incompressible micropolar fluid along a semi-infinite vertical porous plate in a porous medium in presence of heat generation/ absorption and a rotating frame of reference has been performed in the preceding sections. This enables us to carry out the numerical calculations for distribution of the translational velocity, microrotation, temperature and concentration across the boundary layer for different values of the parameters. In the present study we have chosen n = 10, nt = p/2, Pr = 0.71, Sc = 0.16, Gm = 5,
K. Das / International Journal of Heat and Mass Transfer 54 (2011) 3505–3513
3509
Fig. 2. Velocity profiles for various values of D with a = 0.1, R = 0.2, F = 1, K = 0.5, M = 2, Q = 10.
Fig. 4. Velocity profiles for various values of R with a = 0.1, D = 0.2, F = 1, K = 0.5, M = 2, Q = 10.
Fig. 3. Microrotation profiles for various values of D with a = 0.1, R = 0.2, F = 1, K = 0.5, M = 2, Q = 10.
Fig. 5. Microrotation profiles for various values of R with a = 0.1, D = 0.2, F = 1, K = 0.5, M = 2, Q = 10.
Gr = 10, e = 0.01, S = 1 while D, a, R, F, K, M, Q are varied over a range,which are listed in the figures legends.
momentum boundary layer thickness. From Table 1 we show that as an increasing of the rotation parameter the skin friction coefficient and couple stress coefficient decreases but Nusselt number and Sherwood number remain unchanged.
4.1. Effect of viscosity ratio parameter D In Figs. 2 and 3 the effect of D on the translational velocity ,microrotation for a stationary porous plate are shown. It is observed that both V and x decreases with the increase in viscosity ratio parameter D. The magnitude of V and x are maximum near the boundary layer region. Table 1 depicts the effects of D on the skin friction coefficient Cf, couple stress coefficient Cw, Nusselt number Nu and Sherwood number Shx. It is seen that as D increases, the skin friction coefficient and couple stress coefficient decrease whereas Nusselt number and Sherwood number remain unchanged. 4.2. Effect of the rotational parameter R The translational velocity and microrotation profiles against y for different values of R are displayed in Figs. 4 and 5 respectively. It is observed that an increasing in R leads to decreasing in the values of translational velocity and microrotation and so decrease the
4.3. Effect of the thermal radiation parameter F Figs. 6 and 7 show the translational velocity and microrotation distribution across the boundary layer for different values of the thermal radiation parameter F. It is clear as F increases the translational velocity increases but the effect is reverse for microrotation distribution. Typical variations of the temperature proflies along y are shown in Fig. 16 for various values of the thermal radiation parameter F. The results show that as an increasing of the thermal radiation parameter the temperature profiles increases and hence, there would be an increase of thermal boundary layer thickness. Table 1 depicts the effects of F on the skin friction coefficient Cf, couple stress coefficient Cw,Nusselt number Nu and Sherwood number Shx. It is observed that the skin friction coefficient and couple stress coefficient both increases as F increases whereas Nusselt number Nu decreases as F increases.
3510
K. Das / International Journal of Heat and Mass Transfer 54 (2011) 3505–3513
Table 1 Effects of various parameters on Cf, Cw, Nu/Rex and Shx/Rex.
D
a
Q
R
F
K
M
Cf
Cw
Nu/Rex
Shx/Rex
0.2 0.4 0.8 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
0.2 0.2 0.2 0.4 0.6 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
2.5 2.5 2.5 2.5 2.5 4.0 6.0 4.0 6.0 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5
0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.5 0.8 0.2 0.2 0.2 0.2 0.2 0.2
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.2 1.0 0.5 0.5 0.5 0.5
5 5 5 5 5 5 5 5 5 5 5 5 5 2 10 5 5
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
12.0772 11.1418 10.6062 8.8206 6.551 8.1933 6.4255 2.2103 2.6688 5.7365 2.5634 8.7101 6.0090 14.2173 11.0526 10.5074 3.3836
10.5959 3.5879 0.8650 7.2560 4.4247 1.2939 0.1209 11.5164 13.1588 12.6590 9.9259 1.9060 16.6030 27.1554 7.4566 5.8739 61.6692
1.4561 1.4561 1.4561 1.4561 1.4561 1.7768 2.1223 0.2130 0.2130 1.4561 1.4561 1.7128 1.1984 1.4561 1.4561 1.4561 1.4561
0.2760 0.2760 0.2760 0.3453 0.40000 0.2760 0.2760 0.2760 0.2760 0.2760 0.2760 0.2760 0.2760 0.2760 0.2760 0.2760 0.2760
Fig. 6. Velocity profiles for various values of F with a = 0.1, R = 0.2, D = 0.2, K = 0.5, M = 2, Q = 10.
Fig. 7. Microrotation profiles for various values of F with a = 0.1, R = 0.2, D = 0.2, K = 0.5, M = 2, Q = 10.
Fig. 8. Velocity profiles for various values of Q with a = 0.1, R = 0.2, F = 1, K = 0.5, M = 2, D = 0.2.
Fig. 9. Microrotation profiles for various values of Q with a = 0.1, R = 0.2, F = 1, K = 0.5, M = 2, D = 0.2.
K. Das / International Journal of Heat and Mass Transfer 54 (2011) 3505–3513
3511
Fig. 10. Velocity profiles for various values of K with a = 0.1, R = 0.2, F = 1, D = 0.2, M = 2, Q = 10.
Fig. 12. Velocity profiles for various values of M with a = 0.1, R = 0.2, F = 1, K = 0.5, D = 0.2, Q = 10.
Fig. 11. Microrotation profiles for various values of K with a = 0.1, R = 0.2, F = 1, D = 0.2, M = 2, Q = 10.
Fig. 13. Microrotation profiles for various values of M with a = 0.1, R = 0.2, F = 1, K = 0.5, D = 0.2, Q = 10.
4.4. Effect of the heat generation parameter Q
in Figs. 10 and 11. It is obvious that the increased values of K tend to increasing of translational velocity and microrotation on the porous wall and so enhance the momentum boundary layer thikness. Table 1 shows the effect of K on the skin friction coefficient Cf, couple stress coefficient Cw, Nusselt number Nu and Sherwood number Shx. It is seen that both of the skin friction coefficient and couple stress coefficient decreases as K increases. But there are no effect of K on Nusselt number and Sherwood number.
The influence of the heat generation parameter Q on the translational velocity and microrotation distribution across the boundary layer are shown in Figs. 8 and 9. Fig. 8 shows the translational velocity decreases for Q > 0 but the effect is fluctuating nature near the plate and then approach to the boundary layer conditions Q < 0. It is also noted that from Fig. 9 that the microrotation distribution decraeses as Q increases.From Fig. 17, it is appear that the temperature profiles decreases as Q increases. That is, the thickness of the thermal boundary layer is reduce for higher values of the heat generation parameter Q. In Table 1, the effects of the heat generation parameter Q on the skin friction coefficient Cf, couple stress coefficient Cw,Nusselt number Nu and Sherwood number Shx. It is observed that the skin friction coefficient and couple stress coefficient both decreasers as Q increases for Q > 0 but the effect is reverse for Q < 0. Further one can observe that Nusselt number increases as Q(>0) increases but the effect is negligible for Q < 0.
4.6. Effect of magnetic field parameter M Figs. 12 and 13 show the pattern of the translational velocity and microrotation for different values of magnetic field parameter M respectively. Fig. 12 displays that adjacent to the surface of the plate , translational velocity increases as M increases but the effect is opposite far away from the plate. Further more microrotation decreases as M increases. Table 1 shows that the skin friction coefficient Cf decreases as M increases whereas couple stress coefficient Cw increases with increase in magnetic field parameter M.
4.5. Effect of the permeability parameter K
4.7. Effect of chemical reaction parameter a
For different values of the permeability parameter K, the translational velocity and microrotation on the porous wall are plotted
Figs. 14 and 15 illustrate the variation of the translational velocity and microrotation distribution across the boundary layer for
3512
K. Das / International Journal of Heat and Mass Transfer 54 (2011) 3505–3513
Fig. 14. Velocity profiles for various values of a with D = 0.2, R = 0.2, F = 1, K = 0.5, M = 2, Q = 10.
Fig. 17. Temperature profiles for various values of Q with Pr = 0.71, S = 1, F = 2.
Fig. 18. Concentration profiles for various values of a with Sc = 0.16, S = 1. Fig. 15. Microrotation profiles for various values of a with D = 0.2, R = 0.2, F = 1, K = 0.5, M = 2, Q = 10.
various values of the chemical reaction parameter a. It is seen from these figures that the effect of increasing values of the chemical reaction parameter a results in a decreasing translational velocity distribution across the boundary layer whereas microrotation distribution increases as a increases. Fig. 18 shows the variation of concentration profiles for different values of a. It is observed from this figure the concentration profiles decrease with increasing of a. Also it is found from Table 1 that as a increases local skin friction coefficient Cf and couple stress coefficient Cw decraese but an increasing in aleads to increasing in the values of Sherwood number Shx. From the table we also observed that the Nusselt number Nu remain unchanged.
5. Conclusions
Fig. 16. Temperature profiles for various values of F with Pr = 0.71, S = 1, Q = 4.
In this work, we have theoretically studied the effect of chemical reaction and thermal radiation on unsteady MHD free convection heat and mass transfer flow of an incompressible, micropolar fluid along a semi-infinite vertical porous moving plate embedded in a uniform porous medium with heat generation in presence of a rotating frame of reference. The governing equations are solved analytically by using perturbation technique. The results are discussed
K. Das / International Journal of Heat and Mass Transfer 54 (2011) 3505–3513
through graphs and table. The following conclusions can be made from the present investigation: (i) The translational velocity distribution across the boundary layer are decreased with an increasing values of D, R, Q(>0) and a while they show opposite trends with an increasing values of F and K. (ii) The magnitude of microrotation decreases with an increasing of D, R, Q, M and F. Hence the momentum boundary layer thicnness is reduced. But the effect are reverse for a and K. (iii) The temperature profile decreases with an increasing values of Q whereas the effect is opposite for F. Thus the thermal boundary layer thicnness increases for higher values of the thermal radiation parameter F (iv) For an increasing value of a, the concentration decreases slightly. (v) The analytical results obtained in this work are more generalised form of Bakr [27] and can be taken as a limiting case by taking F ? 0 and K ? 1. It is hoped that the results obtained will not only provide useful information for applications but also serve as a complement to the previous studies. Acknowledgement I gratefully acknowledge the referees for their constructive comments which improved the quality of the paper. References [1] A.C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1964) 1–18. [2] R. Bhargava, H.S. Takhar, Numerical study of heat transfer characteristics of the micropolar boundary layer near a stagnation point on a moving wall, Int. J. Eng. Sci. 38 (2000) 383–394. [3] M.M. Rahman, M.A. Sattar, MHD free convection and mass transfer flow with oscillatory plate velocity and constant heat source in a rotating frame of reference, Dhaka Univ. J. Sci. 47 (1999) 63–73. [4] M.M. Khonsari, D.E. Brewe, Effects of viscous dissipation on the lubrication characteristics of micropolar fluids, Acta Mech. 105 (1994) 57–68. [5] Y.J. Kim, J.C. Lee, Analytical studies on MHD oscillatory flow of a micropolar fluid over a vertical porous plate, Surf. Coat Technol. 171 (2003) 187–193. [6] R. Deka, U.N. Das, V.M. Soundalgekar, Effects of mass transfer flow past an impulsively started infinite vertical plate with constant heat flux and chemical reaction, Forsch. Ingenieurwes. 60 (1994) 284–287. [7] R. Muthucumaraswamy, P. Ganesan, Effect of the chemical reaction and injection on flow characteristics in an unsteady upward motion of an isothermal plate, J. Appl. Mech. Tech. Phys. 42 (2001) 665–671. [8] A.J. Chamkha, MHD flow of a uniformly stretched vertical permeable surface in the presence of heat generation/absorption and a chemical reaction, Int. Commun. Heat Mass Transfer 30 (2003) 413–422.
3513
[9] M.M. Rahman, M.A. Sattar, MHD convective flow of a micropolar fluid past a continuously moving vertical porous plate in the presence of heat generation/ absorption, ASME J. Heat Transfer 128 (2006) 142–152. [10] F.S. Ibrahim, A.M. Elaiw, A.A. Bakr, Effect of the chemical reaction and radiation absorption on the unsteady MHD free convection flow past a semi infinite vertical permeable moving plate with heat source and suction, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 1056–1066. [11] M.M. Rahman, M.A. Rahman, M.A. Samad, M.S. Alam, Heat transfer in micropolar fluid along a non-linear stretching sheet with temperature dependent viscosity and variable surface temperature, Int. J. Thermophys. 30 (2009) 1649–1670. [12] M.M. Rahman, M.J. Uddin, A. Aziz, Effects of variable electric conductivity and non-uniform heat source (or sink) on convective micropolar fluid flow along an inclined flat plate with surface heat flux, Int. J. Ther Sci. 48 (2009) 2331– 2340. [13] M.M. Rahman, A. Aziz, M. Al-Lawatia, Heat transfer in micropolar fluid along an inclined permeable plate with variable fluid properties, Int J. Ther Sci. 49 (2010) 993–1002. [14] R.A. Damseh, M.Q. AL-Odat, A.J. Chamkha, A. Benbella Shannak, Combined effect of heat generation or absorption and first order chemical reaction on micropolar fluid flows over a uniform stretched permeable surface, Int. J. Thermal Sci. 48 (2009) 1658–1663. [15] M.M. Rahman, M. Al-Lawatia, Effects of higher order chemical reaction on micropolar fluid flow on a power law permeable stretched sheet with variable concentration in a porous medium, Can. J. Chem. Eng. 88 (2010) 23–32. [16] A.C. Cogley, W.E. Vincenty, S.E. Gilles, Differential approximation for radiation in a non-gray gas near equilibrium, AIAA J. 6 (1968) 551–553. [17] MD. Abdus Sattar, Kalim MD. Hamid, Unsteady free-convection interaction with thermal radiation in a boundary layer flow past a vertical porous plate, J. Math. Phys. Sci. 30 (1996) 25–37. [18] K. Vajravelu, Flow and heat transfer in a saturated porous medium, ZAMM 74 (1994) 605–614. [19] R. Ganapathy, A note on oscillatory Couette flow in a rotating system, ASME J. Appl. Mech. 61 (1994) 208–209. [20] M.A. Hossain, H.S. Takhar, Radiation effect on mixed convection along a vertical plate with uniform surface temperature, Heat Mass Transfer 31 (1996) 243–248. [21] A. Raptis, Radiation and free convection flow through a porous medium, Int. Commun. Heat Mass Transfer 25 (1998) 289–295. [22] O.D. Makinde, Free convection flow with thermal radiation and mass transfer past a moving vertical porous plate, Int. Commun. Heat Mass Transfer 32 (2005) 1411–1419. [23] F.S. Ibrahim, A.M. Elaiw, A.A. Bakr, Influence of viscous dissipation and radiation on unsteady MHD mixed convection flow of micropolar fluids, Appl. Math. Inf. Sci. 2 (2008) 143–162. [24] M.M. Rahman, M.A. Sattar, Transient convective flow of micropolar fluid past a continuously moving vertical porous plate in the presence of radiation, Int. J.Appl. Mech. Eng. 12 (2007) 497–513. [25] F.S. Ibrahim, A.M. Elaiw, A.A. Bakr, Effect of the chemical reaction and radiation absorption on unsteady MHD mixed convection flow past a semi-infinite vertical permeable moving plate with heat source and suction, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 1056–1066. [26] M.M. Rahman, Convective flows of micropolar fluids from radiate isothermal porous surfaces with viscous dissipation and Joule heating, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 3018–3030. [27] A.A. Bakr, Effects of chemical reaction on MHD free convection and mass transfer flow of a micropolar fluid with oscillatory plate velocity and constant heat source in a rotating frame of reference, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 698–710.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Effect of chemical reaction and thermal radiation on heat and mass transfer flow of MHD micropolar fluid in a rotating frame of reference K. Das Department of Mathematics, Kalyani Government Engineering College, Kalyani, Nadia, West Bengal 74125, India
a r t i c l e
i n f o
Article history: Received 14 October 2010 Received in revised form 28 January 2011 Accepted 24 February 2011 Available online 12 April 2011 Keywords: Micropolar fluid Chemical reaction Thermal radiation Rotating frame Permeability parameter
a b s t r a c t This paper studies the effect of first order chemical reaction and thermal radiation on hydromagnetic free convection heat and mass transfer flow of a micropolar fluid via a porous medium bounded by a semiinfinite porous plate with constant heat source in a rotating frame of reference. The plate is assumed to oscillate in time with constant frequency so that the solutions of the boundary layer are the same oscillatory type. The dimensionless governing equations for this investigation are solved analytically using small perturbation approximation. The effect of the various dimensionless parameters entering into the problem on the velocity, temperature and concentration profiles across the boundary layer are investigated through graphs. Also the results of the skin friction coefficient, couple stress coefficient, the rate of heat and mass transfer at the wall are prepared with various values of the parameters. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The analysis of the dynamics of micropolar fluid has been the subject of many research papers in recent years because of its increasing importance in processing industries and elsewhere of materials whose behaviour in shear cannot be characterized by Newtonian relationships. The micropolar fluids which contain micro-constituents and can undergo rotation has been proposed by Eringen [1]. These kind of fluids are utilized in analyzing exotic lubricants, the flow of colloidal suspensions, paints, liquid crystals, animal blood, fluid flowing in brain, turbulent shear flows and body fluids both mathematically and industrially. In addition due to its practical application to boundary layer control and thermal protection in high energy flow by means of wall velocity and mass transfer, considerable attention has been paid to the thermal boundary layer flows over moving boundaries [2]. The oscillatory boundary layer flow with constant heat source in the case of MHD free convection currents and mass transfer has been considered by Rahman and Sattar [3]. Khonsari and Brewe [4] have examined the effect of viscous dissipation on the lubrication characteristics of micropolar fluids. The effect of non zero values of micro gyration vector on semi-infinite moving porous plate with constant velocity when the magnetic field is imposed transversely to the plate and the temperature of the plate is oscillating with time have been analyzed by Kim and Lee [5].
E-mail address: [email protected] 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.03.035
Combined heat and mass transfer problems with chemical reaction are of importance in many processes and have, therefore, received a considerable amount of attention in recent years. Possible applications of this type flow can be found in many industries, like design of chemical processing equipment, damage of crops due to freezing, food processing and cooling towers. Deka et al. [6] investigated the effect of first order homogeneous chemical reaction on the process of an unsteady flow past an infinite vertical plate with a constant heat and mass transfer. Muthucumaraswamy and Ganesan [7] discussed the effect of the chemical reaction and injection on flow characteristics in an unsteady upward motion of an isothermal plate. MHD flow of a uniformly stretched vertical permeable surface in the presence of heat generation/absorption and a chemical reaction has been considered by Chamkha [8]. Rahman and Satter [9] examined MHD convective flow of micropolar fluid past a continuously moving vertical porous plate in the presence of heat generation/absorption. Ibrahim et al. [10] obtained the analytical solution for unsteady MHD free convection flow past a semi-infinite vertical permeable moving plate with heat source and chemical reaction. Rahman et al. [11] studied heat transfer in micropolar fluid with temperature dependent fluid properties along a non-stretching sheet. Rahman et al. [12,13] considered heat transfer in micropolar fluid along an inclined plate with variable fluid properties under different boundary conditions. Damseh et al. [14] have investigated heat and mass transfer free convection flow adjacent to a continuous moving vertical porous plate for incompressible micropolar fluid in the presence of heat generation/absorption and a first order chemical reaction. Rahman
3506
K. Das / International Journal of Heat and Mass Transfer 54 (2011) 3505–3513
and Al-Lawatia [15] developed the problem by considering higher order chemical reaction. In all the previous investigation, the effect of thermal radiation on the flow and heat transfer have not been provided.The effect of radiation on MHD flow and heat transfer problem have become more important industrially. At high operating temperature, radiation effect can be quite significant. Many process in engineering areas occur at high temperature and a knowledge of radiation heat transfer becomes very important for design of reliable equipment, nuclear plants,gas turbines and various propulsion devices or aircraft, missiles, satelites and space vehicles. Based on these applications, Cogley et al. [16] showed that in the optically thin limit, the fluid does not absorb its own emitted radiation but the fluid does absorb radiation emitted by the boundaries. Satter and Hamid [17] investigated the unsteady free convection interaction with thermal radiation in a boundary layer flow past a vertical porous plate. Vajravelu [18] studied the flow of a steady viscous fluid and heat transfer characteristic in a porous medium by considering different heating processes. Hossain and Takhar [20] have considered the radiation effect on mixed convection boundary layer flow of an optically dense viscous incompressible fluid along a vertical plate with uniform surface temperature. Raptis [21]investigate the steady flow of a viscous fluid through a porous medium bounded by a porous plate subjected to a constant suction velocity by the presence of thermal radiation. Makinde [22] examined the transient free convection interaction with thermal radiation of an absorbing emitting fluid along moving vertical permeable plate. Ibrahim et al. [23] discussed the case of mixed convection flow of a micropolar fluid past a semi infinite steady moving porous plate with varying suction velocity normal to the plate in presence of thermal radiation and viscous dissipation. Rahman and Satter [24] studied transient convective flow of micropolar fluid past a continuous moving porous plate in the presence of radiation. The effect of the chemical reaction and radiation absorption on the unsteady MHD free convection flow past a semi-infinite vertical permeable moving plate with heat source and suction has been studied by Ibrahim et al. [25]. Rahman and Sultana [26] examined radiative heat flux with variable heat flux in a porous medium. Recently Bakr [27] presented an analysis on MHD free convection and mass transfer adjacent to moving vertical plate for micropolar fluid in a rotating frame of reference in presence of heat generation/absorption and a chemical reaction. The main purpose of the present investigation is to study the effects of thermal radiation and chemical reaction on unsteady MHD free convection heat and mass transfer flow of a micropolar fluid past a vertical porous plate in a rotating frame of reference.It is assumed that the plate is embedded in a uniform porous medium and oscillates in time with a constant frequency in the presence of a transverse magnetic field. The governing equations are solved analytically using perturbation technique. Numerical results are reported for various values of the physical parameters of interest. The paper has been arranged as follows. The Section 2 deals with the mathematical formulation of the problems. Section 3 contains the closed form solutions of velocity, temperature concentration etc. Numerical results and discussion are presented in Section 4. The conclusions have been summarized in Section 5.
Fig. 1. Physical model and coordinate system of the problem.
that there is no applied voltage which implies the absence of an electric field. The flow is assumed to be in the x-direction which is taken along the plate in the upward direction and z-axis is normal to it. Also it is assumed that the whole system is rotate with a constant frame X in a micropolar fluid about z-axis.The fluid is asumed to be gray, absorbing-emitting but not scattering medium. The radiation heat flux in x-direction is considered negligible in comparison that the z-direction. Due to semi-infinite plate surface assumption, furthermore, the flow variables are functions of z and time t only. Under these assumption, the equations that describe the physical situation are given by
@w ¼ 0; @z @u @u @2u þw 2Xv ¼ ðm þ mr Þ 2 þ gbT ðT T 1 Þ þ gbC ðC C 1 Þ @t @z @z 2 mu rB20 u @x mr ; k q @z
ð1Þ
ð2Þ
1 @v @v @ 2 v mv rB20 v @x þw þ 2Xu ¼ ðm þ mr Þ 2 ; þ mr @t @z @z k q @z
ð3Þ
1 1 1 K @2x @x @x ; þw ¼ @t @z qj @z2
ð4Þ
2 2 2 K @2x @x @x ; þw ¼ @t @z qj @z2
ð5Þ
@T @T j @2T Q 1 @qr ðT T 1 Þ þw ¼ ; @t @z qC p @z2 qC p qC p @z
ð6Þ
ð7Þ
2. Mathematical formulation of the problem
@C @C @2C þw ¼ Dm 2 Rr ðC C 1 Þ @t @z @z
Consider unsteady three dimensional flow of an electrically conducting incompressible micropolar fluid past a semi-infinite vertical permeable moving plate embeded in a uniform porous medium and subjected to a constant transverse magnetic field B0 (see Fig. 1 in the presence of thermal and concentration buoyancy effects with chemical reaction and thermal radiation. It is assumed
where u, v and w are velocity components along x, y and z-axis 1 and x 2 are micro-rotation components along x respectively, x and y-axis respectively. bT and bC are the coefficients of thermal expansion and concentration expansion, q is the density of the fluid, m is the kinematic viscosity, mr is the kinematic micro-rotation viscosity, Cp is the specific heat at constant pressure p, K is the spin gradient velocity, j is the micro-inertia density, g is the acceleration
3507
K. Das / International Journal of Heat and Mass Transfer 54 (2011) 3505–3513 B0 p rm
due to gravity, k is the permeability of porous medium, T is the temperature of the fluid in the boundary layer and T1 is the temperature of the fluid far away from the plate, C is the concentration of the solute and C1 is the concentration of the solute far away from the plate, Q is the additional heat source, Dm is the molecular diffusivity and Rr is the chemical reaction rate constant. The boundary conditions are given by 1 ¼x 2 ¼ 0; T ¼ T 1 ; C ¼ C 1 for t 6 0; u ¼ v ¼ 0; x ð8Þ 9 e int 1 ¼ 12 @@zv ; x 2 ¼ 12 @u ; u ¼ U r 1 þ 2 ðe þ eint Þ ; v ¼ 0; x > @z > > = T ¼ T w ; C ¼ C w at z ¼ 0; for t > 0; ð9Þ > and > > ; 1 ¼x 2 ¼ 0; T ¼ T 1 ; C ¼ C 1 as z ! 1; u ¼ v ¼ 0; x
where R ¼ 2UX2m is the rotational parameter, M ¼ Ur q is the magr lqC netic field parameter, Pr ¼ j p ; Sc ¼ Dmm ; Gr ¼ mgbT ðTUw3 T 1 Þ ; Gm ¼ r mgbC ðC w C 1 Þ are the Prandtl number, Schmidt number, Grashof numU 3r 4T 31 r ber, modified Grashof number respectively, F ¼ jk is the heat 2 r radiation parameter, S ¼ wU0r is the suction parameter, K ¼ kU m2 is the Rr m permeability of the porous medium, a ¼ Ur is the chemical reaction parameter, k ¼ l^j is the dimensionless material parameter, D ¼ mmr is the viscosity ratio. Also the boundary conditions become
where Ur is the uniform reference velocity and e is the small constant quantity. The oscillatory plate velocity assumed in Eq. (9) is based on the suggestion proposed by Ganapathy [19]. The continuity Eq. (1) gives
and
w ¼ w0 ;
ð10Þ
where the w0 represents the normal velocity at the plate which is positive for suction and negative for blowing. The radiative heat flux term by using the Rosseland approximation is given by
qr ¼
4r @T 4 ; 3k @z
ð11Þ
where r⁄ is the Stefan–Boltzmann constant and k⁄ is the mean absorption coefficient. Assuming that the differences in temperature within the flow are such that T4 can be expressed as a linear combination of the temperature, we expand T4 in Taylor’s series about T1 and neglecting higher order terms, we get
T 4 ¼ 4T 31 T 3T 41 :
ð12Þ
Thus we have
1 ¼x 2 ¼ 0; h ¼ 0; / ¼ 0 as z ! 1 u ¼ v ¼ 0; x
ð13Þ
Let us introduce the following dimensionless variables:
v 0 ¼ Uv ; r
z0 ¼ zUm r ;
2
t 0 ¼ tUmr ;
n0 ¼ Unm2 ; r
V ¼ u þ iv ;
2
r
r
: ;
for t > 0: ð22Þ
1 þ ix 2 x¼x
and get
@V @V @2V @x S þ iRV ¼ ð1 þ DÞ 2 N2 V þ Grh þ Gm/ þ iD ; ð23Þ @t @z @z @z 2 @x @x @ x ð24Þ S ¼k 2 ; @t @z @z 2 @h @h 1 4F @ h Q S ¼ 1þ ð25Þ h; @t @z Pr 3 @z2 Pr @/ @/ 1 @ 2 / S ¼ a/: @t @z Sc @z2
ð26Þ
The associated boundary conditions (21) and (22) are written as follows:
h ¼ 0;
V ¼ 1 þ 2e ðeint þ eint Þ; and V ¼ 0; x ¼ 0; h ¼ 0;
/ ¼ 0 for t 6 0;
v ¼ 0; x ¼ 2i @V ; @z
h ¼ 1;
ð27Þ 9 / ¼ 1 at z ¼ 0; > =
/ ¼ 0 as z ! 1:
9
01 ¼ x 12m ; = x U
1 1 02 ¼ x 22m ; h ¼ TTT x ; / ¼ CCC ; Q 0 ¼ UQ2mj w T 1 w C 1 U
> > > ;
ð21Þ
We now simplify Eqs. (15)–(20) by putting the fluid velocity and angular velocity in the complex form as
V ¼ 0; x ¼ 0;
@qr 16T 31 r @ 2 T : ¼ @z2 @z 3k
u0 ¼ Uur ;
1 ¼x 2 ¼ 0; h ¼ 0; / ¼ 0 for t 6 0; u ¼ v ¼ 0; x 9 e int int 1 ¼ 12 @@zv ; x 2 ¼ 12 @u u ¼ 1 þ 2 ðe þ e Þ; v ¼ 0; x ;> @z > > = h ¼ 1; / ¼ 1 at z ¼ 0
> ;
for t > 0:
ð28Þ 3. Analytical solutions
r
ð14Þ Then substituting Eq. (14) into Eqs. (1)–(7)yields the followingdimensionless equations (dropping primes):
@u @u @2u 1 S Rv ¼ ð1 þ DÞ 2 þ Grh þ Gm/ M 2 þ u @t @z @z K 2 @x D ; @z
To find the analytical solutions of the above system of partial differential Eqs. (23)–(26) in the neighbourhood of the plate under the above boundary conditions (27) and (28,we express V, x, h and / as [19]
Vðz; tÞ ¼ V 0 þ ð15Þ
ð16Þ
1 1 1 @x @x @2x ; S ¼k @t @z @z2
ð17Þ
2 2 2 @x @x @2x ; S ¼k @t @z @z2
ð18Þ
@h @h 1 4F @ 2 h Q S ¼ 1þ h; @t @z Pr 3 @z2 Pr
ð19Þ
eint V 1 ðzÞ þ eint V 2 ðzÞ ;
e xðz; tÞ ¼ x0 þ eint x1 ðzÞ þ eint x2 ðzÞ ; 2 e int int hðz; tÞ ¼ h0 þ
@v @v @2v 1 @x v þD 1; S þ Ru ¼ ð1 þ DÞ 2 M2 þ K @t @z @z @z
e 2
2
e h1 ðzÞ þ e
h2 ðzÞ ;
e /ðz; tÞ ¼ /0 þ eint /1 ðzÞ þ eint /2 ðzÞ : 2
ð29Þ ð30Þ ð31Þ ð32Þ
Invoking the above Eqs. (29)–(32) into the Eqs. (23)–(28) and equating the harmonic and non-harmonic terms and neglecting the higher order terms of o(e2), we obtain the following set of equations:
ð1 þ DÞV 000 þ SV 00 a1 V 0 þ Grh0 þ Gm/0 þ iDx00 ¼ 0;
ð33Þ
kx000 þ Sx00 ¼ 0;
ð34Þ
ð3 þ 4FÞh000 þ 3SPrh00 3Q h0 ¼ 0;
ð35Þ
/000 þ SSc/00 aSc/0 ¼ 0;
ð36Þ
2
@/ @/ 1 @ / S ¼ a/; @t @z Sc @z2
ð20Þ
3508
K. Das / International Journal of Heat and Mass Transfer 54 (2011) 3505–3513
ð1 þ DÞV 001 þ SV 01 a2 V 1 þ Grh1 þ Gm/1 þ iDx01 ¼ 0;
ð37Þ
kx001 þ Sx01 inx1 ¼ 0;
ð38Þ
ð3 þ 4FÞh001 þ 3SPrh01 3ðQ þ inPrÞh1 ¼ 0;
ð39Þ
/001 þ SSc/01 ðin þ aÞSc/1 ¼ 0;
ð40Þ
ð1 þ DÞV 002 þ SV 02 a3 V 2 þ Grh2 þ Gm/2 þ iDx02 ¼ 0;
ð41Þ
kx002 þ Sx02 þ inx2 ¼ 0;
ð42Þ
ð3 þ 4FÞh002 þ 3SPrh02 3ðQ inPrÞh2 ¼ 0;
ð43Þ
/002
þ
SSc/02
A8 ¼ 1 A7 ;
þ ðin aÞSc/2 ¼ 0;
ð44Þ
where the primes denote differentiation w.r.t z and a1 ¼ iR þ M2 þ 1 ; a2 ¼ iðR þ nÞ þ M2 þ K1 and a3 ¼ iðR nÞ þ M2 þ K1 . K The corresponding boundary conditions can be written as
V 0 ¼ V 1 ¼ V 2 ¼ 1; h0 ¼ 1;
x0 ¼ 2i V 00 ; x1 ¼ 2i V 01 ; x2 ¼ 2i V 02 ;
h1 ¼ h2 ¼ 0;
/0 ¼ 1;
DSkfA1 ðm1 m3 Þ þ A2 ðm2 m3 Þ þ m3 g ; ð2 þ DÞS2 2kðS2 þ a1 kÞ þ DSm3 k Dm4 m5 ; A5 ¼ ð2 þ DÞm24 2ðSm4 þ a2 Þ þ Dm4 m5 Dm6 m7 A6 ¼ 1 A5 ; A7 ¼ ; ð2 þ DÞm26 2ðSm6 þ a3 Þ þ Dm6 m7
A4 ¼
/1 ¼ /2 ¼ 0 at z ¼ 0
B1 ¼
ð2 þ DÞS2 2kðS2 þ a1 kÞ þ DSm3 k im5 fð1 þ DÞm24 Sm4 a2 g ; B2 ¼ ð2 þ DÞm24 2ðSm4 þ a2 Þ þ Dm4 m5
B3 ¼
ð45Þ Cf ¼
x0 ¼ x1 ¼ x2 ¼ 0; h0 ¼ h1 ¼ h2 ¼ 0;
/0 ¼ /1 ¼ /2 ¼ 0 as z ! 1: ð46Þ Solving Eqs. (33)–(44) under the boundary conditions (45) and (46)we obtain the expression for translational velocity,microrotation,temperature and concentration as Sz
V ¼ A1 em1 z þ A2 em2 z þ A3 em3 z þ A4 e k e þ ðA5 em4 z þ A6 em5 z Þeint þ ðA7 em6 z þ A8 em7 z Þeint ; 2 Sz
x ¼ B1 e k þ
e 2
B2 eðintm4 zÞ þ B3 eðintþm6 zÞ ;
ð47Þ
ð49Þ
m1 z
ð50Þ
;
where
p
fðSScÞ2 þ 4aScg m1 ¼ ; 2 p 2 3SPr þ fð3SPrÞ þ 12Q ð3 þ 4FÞg ; m2 ¼ 2ð3 þ 4FÞ p 2 S þ fS þ 4a1 ð1 þ DÞg ; m3 ¼ 2ð1 þ DÞ p 2 S þ fS þ 4inkg ; m4 ¼ 2k p 2 S þ fS þ 4a2 ð1 þ DÞg m5 ¼ ; 2ð1 þ DÞ p 2 S þ fS 4inkg ; m6 ¼ 2k p 2 S þ fS þ 4a3 ð1 þ DÞg ; m7 ¼ 2ð1 þ DÞ Gm ; A1 ¼ ð1 þ DÞm21 Sm1 a1 Gr ; A2 ¼ ð1 þ DÞm22 Sm2 a1 SSc þ
A3 ¼ 1 A1 A2 A4 ;
im7 fð1 þ DÞm26 Sm6 a3 g : ð2 þ DÞm26 2ðSm6 þ a3 Þ þ Dm6 m7
i V 0 ð0Þ ¼ 1 þ D 1 þ 2
SA4 e A1 m1 þ A2 m2 þ A3 m3 þ þ ðA5 m4 þ A6 m5 Þeint : k 2 þðA7 m6 þ A8 m7 Þeint :
sw jz¼0 i ¼ 1þD 1þ 2 qU 2r
ð51Þ
The couple stress coefficient at the wall Cw is given by
@ x1 @ x2 þ i ¼ x0 ð0Þ @z z¼0 @z z¼0 SB1 e ¼ þ ðm4 B2 eint þ m6 B3 eint : k 2
Cw ¼
ð52Þ
The local Nusselt Nu is given by
ð48Þ
h ¼ em2 z ; /¼e
;
The physical quantities of engineering interest are skin-friction coefficient, couple stress coefficient, Nusselt number and Sherwood number. The local skin friction coefficient Cf is given by
and
V 0 ¼ V 1 ¼ V 2 ¼ 0;
ifA1 ðm3 m1 Þ þ A2 ðm3 m2 Þ m3 gfð1 þ DÞS2 kS2 a1 k2 g
x @T Nu ¼ @z z¼0 ¼ Rex h0 ð0Þ; Tw T1
ð53Þ
where Rex ¼ Umr x is the local Reynolds number. Thus
Nu ¼ h0 ð0Þ ¼ m2 : Rex
ð54Þ
The local Sherwood number Shx is given by
x @C Shx ¼ @z z¼0 ¼ Rex /0 ð0Þ: Cw C1
ð55Þ
Thus
Shx ¼ /0 ð0Þ ¼ m1 : Rex
ð56Þ
4. Numerical results and discussion The formulation of the effect of chemical reaction and thermal radiation on MHD free convection heat and mass transfer flow of an incompressible micropolar fluid along a semi-infinite vertical porous plate in a porous medium in presence of heat generation/ absorption and a rotating frame of reference has been performed in the preceding sections. This enables us to carry out the numerical calculations for distribution of the translational velocity, microrotation, temperature and concentration across the boundary layer for different values of the parameters. In the present study we have chosen n = 10, nt = p/2, Pr = 0.71, Sc = 0.16, Gm = 5,
K. Das / International Journal of Heat and Mass Transfer 54 (2011) 3505–3513
3509
Fig. 2. Velocity profiles for various values of D with a = 0.1, R = 0.2, F = 1, K = 0.5, M = 2, Q = 10.
Fig. 4. Velocity profiles for various values of R with a = 0.1, D = 0.2, F = 1, K = 0.5, M = 2, Q = 10.
Fig. 3. Microrotation profiles for various values of D with a = 0.1, R = 0.2, F = 1, K = 0.5, M = 2, Q = 10.
Fig. 5. Microrotation profiles for various values of R with a = 0.1, D = 0.2, F = 1, K = 0.5, M = 2, Q = 10.
Gr = 10, e = 0.01, S = 1 while D, a, R, F, K, M, Q are varied over a range,which are listed in the figures legends.
momentum boundary layer thickness. From Table 1 we show that as an increasing of the rotation parameter the skin friction coefficient and couple stress coefficient decreases but Nusselt number and Sherwood number remain unchanged.
4.1. Effect of viscosity ratio parameter D In Figs. 2 and 3 the effect of D on the translational velocity ,microrotation for a stationary porous plate are shown. It is observed that both V and x decreases with the increase in viscosity ratio parameter D. The magnitude of V and x are maximum near the boundary layer region. Table 1 depicts the effects of D on the skin friction coefficient Cf, couple stress coefficient Cw, Nusselt number Nu and Sherwood number Shx. It is seen that as D increases, the skin friction coefficient and couple stress coefficient decrease whereas Nusselt number and Sherwood number remain unchanged. 4.2. Effect of the rotational parameter R The translational velocity and microrotation profiles against y for different values of R are displayed in Figs. 4 and 5 respectively. It is observed that an increasing in R leads to decreasing in the values of translational velocity and microrotation and so decrease the
4.3. Effect of the thermal radiation parameter F Figs. 6 and 7 show the translational velocity and microrotation distribution across the boundary layer for different values of the thermal radiation parameter F. It is clear as F increases the translational velocity increases but the effect is reverse for microrotation distribution. Typical variations of the temperature proflies along y are shown in Fig. 16 for various values of the thermal radiation parameter F. The results show that as an increasing of the thermal radiation parameter the temperature profiles increases and hence, there would be an increase of thermal boundary layer thickness. Table 1 depicts the effects of F on the skin friction coefficient Cf, couple stress coefficient Cw,Nusselt number Nu and Sherwood number Shx. It is observed that the skin friction coefficient and couple stress coefficient both increases as F increases whereas Nusselt number Nu decreases as F increases.
3510
K. Das / International Journal of Heat and Mass Transfer 54 (2011) 3505–3513
Table 1 Effects of various parameters on Cf, Cw, Nu/Rex and Shx/Rex.
D
a
Q
R
F
K
M
Cf
Cw
Nu/Rex
Shx/Rex
0.2 0.4 0.8 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
0.2 0.2 0.2 0.4 0.6 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
2.5 2.5 2.5 2.5 2.5 4.0 6.0 4.0 6.0 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5
0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.5 0.8 0.2 0.2 0.2 0.2 0.2 0.2
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.2 1.0 0.5 0.5 0.5 0.5
5 5 5 5 5 5 5 5 5 5 5 5 5 2 10 5 5
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
12.0772 11.1418 10.6062 8.8206 6.551 8.1933 6.4255 2.2103 2.6688 5.7365 2.5634 8.7101 6.0090 14.2173 11.0526 10.5074 3.3836
10.5959 3.5879 0.8650 7.2560 4.4247 1.2939 0.1209 11.5164 13.1588 12.6590 9.9259 1.9060 16.6030 27.1554 7.4566 5.8739 61.6692
1.4561 1.4561 1.4561 1.4561 1.4561 1.7768 2.1223 0.2130 0.2130 1.4561 1.4561 1.7128 1.1984 1.4561 1.4561 1.4561 1.4561
0.2760 0.2760 0.2760 0.3453 0.40000 0.2760 0.2760 0.2760 0.2760 0.2760 0.2760 0.2760 0.2760 0.2760 0.2760 0.2760 0.2760
Fig. 6. Velocity profiles for various values of F with a = 0.1, R = 0.2, D = 0.2, K = 0.5, M = 2, Q = 10.
Fig. 7. Microrotation profiles for various values of F with a = 0.1, R = 0.2, D = 0.2, K = 0.5, M = 2, Q = 10.
Fig. 8. Velocity profiles for various values of Q with a = 0.1, R = 0.2, F = 1, K = 0.5, M = 2, D = 0.2.
Fig. 9. Microrotation profiles for various values of Q with a = 0.1, R = 0.2, F = 1, K = 0.5, M = 2, D = 0.2.
K. Das / International Journal of Heat and Mass Transfer 54 (2011) 3505–3513
3511
Fig. 10. Velocity profiles for various values of K with a = 0.1, R = 0.2, F = 1, D = 0.2, M = 2, Q = 10.
Fig. 12. Velocity profiles for various values of M with a = 0.1, R = 0.2, F = 1, K = 0.5, D = 0.2, Q = 10.
Fig. 11. Microrotation profiles for various values of K with a = 0.1, R = 0.2, F = 1, D = 0.2, M = 2, Q = 10.
Fig. 13. Microrotation profiles for various values of M with a = 0.1, R = 0.2, F = 1, K = 0.5, D = 0.2, Q = 10.
4.4. Effect of the heat generation parameter Q
in Figs. 10 and 11. It is obvious that the increased values of K tend to increasing of translational velocity and microrotation on the porous wall and so enhance the momentum boundary layer thikness. Table 1 shows the effect of K on the skin friction coefficient Cf, couple stress coefficient Cw, Nusselt number Nu and Sherwood number Shx. It is seen that both of the skin friction coefficient and couple stress coefficient decreases as K increases. But there are no effect of K on Nusselt number and Sherwood number.
The influence of the heat generation parameter Q on the translational velocity and microrotation distribution across the boundary layer are shown in Figs. 8 and 9. Fig. 8 shows the translational velocity decreases for Q > 0 but the effect is fluctuating nature near the plate and then approach to the boundary layer conditions Q < 0. It is also noted that from Fig. 9 that the microrotation distribution decraeses as Q increases.From Fig. 17, it is appear that the temperature profiles decreases as Q increases. That is, the thickness of the thermal boundary layer is reduce for higher values of the heat generation parameter Q. In Table 1, the effects of the heat generation parameter Q on the skin friction coefficient Cf, couple stress coefficient Cw,Nusselt number Nu and Sherwood number Shx. It is observed that the skin friction coefficient and couple stress coefficient both decreasers as Q increases for Q > 0 but the effect is reverse for Q < 0. Further one can observe that Nusselt number increases as Q(>0) increases but the effect is negligible for Q < 0.
4.6. Effect of magnetic field parameter M Figs. 12 and 13 show the pattern of the translational velocity and microrotation for different values of magnetic field parameter M respectively. Fig. 12 displays that adjacent to the surface of the plate , translational velocity increases as M increases but the effect is opposite far away from the plate. Further more microrotation decreases as M increases. Table 1 shows that the skin friction coefficient Cf decreases as M increases whereas couple stress coefficient Cw increases with increase in magnetic field parameter M.
4.5. Effect of the permeability parameter K
4.7. Effect of chemical reaction parameter a
For different values of the permeability parameter K, the translational velocity and microrotation on the porous wall are plotted
Figs. 14 and 15 illustrate the variation of the translational velocity and microrotation distribution across the boundary layer for
3512
K. Das / International Journal of Heat and Mass Transfer 54 (2011) 3505–3513
Fig. 14. Velocity profiles for various values of a with D = 0.2, R = 0.2, F = 1, K = 0.5, M = 2, Q = 10.
Fig. 17. Temperature profiles for various values of Q with Pr = 0.71, S = 1, F = 2.
Fig. 18. Concentration profiles for various values of a with Sc = 0.16, S = 1. Fig. 15. Microrotation profiles for various values of a with D = 0.2, R = 0.2, F = 1, K = 0.5, M = 2, Q = 10.
various values of the chemical reaction parameter a. It is seen from these figures that the effect of increasing values of the chemical reaction parameter a results in a decreasing translational velocity distribution across the boundary layer whereas microrotation distribution increases as a increases. Fig. 18 shows the variation of concentration profiles for different values of a. It is observed from this figure the concentration profiles decrease with increasing of a. Also it is found from Table 1 that as a increases local skin friction coefficient Cf and couple stress coefficient Cw decraese but an increasing in aleads to increasing in the values of Sherwood number Shx. From the table we also observed that the Nusselt number Nu remain unchanged.
5. Conclusions
Fig. 16. Temperature profiles for various values of F with Pr = 0.71, S = 1, Q = 4.
In this work, we have theoretically studied the effect of chemical reaction and thermal radiation on unsteady MHD free convection heat and mass transfer flow of an incompressible, micropolar fluid along a semi-infinite vertical porous moving plate embedded in a uniform porous medium with heat generation in presence of a rotating frame of reference. The governing equations are solved analytically by using perturbation technique. The results are discussed
K. Das / International Journal of Heat and Mass Transfer 54 (2011) 3505–3513
through graphs and table. The following conclusions can be made from the present investigation: (i) The translational velocity distribution across the boundary layer are decreased with an increasing values of D, R, Q(>0) and a while they show opposite trends with an increasing values of F and K. (ii) The magnitude of microrotation decreases with an increasing of D, R, Q, M and F. Hence the momentum boundary layer thicnness is reduced. But the effect are reverse for a and K. (iii) The temperature profile decreases with an increasing values of Q whereas the effect is opposite for F. Thus the thermal boundary layer thicnness increases for higher values of the thermal radiation parameter F (iv) For an increasing value of a, the concentration decreases slightly. (v) The analytical results obtained in this work are more generalised form of Bakr [27] and can be taken as a limiting case by taking F ? 0 and K ? 1. It is hoped that the results obtained will not only provide useful information for applications but also serve as a complement to the previous studies. Acknowledgement I gratefully acknowledge the referees for their constructive comments which improved the quality of the paper. References [1] A.C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1964) 1–18. [2] R. Bhargava, H.S. Takhar, Numerical study of heat transfer characteristics of the micropolar boundary layer near a stagnation point on a moving wall, Int. J. Eng. Sci. 38 (2000) 383–394. [3] M.M. Rahman, M.A. Sattar, MHD free convection and mass transfer flow with oscillatory plate velocity and constant heat source in a rotating frame of reference, Dhaka Univ. J. Sci. 47 (1999) 63–73. [4] M.M. Khonsari, D.E. Brewe, Effects of viscous dissipation on the lubrication characteristics of micropolar fluids, Acta Mech. 105 (1994) 57–68. [5] Y.J. Kim, J.C. Lee, Analytical studies on MHD oscillatory flow of a micropolar fluid over a vertical porous plate, Surf. Coat Technol. 171 (2003) 187–193. [6] R. Deka, U.N. Das, V.M. Soundalgekar, Effects of mass transfer flow past an impulsively started infinite vertical plate with constant heat flux and chemical reaction, Forsch. Ingenieurwes. 60 (1994) 284–287. [7] R. Muthucumaraswamy, P. Ganesan, Effect of the chemical reaction and injection on flow characteristics in an unsteady upward motion of an isothermal plate, J. Appl. Mech. Tech. Phys. 42 (2001) 665–671. [8] A.J. Chamkha, MHD flow of a uniformly stretched vertical permeable surface in the presence of heat generation/absorption and a chemical reaction, Int. Commun. Heat Mass Transfer 30 (2003) 413–422.
3513
[9] M.M. Rahman, M.A. Sattar, MHD convective flow of a micropolar fluid past a continuously moving vertical porous plate in the presence of heat generation/ absorption, ASME J. Heat Transfer 128 (2006) 142–152. [10] F.S. Ibrahim, A.M. Elaiw, A.A. Bakr, Effect of the chemical reaction and radiation absorption on the unsteady MHD free convection flow past a semi infinite vertical permeable moving plate with heat source and suction, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 1056–1066. [11] M.M. Rahman, M.A. Rahman, M.A. Samad, M.S. Alam, Heat transfer in micropolar fluid along a non-linear stretching sheet with temperature dependent viscosity and variable surface temperature, Int. J. Thermophys. 30 (2009) 1649–1670. [12] M.M. Rahman, M.J. Uddin, A. Aziz, Effects of variable electric conductivity and non-uniform heat source (or sink) on convective micropolar fluid flow along an inclined flat plate with surface heat flux, Int. J. Ther Sci. 48 (2009) 2331– 2340. [13] M.M. Rahman, A. Aziz, M. Al-Lawatia, Heat transfer in micropolar fluid along an inclined permeable plate with variable fluid properties, Int J. Ther Sci. 49 (2010) 993–1002. [14] R.A. Damseh, M.Q. AL-Odat, A.J. Chamkha, A. Benbella Shannak, Combined effect of heat generation or absorption and first order chemical reaction on micropolar fluid flows over a uniform stretched permeable surface, Int. J. Thermal Sci. 48 (2009) 1658–1663. [15] M.M. Rahman, M. Al-Lawatia, Effects of higher order chemical reaction on micropolar fluid flow on a power law permeable stretched sheet with variable concentration in a porous medium, Can. J. Chem. Eng. 88 (2010) 23–32. [16] A.C. Cogley, W.E. Vincenty, S.E. Gilles, Differential approximation for radiation in a non-gray gas near equilibrium, AIAA J. 6 (1968) 551–553. [17] MD. Abdus Sattar, Kalim MD. Hamid, Unsteady free-convection interaction with thermal radiation in a boundary layer flow past a vertical porous plate, J. Math. Phys. Sci. 30 (1996) 25–37. [18] K. Vajravelu, Flow and heat transfer in a saturated porous medium, ZAMM 74 (1994) 605–614. [19] R. Ganapathy, A note on oscillatory Couette flow in a rotating system, ASME J. Appl. Mech. 61 (1994) 208–209. [20] M.A. Hossain, H.S. Takhar, Radiation effect on mixed convection along a vertical plate with uniform surface temperature, Heat Mass Transfer 31 (1996) 243–248. [21] A. Raptis, Radiation and free convection flow through a porous medium, Int. Commun. Heat Mass Transfer 25 (1998) 289–295. [22] O.D. Makinde, Free convection flow with thermal radiation and mass transfer past a moving vertical porous plate, Int. Commun. Heat Mass Transfer 32 (2005) 1411–1419. [23] F.S. Ibrahim, A.M. Elaiw, A.A. Bakr, Influence of viscous dissipation and radiation on unsteady MHD mixed convection flow of micropolar fluids, Appl. Math. Inf. Sci. 2 (2008) 143–162. [24] M.M. Rahman, M.A. Sattar, Transient convective flow of micropolar fluid past a continuously moving vertical porous plate in the presence of radiation, Int. J.Appl. Mech. Eng. 12 (2007) 497–513. [25] F.S. Ibrahim, A.M. Elaiw, A.A. Bakr, Effect of the chemical reaction and radiation absorption on unsteady MHD mixed convection flow past a semi-infinite vertical permeable moving plate with heat source and suction, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 1056–1066. [26] M.M. Rahman, Convective flows of micropolar fluids from radiate isothermal porous surfaces with viscous dissipation and Joule heating, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 3018–3030. [27] A.A. Bakr, Effects of chemical reaction on MHD free convection and mass transfer flow of a micropolar fluid with oscillatory plate velocity and constant heat source in a rotating frame of reference, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 698–710.