Latin American Applied Research 44:9-17 (2014)
MAGNETOHYDRODYNAMIC HEAT AND MASS TRANSFER FLOW WITH INDUCED MAGNETIC FIELD AND VISCOUS DISSIPATIVE EFFECTS
S. AHMED† and A. BATIN‡
† Fluid Dynamics Research, Department of Mathematics, Goalpara College, Goalpara 783101, Assam, India. [email protected] ‡ Department of Mathematics, Indira Gandhi College, Boitamari, Bongaigaon 783389, Assam, India. [email protected]
Abstract An approximate solution to the prob- and Soundalgekar (1984) investigated the steady lami- lem of steady free convective MHD flow of an in- nar free convection flow of an electrically conducting compressible viscous electrically-conducting fluid fluid along a porous hot vertical plate in presence of over an infinite vertical isothermal porous plate with heat source/sink. The combined heat and mass transfer mass convection is presented here. A uniform mag- of an electrically conducting fluid in MHD natural con- netic field is assumed to be applied transversely to vection adjacent to a vertical surface was analyzed by the direction of the flow, taking into account the in- Chen (2004). The study of heat and mass transfer on the duced magnetic field with viscous and magnetic dis- free convective oscillatory flow of a viscous incom- sipations of energy. The dimensionless governing pressible fluid past an infinite vertical porous plate in equations are solved by using the series solution presence of transverse sinusoidal suction velocity and a method. The induced magnetic field, current density, constant free stream velocity was presented by Ahmed temperature gradient and flow velocity are studied (2009). Seddeek (2004) investigated the effects of heat for magnetohydrodynamic body force, magnetic source in the presence of suction and blowing on mixed Prandtl number, Schmidt number and Eckert num- free-forced convective flow and mass transfer over an ber. It is observed that the induced magnetic field is accelerating surface in the case of variable viscosity. found to increase with a rise in magnetic Prandtl Zueco et al. (2009) studied the problem of number. Current density is strongly reduced with thermophoretic hydromagnetic dissipative heat and increasing magnetic Prandtl number, but enhanced mass transfer by network simulation numerical tech- with Schmidt number. The acquired knowledge in nique. Steven et. al. (2012) studied the magnetic hydro- our study can be used by designers to control MHD dynamic free convective flow past an infinite vertical flow as suitable for a certain applications such as porous plate with the effect of viscous dissipation sub- laminar magneto-aerodynamics, and MHD propul- ject to a constant suction velocity. sion thermo-fluid dynamics. The effects of external temperature-dependent Keywords Newtonian Fluid, Thermo-fluid dy- sources on the unsteady free convective flows past an namics, Heat transfer control, MHD energy genera- infinite vertical porous plate were presented by Pop and tors, Viscous and Magnetic dissipative heat. Soundalgekar (1979). Zueco Jordán (2006) presented a numerical analysis of an unsteady free convective MHD I. INTRODUCTION flow of a dissipative fluid along a vertical plate subject The study of magneto-hydrodynamic flow for an elec- to a constant heat flux. Ahmed and Liu (2010) analyzed trically conducting fluid past a porous vertical surface the effects of mixed convection and mass transfer of has attracted the interest of many researchers in view of three-dimensional oscillatory flow of a viscous incom- its important applications in many engineering problems pressible fluid past an infinite vertical porous plate in such as plasma studies, petroleum industries, MHD presence of transverse sinusoidal suction velocity oscil- power generators, cooling of nuclear reactors, the lating with time and a constant free stream velocity. boundary layer control in aerodynamics, and crystal Chamkha and Camille (2000) investigated the effects of growth. In all these applications understanding the be- heat generation/absorption and thermophoresis on hy- haviour of MHD convective flow and the various prob- dromagnetic flow with heat and mass transfer over a flat lem parameters that influence is a very important asset surface. Free convection flow past a vertical plate with to designers developing applications that aim to control surface temperature oscillations was considered by Li et this flow. In view of these applications, several investi- al. (1982). Ahmed (2010) investigated the effect of pe- gators have made model studies on the effect of magnet- riodic heat transfer on unsteady MHD mixed convection ic field free convection flows. Some of them are flow past a vertical porous flat plate with constant suc- Georgantopoulos (1979), Nanousis et al. (1980) and tion and heat sink when the free stream velocity oscil- Raptis and Singh (1983). Along with the effects of lates in about a non-zero constant mean. Chamkha magnetic field as well as transpiration parameter, being (2003) investigated the chemical reaction effects on heat an effective method of controlling the boundary layer and mass transfer laminar boundary layer flow in pres- has been considered by Kafoussias et al. (1979). Raptis ence of heat generation/absorption. The effect of the lo-
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cal acceleration term on the MHD transient free convec- rous plate taking into account the induced magnetic tion flow over a vertical plate was studied by Aldoss field, viscous and magnetic dissipations of energy in and Al-Nimir (2005). Muthucumaraswamy et al. (2001) presence of chemical reaction of first order and heat studied the heat and mass transfer effects on flow past generation/absorption, and the non-linear coupled equa- an impulsively started infinite vertical plate. Asghar et tions are solved by network simulation technique. Ah- al. (2005) investigated the effects of Hall current and med and Chamkha (2010) investigated the effects of ra- heat transfer on the steady flow of an electrically con- diation and chemical reaction on steady mixed convec- ducting, Oldroyd- B and incompressible fluid. The fluid tive heat and mass transfer flow of an optically thin gray is between two electrically insulating disks maintained gas over an infinite vertical porous plate with constant at two constant but different temperatures. The flow is suction taking into account the induced magnetic field, due to a pull with constant velocities of eccentric rotat- and viscous dissipation of energy, and the non-linear ing infinite disks and an external uniform magnetic field coupled equations are solved by series solution method. is applied perpendicular to the disks. The magnetic The steady magnetohydrodynamic (MHD) mixed con- Reynolds number is assumed small so that the induced vection stagnation point flow over a vertical flat plate is magnetic field is neglected. investigated by Ali et al. (2011). Ahmed et al. (2012) The above studies have generally been confined to investigated the effect of the transverse magnetic field very small magnetic Reynolds numbers, allowing mag- on a transient free and forced convective flow over an netic induction effects to be neglected. Such effects infinite vertical plate impulsively held fixed in free must be considered for larger values of magnetic Reyn- stream taking into account the induced magnetic field. olds number. Glauert (1962) presented a seminal analy- The present paper aims to study the problem of a sis for hydromagnetic flat plate boundary layers along a steady magnetohydrodynamic free convective boundary magnetized plate with uniform magnetic field in the layer flow over a porous vertical isothermal flat plate streamwise direction at the plate. He obtained series ex- with constant suction where the effects of the induced pansion solutions (for both large and small values of the magnetic field as well as viscous and magnetic dissipa- electrical conductivity parameter) for the velocity and a tions of energy has been considered. The present study magnetic field, indicating that for a critical value of ap- may have useful applications to several transport pro- plied magnetic field, boundary-layer separation arises. cesses as well as magnetic materials processing. The Dorrepaal and Moosavizadeh (1998) studied the oblique analytical results for some particular cases are matched stagnation-point magnetohydrodynamic flow in the vi- with Raptis et al. (2003) and found them in excellent cinity of a location where a separation vortex reattaches agreement. itself to a rigid boundary, obtaining a similarity solution II. MATHEMATICAL ANALYSIS for the velocity and induced magnetic field distributions The steady MHD mixed convective heat and mass trans- near the point of reattachment. Raptis and Massalas fer flow of an electrically conducting, viscous incom- (1998) investigated the effects of radiation on the oscil- pressible fluid over a porous vertical infinite isothermal latory flow of a gray gas, absorbing-emitting in pres- plate with viscous dissipative heat and magnetic dissipa- ence of induced magnetic field and analytical solutions tion of energy has been considered (Fig. 1) by making were obtained with the help of perturbation technique. the following assumptions: The hydrodynamic free convective flow of an optically thin gray gas in the presence of radiation, when the in- All the fluid properties except the density in the duced magnetic field is taken into account was studied buoyancy force term are constant; by Raptis et al. (2003) using perturbation technique. The Eckert number Ec is small; More recently Bég et al. (2009) obtained local non- The plate is subjected to a constant suction veloci- similarity numerical solutions for the velocity, tempera- ty; ture and induced magnetic field distributions in forced A magnetic field of uniform strength (H0) is applied convection hydromagnetic boundary layers, over an ex- transversely to the direction of the main stream tak- tensive range of magnetic Prandtl numbers and Hart- ing into account the induced magnetic field; mann numbers. Bég et al. (2009) have also derived closed-form solutions for the velocity, temperature and magnetic induction functions in Rayleigh free convec- tion hydromagnetic flow. Alom et al. (2008) investigat- ed the steady MHD heat and mass transfer by mixed convection flow from a moving vertical porous plate with induced magnetic, thermal diffusion, constant heat and mass fluxes, and the non-linear coupled equations are solved by shooting iteration technique. Ahmed and Zueco (2010) studied the effect of the transverse mag- netic field on a steady mixed convective heat and mass transfer flow of an incompressible viscous electrically conducting fluid past an infinite vertical isothermal po- Fig. 1: Physical configuration.
10 S. AHMED, A. BATIN
2 2 The magnetic Prandtl number is so chosen in com- d T d 2T d u 2 d H parison to the Hartmann number such that the in- v x (4) d y C 2 C d y C d y duced magnetic is not negligible; P d y P P d 2 H d u d H The concentration of the diffusing species in the bi- 0 x H v x (5) nary mixture is assumed to be very small in com- d y2 0 d y d y parison with the other chemical species, which are 2 d C d C (6) present, and hence the Soret and Dufour effects are v D 2 negligible; d y d y The equation of conservation of electric charge is Since there is no large velocity gradient here, the vis- cous term in the Eq. (3) vanishes for small and hence J=0 where J=(Jx,Jy,Yz). The direction of propaga- tion is considered only along the y-axis and does for the outer flow, beside there is no induced magnetic not have any variation along the y-axis and so field along x-direction gradient, so we have J /y=0 which gives J =constant. Since the plate is d p (7) y y 0 g electrically non-conducting, this constant is zero d x and hence Jy=0 everywhere in the flow; By eliminating the pressure term from the Eqs. (3) and The wall is maintained at constant temperature T w (7), we obtain 2 and concentration Cw higher than the ambient d u d u d H x . (8) v ( )g 2 H0 temperature T and concentration C respective- d y d y d y ly; The Boussinesq approximation gives
The fluid is non-magnetic, neglecting the thermoe- (T T ) (C C ) . (9) lectric effect together with the short circuit condi- On using (1) and (9) in the Eq. (8) and noting that tion. is approximately equal to 1, the momentum equation re- In case of electrically conducting fluid, the flow and duces to heat transfer are induced by an imposed magnetic field. d u An e.m.f. is produced in the fluid flowing across the v0 g (T T ) g (C C ) d y . (10) transverse magnetic field. The resultant effect of current d 2 u H d H and magnetic field produces a force, resisting the fluid 0 x motion. The fluid velocity also affects the magnetic d y2 d y field by producing an induced magnetic field which per- The boundary conditions are: turbs the original field. d T q We introduce a coordinate system (x, y, z) with x - y 0:u 0, , H x 0, C Cw d y . (11) axis vertically upwards along the plate, y -axis normal d H to the plate into the fluid region and z -axis along the y :u 0, T T , x 0, C C d y width of the plate. Let the plate be long enough in - The non-dimensional quantities are: direction for the flow to be parallel. Let (u,v,0) be the , , , , y v0 y / v u u /v0 / Pr CP / fluid velocity and (H x (y),H y ,0) be the magnetic induc- Pr v/ , v (T T ) /( q) , Dv (C C ) /(vm), tion vector at a point in the fluid. Since the m 0 0 Gr 2 gq /(v4 ) , Gr 2 gm /(Dv4 ) , Sc v / D, plate is infinite in length in -direction, therefore all the 0 m 0 3 , 2 2 , . physical quantities except possibly the pressure are as- Ec v0 /( q CP ) M H0 /( v0 ) Hx Hx / H0 sumed to be independent of . The non-dimensional forms of (4) to (6) and (10) are Within the frame of such assumptions and under the 2 du d u M dH x . (12) Boussinesq’s approximation and in consistency with Gr Grm dy dy 2 Pr dy boundary layer theory, the governing equations relevant m 2 2 to the problem are: d 1 d 2 du M Ec dH Ec x . (13) d v 2 2 0 (1) dy Pr dy dy Prm dy d y 2 d 1 d . (14) which is satisfied with v v = a constant. 0 dy Sc dy 2 d H y (2) 2 0 d Hx du dHx . (15) d y Prm Prm 0 dy2 dy dy which holds for = a constant=strength from H y H0 The corresponding boundary conditions are: applied magnetic field. d y 0 :u 0, 1, H x 0, 1 d u d p d 2u d H dy (16) v g H x (3) d y d x d y2 0 d y dH y :u 0, 0, x 0 0 dy
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III. METHOD OF SOLUTION y Pr y y 2 Pr y u (y) e e e e Perturbation theory leads to an expression for the de- 1 13 14 15 sired solution in terms of a power series in some 2 y 2Sc y (Pr ) y (30) 16 e 17 e 18e “small” parameter that quantifies the deviation from the (Pr Sc) y (Sc ) y (Pr Prm ) y exactly solvable problem. The leading term in this pow- 19 e 20 e 21 e er series is the solution of the exactly solvable problem, 2 ( Prm ) y 2Prm y (Pr Prm ) y while further terms describe the deviation in the solu- 22 e 23 e 24e tion, due to the deviation from the initial problem. Per- H (y) e y e Pr y e 2 y e 2 Pr y turbation theory is applicable if the problem at hand can x1 21 22 23 be formulated by adding a “small” term (Eckert number 2 Sc y (Pr ) y (Pr Sc) y (31) 24 e 25 e 26 e “Ec” in this work) to the mathematical description of (Sc ) y (Pr Prm ) y ( Prm ) y the exactly solvable problem. 27 e 28 e 29 e The solution of the Eq. (14) subject to the boundary 2Pr y (Sc Pr ) y e m e m condition (16) is 30 31 eScy (17) These analytical approximate solutions under per- Now, in order to solve the Eqs. (12), (13) and (15) turbation technique give a wider applicability in under- under the boundary condition (16), we note that standing the basic physics and chemistry of the prob- EC 1 for all the incompressible fluids and it is as- lem, which are particularly important in industrial and technological fields. sumed the solutions of the equations to be of the form 2 f (y) f0 (y) Ec f1(y) O (Ec ) (18) IV. RATE OF HEAT TRANSFER The Nusselt number is often used to determine heat where f stands for u, or Hx. Substituting (18) into the Eqs. (12), (13) and (15) transfer. The non-dimensional heat flux at the plate y=0 and equating the coefficients of the same degree terms in terms of Nusselt number Nu is given by and neglecting terms of O(Ec2), the following differen- d Nu Nu Ec Nu (32) tial equations are obtained: 0 1 dy y0 M u u Gr Gr H (19) 0 0 0 m 0 x 0 where Nu0 0(0) 1 and Prm Nu (0) Pr 2 2Pr M (20) 1 1 11 12 u1 u1 Gr1 Grm 1 H x 1 2Sc (Pr ) (Pr Sc) Prm 13 14 15 (Sc ) (Pr Pr ) 0 Pr0 0 (21) 16 m 17 ( Pr ) 2Pr (Sc Pr ) 2 Pr M 2 (22) m 18 m 19 m 20 1 Pr1 Pr (u0 ) (H x ) Pr 2 0 m V. CURRENT DENSITY H Pr H Pr u (23) The non-dimensional current density at the plate y=0 is x 0 m x 0 m 0 given by (24) H x Prm H x Prm u1 1 1 dH x (33) The boundary conditions (16) reduce to J J 0 Ec J1 dy y0 y 0: u0 0, u1 0, 0 1, 1 0, where H x 0, H x 0 0 1 (25) J H (0) (P Pr Sc ) 0 x 0 m 3 4 5 y 0: u0 0, u1 0, 0 0, 1 0, J1 H x (0) Pr 21 2 22 2Pr 23 H 0, H 0 1 x 0 x 1 2Sc (Pr ) (Pr Sc) The solutions of the Eqs. (19) to (24) subject to the 34 25 26 (Sc ) (Pr Pr ) ( Pr ) boundary conditions (25) are: 27 m 28 m 29 2Pr (Sc Pr ) 1 Pr y (26) m 30 m 31 0 (y) e Pr VI. VALIDATION y Pr y Sc y u0 (y) e 1e 2 e (27) The method is validated by directly comparing its re- Pr y sults with those of Raptis et al. (2003) without radiation, H (y) e m e y e Pry e Scy (28) x0 3 4 5 for Prm=3, M=4, Ec=0=Grm as shown in Table 1, where
Prm=3>1 which implies that viscous diffusion rate is (y) e Pr y e 2 y e 2 Pr y e 2 Sc y 1 11 12 13 dominant over the magnetic diffusion rate. From Table 2(Pr ) y 2(Pr Sc) y 2(Sc ) y (29) 1, it is seen that the results for flow velocity are in ex- 14 e 15e 16 e cellent agreement. Therefore, this leads to confidence in 2(Pr Prm ) y 2( Prm ) y 2Prm y 17 e 18 e 19 e the numerical results. In Table 1, it is observed that the flow velocity is accelerated when free convection pa- 2(Sc Prm ) y 20 e rameter increases from 2.0 through 5.0 to 10.0.
12 S. AHMED, A. BATIN
Table 1: Comparison of values of the flow velocity (u) with ues of M and Prm, i.e., the induced magnetic flux rever- those obtained by Raptis et al. (2003) for Prm=3, M=4, sal arises for y0 into the boundary layer, and trans- Ec=0=Grm: verse to the plate. In all the cases, Hx peaks a short dis- Raptis et al. (2003) without radiation tance from the plate, and then decays to be zero in the Y Gr = 2 Gr = 5 Gr = 10 free stream. 0.0 0.00000 0.00000 0.00000 2.0 5.93172 7.43029 9.75438 Figure 3 shows the distribution of current density at 4.0 2.58730 3.58741 4.83512 the plate for various magnetic Prandtl number and 6.0 1.37572 1.87310 2.31820 Schmidt number over Hartmann number. A rise in Prm 8.0 0.62753 0.07595 0.08375 from 4.0 through 5.0 to 6.0 serves to depress strongly 10.0 0.01511 0.02531 0.03914 current density magnitudes throughout the regime, but Present work the current density rises with Schmidt number. These Gr = 2 Gr = 5 Gr = 10 effects are very useful for small Hartmann numbers and 0.0 0.00000 0.00000 0.00000 insignificant in the free stream. It is also found that the 2.0 5.93189 7.43103 9.75452 current density has a large negative value throughout the 4.0 2.58785 3.58773 4.83530 flow field. Thus it is concluded that the magnetic 6.0 1.37632 1.87345 2.31843 Prandtl number or Schmidt number has a strong role on 8.0 0.62774 0.07610 0.08383 10.0 0.01547 0.02563 0.03935 the current density. In Fig. 4, the distribution of temperature gradient distribution (/y) with Hartmann, magnetic Prandtl and Prandtl number, is presented for diffusing species helium. An increase in M, Prm and Pr leads to reduce sharply /y and these results are very much effective near the plate, but insignificant far away from the plate.
Fig. 2: Induced magnetic field distribution with Hartmann number and Magnetic Prandtl number
VII. RESULTS AND DISCUSSION In order to get physical insight into the problem, we Fig. 3: Current density distribution with Magnetic Prandtl have carried out the numerical calculations for the ve- number and Schmidt number. locity field, induced magnetic field, temperature gradi- ent and current density at the plate and the values are demonstrated in graphs and Tables. The Prandtl number Pr is taken for air (Pr=0.71) at 20oC, water (Pr=7.0) and water at 4oC (Pr=11.4). The thermal Grashof number Gr and the Eckert number (Ec) take positive values corre- sponds to cooling of the plate by free convection cur- rents. Also the values of the Schmidt number (Sc), so chosen to represent the presence of various species Hy- drogen (Sc=0.20), Helium (Sc=0.30), Steam (Sc=0.60) and Oxygen (Sc=0.66). Figure 2 illustrates the response of induced magnetic field to Hartmann number and magnetic Prandtl number for conducting air (Pr = 0.71) and diffusing species he- lium (Sc = 0.30). Increase in Hartmann number clearly Fig. 4: Temperature gradient distribution with Hartmann, Magnetic Prandtl and Prandtl number. reduces the induced magnetic field magnitudes, whereas increasing in Pr boosts the values of H through the In Fig. 5, the influence of Hartmann number and m x magnetic Prandtl number on the flow velocity distribu- boundary layer when y>1. In this figure, Prm is set as greater than 1.0 and implies that the viscous diffusion tions is shown for conducting air and diffusing species rate exceeds the magnetic diffusion. In the vicinity of helium. With an increase in M from 0.5 through 1.0 to the plate, the values of H remains negative for all val- 3.0, there is clear decrease in flow velocity i.e. flow is x strongly decelerated. It is evident that the presence of
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the transverse magnetic field produces a resistive force viscous diffusion rate is five times the magnetic diffu- on the flow velocity. This force is called the Lorentz sion rate. Increasing Eckert number clearly increases the force acting on the flow velocity, which leads to slow induced magnetic field magnitudes when y>1, but this down the motion of electrically conducting fluid. In no trend is reversed in the vicinity of the wall. Also it is case however, there is a flow reversal i.e. velocity re- found that, the induced magnetic field decrease strongly mains positive through the boundary layer. On the other with an increase in Schmidt number for y>1. Also for hand, the flow velocity is accelerated when magnetic different values of Ec and Sc, the induced magnetic flux Prandtl number rises from 2.0 through 3.5 to 4.0. reversal arises for the region 0 Table 2: Induced Magnetic distribution for different values of Sc and Ec over y, for Gr=2=Grm, Pr=0.71, Prm=5 and M=6. y Sc = 0.22 Sc = 0.60 Sc = 0.66 0 -1.8122 -0.3437 -0.2341 1 29.9975 1.4738 -0.6493 2 20.1514 0.8814 -0.2569 3 13.7878 0.4850 -0.1165 4 9.7179 0.2675 -0.0500 5 7.0248 0.1478 -0.0248 y Ec = 0.05 Ec = 0.06 Ec = 0.07 0 -4.4541 -5.3450 -6.2358 1 23.7062 26.8368 29.9674 2 15.0347 16.8893 18.7440 Fig. 5: Flow velocity distribution with Magnetic Prandtl num- 3 9.4739 10.5562 11.6385 ber and Schmidt number 4 6.0377 6.1637 7.2898 5 3.9015 4.2611 4.6206 Table 3: Current density distribution for different values of Ec and Pr over M, for Gr=2=Grm, Pr=0.71, Prm=5 and Sc=0.30 M Ec = 0.04 Ec = 0.05 Ec = 0.07 0 -6.5438 -8.1782 -11.4507 1 -0.6451 -0.8063 -1.1294 2 -0.1877 -0.2344 -0.3286 3 -0.0820 -0.1020 -0.1433 4 -0.0447 -0.0551 -0.0773 5 -0.0271 -0.0347 -0.0471 M Pr = 0.71 Pr = 7.0 Pr = 11.4 0 -15.0734 -1.6362 -1.5352 1 -1.6511 -0.1617 -0.1517 2 -0.4302 -0.0471 -0.0449 Fig. 6: Temperature distribution with Magnetic Prandtl num- 3 -0.1647 -0.0209 -0.0193 ber and Hartmann number 4 -0.0755 -0.0119 -0.0103 5 -0.0408 -0.0074 -0.0061 The effects of Hartmann number M, and magnetic The numerical values of current density for Eckert Prandtl number Prm on the fluid temperature for the mercury flow are illustrated in Fig. 6. The trend is very number and Prandtl number over Hartmann number are consistent for all M (or Pr ) i.e. a monotonic decay presented in Table 3 for Gr==5, Grm=5, Pr=0.71, Prm=5 m and Sc=0.30. An increase in viscous dissipative heat re- from a maximum value ( = 1) at the wall (y=0) to the duces the current density, whereas current density rises free stream. Temperatures are strongly enhanced with with the increase of Prandtl number. It is observed that, increasing transverse magnetic field (or Magnetic Prandtl number). The presence of transverse magnetic when Hartmann number, M1, maximum effects on field produces a resistive force on the fluid flow. This current density occurred, but these effects are insignifi- force is called the Lorentz force, which leads to slow cant for large values of Hartmann number. down of the motion of electrically conducting fluid and VIII. CONCLUSIONS therefore, this leads to decrease in the thickness of In this paper, it was demonstrated that the perturbation boundary layer and hence to increase in the temperature technique can be extended and applied to obtain a of fluid flow. closed form expression for the problem of a steady The response of induced magnetic field (Hx) to vis- mixed convective MHD flow of an incompressible vis- cous dissipative heat (Ec) as well as Schmidt number cous electrically conducting fluid past an infinite verti- (Sc) is shown in Table 2 for Gr=2, Grm=2, Pr=0.71, cal porous plate in presence of transverse magnetic field Prm=5 and M=6. Here magnetic Prandtl number, Prm, is with combined heat and mass transfer, taking into ac- set as 5 and implies the ratio of momentum diffusivity count the induced magnetic field with viscous and mag- (viscosity) is five times the magnetic diffusivity i.e. the netic dissipations of energy. The present analysis brings 14 S. AHMED, A. BATIN out the following results of physical parameters of the u Dimensionless velocity component in x-direction flow field: (m. s) A velocity overshoot is observed with lower values v0 Dimensionless suction velocity (m. s ) of magnetic field effects but vanishes for very high J Current density values of the Hartmann number in the flow regime; Greek symbols Increasing magnetic Prandtl number (Pr ) or mag- m Coefficient of volume expansion for heat transfer netic field (M) serves to decelerates the flow veloci- ty significantly for the case of conducting air. (K ), Coefficient of volume expansion for mass transfer With an increase in Prm, the induced magnetic field along x-direction is progressively accelerated, but () the magnetic field depress the Hx. Magnetic diffusivity Increasing Schmidt number (Sc) serves to escalate Dimensionless fluid temperature (K), the Current density at the plate, but the current den- Thermal conductivity (W. m. K), sity decreases for the effect of Prm. Coefficient of viscosity (kg. m ) The temperature component (θ) increases with an Kinematic viscosity (m2.s), increase of Magnetic field or Magnetic Prandtl Density (kg. m), number. Electrical conductivity The induced magnetic flux reversal arises for both Shearing stress (N. m) the effects of Ec and Sc. Dimensionless species concentration (Kg. m All the temperature gradients (/y) have maxi- mum values in the neighbourhood of y=1 compari- Subscripts son to the other values of distance and it is ob- w Conditions on the wall Free stream conditions served to decrease /y for M or Prm or Pr. The study has important applications in nuclear heat APPENDIX transfer control, and MHD energy generators. Currently 1 Pr (1 Pr )2 4M the authors are exploring extensions of this work to m m , study rheological (non-Newtonian) working fluid ef- 2 fects, the results of which will be reported in the near Gr(1 Prm / Pr) 1 3 2 , future. - Pr (1 Prm)Pr (M Prm)Pr Gr (Sc Pr ) ACKNOWLEDGEMENTS m m 2 , The authors acknowledge the important comments of - Sc 3 (1 Pr )Sc2 (M Pr )Sc the reviewers which have helped to improve the present m m Pr article. ( ), Prm 1 m 1 2 3 , 4 , NOMENCLATURE Prm Prm Pr H Uniform magnetic field Pr 0 2 m ( ), H Induced magnetic field along x-direction 5 , 3 4 5 x Sc - Prm C Species concentration (Kg. m) M Pr 2 2 C Specific heat at constant pressure (J. kg. K) 2 2 3 P 1 Pr 2 , Prm C Species concentration in the free stream (Kg. m ) 2 2 M Pr C Species concentration at the surface (Kg. m ) 3 2 4 w 2 Pr 1 , 2 2 D Chemical molecular diffusivity (m .s ) Prm Ec Eckert number/dissipative heat M Pr Sc 2 2 Pr Sc 22 5 , g Acceleration due to gravity (m.s ) 3 2 2 Gr Thermal Grashof number Prm Gr Mass Grashof number 2M Pr 2 m 2Pr 2 3 4 , M Hartmann number/Magnetic parameter 4 1 2 Prm m Mass flux per unit area 2M Sc Pr 2 Nu Nusselt number 2Pr 3 Sc 4 5 , Pressure (Pa) 5 1 2 2 p Prm Prm Magnetic Prandtl number 2M Pr Sc 3 5 Pr Prandtl number 6 2ScPr 2 , Pr 2 q Heat flux per unit area m 2 Sc Schmidt number 2M Pr 4 2M Pr 3 7 , 8 , T Dimensional Temperature Prm Prm Tw Dimensional Fluid temperature at the surface T Dimensional Fluid temperature in the free stream 15 Latin American Applied Research 44:9-17 (2014) 2M Pr Sc M ( Sc ) M Pr 2 , 5 27 9 10 , 8 Gr 16 , Prm Prm M ( Pr Pr ) 1 2 m 28 11 , 12 2 , 9 Gr 17 , 2 (2 Pr) 2Pr Prm 3 4 M ( Pr ) , , Gr m 29 , 13 2Sc(2Sc Pr) 14 (Pr )( Pr) 10 18 Prm 5 4 , , 11 Gr 19 2M 30 , 15 (Pr Sc)(Sc Pr) 16 (Sc )( Sc 2Pr) M ( Sc Prm ) Gr 31 , 4 , 12 20 Pr 17 (Pr Pr )(Pr Pr) m m m 1 2 4 4 , 13 , 14 , 18 , 19 Pr(Pr -1) ( -1) ( Prm )( Prm 2Pr) 2Prm(2Prm 1) 1 3 Pr 2 2Pr 2Sc (Pr ) 15 , Pr 11 12 13 14 2Pr(2Pr -1) (Pr Sc ) (Sc ) (Pr Pr ) ( Pr ) , 15 16 m 17 m 18 4 , 5 , 16 2 (2 -1) 17 2Sc(2Sc -1) 10 , 20 (Sc Pr )(Sc Pr 2Pr) m m 6 , 7 , GrPr 18 (Pr )(Pr -1) 19 (Pr Sc)(Pr Sc -1) , 21 Pr 3 (1 Pr )Pr 2 (M Pr )Pr m m 8 , 9 , Gr Pr 20 (Sc )(Sc -1) 21 (Pr Pr )(Pr Pr -1) 11 m , m m 22 8 3 4(1 Pr )r 2 2(M Pr ) m m 10 , 11 , Gr Pr 22 23 12 m ( Prm )( Prm -1) 2Prm (2Prm -1) 23 3 2 , 8Pr 4(1 Prm )Pr 2(M Prm )Pr 12 , Gr13Prm 24 , (Sc Prm )(Sc Prm -1) 24 8Sc3 4(1 Pr )Sc 2 2(M Pr )Sc m m ( Gr Pr 13 14 15 16 17 18 14 m , 25 3 2 19 20 21 22 23 24). 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