MHD Free Convection and Mass Transfer Flow past a Vertical Flat Plate
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MHD Free Convection and Mass Transfer Flow past a Vertical Flat Plate

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MHD Free Convection and Mass Transfer Flow past a Vertical Flat Plate MHD Free Convection and Mass Transfer Flow past a Vertical Flat Plate Document Transcript

  • International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.3, Issue.1, Jan-Feb. 2013 pp-75-82 ISSN: 2249-6645 MHD Free Convection and Mass Transfer Flow past a Vertical Flat Plate S. F. Ahmmed1, S. Mondal2, A. Ray3 1, 2, 3 Mathematics Discipline, Khulna University, BangladeshABSTRACT: We investigate the two dimensional free water purification processes, the motion of natural gases andconvection and mass transfer flow of an incompressible, oil through oil reservoirs in petroleum engineering and soviscous and electrically conducting fluid past a continuously on. The porous medium is in fact a non-homogeneousmoving vertical flat plate in the presence of heat source, medium. The velocity is usually so small and the flowthermal diffusion, large suction and the influence of uniform passages are so narrow that laminar flow may be assumedmagnetic field applied normal to the flow. Usual similaritytransformations are introduced to solve the momentum, without hesitation. Rigorous analysis of the flow is notenergy and concentration equations. To obtain the solutions possible because the shape of the individual flow passages isof the problem, the ordinary differential equations are solved so varied and so complex. Poonia and Chaudhary [2] studiedby using perturbation technique. The expressions for velocity about the flows through porous media. Recently researchersfield, temperature field, concentration field, skin friction, like Alam and Rahman [3], Sharma and Singh [4]rate of heat and mass transfer have been obtained. The Chaudhary and Arpita [5] studied about MHD freeresults are discussed in detailed with the help of graphs and convection heat and mass transfer in a vertical plate ortables to observe the effect of different parameters. sometimes oscillating plate. An analysis is performed to study the effect of thermal diffusing fluid past an infiniteKeywords: Free convection, MHD, Mass transfer, Thermal vertical porous plate with Ohmic dissipation by Reddy anddiffusion, Heat source parameter. Rao [6]. The combined effect of viscous dissipation, Joule heating, transpiration, heat source, thermal diffusion and I. INTRODUCTION Hall current on the hydro-Magnetic free convection and Magneto-hydrodynamic (MHD) is the branch of mass transfer flow of an electrically conducting, viscous,continuum mechanics which deals with the flow of homogeneous, incompressible fluid past an infinite verticalelectrically conducting fluids in electric and magnetic fields. porous plate are discussed by Singh et. al. [7]. Singh [8] hasMany natural phenomena and engineering problems are also studied the effects of mass transfer on MHD freeworth being subjected to an MHD analysis. Furthermore, convection flow of a viscous fluid through a vertical channelMagneto-hydrodynamic (MHD) has attracted the attention walls. An extensive contribution on heat and mass transferof a large number of scholars due to its diverse applications. flow has been made by Gebhart [9] to highlight the insight In engineering it finds its application in MHD on the phenomena. Gebhart and Pera [10] studied heat andpumps, MHD bearings etc. Workers likes Hossain and mass transfer flow under various flow situations. ThereforeMandal [1] have investigated the effects of magnetic field several authors, viz. Raptis and Soundalgekar [11], Agrawalon natural convection flow past a vertical surface. Free et. al. [12], Jha and Singh [13], Jha and Prasad [14] haveconvection flows are of great interest in a number of paid attention to the study of MHD free convection andindustrial applications such as fiber and granular insulation, mass transfer flows. Abdusattar [15] and Soundalgekar ET.geothermal systems etc. Buoyancy is also of importance in al. [16] also analyzed about MHD free convectionan environment where differences between land and air through an infinite vertical plate. Acharya et. al.[17] havetemperatures can give rise to complicated flow patterns. presented an analysis to study MHD effects on free Convection in porous media has applications in convection and mass transfer flow through a porous mediumgeothermal energy recovery, oil extraction, thermal energy with constant auction and constant heat flux consideringstorage and flow through filtering devices. The phenomena Eckert number as a small perturbation parameter. This is theof mass transfer are also very common in theory of stellar extension of the work of Bejan and Khair [18] under thestructure and observable effects are detectable, at least on influence of magnetic field. Singh [19] has also studiedthe solar surface. The study of effects of magnetic field on effects of mass transfer on MHD free convection flow of afree convection flow is important in liquid-metal, viscous fluid through a vertical channel using Laplaceelectrolytes and ionized gases. The thermal physics of transform technique considering symmetrical heating andhydro-magnetic problems with mass transfer is of interest in cooling of channel walls. A numerical solution of unsteadypower engineering and metallurgy. The study of flows free convection and mass transfer flow is presented by Alamthrough porous media became of great interest due to its and Rahman [20] when a viscous, incompressible fluidwide application in many scientific and engineering flows along an infinite vertical porous plate embedded in aproblems. Such type of flows can be observed in the porous medium is considered.movement of underground water resources, for filtration and www.ijmer.com 75 | Page
  • International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.3, Issue.1, Jan-Feb. 2013 pp-75-82 ISSN: 2249-6645 In view of the application of heat source andthermal diffusion effect, the study of two dimensional MHDfree convection and mass transfer flow past an infinitevertical porous plate taking into account the combined effectof heat source and thermal diffusion with a great agreementwas conducted by Singh ET. al.[21]. In our present work, the effect of large suction onMHD free convection heat and mass transfer flow past avertical flat plate has been investigated by similaritysolutions which are obtained by employing the perturbationtechnique. We have used MATHMATICA to draw graphand to find the numerical results of the equation. II. THE GOVERNING EQUATIONS Continuity equation Consider a two dimensional steady free convectionheat and mass transfer flow of an incompressible, u v (1)electrically conducting and viscous fluid past an electrically  0 x ynon-conducting continuously moving vertical flat plate. Introducing a Cartesian co-ordinate system, x-axisis chosen along the plate in the direction of flow and y-axis Momentum equationnormal to it. A uniform magnetic field is applied normally tothe flow region. The plate is maintained at a constant u v  2u u  v   2  g  (T  T )temperature Tw and the concentration is maintained at a x y yconstant value Cw. The temperature of ambient flow is T∞ (2)  B0 2 ( x)and the concentration of uniform flow is C∞.  g  (C  C )  * u Considering the magnetic Reynold’s number to be very small, the induced magnetic field is assumed to be negligible, so that B = (0, B0(x), 0).The equation of Energy equation  conservation of electric charge is. J Where J = (Jx, Jy, T T K  2T (3)Jz) and the direction of propagation is assumed along y-axis u v   Q(T  T ) x y C p y 2so that J does not have any variation along y-axis so that the  J yy derivative of J namely  0 resulting in Jy=constant. Concentration equation y Also the plate is electrically non-conducting Jy=0 C C  2C  2T (4) u v  DM  DTeverywhere in the flow. Considering the Joule heating and x y y 2 y 2viscous dissipation terms to negligible and that the magnetic The boundary conditions relevant to the problem arefield is not enough to cause Joule heating, the term due to u  U0 , v  v0 ( x), T  Tw, C  Cw at y  0 (5)electrical dissipation is neglected in the energy equation. u  0, v  0, T  T, C  C at y  The density is considered a linear function oftemperature and species concentration so that the usual Where u and v are velocity components along x-Boussinesq’s approximation is taken as ρ=ρ0 [1-{β (T-T∞) axis and y-axis respectively, g is acceleration due to gravity,+β*(C-C∞)}]. Within the frame work of delete such T is the temperature. K is thermal conductivity, σ′ is theassumptions the equations of continuity, momentum, energy electrical conductivity, DM is the molecular diffusivity,U0 isand concentration are following [21] the uniform velocity, C is the concentration of species, B0(x) is the uniform magnetic field, CP is the specific heat at constant pressure, Q is the constant heat source(absorption type), DT is the thermal diffusivity, C(x) is variable concentration at the plate, v0(x) is the suction velocity, ρ is the density,  is the kinematic viscosity, β is the volumetric coefficient of thermal expansion and β* is the volumetric coefficient of thermal expansion with concentration and the other symbols have their usual meaning. For similarity solution, the plate concentration C(x) is considered to be C(x)=C∞+(Cw-C∞)x. www.ijmer.com 76 | Page
  • International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.3, Issue.1, Jan-Feb. 2013 pp-75-82 ISSN: 2249-6645 III. MATHEMATICAL ANALYSIS suction We introduce the following local similarity X ( )  1  X 1 ( )   2 X 2 ( )   3 X 3 ( )  ......... (16)variables of equation (6) into equation (2) to (4) andboundary condition (5) and get the equation (7) to (9) and Y ( )  Y1 ( )   2Y2 ( )   3Y3 ( )  ......... (17)boundary condition (10) Z ( )  Z1 ( )   2 Z 2 ( )   3 Z 3 ( )  .......... (18) Using equations (16)-(18) in equations (12)-(14) and U0 considering up to order O(ε3), we get the following three   2 xU 0 f ( ), y , (6) 2 x sets of ordinary differential equations and their T  T C ( x)  C corresponding boundary conditions  ( )  ,  ( )  Tw  T Cw  C First order O(ε): f   ff   Mf   Gr  Gm  0 (7) X 1  X 1  0   (19)    Pr f   S Pr  0 (8) Y1 Pr Y1  0 (20)    2Scf   Scf   S0 Sc   0  (9) Z1  ScZ1  ScS 0Y1  0   (21) st  Cp The boundary conditions for 1 order equations arewhere Pr  is the Prandtl number, k X1  0, X   1, Y  1, Z  1 at   0 1 1 (22) 1 2 xg  (Tw  T )Gr  is the Grashof number, X 1  1, Y1  0, Z1  0 as   0 U 02 Second order O(ε2): 2 xg   (Cw  C )Gm  is Modified Grashof X 2  X 2  X1 X1  MX1  GrY1  GmZ1     (23) U 02 number, Y2  Pr X1Y1  Pr Y2  S Pr Y1   (24) 2 xQ Z 2  ScZ 2  2ScX1Z1  ScX 1Z1  ScS 0Y2      (25)S is the Heat source paramater, nd U0 The boundary conditions for 2 order equations are 2 x  0 ( x) 2  X 2  0, X 2  0, Y2  0, Z 2  0 at   0 (26)M is Magnetic parameter, U0  X 2  0, Y2  0, Z 2  0 as    Third order O(ε ): 3Sc  is the Schmidt number and DM y X 3  X 3  X1X 2 X1 X 2  MX 2  GrY2  GmZ2     (27) Tw  T Y3  Pr X 1Y2  Pr X 2Y1  Pr Y3  S Pr Y2    (28)S0  is the Soret number Cw  C Z3  ScZ3  2ScX 1Z 2  2ScX 2 Z1  ScX 2 Z1    (29)with the boundary condition,  ScX1Z 2  ScS0Y3  f  f w , f   1,   1,   1 at   0 (10) The boundary conditions for 3rd order equations are  X 3  0, X 3  0, Y3  0, Z3  0 at   0 (30) f   0,   0,   0 as     X 3  0, Y3  0, Z3  0 as   2x where f w  vo ( x) and primes denotes the The solution of the above coupled equations (19) to (30) U 0 under the prescribed boundary conditions are as follows derivatives with respect to η. First order O(ε):    f w , f ( )  f w X ( ), X  1  e  (31) 1  ( )  f w2Y ( ),  ( )  f w2 Z ( ) (11) Y1  e  Pr (32) By using the above equation (11) in the equations (7) - (9) with boundary condition (10) we get, Z1  (1  A1 )e  Sc  A1e  Pr (33) X   XX    (MX   GrY  GmZ ) (12) Second order O(ε2): 1 2 (34) Y   Pr XY   S Pr Y (13) X2  e  ( A7  A2 )e  A8 e Pr   A9 e Sc  A10 4 Z   2ScZX   ScXZ   ScS 0Y   0 (14)  Pr (1Pr) (35) Y2  ( A11  A12 )e  A11e The boundary conditions (10) reduce to Z 2  ( B5  B1  B2  B3  B6 )e Sc  ( B3  B5  B4 ) (36) X  1, X    , Y   and Z   at   0 (15)  Pr   (1 Pr)  (1 Sc ) e  B1e  B2 e X   0 , Y  0, Z  0 as    Third order O(ε ): 3 Now we can expand X, Y and Z in powers of  as follows since   1 is a very small quantity for large 2 fw www.ijmer.com 77 | Page
  • International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.3, Issue.1, Jan-Feb. 2013 pp-75-82 ISSN: 2249-6645 B9 2  B A asX 3  B31  ( B24  B32   )e  ( 7  A2  2  ) 2 4 2 (37)  C  (47) e 2  ( B25  B26  B18 )e  Pr   [ B27  B28  B21 ] Sh      y  y  0 1 3 e  Sc  B29 e  (1 Pr)  B30 e  (1 Sc )  e 24 Hence from equation (42), we can write B34 2  Pr  Sh   ( )   0   A1 Pr  Sc(1  A1 )   {Sc( B1  B2Y3  [ B40  B41   ]e  [ B39  B35 2 (38)  B3  B5 )  B6  Pr( B3  B5 )  B4  B36 ]e  (1 Pr)  B37 e  (2  Pr)  B38 e  (Pr  Sc ) (48)  B1 (1  Pr)  B2 (1  Sc)  A8 2 Pr  e   2 { B122 Sc  (1  Sc) B116  B97 2  (1  Pr) B117  B100  B102 (2  Pr)Z 3  B122 e  Sc  ( B116  B97 )e  (1 Sc )  B103 (2  Sc)  B104 (Pr  Sc) (39)  ( B117  B100 )e  (1 Pr)  B102 e  (2  Pr)  2 B105 Pr  2 B106 Sc  B118 Pr  B119 }  B103e  (2  Sc )  B104 e  (Pr  Sc )  B105e 2 Pr   B106 e 2 Sc  ( B118  B119  B110 2 )e  Pr  IV. RESULTS AND DISCUSSION To observe the physical situation of the problemUsing equations (16) to (18) in equation (11) with the under study, the velocity field, temperature field,help of equations (31) to (39) we have obtained the concentration field, skin-friction, rate of heat transfer andvelocity, the temperature and concentration fields as rate of mass transfer are discussed by assigning numericalfollows values to the parameters encountered into the correspondingVelocity distribution equations. To be realistic, the values of Schmidt number (Sc) are chosen for hydrogen (Sc=0.22), helium (Sc=0.30),u  U 0 f ( )  U 0 f w2 X ( ) (40) water-vapor (Sc=0.60), ammonia (Sc=0.78), carbondioxeid  U 0 [ X 1( )   X 2 ( )   2 X 3 ( )]   (Sc=0.96) at 25°C and one atmosphere pressure. The valuesTemperature distribution of Prandtl number (Pr) are chosen for air (Pr=0.71), ammonia (Pr=0.90), carbondioxied (Pr=2.2), water ( )  f w2Y ( ) (41) (Pr=7.0). Grashof number for heat transfer is chosen to be  Y1 ( )   Y2 ( )   Y3 ( ) 2 Gr=10.0, 15.0, -10.0, -15.0 and modified Grashof number for mass transfer Gm=15.0, 20.0, -15.0, -20.0. The valuesConcentration distribution Gr>0 and Gm>0 correspond to cooling of the plate while the ( )  f w Z ( )  Z1 ( )  Z 2 ( )   2 Z 3 ( ) 2 (42) value Gr<0 and Gr<0 correspond to heating of the plate.The main quantities of physical interest are the local skin- The values of magnetic parameter (M=0.0, 0.5, 1.5,friction, local Nusselt number and the local Sherwood 2.0), suction parameter (fw =3.0, 5.0, 7.0, 9.0), Soret number (S0=0.0, 2.0, 4.0, 8.0) and heat source parameter (S=0.0, 4.0,number. The equation defining the wall skin-friction as 8.0, 12.0) are chosen arbitrarily.  u  (43) The velocity profiles for different values of the     y    y 0 above parameters are presented from Fig.-1 to Fig.-11. All the velocity profiles are given against η.Thus from equation (40) the skin-friction may be written In Fig.-1 the velocity distributions are shown foras different values of magnetic parameter M in case of cooling   f ( )   0  [1   {1  A7  2 A2  A8 pr 2 of the plate (Gr>0). In this figure we observe that the velocity decreases with the increases of magnetic parameter.  A9 Sc 2 }   2 {B9  2 B32  B24 The reverse effect is observed in the Fig.-2 in case (44)  B7  2 A2  2 B18 Pr  Pr 2 ( B25 of heating of the plate (Gr<0). The velocity distributions are given for different values of heat source parameter S in  B26 ) 2 B21 Sc  Sc 2 ( B27  B28 ) case of cooling of the plate in the Fig.-3. From this figure 3 we further observe that velocity decreases with the increases  B29 (1  Pr) 2  B30 (1  Sc) 2  }] of heat source parameter. The opposite phenomenon is 8 observed in case of heating of the plate in Fig.-4. In Fig.-5The local Nusselt number denoted by Nu and defined as the velocity distributions are depicted for different values of  T  (45) suction parameter fw in case of cooling of the plate. Here weNu     y  see that velocity decreases with the increases of suction   y 0 parameter. The reverse effect is seen in case of heating ofTherefore using equation (41), we have the plate in Fig.-6.Nu   ( )   0   Pr   ( A11  A12 )   2 [( B41 The Fig.-7 represents the velocity distributions for (46) different values of Schmidt number Sc in case of cooling of  Pr B40  (1  Pr)( B39  B35 ) the plate. By this figure it is noticed that the velocity for  B36  B37 (2  Pr)  B38 (Pr ammonia (Sc=0.78) is less than that of hydrogen (Sc=0.22)  Sc)  A8 Pr] decreases with the increases of Schmidt number. The reverse effect is obtained in case of heating of the plate inThe local Sherwood number denoted by Sh and defined Fig.-8. The Fig.-9 exhibits the velocity distributions for www.ijmer.com 78 | Page
  • International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.3, Issue.1, Jan-Feb. 2013 pp-75-82 ISSN: 2249-6645different values of Soret number S0 in case of cooling of the the skin-friction in comparison to their absence. Theplate. Through this figure we conclude that the velocity increase of M, Sc and fw decreases the skin-friction while anincreases increasing values of Soret number. The opposite increase in Pr, Gr and Gm increase the skin-friction in thephenomenon is seen in case of heating of the plate from absent of S and S0.Fig.-10. Table-2 represents the skin-friction for heating of In Fig.-11 the velocity distributions are presented the plate and in this table it is clear that all the reversesfor different values of M, S, Sc, Pr and fw in both case of phenomena of the Table-1 are happened.cooling and heating of the plate where S0 =0.0 and U0=1.0. The numerical values of the rate of heat transfer in It is noted that for externally cooled plate (a) an terms of Nusselt number (Nu) due to variation in Pandltincrease in M decreases the velocity field is observed by number (Pr), heat source parameter (S) and suctioncurves (i) and (ii); (b) an increase in fw decreases the parameter (fw) is presented in the Table-3. It is clear that invelocity field observed by curves (i) and (iii); (c) an increase the absence of heat source parameter the increase in Prin S decreases the velocity field observed by curves (iii) and decreases the rate of heat transfer while an increase in fw(iv); (d) an increase in Sc decreases the velocity field increases the rate of heat transfer. In the presence of heatobserved by curves (i) and (v); (e) an increase in Pr source parameter the increasing values of fw also increasesdecreases the velocity field is seen in the curves (i) and (vi); the rate of heat transfer while an increase in Pr decreases the(f) all these effects are observed in reverse order for rate of heat transfer. It is also clear that the rate of heatexternally heated plate. All the velocity profiles attain their transfer decreases in presence of S than the absent of S.peak near the surface of the plate and then decrease slowly Table-4 depicts the numerical values of the rate ofalong y-axis. i.e. far away from the plate. It is clear that all mass transfer in terms of Sherwood number (Sh) due tothe profiles for velocity have the maximum values at η=1. variation in Schmit number (Sc), heat source parameter (S), The temperature profiles are displayed from Fig.- Soret number (S0), suction parameter (fw) and Pandlt number12 to Fig.-14 for different values of Pr, S and fw against η. We see in the Fig.-12 the temperature distributions (Pr). An increase in Sc or Pr increases the rate of massfor different values of Prandtl number Pr. In this figure we transfer in the presence of S and S0 while an increase in fwconclude that the temperature decreases with the increases decreases the rate of mass transfer.of Prandtl number. This is due to fact that there would be adecrease of thermal boundary layer thickness with the V. FIGURES AND TABLESincrease of prandtl number. In Fig.-13 displays thetemperature distributions for different values of heat sourceparameter S. In this figure it is clear that the temperaturefield increases with the increases of heat source parameter. Fig.-14 is the temperature distributions for differentvalues of Pr, S, fw for Gr=10.0, Gm=15.0, M=0.0, S0 =0.0and Sc=0.22. In this figure it is obtained that an increases inPr and S decreases the temperature field but an increases inof fw increases the temperature field. The profiles for concentration are presented fromFig.-15 to Fig.-17 for different values of S0, Sc and fw. The Fig.-15 demonstrates the concentrationdistributions for different values of Soret number. It isnoticed that the concentration profile increases with theincreases of Soret number. Fig.-16 gives the concentrationdistributions for different values of suction parameter fw. We Fig.-1. Velocity profiles when Gr= 10.0, Gm= 15.0,are also noticed that the effects of increasing values of f w S=0.0, S0 =0.0, Sc=0.22, Pr=0.71, fw=5.0 and U0=1.0decrease the concentration profile in the flow field. In Fig.- against η.17 the concentration distributions are shown for differentvalues of Sc, S0 and fw for Gr=10.0, Gm=15.0, M=0.0, S=2.0and Pr=0.71. In this figure it is seen that the concentrationfield increases with the increases of Schmidt number Sc. The concentration field also increases with theincreases of Soret number S0. The concentration fielddecreases with the increases of suction parameter fw. All theconcentration profiles reach to its maximum values near thesurface of the vertical plate i.e. η=1 and then decreaseslowly far away the plate i.e. as η→∞ The numerical values of skin-friction (τ) at theplate due to variation in Grashof number (Gr), modifiedGrashof number (Gm), heat source parameter (S), Soretnumber (S0), magnetic parameter (M), Schmit number (Sc),suction parameter (fw), and Prandtl number (Pr) forexternally cooled plate is given in Table-1. It is observedthat both the presence of S and S0 in the fluid flow decrease www.ijmer.com 79 | Page
  • International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.3, Issue.1, Jan-Feb. 2013 pp-75-82 ISSN: 2249-6645Fig.-2. Velocity profiles when Gr= -10.0, Gm= -15.0, Fig.-6. Velocity profiles when Gr= -10.0, Gm= -15.0,S=0.0, S₀ =0.0, Sc=0.22, Pr=0.71, fw=5.0 and U₀=1.0 M=0.0, S0 =0.0, S=0.0, Sc=0.22, Pr=0.71 and U0=1.0 against η against η. Fig.-3. Velocity profiles when Gr=10.0, Gm= 15.0,M=0.0, S0=0.0, Sc=0.22, Pr=0.71, fw=5.0 and U0=1.0 Fig.-7. Velocity profiles when Gr= 10.0, Gm= 15.0, against η. M=0.0, S0 =0.0, S=0.0, fw =3.0, Pr=0.71 and U0=1.0 against η.Fig.-4. Velocity profiles when Gr=-10.0, Gm=-15.0, Fig.-8. Velocity profiles when Gr= -10.0, Gm= -15.0,M=0.0, S0=0.0, fw=5.0, Sc=0.22, Pr=0.71 and U0=1.0 M=0.0, S0=0.0, S=0.0, fw =3.0, Pr=0.71 and U0=1.0 against η. against ηFig.-5. Velocity profiles when Gr= 10.0, Gm= 15.0,M=0.0, S0 =0.0, S=0.0, Sc=0.22, Pr=0.71 and U0=1.0 Fig.-9. Velocity profiles when Gr= 10.0, Gm= 15.0, against η. M=0.0, S=0.0, Sc =0.22, fw =5.0, Pr=0.71 and U0=1.0 against η. www.ijmer.com 80 | Page
  • International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.3, Issue.1, Jan-Feb. 2013 pp-75-82 ISSN: 2249-6645 Fig.-10. Velocity profiles when Gr= -10.0, Gm= -15.0, Fig.-14. Temperature profiles when Gr=10.0, Gm=15.0, M=0.0, S=0.0, fw=5.0 , Sc =0.22 M=0.0, S0 =0.0 and Sc=0.22 against η.Fig.-11. Velocity profiles for different values of M, S, Sc, Fig.-15. Concentration profiles when Gr=10.0, Gm=15.0, Pr and fw when S0 =0.0 and U0=1.0 in both heating and S=0.0, M=0.5, Sc=0.22, Pr=0.71 and fw=5.0 against η cooling plate against η.Fig.-12. Temperature profiles when Gr=10.0, Gm=15.0, Fig.-16. Concentration profiles when Gr=10.0, Gm=15.0, M=0.5, S=1.0, S0 =4.0, Sc=0.22 and fw=3.0 against η. S=4.0, M=0.0, S0=4.0, Pr=0.71 and Sc=0.22 against ηFig.-13. Temperature profiles when Gr=10.0, Gm=15.0,M=0.0, S0 =1.0, Sc=0.22, Pr= 0.71 and fw=3.0 against η. Fig.-17. Concentration profiles for different values of Sc, S₀ and fw when Pr=0.71, M=0.0 and S =2.0 against η. Table-1: Numerical values of Skin-Friction (τ) due to cooling of the plate. S. Gr Gm S S0 M Sc fw Pr Τ No 1. 10.0 15.0 0.0 0.0 0.5 0.22 5 0.71 3.9187 2. 15.0 15.0 0.0 0.0 0.5 0.22 5 0.71 4.3051 3. 10.0 20.0 0.0 0.0 0.5 0.22 5 0.71 4.6694 4. 10.0 15.0 4.0 0.0 0.5 0.22 5 0.71 3.7917 5. 10.0 15.0 0.0 2.0 0.5 0.22 5 0.71 4.6313 6. 10.0 15.0 0.0 0.0 2.0 0.22 5 0.71 3.2845 7. 10.0 15.0 0.0 0.0 0.5 0.60 5 0.71 2.5551 8. 10.0 15.0 0.0 0.0 0.5 0.22 7 0.71 2.5571 9. 10.0 15.0 0.0 0.0 0.5 0.22 5 0.90 4.1025 www.ijmer.com 81 | Page
  • International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.3, Issue.1, Jan-Feb. 2013 pp-75-82 ISSN: 2249-6645 Table-2: Numerical values of Skin-Friction (τ) due to parameter decreases the skin-friction. While an increase cooling of the plate of Grashof number, Modified Grashof number and S. Gr Gm S S0 M Sc fw Pr τ Prandtl number increases the skin-friction. In case of No. heating of the plate (Gr<0) all the effects are in reverse 1. -10 -15 0.0 0.0 0.5 0.22 5 0.71 -2.131 -15 -15 0.0 0.0 0.5 0.22 5 0.71 -2.517 order. 2. 3. -10 -20 0.0 0.0 0.5 0.22 5 0.71 -2.881 (6) The rate of heat transfer or Nusselt number (Nu) 4. -10 -15 4.0 0.0 0.5 0.22 5 0.71 -2.004 increase with the increase of suction parameter while an 5. -10 -15 0.0 2.0 0.5 0.22 5 0.71 -2.843 increase in Prandtl number it decreases. It also 6. -10 -15 0.0 0.0 2.0 0.22 5 0.71 -1.809 -10 -15 0.0 0.0 0.5 0.60 5 0.71 -0.767 decreases in presence of S in comparison to absence of 7. 8. -10 -15 0.0 0.0 0.5 0.22 7 0.71 -0.662 S. 9. -10 -15 0.0 0.0 0.5 0.22 5 0.90 -2.314 An increase in Schmidt number or Prandtl number increases the rate of mass transfer (Sherwood number) in theTable-3: Numerical values of the Rate of Heat Transfer (Nu) presence of heat source parameter and Soret number while S. No. Pr S fw Nu an increase in suction parameter fw decreases the rate of 1. 0.71 0.0 3 -1.3753 mass transfer. We also see that both the increase in source 2. 0.71 4.0 3 -1.4886 parameter and Soret number increases the rate of mass 3. 7.0 0.0 3 -7.2539 4. 7.0 4.0 3 -7.6531 5. 0.71 0.0 5 -0.8068 REFERENCES [1] M. A. Hossain and A. C. Mandal, Journal of Physics 6. 0.71 4.0 5 -0.9232 D: Applied Physics, 18, 1985, 163-9 [2] H. Poonia and R. C. Chaudhary, Theoret. AppliedTable-4: Numerical values of the Rate of Mass Transfer (Sh) Mechanics, 37 (4), 2010, 263-287 [3] M. S. Alam and M. M. Rahaman, Int. J. of Science S. Pr S S0 fw Sc Sh and Technology, II (4), 2006 No. [4] P.R. Sharma and G. Singh, Int. J. of Appl. Math and 1. 0.71 0.0 0.0 3 0.22 -0.5214 Mech, 4(5), 2008, 1-8 2. 0.71 4.0 2.0 3 0.22 7.4912 [5] R. C. Chaudhary and J. Arpita, Roman J. Phy., 52( 5- 3. 0.71 0.0 2.0 3 0.22 6.5525 7), 2007, 505-524 4. 0.71 4.0 2.0 3 0.60 111.935 [6] B. P. Reddy and J. A. Rao, Int. J. of Applied Math and 5. 0.71 4.0 2.0 5 0.22 1.0846 Mech. 7(8), 2011, 78-97 6. 7.0 4.0 2.0 3 0.22 662.888 [7] A. K. Singh, A. K. Singh and N.P. Singh, Bulletin of 0.71 8.0 4.0 3 0.22 25.3419 the institute of mathematics academia since, 33(3), 7. 2005 [8] A. K. Singh, J. Energy Heat Mass Transfer, 22, 2000, VI. CONCLUSION 41-46 In the present research work, combined heat and [9] B. Gebhart, Heat Transfer. New York: Mc Graw-Hillmass transfer effects on MHD free convection flow past a Book Co., 1971flat plate is presented. The results are given graphically to [10] B. Gebhart and L. Pera, Ind. J. Heat Mass Transfer, 14,illustrate the variation of velocity, temperature and 1971, 2025–2050concentration with different parameters. Also the skin- [11] A. A. Raptis and V. M. Soundalgekar, ZAMM, 64,friction, Nusselt number and Sherwood number are 1984, 127–130presented with numerical values for various parameters. In [12] A. K. Agrawal, B. Kishor and A. Raptis, Warme undthis study, the following conclusions are set out Stofubertragung, 24, 1989, 243–250(1) In case of cooling of the plate (Gr>0) the velocity [13] B. K. Jha and A. K. Singh, Astrophysics. Space decreases with an increase of magnetic parameter, heat Science, pp. 251–255 source parameter, suction parameter, Schmidt number [14] B. K. Jha and R. Prasad, J. Math Physics and Science, and Prandtl number. On the other hand, it increases 26, 1992, 1–8 with an increase in Soret number. [15] M. D. Abdusattar, Ind. J. Pure Applied Math., 25,(2) In case of heating of the plate (Gr<0) the velocity 1994, 259–266 increases with an increase in magnetic parameter, heat [16] V. M. Soundalgekar, S. N. Ray and U. N Das, Proc. source parameter, suction parameter, Schmidt number Math. Soc, 11, 1995, 95–98 and Prandtl number. On the other hand, it decreases [17] M. Acharya, G. C. Das and L. P. Singh, Ind. J. Pure with an increase in Soret number. Applied Math., 31, 2000, 1– 18(3) The temperature increase with increase of heat source [18] A. Bejan and K. R. Khair, Int. J. Heat Mass Transfer, parameter and suction parameter. And for increase of 28, 1985, 909–918 Prandtl number it is vice versa. [19] A. K. Singh, J. Energy Heat Mass Transfer, 22, 2000,(4) The concentration increases with an increase of Soret 41–46 number and Schmidt number. Whereas it decreases with [20] M. S. Alam and M. M. Rahman, BRACK University J. an increase of suction parameter. II (1), 2005, 111-115(5) In case of cooling of the plate (Gr>0) the increase of [21] N. P. Shingh, A. K. Shingh and A. K. Shingh, the magnetic parameter, Schmidt number and suction Arabian J. For Science and Engineering, 32(1), 2007. www.ijmer.com 82 | Page