Radiation Effects on MHD Free Convective Heat and Mass Transfer Flow Past a Vertical Porous Flat Plate with Suction

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1 International Journal o Science, Engineering and Technology Research (IJSETR), Volume 3, Issue 5, May 4 Radiation Eects on MHD Free Convective Heat and Mass Transer Flow Past a Vertical Porous Flat Plate with Suction R. Chandrasekhar Reddy, K. Jayarami Reddy, S.V. Subrahmanyam 3, M. Suryanarayana Reddy 4 Researach Scholar, JNTUA, Anantapur, A.P., India,3 Department o Mathematics, K.L. University, Guntur-55, A.P., India. 4 Department o Mathematics, JNTU College o Engineering, Pulivendula, A.P., India Abstract:- The present paper deals with the radiation eects on MHD ree convective heat and mass transer low past a vertical porous lat plate with suction. The governing equations are transormed by using similarity transormation and the resultant dimensionless equations are solved numerically using the Runge-Kutta ourth order method with shooting technique. The eects o various governing parameters on the velocity, temperature, concentration, skin-riction coeicient, Nusselt number and Sherwood number are computed and discussed in detail. Key words : MHD, Heat and Mass Transer, Suction, Porous medium, Radiation. I. INTRODUCTION Study o luid low in porous medium is based upon the empirically determined Darcy s law. Such lows are considered to be useul in diminishing the ree convection, which would otherwise occur intensely on a vertical heated surace. In addition, recent developments in modern technology have intensiied more interest o many researchers in studies o heat and mass transer in luids due to its wide applications in geothermal and oil reservoir engineering as well as other geophysical and astrophysical studies. Yamamoto et al [] investigated the acceleration o convection in a porous permeable medium along an arbitrary but smooth surace. Suction/blowing on convective heat transer over a vertical permeable surace embedded in a porous medium was analyzed by Cheng []. Raptis and Singh [3] studied low past an impulsively started vertical plate in a porous medium by a inite dierence method. Kim and Vaai [4] have analyzed the buoyancy driven low about a vertical plate or constant wall temperature and heat lux. Sattar [5] obtained analytical solution by the perturbation technique adopted by Singh and Dikshit [6]. Sattar et al [7] studied unsteady ree convection low along a vertical porous plate embedded in a porous medium. Some important applications o MHD low with heat and mass transer are cooling o nuclear reactors, liquid metals luid, power generation system and aerodynamics. MHD inds applications in electromagnetic pumps, crystals growing, MHD couples and bearing, plasma jets and chemical synthesis. The eects o magnetic ield on ree convection low o electrically conducting luids past a plate has been studied by many authors such as Soundalgekar [8]. Raptis [9] studied mathematically the case o unsteady two-dimensional natural convective heat transer o an incompressible, electrically conducting viscous luid via a highly porous medium bound by an ininite vertical porous plate. Kim [] studied unsteady MHD convection low o polar luids past a semi-ininite vertical-moving porous plate in a porous medium. Ahmed [] looked the eects o unsteady ree convective MHD low through a porous medium bounded by an ininite vertical porous plate. Chaudhary and Arpita Jain [] have discussed the MHD heat and mass diusion low by natural convection past a surace embedded in a porous medium. All the above investigations are restricted to MHD low and heat transer problems only. However, o late, the radiation eects on MHD low and heat transer problems have become more important, industrially. At high operating temperature, radiation eects can be quite signiicant. Radiation eect on mixed convection along a isothermal vertical plate were studied by Hossain and Takhar [3]. The radiation eects on boundary layer low with and without applying a magnetic ield under dierent situations has been studied by many ISSN: All Rights Reserved 4 IJSETR 498

2 International Journal o Science, Engineering and Technology Research (IJSETR), Volume 3, Issue 5, May 4 investigators [4-6]. Prasad et al. [7] studied the radiation and mass transer eects on unsteady MHD ree convection low past a vertical porous plate embedded in porous medium. Mohammed Ibrahim et.al. [8] proposed the radiation and chemical reaction eects on MHD ree convection low past a moving vertical plate. In many engineering and physical problems, it is highly important to study the eect o heat generation and absorption. Possible heat generation eect may alter the temperature distribution; consequently, the particle deposition rate in nuclear reactors, electronic chips and semi conduction waers. Rahman and Sattar [9] presented magnetohydrodynamic convective low o a micropolar luid past a vertical porous plate in the presence o heat generation/absorption. Sharma and Singh [] have discussed the unsteady MHD ree convective low and heat transer along a vertical porous plate with variable suction and internal heat generation. Sharma et al. [] have analyzed the heat and mass transer eects on unsteady MHD ree convective low along a vertical porous plate with internal heat generation and variable suction. The present paper investigates the radiation eects o an unsteady electrically conducting, viscous, incompressible luid interaction with heat generation on a ree convective low past a vertical porous lat plate embedded in a porous medium with suction in presence o heat and mass transer permitted by a transversely applied uniorm magnetic ield. Skin-riction coeicient, Nusselt number and Sherwood number are also discussed. The similarity solutions are then obtained numerically or various parameters entering into the problem and discussed them rom the physical point o view. T() t T and C() t C. The plate is considered to be o ininite length, all derivatives with respect to x vanish and so the physical variables are unctions o y and t only. The luid is assumed to have constant properties except that the inluence o the density variations with temperature and concentration, which are considered only in the body orce term, and is considered to be gray, absorbing emitting radiation but non-scattering medium and the Rosseland approximation is used to describe the radioactive heat lux in the energy equation. A uniorm magnetic ield o strength B is applied normal to the plate parallel to y-direction. Under the usual boundary layer and Boussinesq approximation and using the Darcy-Forchhemier model, the low and heat transer in the presence o radiation are governed by the ollowing equations. Continuity Equation v y Momentum Equation B t y y k k u u u b v g T T g C C u u u Energy Equation q T T T r v T T t y y cp y cp Q () () (3) II. MATHEMATICAL FORMULATION Concentration Equation Let us consider the problem o an unsteady MHD ree convection low o a viscous, incompressible and electrical conducting luid along a vertical porous lat plate under the inluence o a uniorm magnetic ield. The low is assumed to be in the x direction, which is taken along the plate in the upward direction and y axis normal to the plate. Initially it is assumed that the plate and the luid are at a constant temperature T in a stationary condition with concentration level C at all points. At time t the plate is assumed to be moving in the upward direction with the velocity U(t) and there is a suction velocity v () t taken to be a unction o time, the temperature o the plate raised to T(t) and the concentration level at the plate is raised to C(t) where ISSN: t y y C v C D C m All Rights Reserved 4 IJSETR where u and v are the velocity components along x- and y- directions respectively, t is time, is the kinematic viscosity, is the density o the luid, g is the (4) acceleration due to gravity, is the coeicient o volume expansion, is the volumetric coeicient o expansion with concentration, is the thermal diusivity, is the electric conductivity, B is the uniorm magnetic ield induction, T and T are the temperature o the luid within the boundary layer and in the ree stream respectively, while C and C are the 499

3 International Journal o Science, Engineering and Technology Research (IJSETR), Volume 3, Issue 5, May 4 corresponding concentrations, cp is the speciic heat at constant pressure, k is the permeability o the porous medium, Q is the heat generation and D m is the coeicient o mass diusivity. Initially ( t = ) the luid and the plate are at rest. Thus the no slip boundary conditions at the surace o the plate or the above problem or t > are: u U( t), v v ( t), T T ( t), C C ( t) at y = u, T T, C C as y (5) By using Rosseland approximation q r takes the orm q r 4 T 3k y 4 where is the Stean-Boltzmann constant and k is the mean absorption coeicient. It is assumed that the temperature dierences within the low are suiciently 4 small such that T may be expressed as a linear unction 4 o temperature. This is accomplished by expanding T in a Taylor series about T and neglecting higher-order terms, thus (6) T 4T T 3T (7) Using (6) and (7) in equation (3) we have 4 T 3 T T T T v T T t y y cpk y cp Q (8) In order to obtain a similarity solution in time o the problem, we introduce a similarity parameter as () t (9) such that is a length scale. With this similarity parameter, a similarity variable is then introduced as y () In terms o this length scale, a convenient solution o the equation () can be taken as v v() t v () where v is the mass transer parameter, which is positive or suction and negative or injection. Following Samad and Rahman [], we see that U(t), T(t) and C(t) are now considered to have the ollowing orm: U () t U n T () t T T T n C() t C C C n () where n is a non-negative integer and, U, T and C are respectively the ree stream velocity, mean temperature and concentration. Here, where is the value o at t t. Now to make the equations (), (4) and (8) dimensionless, we introduce the ollowing transormations: u U ( t) ( t) U ( ) T T T T n n C C C C n ( ) ( ) (3) Using equations (9), () and (3) the equations (), (4) and (8) are become (using the analysis o Hashimoto [3], Sattar et al. [4] and Sattar and Maleque [5] ) Fs v n M Gr Gc M Da ( ) 4 4 (4) 3R Pr nr Pr 3R Pr ( v ) Q (5) 3R 4 3R 4 3R 4 ( v ) Sc 4nSc (6) where number, g( T T ) Gr is the local Grasho U Grasho number, g ( C C ) Gc is the modiied U B M is the local magnetic ISSN: All Rights Reserved 4 IJSETR 5

4 International Journal o Science, Engineering and Technology Research (IJSETR), Volume 3, Issue 5, May 4 b ield parameter, Fs is the Forchhemier number, n b Fs Re is the modiied Forchhemier v number, Re is the local Reynolds number, k Da is the Darcy number, Pr is the Prandtl k number, R is the thermal radiation parameter, 3 4 T ( is the thermal conductivity), generation parameter and number. Q Q c is the heat m p Sc is the Schmidt D The corresponding boundary conditions or t > are given by,, at,,, as (7) A. Skin-Friction Coeicient, Nusselt Number And Sherwood Number The parameters o engineering interest or the present problems are the skin-riction coeicient, local Nusselt number and local Sherwood number which indicate physically wall shear stress, rate o heat transer and rate o mass transer respectively. The skin-riction coeicient is given by C Re x (), the local Nusselt number may be written as Nux Re x () and the local Sherwood number may be written as Shx Re x () (8) (9) () Thus the values proportional to the skin-riction coeicient, Nusselt number and Sherwood number are (), () and () respectively. III. NUMERICAL COMPUTATION The numerical solutions o the nonlinear dierential equations (4) (6) under the boundary conditions (7) have been perormed by applying ourth order Runge- Kutta iteration technique along with shooting method. We have chosen a step size o. to satisy the 6 convergence crite- rion o in all cases. The value o was ound to each iteration loop by. The maximum value o to each group o parameters v, Gr, Gc, M, Da, Fs, n, Pr, R, Q, and Sc determined when the value o the unknown boundary conditions at not change to successul loop with error less than 6. Figures - show the velocity, temperature and concentration proiles or dierent step sizes respectively considering Gr., Gc 6., M.5, Da.5, Fs.,Pr.7, R.5, Q.5, Sc.6, n. and v =.5. IV. RESULTS AND DISCUSSION For the purpose o discussing the results, the numerical calculations are presented in the orm o nondimensional velocity temperature and concentration proiles. Numerical computations have been carried out or dierent values o the magnetic ield parameter (M), radiation parameter (R), suction parameter ( v ), modiied Forchhemier number (Fs), and Schmidt number (Sc). The eect o radiation parameter R on the velocity proiles is shown in Figure. This igure shows that velocity decreases with the increase o the radiation parameter R. Figure. show the eect o radiation parameter R on the temperature proiles. For large R, it is clear that temperature decreases more rapidly with the increase o radiation parameter R. thereore using radiation we can control the low characteristic and temperature distribution. The eect o magnetic ield parameter on the velocity proiles are shown in Figure. 3. It is observed rom this igure that the magnetic ield has decreasing eect on the velocity ield increases. There is no outcome on the temperature and concentration proiles due the distinction o the values o magnetic ield parameter M. ISSN: All Rights Reserved 4 IJSETR 5

5 International Journal o Science, Engineering and Technology Research (IJSETR), Volume 3, Issue 5, May 4 Figures 4 & 5 display the eects o the suction parameter v on the velocity, temperature and concentration proiles respectively. It is observed that, when suction v increases, all the proiles i.e. velocity, temperature and concentration are decrease. The eect o the modiied Forchhemier number Fs on the velocity ield is shown on Figure. 6. It is observed rom this igure that modiied Forchhemier number has slightly decreasing eect on the velocity ield. The inluence o the Schmidt number Sc on the dimensionless velocity ( ) and concentration ( ) proiles are plotted in Figs. 7 and 8 respectively. As the Schmidt number Sc increases the concentration decreases. This causes the concentration buoyancy eects to decrease yielding a reduction in the luid velocity. The reductions in the velocity and concentration proiles are accompanied by simultaneous reductions in the velocity and concentration boundary layers. These behaviors are clear rom Figs. 7 and 8. Finally, the eects o various parameters on the skin-riction coeicient C, local Nusselt number Nu and local Sherwood number Sh are shown in Tables 7. Tabel.. Skin-riction coeicient, local Nuselt number and local Sherwood number or dierent values o M. M C Nu Sh Tabel.. Skin-riction coeicient, local Nuselt number and local Sherwood number or dierent values o Da Da C Nu Sh Tabel. 3. Skin-riction coeicient, local Nuselt number and local Sherwood number or dierent values o n N C Nu Sh Tabel. 4. Skin-riction coeicient, local Nuselt number and local Sherwood number or dierent values o R R C Nu Sh Tabel. 5. Skin-riction coeicient, local Nuselt number and local Sherwood number or dierent values o Q Q C Nu Sh Tabel. 6. Skin-riction coeicient, local Nuselt number and local Sherwood number or dierent values o Sc Sc C Nu Sh Tabel. 7. Skin-riction coeicient, local Nuselt number and local Sherwood number or dierent values o v v C Nu Sh V. CONCLUSIONS In this paper we have investigated the thermal radiation interaction with unsteady MHD ree convective heat and mass transer low past a vertical porous lat plate embedded in porous medium under the inluence o heat source. From the present study we can make the ollowing conclusions: Radiation has signiicant eects on the velocity as well as temperature distributions. i.e. velocity and temperature proiles reduce with the increase o thermal radiation. Magnetic ield has signiicant eect on velocity ield and retards the motion o the luid. Using suction boundary layer growth can be controlled. Suction stabilizes the hydrodynamic, thermal as well as concentration boundary layers growth. The skin-riction coeicient, local Nusselt number and local Sherwood number increase with an increase o suction parameter or constant parameter. ISSN: All Rights Reserved 4 IJSETR 5

6 International Journal o Science, Engineering and Technology Research (IJSETR), Volume 3, Issue 5, May 4 As radiation increases, the skin-riction coeicient and local Nusselt number are also increase. The skin-riction coeicient and local Nusselt number are reduces with an increase o heat generation parameter. REFERENCES. Yamamoto, K. and Iwamura, N., Flow with Convection Acceleration Through a Porous Medium, J. Eng. Math., Vol., No., pp.4-54, Cheng, P., The inluence o lateral mass lux on ree convection boundary layers in a saturated porous medium, Int.J. Heat Mass Transer, Vol.,pp. - 6, Raptis. A and Singh. A. K., Free convection low past an impulsively started vertical plate in a porous medium by inite dierence method. Astrophys. Space Sci., Vol., pp , Kim, S. and Vaai, K., Analysis o natural convection about a vertical plate embedded in a porous medium, Int. J. Heat Mass Transer, Vol.3, pp , Sattar, M. A., Free and Forced Convection Flow Through a Porous Medium Near the Leading Edge, Astrophys. Space Sci., Vol.9, pp.33-38, Singh, A. K. and Dikshit, C. K., Hydromagnetic Flow Past a Continuously Moving Semi-Ininite Plate or Large Suction, Astrophys. Space Sci., Vol.48, pp.49-56, Sattar, M. A., Rahman, M. M. and Alam, M. M., Free Convection Flow and Heat Transer through a Porous Vertical Flat Plate Immersed in a Porous Medium, J. Energy Res., Vol.(), pp.7-,. 8. Sondalgekar, V.M, Unsteady MHD ree convection low past an ininite vertical lat plate with variable suction, Indian J.Pure Appl. Math, Vol. 3, pp , A. A. Raptis, Flow through a porous medium in the presence o a magnetic ield, International Journal o Energy Research, vol., no., pp. 97, Y. J. Kim, Unsteady MHD convection low o polar luids past a vertical moving porous plate in a porous medium, International Journal o Heat and Mass Transer, vol. 44, no. 5, pp ,.. Ahmed S, Eects o unsteady ree convective MHD low through a porous medium bounded by an ininite vertical porous plate, Bull. Cal. Math. Soc, Vol. 9, pp.57-5, 7.. Chaudhary R C and Arpita Jain, MHD heat and mass diusion low by natural convection past a surace embedded in a porous medium, Theoretical Applied Mechanics, Vol. 36, No., pp.-7, Hossain, M.A and Takhar H.S, Radiation eect on mixed convection along a vertical plate with uniorm surace temperature, Heat and Mass Transer, Vol.3, pp.43-48, Israel-cookey, C.,A. Ogulu and V.B. Omubo- Pepple, Inluence o viscous dissipation and radiation on unsteady MHD ree-convection low past an Ininite heated vertical plate in a porous medium with time-dependent suction, Int.J. Heat Mass Transer, Vol. 64, pp , Mahmoud, M.A.A, Thermal radiation eects on MHD low o a micropolar luid over a stretching surace with variable thermal conductivity, Physica A, Vol. 375, pp. 4-4, Hayat, T., Z. Abbas, M.Sajid and S.Asghar, The inluence o thermal radiation on MHD low o a second grade luid, Int. J. Heat Mass Transer,Vol. 5, pp , Prasad, V. R., R. Muthucumaraswamy and B. Vasu, Radiation and Mass transer eects on unsteady MHD ree convection low past a vertical porous plate embedded in porous medium: a numerical study, Int. J. o Appl. Math and Mech., Vol. 6, No. 9, pp.,. 8. S Mohammed Ibrahim., T Sankar Reddy and N Bhaskar Reddy, Radiation and chemical reaction eects on MHD convective low past a moving vertical porous plate, International Journal o Applied Mathematical Analysis and Applications, Vol. 7, No., pp. -6,. 9. Rahman M.M. and Sattar M.A, Magneto hydrodynamic convective low o a micropolar luid past a continuously moving vertical porous plate in the presence o heat generation/absorption, ASME J. H. Transer, Vol. 8, 4 5, 6.. Sharma PR and Singh G, Unsteady MHD ree convective low and heat transer along a vertical porous plate with variable suction and internal heat generation, Int. J. Appl. Math and Mech., 4, pp. -8, 8. ISSN: All Rights Reserved 4 IJSETR 53

7 () () () International Journal o Science, Engineering and Technology Research (IJSETR), Volume 3, Issue 5, May 4. Sharma PR Dadkeech IK and Gurminder Singh, Heat and mass transer eects on unsteady MHD ree convective low along a vertical porous plate with internal heat generation and variable suction, Int. J. o Math Archive, 3(5), pp. 63-7,.. M. A. Samad and M. M. Rahman, Theramal radiation interaction and unsteady MHD low past a vertical porous plate immersed in a porous medium, Journal o Naval Architechture and Marine Engineering, Vol. 3, No., pp. 7-4, Hashimoto, H., Boundary Layer Growth on a Flat Plate with Suction or Injection, J. Phys. Soc. Japan, Vol., pp.7-, Sattar, M. A., Rahman, M. M. and Alam, M. M., Free Convection Flow and Heat Transer Through a Porous Vertical Flat Plate Immersed in a Porous Medium, J. Energy Res., Vol.(), pp.7-. Siegel, R. (958): Trans. Amer. Soc. Mech. Eng., Vol.8, pp.347,. 5. Sattar, M. A. and Maleque, M. A., Unsteady MHD natural convection low along an accelerated porous plate with hall current and mass transer in a rotating porous medium, J. Energy, Heat and Mass Transer., Vol., pp.67-7, R =.,.5,., Figure. Temperature proiles or dierent values o radiation parameter (R) M =.,.5,.,..6 R =.,.5,., Figure 3. Velocity proiles or dierent values o Magnetic ield parameter (M) Figure. Velocity proiles or dierent values o radiation parameter (R) ISSN: All Rights Reserved 4 IJSETR 54

8 () () () () International Journal o Science, Engineering and Technology Research (IJSETR), Volume 3, Issue 5, May v =., 5,.,..4 Fs =.,.5,., Figure 4. Velocity proiles or dierent values o suction parameter ( v ) Figure 6. Velocity proiles or dierent values o modiied Forchhemier number (Fs) v =.,.5,.,..4 Sc=.,.6,.78, Figure 5. Temperature proiles or dierent values o suction parameter ( v ) Figure 7. Velocity proiles or dierent values o Schmidt number (Sc) ISSN: All Rights Reserved 4 IJSETR 55

9 () International Journal o Science, Engineering and Technology Research (IJSETR), Volume 3, Issue 5, May Sc =.,.6,.78, Figure 8. Temperature proiles or dierent values o Schmidt number (Sc) ISSN: All Rights Reserved 4 IJSETR 56