Journal of dvances in pplied Mathematics Vol. No. Januar 6 https://dx.doi.org/.66/jaam.6.3 7 Influence of Thermo-Diffusion and Heat Source on MHD Free Convective Radiating Dissipative Boundar Laer of Chemicall Reacting Fluid Flow in a Porous Vertical Surface Mohammed Ibrahim Shaik * and Suneetha Karna Dept. of Mathematics Gitam Institute of Technolog GITM niversit Vishakhapatnam ndhra Pradesh 5345 India. Dept. of Mathematics Priadarshini College of Engineering & Technolog Nellore ndhra Pradesh 544 India Email: ibrahimsvu@gmail.com bstract. The article is focused on the effects of Soret and chemical reaction on MHD free convection flow through a vertical porous surface in the presence of heat source viscous dissipation and radiation. The dimensionless governing equations for this investigation are solved analticall using perturbation technique. Numerical evaluation of the numerical results is performed and the results for velocit temperature and concentration profiles within the boundar laer are discussed and shown graphicall. lso skin-friction coefficient due to velocit Nusselt number coefficient due to temperature and Sherwood number coefficient due to concentration fields are evaluated. Kewords: Boundar laer; MHD; Radiation; Viscous dissipation; Chemical reaction; Permeabilit of porous; Heat source. Introduction Owing to their numerous applications in the industrial manufacturing process the problem of heat and mass transfer in the boundar laers of a continuousl moving semi-infinite flat plate has attracted the attention of researchers for the past three decades. Some of the application areas are hot rolling stud production metal spinning drawing plastic films glass blowing continuous casting of metals and spinning of fibers. nnealing and thinning of copper wire is another example. In all these cases the qualit of the final product depends on the rate of heat and mass transfer at the moving surface. B drawing the strips in an electricall conducting fluid subjected to a magnetic field the rate of cooling can be controlled and the final products of the desired characteristic might be achieved. Sakiadis [] studied the boundar laer of viscous fluid on the continuous moving solid surface. Gribben [] investigated the MHD boundar laer flow over a semi-infinite plate with an aligned magnetic field in the presence of pressure gradient. Free convection boundar laers and the stud on a uniforml heated vertical plate was presented b Goldstein and Eckert [3]. Beard and Walters [4] proposed the two-dimensional boundar laer flow for an elastico-viscous fluid near a stagnation point. In the present ears the free convective flow of heat and mass transfer problems in the presence of magnetic field through a porous medium have attracted the attention of a number of research scholars because of their possible application in man branches of science and technolog such as fibre and granular insulation a geothermal sstem etc. In engineering sciences it finds man applications in MHD pumps MHD bearing MHD power generators etc. The effects of free convection on stokes problem for a vertical plate under the action of transversel applied magnetic field with heat and mass transfer was discussed b Soundalgekar et al.[5]. Thermal diffusion and natural convection between inclined plates in a porous medium with MHD was analzed b Raju et al. [6]. hmed [7] discovered the radiating fluid and an induced magnetic field over a porous vertical plate. Samira et al. [8] derived free convection MHD flow of an elastoviscous fluid past an infinite vertical plate. Chowdhur and Islam [9] presented the effects of heat and mass transfer on MHD free convective flow of visco-elastic fluid past an infinite vertical porous plate. Copright 6 Isaac Scientific Publishing JM
8 Journal of dvances in pplied Mathematics Vol. No. Januar 6 Radiative heat and mass transfer pla an important role in manufacturing industries for the design of fins steel rolling nuclear power plants gas turbines and various propulsion devices for aircraft missiles satellites and space vehicles are examples of such engineering applications. The effect of radiation on MHD free convective heat and mass transfer flow becomes more industriall important. Radiation effect on the natural convective flow of a gas past a semi infinite plate was studied b Soundalgekhar and Takhar []. Raju et al. [] investigated the combined effect of heat and mass transfer on a free convective flow through a porous medium bounded b a vertical surface with radiation. Radiation effect on free convective flow over a porous vertical plate was studied b Hossain et al.[]. Radiation effect on MHD free convective flow on the vertical porous plate using finite difference method was proposed b bd E Nab et al. [3]. Radiation effect on the three-dimensional flow in a vertical channel subjected to a periodic suction was discovered b Guria et al. [4]. Rajesh and Vijaa Kumar Varma [5] studied heat and mass transfer effects on radiating MHD free convection flow past an exponentiall accelerated vertical plate with variable temperature. The heat transfer in a porous medium with high porosit in the presence of radiation was analzed b Raptis [6]. Convective flows with simultaneous heat and mass transfer under the influence of magnetic field and chemical reaction arises in man natural and artificial transfer process in man branches of science and engineering applications. This phenomenon plas an important role in the chemical industr power and cooling industr for dring chemical vapor decomposition on surfaces cooling nuclear reactor and petroleum industr. The effects of chemical reaction and thermal radiation on stead MHD free convective boundar laer flow through a porous vertical surface with constant suction was investigated b Raju et al. [7]. Sata Naraana and Sravanthi [8] proposed the influence of variable permeabilit on unstead MHD convection flow past a semi-infinite inclined plate with thermal radiation and chemical reaction. The effect of the chemical reaction on the process of an unstead flow past a vertical plate with a constant heat flux was proposed b Das et al. [9]. Sata Naraana [] analzed the effects of chemical reaction and thermal radiation on MHD free convection flow past an infinite vertical plate with variable suction and heat source. The stud of heat generation or absorption effects in moving fluids is important in view of several phsical problems such as fluids undergoing exothermic or endothermic chemical reaction. Mohamed et al. [] presented the effects of radiation and chemical reaction on MHD free convective flow of a polar fluid through a porous medium with heat generation. The problem of free convective flow along a vertical wav surface embedded in a saturated porous media in the presence of internal heat generation or absorption effect was presented b Had et al. []. lam et al. [3] noticed the heat and mass transfer effects on MHD free convective flow past an inclined semi-infinite heated surface in the presence of heat generation. The heat and mass transfer effects on MHD flow over a moving permeable clinder with heat generation or absorption and the chemical reaction was discovered b Chamkha [4]. In all the studies mentioned above viscous dissipation is neglected. But viscous dissipation in free convective flow is an important factor when the flow field is of extreme size or in the high gravitational field. Viscous dissipation effect on stead free convective heat and mass transfer flow past a semi-infinite flat plate was studied b Geetha and Moorth [5]. The thermal radiation effect on hdromagnetic free convection flow past an impulsivel started vertical plate with variable surface temperature and concentration b taking into account the heat due to viscous dissipation was presented b Suneetha et al. [6]. Sata Naraana [7] reported the effects of thermal radiation and the viscous dissipation on an unstead MHD convection flow past a semi-infinite vertical permeable moving porous plate. The influence of viscous dissipation effect in natural convective flows showing that the heat transfer rates are reduced b an increase in the dissipation parameter was noticed b Mahajan et al. [8]. The Soret effect arises when the mass flux contains a term that depends on the temperature gradient. The major focus of our investigation is the effect on free convection of the addition of a second fluid. of a second additional fluid on free convection. Convection in binar fluids is considerabl more complicated than that in pure fluids. Both temperature and concentration gradient contribute to the initiation of convection and each ma be stabilizing or destabilizing. Even when a concentration gradient is not externall imposed (the thermo solutal problems) it can be created b the applied thermal gradient via the Soret effect. Mohammed Ibrahim and Suneetha [9] proposed the Soret and chemical reaction effects on MHD flow of a visco-elastic fluid through an impulsivel started infinite vertical plate in the presence of heat source. Bhavana et al. [3] studied the effect of the Soret on MHD free convective flow over a vertical plate in the presence of heat generation. Soret and chemical reaction JM Copright 6 Isaac Scientific Publishing
Journal of dvances in pplied Mathematics Vol. No. Januar 6 9 effects on hdromagnetic free convective Walter s memor flow in the presence of constant suction and heat sink was investigated b Pavan kumar et al.[3]. hmed [3] discussed the heat and mass transfer effects on Hartman flow in the presence of constant heat source and Soret. The same author hmed [33] explained the effects of Soret and radiation on transient MHD free convection from an impulsivel started infinite vertical plate. However to the best of the authors knowledge no attempt so far has been made to analze the heat source and Soret effects on a stead MHD flow of viscous fluid through a porous medium bounded b a porous surface subjected to suction with a constant viscosit in the presence of radiation and chemical reaction. Mathematical Model We consider a viscous incompressible electricall conducting and radiating fluid through a porous medium occuping a semi-infinite region of the space bounded b a vertical infinite surface. The x axis is taken along the surface in an upward direction and the axis is normal to it. uniform magnetic field B is assumed to be applied in a direction perpendicular to the surface. The properties of a fluid are assumed to be constant except for the densit in the bod force term. In addition a chemicall reactive species is assumed to be emitted from the vertical surface into a hdrodnamic flow field. It diffuses into the fluid where it undergoes a homogenous chemical reaction. The reaction is assumed to take place entirel in the stream. Then the full developed flow under the above assumptions through a porous medium is governed b the following set of equations: * v () * u u B u v g T T T gc C C u () k q Q T k T u r C C C C p p p p v T T C C T v D k C C D (4) c The relevant boundar conditions are given as follows u T T C C at w w (5) u T T C C as Eq. () gives that v constant v (6) In the opticall thick limit the fluid does not absorb its own emitted radiation in which there is no self-absorption but it does absorb radiation emitted b the boundaries. Cogle et al. [34] have shown that in the opticall thin limit for a non-gra gas near equilibrium the radiative heat flux is represented b the following form: q de r b 4T T K d 4I w T T (7) dt w On introducing the following non-dimensional quantities p (3) Copright 6 Isaac Scientific Publishing JM
Journal of dvances in pplied Mathematics Vol. No. Januar 6 u v T T C C c B u C Sc M v k D p Pr T T C C v w w g ( T T ) g ( C C ) v v k T w c w p k c 3 3 v v v C T T p w w Gr Gm E k Kr 4I Q D ( T T ) w F Q S kv c v pr ( C C ) p w the non-dimensional form of the governing Eqs. ()- (4) reduce to u u M u Gr GmC (9) Pr Pr Eu ( ) M () C ScC ScKrC SSc () where M M M F Q. k (8) The corresponding boundar conditions are given b u C at u C as () 3 Solution of the problem In order to solve the coupled nonlinear sstem of Eqs. (9)-() with the boundar conditions Eq. () the following simple perturbation is used. The governing Eqs. (9)-() are expanded in powers of Eckert number E(<). u u Eu O( E ) E O( E ) (3) C C EC O( E ) Substituting Eq. (3) into Eq. (9)-() and equating the coefficient at the terms with the same powers of E and neglecting the terms of the higher order the following equations are obtained. Zero order terms: u u M u Gr GmC (4) First order terms: Pr M (5) C ScC KrScC ScS (6) u u M u Gr GmC (7) u Pr M Pr (8) C ScC KrScC ScS (9) The corresponding boundar conditions are u u C C at () u u C C as Solving Eqs. (4)-(9) under the boundar conditions () the following solutions are obtained. m e () m m C e e m 3 m u e e e m 5 4 3 m 4 m 3 m m m3m mm mm3 e e e e e e e 6 7 8 9 () (3) (4) JM Copright 6 Isaac Scientific Publishing
Journal of dvances in pplied Mathematics Vol. No. Januar 6 m m m 4 m m 5 3 C e e e e e 3 4 5 6 m m 3 m m m m 3 e e e 7 8 9 m m m m m 6 5 4 3 u e e e e e 9 3 4 m 3 3 e e e e m m m m m m 5 6 7 8 Substituting Eqs. ()-(6) into Eq. (3) we obtain the velocit temperature and concentration distribution in the boundar laer as follows: m 3 m m t e e e 5 4 3 m m 6 5 m m 4 3 m e e e e e (7) 9 3 4 E m m m 3 m m m m 3 e e e e 5 6 7 8 e e e e m m m m 4 3 m 6 7 8 t e E e e e m m m m m m 3 3 9 m 5 m m 3 m m 4 e e e e e m m 3 4 5 6 C t e e E (9) m m 3 m m m m 3 e e e 7 8 9 The expressions for the constants in the above equations are given in the ppendix. The skin-friction Nusselt number and Sherwood number are important phsical parameters for this tpe of boundar laer flow. Knowing the velocit field the skin-friction at the plate can be obtained which in non-dimensional u form is given b m m m m m m 6 9 5 4 3 3 4 5 m m m 3 5 4 3E (3) ( m m ) ( m m ) ( m m ) 3 6 7 3 8 Knowing the temperature field the rate of heat transfer coefficient can be obtained which in the non-dimensional form in terms of the Nusselt number is given b Nu m m m m 4 6 3 7 8 Nu me m m m m m m 3 9 3 Knowing the concentration field the rate of mass transfer coefficient can be obtained which in the C non-dimensional form in terms of the Sherwood number is given bsh m m m m m 5 4 3 4 3 5 6 Sh m m E m m m m m m 4 Results and Discussion 3 7 8 3 9 In the preceding section the problem of a stead free-convective flow of a viscous incompressible thermall radiating and chemicall reacting fluid past a semi infinite plate in the presence of heat generation and Soret was formulated and solved b means of perturbation method. The expressions for the velocit temperature and concentration profiles were obtained. To illustrate the behavior of these phsical quantities numerical values of these quantities were computed with respect to the variations in the governing parameters namel thermal Grashof number Gr solutal Grashof number Gm Prandtl (5) (6) (8) (3) (3) Copright 6 Isaac Scientific Publishing JM
Journal of dvances in pplied Mathematics Vol. No. Januar 6 number Pr Schmidt number Sc radiation parameter F magnetic parameter M permeabilit of porous medium k heat generation parameter Q chemical reaction Kr and Soret number So. In the present stud the following default parametric values are adopted: Gr=5. Gm=3. M=. k=.5 Pr=.7 E=. F=.5 Q=. Sc=. Kr=. and So=.. The numerical results are demonstrated through different graphs as well as tables and their results are interpreted phsicall. Fig. indicates the fact that an increase in Grashof number Gr for heat transfer leads to a rise in the value of velocit due to enhancement in buoanc force. The plot of velocit profiles for different values of Grashof number for mass transfer Gm is given in Fig.. It is observed that velocit increases with Grashof number for mass transfer. For various values of the magnetic parameter M the velocit profiles are plotted in Fig. 3. It is obvious that the existence of the magnetic field decreases the velocit. Fig. 4 shows the velocit profiles for different values of the permeabilit of the porous medium parameter k. Clearl as k increases the peak values of the velocit tends to increase. Fig. 5(a) displas Prandtl number Pr effect on the velocit field it is noticed that velocit decreases as Prandtl number Pr increases. The influence of Prandtl number Pr on temperature field is shown in Fig. 5(b) it is observed that temperature profiles decrease as Prandtl parameter Pr increases. For different values of the radiation parameter F the velocit profiles and temperature profiles are plotted in Figs.6(a) and 6(b). The radiation parameter F defines the relative contribution of conduction heat transfer to thermal radiation transfer. It is obvious that increases in the radiation parameter F result in a decrease in the velocit and temperature within the boundar laer. The effects of heat generation parameter Q on the velocit profiles and temperature profiles are shown in Figs. 7(a) and 7(b). From these curves we see that the heat is generated the buoanc force increases which induces the flow rate to increase giving rise to the increase in the velocit and temperature profiles.. Fig. 8 depicts the effect of Schmidt number Sc on velocit field. This shows that velocit decreases as the Schmidt number Sc increases. Phsicall this is true because along with the increase in Schmidt number Sc the viscosit of the fluid increases and hence velocit decreases. similar effect is noticed in the presence of chemical reaction parameter Kr in Fig. 9 and it is quite interesting to observe the opposite reaction near the plate. Fig. shows the concentration profiles across the boundar laer for various values of Schmidt number Sc. It is found that an increase in Sc results in a decrease in the concentration distribution. Because the smaller values of Sc are equivalent to increasing the chemical molecular diffusivit. The effect of concentration is presented in Fig. and from this figure it is noticed that concentration decreases with the increase in chemical reaction parameter Kr. The effect of Soret number So on the concentration profiles is shown in Fig.. From this figure we see that the concentration profile decreases with the increase of So. The effects of various phsical parameters on skin-friction coefficient Nuselt number and Sherwood number are shown in Table. - 3. The behavior of these parameters is self-evident from Table. to 3 and hence the are not discussed an further in order for brevit..5 -- -- -- --Gr=5. ----------Gr=7....Gr=9..8.6 -- -- -- --Gm=3. -----------Gm=4....Gm=5..4.5. Gm=4M=.k=.5Pr=.7E=. F=.3Q=.5Sc=.Kr=.S=.5.8.6.5.4. Gr=7M=.k=.5Pr=.7E=. F=.3Q=.5Sc=.Kr=.S=.5 3 4 5 6 Fig. Velocit profiles for various values of Gr. 3 4 5 6 Fig.Velocit profiles for various values of Gm. JM Copright 6 Isaac Scientific Publishing
Journal of dvances in pplied Mathematics Vol. No. Januar 6 3.6.8.6.4 -- -- -- --M=. -----------M=.5...M=..4. -- -- -- --k=. ----------k=.3...k=.5..8.8.6 Gr=7Gm=4M=.Pr=.7E=. F=.3Q=.5Sc=.Kr=.S=.5.6.4. Gr=7Gm=4k=.5Pr=.7E=. F=.3Q=.5Sc=.Kr=.S=.5.4. 3 4 5 6 3 4 5 6 Fig 3. Velocit profiles for various values of M. Fig 4. Velocit profiles for various values of k..9.8 -- -- -- --Pr=.7 ----------Pr=....Pr=.5.9.8 -- -- -- --Pr=.7 ----------Pr=....Pr=.5.7.7.6.6.5 T.5.4.3.4.3 Gr=5Gm=3M=.k=.5E=. F=.5Q=.Sc=.Kr=.S=... Gr=5gm=3M=.k=.5E=. F=.5q=.Sc=.Kr=.S=. 3 4 5 6 Fig 5(a).Velocit profiles for various values of Pr... 3 4 5 6 Fig 5(b).Temperature profiles for various values of Pr..9.8 -- -- -- --F=.5 ----------F=....F=.5.9.8 -- -- -- --F=.5 ----------F=....F=.5.7.7.6.6.5 T.5.4.3.. Gr=5Gm=3M=.k=.5Pr=.7 E=.Q=.Sc=.Kr=.S=. 3 4 5 6 Fig 6(a). Velocit profiles for various values of F..4.3.. Gr=5Gm=3M=.k=.5Pr=.7E=. Q=.Sc=.Kr=.S=. 3 4 5 6 Fig 6(b).Temperature profiles for various values of F. Copright 6 Isaac Scientific Publishing JM
4 Journal of dvances in pplied Mathematics Vol. No. Januar 6.5 -- -- -- --Q=. ----------Q=.3...Q=.5.9.8 -- -- -- --Q=. ----------Q=....Q=.3.7.6 Gr=5Gm=4M=.k=.5Pr=.7 E=.F=.3Sc=.Kr=.S=. T.5.4.5.3.. Gr=5Gm=3M=.k=.5Pr=.7 E=.F=.5Sc=.Kr=.S=. 3 4 5 6 Fig 7(a). Velocit profiles for various values of Q..8 3 4 5 6 Fig 7(b). Temperature profiles for various values of Q.5.6.4 -- -- -- --Sc=. ----------Sc=....Sc=.3 -- -- -- --Kr=.5 ----------Kr=....Kr=.5..8.6.5.4. Gr=5Gm=4M=.k=.5Pr=.7 E=.F=.3Sc=.Kr=.S=.5 Gr=5Gm=4M=.k=.5Pr=.7 E=.F=.3Q=.5Sc=.S=. 3 4 5 6 Fig 8. Velocit profiles for various values of Sc. 3 4 5 6 Fig 9. Velocit profiles for various values of Kr.. -- -- -- --Sc=. ----------Sc=.5...Sc=..9.8 -- -- -- --Kr=. ----------Kr=.5...Kr=3..8.7.6 C.6 C.5.4. Gr=5Gr=3M=.k=.5Pr=.7 E=.F=.5Q=.Kr=.S=..4.3.. Gr=5Gm=3M=.k=.5Pr=.7 E=.F=.5Q=.Sc=.S=. 3 4 5 6 Fig. Concentration profiles for various values of Sc. 3 4 5 6 Fig. Concentration profiles for various values of Kr JM Copright 6 Isaac Scientific Publishing
Journal of dvances in pplied Mathematics Vol. No. Januar 6 5.9.8 -- -- -- --S=.5 ----------S=....S=.5 C.7.6.5.4.3 Gr=5Gm=3M=.k=.5Pr=.7 E=.F=.5Q=.Sc=.Kr=... 3 4 5 6 Fig. Concentration profiles for various values of So. Table : Effects of various phsical parameters on Skin friction for Pr=.7 E=. F=.5 Q=. Sc=. Kr=. So=.. Gr Gm M K Cf Nu Sh 5. 3...5 3.834.654.48 7. 3...5 4.6475.584.497 9. 3...5 5.494.5.45 5. 4...5 4.377.63.493 5. 5...5 4.94.545.45 5. 3. 3..5 3.793.77.468 5. 3. 5..5.664.745.46 5. 3....8483.733.465 5. 3...3 3.76.73.47 Table : Effects of various phsical parameters on Nusselt number for Gr=5 Gm=3 M= k=.5 Sc=. Kr= So=.. Pr E F Q Cf Nu Sh.7..5. 3.834.654.48...5. 3.646.873.37.5..5. 3.45.77.835.7..5. 3.877.55.44.7.3.5. 3.8.356.446.7..3. 3.9357.99.45.7..4. 3.8634.999.433.7..5.3 3.9357.99.45.7..5.5 4.565.689.4937 Table 3: Effects of various phsical parameters on Sherwood number for Gr=5 Gm=3 M= k=.5 Pr =.7 E=. F=.5 Q=.. Sc Kr So Cf Nu Sh... 3.834.654.48.5.. 3.66.673.637... 3.44.689.947 Copright 6 Isaac Scientific Publishing JM
6 Journal of dvances in pplied Mathematics Vol. No. Januar 6.66 3.. 3.564.68.7889.66 5.. 3.38.693.4.66. 3. 4.38.68.76.66. 5. 4.74.6.74 4 Conclusions This paper studies the effects of the Soret heat source and chemical reaction on stead MHD free convective boundar laer flow past a porous medium in the presence of thermal radiation and constant suction. The following conclusions can be drawn from the present investigation: The velocit profiles increase with the increase of buoanc forces permeabilit parameter and heat source. On the other hand it decreases with the increase of magnetic parameter and radiation parameter. The temperature distributions increase with the increase of heat source. The concentration field decreases with the increase of Schmidt number chemical reaction parameter while it increases with an increase of Soret number. It is observed that the increase in the magnetic field parameter is to decrease the skin-friction coefficient whereas the effect of increasing the value of buoanc parameter and permeabilit of porous medium is to increase the value of the skin-friction coefficient. Nusselt number decreases with the increase of heat generation parameter and Eckert number. Sherwood number increases with the increase of Schmidt number chemical reaction and Soret number. ppendix m m Sc Sc 4KrSc 4M m m m 3 S Sc 4ScKr 4M SScm m 6 m Scm KrSc Pr Pr 4M 5 Gm Gr Gm 3 4 m m M m m M Prm Prm 4 3 7 8 4m Prm M 4m Prm M Prmm 3 4 m m m m M 3 Pr 9 8 7 6 4SScm 7 m m Sc KrSc 5 4 SScm m 3 9 3 3 7 5 4 3 Pr Pr 4M 4 Prm 3 5 6 4m Prm M 3 3 Prmm 3 5 4 9 m m m m M 3 3 Prmm Pr Pr 3 5 3 m m m m M 3 3 SScm 4 m m Sc KrSc 3 4 4 4SScm 6 5 4 m m Sc m m KrSc m m Sc KrSc 4SScm 3 6 m m Sc KrSc 4 4 3 3 4SScm 6 6 4 SScm m 8 m msc KrSc m m Sc m m KrSc JM Copright 6 Isaac Scientific Publishing
Journal of dvances in pplied Mathematics Vol. No. Januar 6 7 SScm m 3 3 3 Gm 9 m m Sc m m KrSc m m M 5 5 Gr Gm 8 6 5 4m m M GmGr 3 m m M 4 4 GrGm 9 8 3 4 5 6 7 8 9 GmGr 7 9 6 m m m m M 3 3 Gr Gm 6 4 3 4m m M 3 3 m m m m M 3 3 Gr Gm 7 5 4 4m m M GmGr 8 7 m m m m M 9 3 4 5 6 7 8 References. B. C. Sakiadis Boundar laer behavior on continuous moving solid surfaces-i. boundar laer equations for two-dimensional and axis-smmetric flow IChE J Vol. 7 pp.6-8 96.. R.J. Gribben proc.roel.soclondon Vol. 87pp.3-4 965. 3. R. J. Goldstein and E. R. G Eckert The Stead and Transient Free Convection Boundar Laers on a niforml Heated Vertical Plate Int. J. Heat Mass Transfer Vol. pp.8-8 96. 4. D. M Beard and K. Walters Elastico-viscous boundar laer flows two dimensional flow near a stagnation point Proc.camber.Phil.Soc Vol. 6 pp.667-674 964. 5. V. M. Soundalgekar S. K Gupta and N. S Birajdra Nucl.Eng.Design Vol.53 pp.39-346 979. 6. M. C Raju S. V. K. Varma and N. Redd MHD thermal diffison naturel convection between inclined plates in porous medium Journal of future Engineering and Technolog Vol.6 No. pp.45-48. 7. S. hmed Induced magnetic field with radiating fluid over a porous vertical plate analtical stud J Naval rchit Mar Eng Vol. 7 No. pp. 83-94. 8. N. K Samira R. Prasad and M.. S Redd MHD free convection flow of an elastoviscous fluid past an infinite vertical plate stor. Phsics and space Science Vol. 8 pp.5-34 99. 9. M. K. Chowdhur and M. N Islam MHD free convection flow of visco-elastic fluid past an infinite vertical porous plate Heat and Mass Transfer Vol.36 pp.439.. V. Soundalgekhar and H. S Takhar Radiative convective flow past a semi-infinite vertical plate Modeling Measure and control Vol.5 pp.3-4 99.. M. C Raju S. V. K. Varma and N. Redd Radiation and mass transfer effects on a free convection flow through a porous medium bounded b a vertical surface Journal of future Engineering and Technolog Vol.7 No. pp.7-... M. Hossain M. lim and D.. S. Rees Effect of Radiation on free convection from a porous a vertical plate International Journal of Heat and Mass transfer Vol.4 pp.8-9 999. 3. M.. bd E - Nab E. M. E. EI-Barbar and N. Y. bdelazen Finite difference solution of radiation effects on MHD free convection flow on vertical porous plate ppl.math.comput Vol.5 No. pp.37-346 4. 4. M. Guria N. Ghara and R. N. Jana Radiation effect on three dimensional vertical channel flow International Journal of pplied Mechanics and Engineering Vol. 5 No. 4 pp. 65-8. 5. V. Rajesh and S. V. K. Varma Radiation and mass transfer effects on MHD free convection flow past an exponentiall accelerated vertical plate with variable temperature RPN Journal of Eng. nd pp. Sci Vol. 4 No. 6 pp. -6 9. 6.. Raptis Radiation and flow through a porous medium Journal of Porous Media Vol. 4 No. 3 pp. 7 73. 7. M. C Raju N. nanda Redd and S. V. K. Varma naltical stud of MHD free convective dissipative boundar laer flow past a porous vertical surface in the presence of thermal radiation chemical reaction and constant suction in Shams Engineering Journal Vol. 5 pp.36-369 4. Copright 6 Isaac Scientific Publishing JM
8 Journal of dvances in pplied Mathematics Vol. No. Januar 6 8. P. V. Sata Naraana and S. Sravanthi Influence of variable permeabilit on unstead MHD convection flow past a semi-infinite inclined plate with thermal radiation and chemical reaction Journal of Energ Heat and Mass Transfer Vol.34 pp 43-6. 9.. N Das R. Deka and V. M Soundalgekar Effects of mass transfer on flow past an impulsivel started infinite vertical plate with constant heat flux and chemical reaction ForschungimIngenieurwesen Vol. 6 No. pp. 84 87 994.. P. V Sata Naraana Chemical reaction and thermal radiation effects on an stead MHD free convection flow past an infinite vertical plate with variable suction and heat source or sink IJMMS Vol. 4 pp. 7-45... Mohamed S. M bo-dahab and T.. Nofal Thermal radiation and MHD effects on free convective flow of a polar fluid through a porous medium in the presence of internal heat generation and chemical reaction Mathematical Problems in Engineering Vol rticle ID 8479 7 pages.. F. M. Had R.. Mohamed and. Mahd MHD free convection flow along a vertical wav surface with heat generation or absorption effect International Communications in Heat and Mass Transfer Vol. 33 No. pp. 53 63 6. 3. M. S. lam M. M Rahman and M. Sattar MHD Free convection heat and mass transfer flow past an inclined surface with heat generation Thammasat International Journal of Science and Technolog Vol. No. 4 pp. 8 6. 4.. J Chamkha Heat and mass transfer from MHD flow over a moving permeable clinder with heat generation or absorption and chemical reaction Commun. in Numerical nalsis rticle ID cna-9 pages. 5. P. Geetha and M. B. K. Moorth Viscous dissipation effect on stead free convection and mass transfer flow past a semi-infinite flat plate Journal of Computer Science Vol. 7 No. 7 pp. 3 8. 6. S. Suneetha N. Bhaskar Redd and V. Ramachandra Prasad The thermal radiation effects on MHD free convection flow past an impulsivel started vertical plate with variable surface temperature and concentration Journal of Naval rchitecture and Marine engineering Vol. pp. 57 7 8. 7. P. V. Satanaraana D. Chenna Kesavaiah and S. Venkataramana Viscous dissipation and thermal radiation effects on stead MHD convection flow past a semi-infinite vertical permeable moving porous plate IJM 476-487. 8. R. L. Mahajan and B. B. Gebhart Viscous dissipation effects in Buoanc Induced flows Int. J. Heat Mass Transfer Vol. 3 No. 7 pp. 38 38 989. 9. S.Mohammed Ibrahim and K.Suneetha Chemical reaction and Soret effects on unstead MHD flow of a viscoelastic fluid past an impulsivel started infinite vertical plate with Heat source International Journal of Mathematics and Computational Science Vol. No. pp.5-4 5. 3. M. Bhavana D. Chenna Kesavaiah and. Sudhakaraiah The Soret effect on free convective unstead MHD flow over a vertical plate with heat source International Journal of Innovative Research in Science Engineering and Technolog Vol. No.5 pp.67-68 3. 3. C. Pavan Kumar. Rajeswara Rao and D. R. Prasad Rao Chemical reaction and thermo-diffusion effects on Hdromagentic free convective Walter s Memor flow which constant suction and heat sink International Journal of Mathematical rchive Vol.4 No.7 pp. 94-4 3. 3. N. hmed Heat and mass transfer in Hartman flow with Soret effect in presence of a constant heat source Turkish J of ph Vol. 33 pp.446-46. 33. N. hmed Soret and radiation effects on transient free convection from an impulsivel started infinite vertical plate Journal of Heat transfer (SME) Vol. 34 pp.-9. 34.. C. Cogle W. G. Vincenti and S. E. Gill Differential approximation for radiative transfer in a non-gra gas near equilibrium I J Vol. 6 pp. 55-553 968. JM Copright 6 Isaac Scientific Publishing