Magneto Convective Flow of a Non-Newtonian Fluid through Non-Homogeneous Porous Medium past a Vertical Porous Plate with Variable Suction

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1 Journal of lied athematics and Phsics Published Online Februar 6 in SciRes. htt:// htt://dx.doi.org/.36/jam.6.3 agneto Convective Flow of a Non-Newtonian Fluid through Non-Homogeneous Porous edium ast a Vertical Porous Plate with Variable Suction S. Harinath Redd. C. Raju E. Keshava Redd Deartment of Humanities and Sciences nnamachara Institute of Technolog and Sciences (utonomous) Rajamet India Deartment of athematics Jawaharlal Nehru Technological Universit Hderabad India Received 9 December 5; acceted 9 Februar 6; ublished Februar 6 Coright 6 b authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons ttribution International License (CC BY). htt://creativecommons.org/licenses/b/./ bstract Radiation absortion and chemical reaction effects on unstead HD free convective flow of a visco-elastic fluid ast a vertical orous late in the resence of variable suction and heat source is considered. uniforagnetic field is assumed to be alied in the transverse direction of the flow. The set of non-linear artial differential equations is transformed into a set of ordinar differential equations b suer imosing a solution with stead and unstead art. The set of ordinar differential equations is solved b using regular erturbation scheme. The exressions for velocit temerature and secies concentration fields are obtained and the exressions for Skin friction Nusselt number and Sherwood number are also derived. The effects of numerous hsical arameters on the above flow quantities are studied with the hel of grahs and tables. Kewords HD Visco-Elastic Fluid Unstead Flow Chemical Reaction Radiation bsortion Suction. Introduction HD free convection fluid flows frequentl occur in natural world. Fluid asses through orous medium are of great interest nowadas and man researchers attract towards the alications in the fields of science and technolog namel in the area of agriculture engineering to know about the ground water resources in fuel technol- How to cite this aer: Redd S.H. Raju.C. and Redd E.K. (6) agneto Convective Flow of a Non-Newtonian Fluid through Non-Homogeneous Porous edium ast a Vertical Porous Plate with Variable Suction. Journal of lied athematics and Phsics htt://dx.doi.org/.36/jam.6.3

2 S. H. Redd et al. og to stud the moment of natural gas oil and water through the oil reservoirs. Chaudhar et al. [] considered Hall effect on HD mixed convection flow of a visco-elastic fluid ast an infinite vertical orous late with mass transfer and radiation. Satanaraana et al. [] discussed HD free convective heat and mass transfer ast a vertical orous late with variable temerature. Kesavaiah et al. [3] studied and resented effects of chemical reaction and radiation absortion on unstead HD convection heat and mass transfer flow ast a semi-infinite vertical ermeable moving late embedded in orous medium with heat source and suction. Redd et al. [] considered effects of chemical reaction and radiation absortion on unstead magnetohdrodnamic double diffusive convective flow of viscous fluid ast a semi-infinite orous late. Raju et al. [5] investigated radiation and mass transfer effects on a free convection flow through orous medium bounded b a vertical surface. uthukumaraswam et al. [6] analzed first order chemical reaction on flow ast an imulsivel started vertical late with uniform heat and mass flux. Das et al. [7] studied the effects of mass transfer on a flow ast an imulsivel started infinite vertical late with constant heat flux and chemical reaction. Kandaswam et al. [8] focused on the roblem of chemical reaction heat and mass transfer on magnetohdrodnamic flow over a vertical stretching surface with heat source and thermal stratification effects. Effects of chemical reaction and thermohoresis on HD mixed convective heat and mass transfer flow along an inclined late in the resence of heat generation/absortion with viscous dissiation and joule heating was considered b lam et al. [9]. uthucumaraswam et al. [] analzed the effects of chemical reaction on moving infinite vertical late with uniform heat flux and variable mass diffusion. ahaatra et al. [] investigated the effect of chemical reaction on free convection flow through a orous medium bounded b a vertical surface. ishra et al. [] considered the effect of mass and heat transfer on magnetohdrodnamic flow of a visco-elastic fluid through orous medium with oscillator suction and heat source. Beget et al. [3] had established comutational fluid dnamics modeling of bounc induced visco-elastic flow in a orous medium with magnetic field effect. Soundalgekar et al. [] investigated effects of mass transfer and natural convection effects on HD stokes roblem for a vertical late. Kandasam et al. [5] studied the effects of chemical reaction heat and mass transfer along a wedge with heat source and concentration in the resence of source or injection. Gurmindersingh et al. [6] analzed the mass transfer with chemical reaction in HD mixed convection flow along a vertical stretching sheet. Radiation effects on HD free convection flow over a vertical late with heat and mass flux was considered b Sivaiah et al. [7]. Sahim et al. [8] considered Lalace technique on HD radiating and chemicall reacting fluid over an infinite vertical surface. Singh et al. [9] investigated heat and mass transfer in HD flow of a viscous fluid ast a vertical late under oscillator suction velocit. Redd et al. [] had resented thermal radiation and chemical reaction effects on magnetohdrodnamic mixed convective boundar laer sli flow in a orous medium with heat source and Ohmic heating. Rout et al. [] studied effect of radiation and chemical reaction on free convective HD flow through a orous medium with double diffusion. Effects of chemical reaction and radiation absortion on HD flow of dust visco-elastic fluid were considered b Prakash et al. []. Damala et al. [3] discussed radiation absortion chemical reaction and magnetic field effects on the free convection and mass transfer flow through orous medium with constant suction and constant heat flux. Raju et al. [] studied unstead HD free convection and chemicall reactive flow ast an infinite vertical orous late. Umamaheswar et al. [5] investigated unstead HD free convective visco-elastic fluid flow bounded b an infinite inclined orous late in the resence of heat source viscous dissiation and Ohmic heating. Reed et al. [6] considered chemical reaction and radiation effects on unstead HD free convection flow near a moving vertical late. Rajesh et al. [7] studied radiation effects on HD flow through a orous medium with variable temerature or variable mass diffusion. Kesavaiah et al. [8] investigated radiation and mass transfer effects on moving vertical late with variable temerature and viscous dissiation. Recentl Ravikumar et al. [9] investigated combined effects of heat absortion and HD on convective Rivlin-Ericksenflow ast a semi-infinite vertical orous late. nd Venkateswarlu et al. [3] had resented chemical reaction and radiation absortion effects on the flow and heat transfer of a Nano fluid in a rotating sstem. The objective of the resent aer is to analze radiation absortion and chemical reaction on HD visco-elastic free convection flow through orous medium bounded b a vertical surface with constant heat and mass flux in the resence of homogeneous chemical reaction. The dimensionless equations of continuit linear momentum energ and diffusion which governed the flow field were solved using erturbation technique. The behavior of velocit temerature concentration and skin friction coefficient was discussed for various arameters involved in the governing equations the alicable criteria that follow. 3

3 S. H. Redd et al.. Formulation of the Problem The unstead free convection viscous incomressible electricall conducting flow of a radiation absortion chemicall reacting and visco-elastic (Walters B * ) fluid ast asemi-infinite vertical orous late in a orous medium with variable suction as well as ermeabilit in resence of a transverse magnetic field is considered. Let x * -axis be along the late in the direction of the fluid flow and * -axis erendicular to it. It is assumed that magnetic Renolds number is much less than unit so that the induced magnetic field is neglected in comarison with the alied transverse magnetic field. The basic flow in the medium is therefore entirel due to the buoanc force caused b the temerature difference between the wall and the medium. This assumed that initiall at t * the late as well as fluids are at the same temerature and concentration. s the concentration of the secies is ver low so that the Soret and Dofour effects are neglected. When t * > the temerature of the late is instantaneousl raised to * * T w and the concentration of the secies is set to C w (see Figure ). It is considered that the ermeabilit of the orous medium in the following form ** ( ) ( ) * * K t K * + εe nt () * * where K ( t ) is K * is orosit ω * is frequenc of oscillation t * is time ε is a small ositive constant. The suction velocit is assumed to be time varing and it takes the following form ** * ( ) ( ) ent vt v + ε () Here v > and ε are ositive constants. Under the above assumtions with usual Boussineq s aroximation (ishra et al. [3] Raju and Varma [3] [33]) the governing equations and boundar conditions are given b * * u u u σ B + v υ + gβ * * ( T T ) + gβ ( C C ) u t ρ ( e ) iw ε t * * * * 3 3 υu k u u + v 3 κ ρ + t T T T + v κ + s T T R C C * * * t ( α ) ( α ) C C C + v D k C C * * * t ( α ) (3) () (5) Figure. Phsical configuration and coordinate geometr. 35

4 S. H. Redd et al. nt ( ω ) ε( ω ) * ** * nt ε ** u T T + T T e C C + C C e at * * as u T T C C Introducing the non-dimensional quantities v vt T T C C t u T C v T T C C υω u α ω υ υ v w w * S σ B υ β w κ r c 3 υ ρ k ( ) υ vk υ υ g C C S P G v v v ( w ) υ kv k υ υ( w ) κ 3 v D ρυ v v ( T T ) υgβ T T C C R G S R R r c c r The Equations (3) to (5) are reduced to the following dimensionless equations 3 3 u nt u u u u nt u ( + εe ) + G rt + CGc u Rc ( + εe ) nt 3 t K ( e ) + ε t T nt T T ( + εe ) + ST R C t P C nt C C ( + εe ) C κr t Sc nt nt u T + εe C + εe at u T C as r w (6) (7) (8) (9) () () 3. ethod of Solution In view of transient suction temerature and concentration at the late let us assume the velocit temerature concentration in the neighborhood of the late. ( ) ( ) ε ( ) e nt ( ε ) ( ) ( ) ε ( ) e nt ( ε ) ( ) ( ) ε ( ) e nt ( ε ) u t u + u + o () T t T + T + o (3) C t C + C + o () Substituting above Equations ()-() into the Equations (8)-() and equating the ε coefficient and coefficient of ε we get the following equations. R u + u + u + u CG G T c c r κ nr n u Ru + u + u + + u u GT CG Ru c c r c c κ r r r (5) (6) T + PT + SPT R PC (7) n T PT P + + S T P RC T ( ) r r r c cκr (8) C + SC S C (9) n C + SC S κ + C SC c c r c () 36

5 S. H. Redd et al. Now the boundar conditions are reduced to the following forms u u T T C C as u u T T C C as The Equations (5) and (6) are not solvable b using the given boundar conditions (). Hence the erturbation method has been alied using R c (R c < ) the elastic arameter as the erturbation arameter. u u ( ) + Ru c ( ) + o( Rc ) () u u + Ru + o R ( ) c ( ) ( c ) Substituting Equation () into Equations (5) and (6) equating the coefficients of R c and R C to zero we get the following set of equations. Zeroth order equations () First order equations u + u + u GC GT c r κ u + u + u u κ u + u + n + u u u GT GC r c κ κ (3) () (5) u + u + n + u u u u n + u u κ κ Using the erturbation the boundar conditions are reduced as follows: u u u u as u u u u as Solving these differential equations b using the boundar conditions we get the following results (endix) u e + e + e + e m m9 m5 m m5 m ( ) m m m m ε e + e + e + e + e + e + e + e e m m nt m ( ) { ( )) } m m m ε 8 T e + e e + e + e + e e nt m3 {( ) } C + ε + m e m e e nt e The skin friction Nusselt number and Sherwood number at the late are defined as follows: u τ at T Nu at C Sh at (6) (7) 37

6 S. H. Redd et al.. Results and Discussion The resent stud considers the effects of radiation absortion and chemical reaction effect on transient free convection flow of a non-newtonian fluid through non-homogeneous orous medium ast a vertical orous late with magnetic field and variable suction. Solutions for velocit temerature and concentration field are obtained b using erturbation technique. The effects of various arameters like Grashof number for heat and mass transfer Gr and Gc chemical reaction Kr Radiation absortion R Prandtl number Pr on velocit temerature and concentration have been studied analticall and effects are executed with the hel of Figures -5. lso the behavior of skin friction rate of heat transfer and rate of mass transfer with resect to various arameters have been studied and results were resented in Tables -. C Pr.7; Gr; Gc; t7; z.; Sc.; n.; Kr. Kr. Kr.6 Kr Figure. Effect of Kr on concentration. C t; z.; n.; kr. Sc Sc. Sc.3 Sc Figure 3. Effects of Sc on concentration. 38

7 S. H. Redd et al. T.8.6. t; z.; Sc.; n.; kr.; s; R. s - s s.5 s Figure. Effects of S on temerature. T Pr.7; t; z. ; n.; kr.; s.; R.; Rc.; Sc. Sc. Sc.6 Sc Figure 5. Effects of Sc on temerature. T.8.6. t; z.; Sc.; n.; s.; R.; Kr. Kr. Kr.3 Kr Figure 6. Effects of Kr on temerature. 39

8 S. H. Redd et al. T.8.6. t; z.; Sc.; n.; kr.; s.; R.; Pr.3 Pr.5 Pr.7 Pr Figure 7. Effects of Pr on temerature. U 8 6 Pr.7 ; z.; n.; s; K; Gr; t; Sc.; kr.; R.; Rc.; Gc5 Gc Gc5 Gc Figure 8. Effects of Gc on velocit. U Pr7 Gc; ; t; z.; Sc.; n.; kr.; s; R.; K; Rc.; Gr5 Gr Gr5 Gr 6 8 Figure 9. Effects of Gr on velocit.

9 S. H. Redd et al. U 8 Pr.7 Gc; z.; n.; s; K; Gr; t; Sc.; ; R.; Rc.; Kr. Kr.3 Kr.5 Kr Figure. Effects of Kr on velocit. U Pr.7 Gc; z.; n.; s; K; Gr; t; Sc.; kr.; R.; Rc.; Figure. Effects of on velocit. U 5 Gr; ; z.; n.; s; K; Gc; t; Sc.; kr.; R.; Rc.; Pr.5 Pr.7 Pr Figure. Effects of Pr on velocit.

10 S. H. Redd et al. U 8 6 Pr.7; Gc; t; Sc.; kr.7; K; Gr; ; z.; n.; s; Rc.; R. R.3 R Figure 3. Effects of R on velocit. U 8 6 Pr.7; Gr; Gc; ; t; z.; Sc.; n. ; kr.; R.; K; Rc.; s - s s Figure. Effects of S on velocit. U 8 6 Pr.7; Gr Gc; ; t; z.; n.; kr.; s; R.; K; Rc.; Sc. Sc.6 Sc Figure 5. Effects of Sc on velocit.

11 S. H. Redd et al. Table. Effect of Sc on skin friction Sherwood number Nusselt number z. t n.. Pr Gr Gc Sc Kr R K Rc S ττ Sh Nu Table. Effect of Pr on skin friction Sherwood number Nusselt number z. t n.. Pr Gr Gc Sc Kr R K Rc S ττ Sh Nu Table 3. Effect of Gr on skin friction Sherwood number Nusselt number z. t n.. Pr Gr Gc Sc Kr R K Rc S ττ Sh Nu Table. Effect of Gc on skin friction Sherwood number Nusselt number z. t n.. Pr Gr Gc Sc Kr R K Rc S ττ Sh Nu Table 5. Effect of on skin friction Sherwood number Nusselt number z. t n.. Pr Gr Gc Sc Kr R K Rc S ττ Sh Nu Table 6. Effect of Kr on skin friction Sherwood number Nusselt number z. t n.. Pr Gr Gc Sc Kr R K Rc S ττ Sh Nu Table 7. Effect of K on skin friction Sherwood number Nusselt number z. t n.. Pr Gr Gc Sc Kr R K Rc S ττ Sh Nu

12 S. H. Redd et al. Table 8. Effect of S on skin friction Sherwood number Nusselt number z. t n.. Pr Gr Gc Sc Kr R K Rc S ττ Sh Nu Table 9. Effect of Rc on skin friction Sherwood number Nusselt number z. t n.. Pr Gr Gc Sc Kr R K Rc S ττ Sh Nu Table. Effect of R on skin friction Sherwood number Nusselt number z. t n.. Pr Gr Gc Sc Kr R K Rc S ττ Sh Nu Figure exhibits the effect of chemical reaction (Kr) on concentration it is noticed that the concentration of the fluid decreases as chemical reaction arameter increases. Figure 3 shows the effect of Schmidt number Sc on concentration when the values of Sc increases the concentration value decreases. From Figure we have noticed that temerature of the fluid increases as source or sink increases. Figure 5 shows the temerature rofile for the different values of Sc from this figure we have noticed that increase in the value of Sc results in increase in the temerature rofile. Figure 6 deicts the effect of Kr on temerature when the values of Kr increases the temerature value increases. Figure 7 shows the effect of Prandtl number on temerature when Pr value increases the temerature decreases similar te of results are noticed with Satanaraana et al. []. Figure 8 deicts the effect of Gc on velocit the velocit of the fluid increases when Gc increases. Figure 9 exhibits that the effect of Gr on velocit from this figure we observed that the velocit of the fluid increases when Gr increases. Figure shows the effect of Kr on velocit the velocit of the fluid decreases in the increase of Kr. Figure illustrates velocit rofiles for different values of from this figure we have observed that velocit of the fluid decreases when an increase in the values of. Figure shows the velocit rofile for different values of Pr. It is observed that increase in the value of Pr results in decrease in the velocit rofile. Figure 3 shows the velocit rofile for different values of radiation absortion from this figure it is noticed that an increase in the value of R results a decrease in the velocit rofile. Figure illustrates the effect of source/sink on velocit from this figure it is noticed that velocit of the fluid increases for decreasing values of source/sink. Figure 5 shows the effect of Sc on velocit from this figure it is noticed that when Sc values increases the velocit of the fluid decreases. On the other hand Tables - show the effect of Sc Pr Gr and Gc on the arameters skin friction Sherwood number Nusselt number. It can be observed that skin friction coefficient increased with the increase in Sc Pr Gr Gc. It can be clearl observed that rate of heat transfer of the fluid increases for increase in Sc and it is not shown an effect in case of Pr Gr Gc. The Nusselt number increased as increase in the Sc and Pr but it is constant in the case of Gr and Gc. Further Tables 5- show the effect of Kr K S Rc and R on the arameters skin friction Sherwood number Nusselt number. It can be observed that skin friction coefficient increased with an increase in Kr K Rc and R whereas decreased in the increase of and S. The rate of the heat transfer of the fluid increases with an increase in Kr but it not shown an effect in case of K S Rc and R. The Nusselt number increased with an increase in Kr S and it is decreased with an increase in R.

13 S. H. Redd et al. 5. Conclusions The resent stud is carried out to investigate the magneto convective flow of a non-newtonian fluid through non-homogeneous orous medium ast a vertical orous late with variable suction. The dimensionless governing equations are solved b using the erturbation technique. The results for velocit and temerature are obtained and lotted grahicall. The numerical results for skin friction and Nusselt number are comuted in tables. The main conclusions of this stud are as follows:. Velocit of the fluid increases with an increasing values of S Gc Gr. nd it decreases in the case of Kr Sc Pr and R.. Temerature of the fluid increases with an increasing values of Kr Sc and S whereas decreased in the case of Pr. 3. Kr and Sc show negative imact on the concentration of the fluid.. Coeffecient of skin friction receives ositive imact in case of Sc Pr Gr Gc Kr K Rc while negative effect in the case of and S. Sherwood number increases for increasing values of Sc and Kr. Coefficient of rate of heat transfer increases with an increase in Sc Pr Kr and S. References [] Chaudhar R.C. and Jain P. (6) Hall Effect on HD ixed Convection Flow of a Visco-Elastic Fluid ast an Infinite Vertical Porous Plate with ass Transfer and Radiation. Ukrainian Journal of Phsics [] Satanaraana P.V. (3) HD Free Convective Heat and ass Transfer ast a Vertical Porouse Plate with Variable Temerature. International Journal of lied athematics and echanics [3] Kesavaiah D.C. Satanaraana P.V. and Venkataramana S. () Effects of the Chemical Reaction and Radiation bsortion on Unstead HD Convective Heat and ass Transfer Flow ast a Semi-Infinite Vertical Permeable oving Plate Embedded in Porous edium with Heat Source and Suction. International Journal of lied athematics and echanics [] Venkata S. Redd R. and Redd G.V. (3) Effect of Chemical Reaction and Radiation bsortion on Unstead HD Double Diffusive Convective Flow of Viscous Fluid ast a Semi-Infinite Porous Plate. International Journal of dvanced Engineering Technolog IV 37-. [5] Raju.C. Varma S.V.K. and Redd N.. () Radiation and ass Transfer Effects on a Free Convection Flow through Porous edium Bounded b a Vertical Surface. I-anager Journal of Future Engineering and Technolog [6] uthucumarswam R. and Ganesan P. () First Order Chemical Reaction on Flow ast an Imulsivel Started Vertical Plate with Uniform Heat and ass Flux. cta echanica htt://dx.doi.org/.7/bf835 [7] Das U.N. Deka R. and Soundalgekar V.. (995) Effects of ass Transfer on a Flow ast an Imulsivel Started Infinite Vertical Plate with Constant Heat Flux and Chemical Reaction. Forschung im Ingenieurwesen [8] Kandasam R. Perisam K. and Prabhu S. (5) Chemical Reaction Heat and ass Transfer on HD Flow over a Vertical Stretching Surface with Heat Source and Thermal Stratification Effects. International Journal of Heat and ass Transfer htt://dx.doi.org/.6/j.ijheatmasstransfer [9] lam.s. Rahman.. and Sattar.. (8) Effects of Chemical Reaction and Thermohoresis on HD ixed Convective Heat and ass Transfer Flow along an Inclined Plate in the Presence of Heat Generation/bsortion with Viscous Dissiation and Joule Heating. Canadian Journal of Phsics [] uthucumaraswam R. and Kulaivel T. (3) Chemical Reaction Effects on oving Infinite Vertical Plate with Uniform Heat Flux and Variable ass Diffusion. Forschung im Ingenieurwesen [] ahaatra. Dash G.C. Panda S. and chara. () Effects of Chemical Reaction on Free Convection Flow through a Porous edium Bounded b a Vertical Surface. Journal of Engineering Phsics and Thermohsics htt://dx.doi.org/.7/s [] ishra S.R. and Dash G.C. (3) ass and Heat Transfer Effect on HD Flow of a Viscoelastic Fluid through Porous edium with Oscillator Suction and Heat Source. International Journal of Heat and ass Transfer htt://dx.doi.org/.6/j.ijheatmasstransfer...53 [3] Beg O.. Takhar H.S. Kumari. and Nath G. () Comutational Fluid Dnamics odelling of Buoanc Induced Viscoelastic Flow in a Porous edium with agnetic Field Effects. International Journal of lied echanics and Engineering [] Soundalgekar V.. Guta S.K. and Birajdar N.S. (979) Effects of ass Transfer and Free Convection Effects on HD Stokes Problem for a Vertical Plate. Nuclear Engineering and Design

14 S. H. Redd et al. [5] Kandasam R. Perisam K. and Sivagnana Prabhu K.K. (5) Effects of Chemical Reaction Heat and ass Transfer along a Wedge with Heat Source and Concentration in the Presence of Suction or Injection. International Journal of Heat and ass Transfer [6] Singh G. () ass Transfer with Chemical Reaction in HD ixed Convection Flow along a Vertical Stretching Sheet. International Journal of Engineering & Technolog -. [7] Sivaiah. () Radiation Effects on HD Free-Convection Flow over a Vertical Plate with Heat and ass Flux. Emirates Journal for Engineering Research [8] hmed S. () Lalace Technique on HD Radiating and Chemicall Reacting Fluid over an Infinite Vertical Surface. International Journal of Engineering & Technolog [9] Singh.K. and Singh N.P. (3) Heat and ass Transfer in HD Flow of a Viscous Fluid Past a Vertical Plate Under Oscillator Suction Velocit. Indian Journal of Pure and lied athematics [] Redd.G. () Thermal Radiation and Chemical Reaction Effects on HD ixed Convective Boundar Laer Sli Flow in a Porous edium with Heat Source and Ohmic Heating. The Euroean Phsical Journal Plus 9. htt://dx.doi.org/./ej/i--3 [] Rout B.R. () Effect of Radiation and Chemical Reaction on Natural Convective HD Flow through a Porous edium with Double Diffusion. Journal of Engineering Thermohsics htt://dx.doi.org/.3/s8387x [] Prakash J. () Effects of Chemical Reaction and Radiation bsortion on HD Flow of Dust Viscoelastic Fluid. lications and lied athematics [3] Kesavaiah C.D. () Radiation bsortion Chemical Reaction and agnetic Field Effects on Free Convection and ass Transfer Flow through Porous edium with Constant Suction and Constant Heat Flux. International Journal of Scientific Engineering and Technolog 7-8. [] Raju.C. Varma S.V.K. and Rao R.R.K. (3) Unstead HD Free Convection and Chemicall Reactive Flow ast an Infinite Vertical Porous Plate. Journal of Future Engineering and Technolog [5] Umamaheswar. Varma S.V.K. and Raju.C. (3) Unstead HD Free Convective Visco-Elastic Fluid Flow Bounded b an Infinite Inclined Porous Plate in the Presence of Heat Source Viscous Dissiation and Ohmic Heating. International Journal of dvanced Science and Technolog htt://dx.doi.org/.57/ijast [6] Redd T.S. Raju.C. and Varma S.V.K. () Chemical Reaction and Radiation Effects on Unstead HD Free Convection Flow near a oving Vertical Plate. Journal of Future Engineering & Technolog 7 -. [7] Rajesh V. and Varma S.V.K. () Radiation Effects on HD Flow through a Porous edium with Variable Temerature or Variable ass Diffusion. International Journal of lied athematics and echanics [8] Kesavaiah C.D. and Satanaraana P.V. () Radiation and ass Transfer Effects on oving Vertical Plate with Variable Temerature and Viscous Dissiation. International Journal of athematical rchive [9] Ravikumar V. Raju.C. and Raju G.S.S. () Combined Effects of Heat bsortion and HD on Convective Rivlin-Ericksen Flow ast a Semi-Infinite Vertical Porous Plate. in Shams Engineering Journal htt://dx.doi.org/.6/j.asej.3.. [3] Venkateswarlu B. and Sata Naraana P.V. (5) Chemical Reaction and Radiation bsortion Effects on the Flow and Heat Transfer of a Nanofluid in a Rotating Sstem. lied Nanoscience [3] ishra S.R. Das G.C. and chara. (3) ass and Heat Transfer Effect on HD Flow of a Visco-Elastic Fluid through Porous edia with Oscillator Suction and Heat Source. International Journal of Heat and ass Transfer htt://dx.doi.org/.6/j.ijheatmasstransfer...53 [3] Raju.C. and Varma S.V.K. () Unstead HD Free Convection Oscillator Couette Flow through a Porous edium with Periodic Wall Temerature. Journal on Future Engineering and Technolog [33] Raju.C. and Varma S.V.K. () Soret Effects Due to Natural Convection in a Non-Newtonian Fluid Flow in Porous edium with Heat and ass Transfer. Journal of Naval architecture and arine Engineering htt://dx.doi.org/.339/jname.vi

15 S. H. Redd et al. endix S + S + S κ c c c r m m 3 n + + κ + S S S c c c r Sm c n m ms S κ c c + r RP m m P + SP r r r ( r ) ( ) ( r ) RP 3 R+ m P mp 5 5 r P + P SP r r r m 5 m 7 n + Pr P P r r S m κ 3 6 n m mp+ P S 3 3 r r 7 n m mp + P S r r 5 8 n m mp+ P S 5 5 r r ( c r) ( ) r G + G 9 G m m + 9 m m ( ) + 3 m m 9 m m m m 3 m m + 6 m 5 m m ( ) m κ ( ) Gr m G κ ( ) 5 8 r G + G 6 r c( ) m G G κ 7 r c 3 8 n n n 5 5 m 3 κ n 6 n n ( ) n 9 3 m + m

16 S. H. Redd et al. 7 m 3 7 K n κ n 3 3 m + m n m + m + m + m κ n 3 m + m m m n m + m + m + m K 36 9 n n K 38 3 n n n 7 7 κ 3 n n R c 7 ( ) R 5 3 c + R + R c 6 R 8 c 3 + R c c + R R c 36 R + 5 c 39 c + R + R + R c c c Nomenclature C: Secies concentration R : Radiation absortion. C: Non-dimensional secies concentration Kr: Chemical reaction D: olecular diffusivit Gc: Grashof number for mass transfer : agnetic arameter Gr: Grashof number for heat transfer B : agnetic field of uniform strength g: cceleration due to gravit σ: Electrical conductivit K: Permeabilit of the medium ρ: Densit of the fluid K: Permeabilit/orosit arameter t: Time k: Thermal diffusivit β: Volumetric coefficient of exansion for heat transfer : agnetic arameter β * : Volumetric coefficient of exansion with secies concentration N: Nusselt number Pr: Prandtl number ε: a small ositive constant S: Heat source arameter R c : Elastic arameter. Sc: Schmidt number υ : Kinematic coefficient of viscosit. Sh: Sherwood number v : Constant suction velocit. 8