International Journal of Mathematical Archive-9(5), 2018, Available online through ISSN

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1 Intenational Jounal of Mathematical Achive-9(5), 18, -3 Available online though ISSN MHD FLOW OVER A MOVING INFINITE VERTICAL POROUS PLATE WITH UNIFORM HEAT FLUX IN THE PRESENCE OF THERMAL RADIATION 1 E. RAGHUNANDANA SAI Reseach Schola, Depatment of Mathematics Kishna Univesity, Machilipatnam 51 1 (A. P), India. Ch. V. RAMANA MURTHY* Si Vasavi Institute of Engineeing and Technology, Nandamuu, Pedana (A.P.), India. (Received On: ; Revised & Accepted On: ) ABSTRACT MHD Flow Ove a Moving Infinite Vetical Poous Plate With Unifom Heat Flux In The Pesence Of Themal Radiation With efeence to all citical paametes that appea in the field equations wee was studied in this pape. In most of the liteatue, the flow paametes wee neithe examined no commented in detail in any of the above investigatos. This is a point of geat concen and is if academic inteest to study. Also, not much of pime impotance was given to the bounding suface and factos influencing the skin fiction. Theefoe, an attempt has been made to study the influence of such citical paametes that appea in the field equations and thei effects on vaious flow entities. It is noticed that as the fequency of excitation deceases the velocity deceases. Futhe, as we move away fom the bounday of the suface the velocity deceases. The effect of the fequency of excitation and fo a constant poe size has been illustated. In this case, it is noticed that as the fequency of excitation deceases the velocity deceases. Futhe, as we move away fom the bounday of the suface the velocity deceases. it is epoted that, as the fequency of excitation deceases the velocity also deceases. In addition to the above, as we move fa away fom the bounday of the suface the velocity deceases. Also, it is has been epoted that, as the poe size of the bounday suface deceases, the velocity also deceases. Futhe, as we away fom the bounding suface the velocity deceases. Fo diffeent values fequency of excitation, the pofiles of skin fiction ae noticed to be linea and of couse with negative slope. It is noticed that as the fequency of excitation deceases, the skin fiction on bounday suface found to be inceasing. Key wods: Impulsively stated vetical plate, MHD flow, Heat and Mass, Radiation tansfe, Heat flux. NOMENCLATURE C : Specific heat at constant pessue P g : Acceleation due to gavity G : Themal Gashoff numbe κ : Themal conductivity of the fluid P : Pandtl numbe p : Pessue q : Radiative heat flux in the y-diection k : Themal diffusivity K : Poosity M : Magnetic field N : Radiation paamete t : Time T : Tempeatue of the fluid nea the plate T W : Tempeatue of the plate T : Tempeatue of the fluid fa away fom the plate u : Velocity of the fluid in the x-diection Intenational Jounal of Mathematical Achive- 9(5), May-18

2 1 E. Raghunandana Sai and Ch. V. Ramana Muthy* / MHD Flow Ove A Moving Infinite Vetical Poous Plate with Unifom Heat Flux in The Pesence of / IJMA- 9(5), May-18. u : Velocity of the fluid plate U : Dimensionless velocity y : Coodinate axis nomal to the plate y : Dimensionless coodinate axis nomal to the plate * k : Mean absoption coefficient α : Themal diffusivity β : Volumetic coefficient of themal expansion µ : Coefficient of viscosity ν : Kinematic viscosity ρ : Density σ : Stefan-Boltzmann constant τ : Dimensionless skin-fiction θ : Dimensionless tempeatue INTRODUCTION Radiative convective flow in seveal industial and envionmental situations occus moe fequently in many situations. The applications ae moe found in, fossil fuel combustion and in cooling chambes and moe so in enegy pocesses, astophysical flows, sola powe technology and space vehicle e-enty. An impotant ole in manufactuing sectos fo the design of highly pecision equipment in adiative heat tansfe is found to have seveal applications: geneally, nuclea powe plants, gas tubines and populsion devises fo ai caft, missiles and space vehicles ae few such examples. Stokes [1] initially, studied the poblem of viscous incompessible fluid past an impulsively stated infinite hoizontal plate which moves in its own plane. Subsequently, Binkman [], examined the viscous foce impated by a flowing fluid in a dense swam of paticles. Late, Stewatson [3] studied an analytical solution fo a viscous flow past an impulsively stated semi-infinite hoizontal plate. Subsequently, the case of two dimensional steady state flow of an incompessible fluid with paallel igid poous walls, with the flow being influenced by unifom suction o injection was investigated by Beman [4]. Late, Macy [6] and Moi [5] studied the flow between two vetical plates, whee the plates ae electically non-conducting and unde the assumption that the wall tempeatue influences linealy in the diection of the flow and existence of heat souce in the vetical channel. Subsequently, the flow in the enal tubules as viscous flow though cicula tube of unifom coss section with a pemeable bounday by pescibing thei adial velocity at the wall as exponentially deceasing function as axial distance was studied by Macy [6]. Hall [7] examined simila such poblem by using finite diffeences method of a mixed explicit and implicit time fo the stability of the solution. The effects of adioactive heat tansfe of fee convection egimes in an enclosed with specialized applications which occus geophysics and geothemal esevois was analysed by Chang et al. [8]. Theeafte, Mahajan et al. [9] analysed, the influence of viscous heat dissipating effect in natual convective flows. Late, the themal and adiation effects of an optionally thin gay gas bounded by a stationay vetical plate was examined by Soundalgeka and Thake [1]. By applying Rossland s appoximation, Hossain et al. [11] examined the adiation effects on a mixed convection along a vetical plate with a unifom suface tempeatue. Subsequently, the effects of themal adiation and convective flow past a moving infinite vetical plate was discussed and pesented by Raptis and Pedikis [1]. Theeafte, the effects of themal adiation on the flow past a semi-infinite vetical isothemal plate with unifom heat flux in the pesence of tansvesely applied magnetic field was analysed by Antony Raj et al. [13]. The natue of velocity with efeence to all citical paametes that appea in the field equations wee neithe examined no commented in detail in any of the above investigatos. This is a point of geat concen and is if academic inteest to study. Also, not much of pime impotance was given to the bounding suface and factos influencing the skin fiction. Theefoe, an attempt has been made to study the influence of such citical paametes that appea in the field equations and thei effects on vaious flow entities. MATHEMATICAL FORMULATION Flow of an incompessible viscous adiating fluid past an impulsively stated infinite vetical plate with unifom heat flux is consideed. The x-axis is taken along the plate in the vetical diection and the y-axis is taken nomal to the plate. The Flow geomety is as shown below Coesponding Autho: Ch. V. Ramana Muthy* Si Vasavi Institute of Engineeing and Technology, Nandamuu, Pedana (A.P.), India. 18, IJMA. All Rights Reseved 3

3 1 E. Raghunandana Sai and Ch. V. Ramana Muthy* / MHD Flow Ove A Moving Infinite Vetical Poous Plate with Unifom Heat Flux in The Pesence of / IJMA- 9(5), May-18. Schematic epesentation of the poblem Initially, the plate and fluid ae at the same tempeatue in a stationay condition. The plate is given an impulsive motion in the vetical diection against the gavitational field with constant velocityu when t >. At the same time, the heat is supplied fom the plate to the fluid at unifom ate. The fluid exhibits the popeties of gey, absobingemitting adiation but a non-scatteing medium. Then by usual Bossiness s appoximation, the unsteady flow is govened by the following equations. u u = gβ( T T ) + ν t ρc p T T q = k t y y (1) () In view of Rossland appoximation q is given by: q 4σ T = * 3k 4 (3) While, the initial and bounday conditions ae: t : u =, T = T fo all y T q t > : u = u, = at = k u =,T T as y y (4) 4 Unde the assumption that, the tempeatue diffeences within the flow ae sufficiently small such that T may be 4 expessed as a linea function of the tempeatue. This is accomplished by expanding T in a Taylo seies about T and neglecting highe ode tems, thus T T T 3T (5) 4 By using equations (4) and (5), equation () educes to C p T T = k t 16σT + 3k 3 * T ρ (6) On intoducing the following non-dimensional quantities u tu yu T T U =, t =, y =, θ =, u ν ν T T gβν ( T G = u T) (7) µ C w p,p =, N = 3 k 4 w * κ k σt 3 18, IJMA. All Rights Reseved 4

4 1 E. Raghunandana Sai and Ch. V. Ramana Muthy* / MHD Flow Ove A Moving Infinite Vetical Poous Plate with Unifom Heat Flux in The Pesence of / IJMA- 9(5), May-18. By consideing the magnetic intensity as M and the pemeability of the bounday as K Eqs. (1) to (6), can be e defined as: U U 1 = Gθ + + M u + (8) t K θ θ 3N P = (3N + P) (9) t The initial and bounday conditions in non-dimensionless fom ae u =, θ =, fo all y, t θ t > : u = 1, = 1 at y = (1) u =, θ, as y Methodology fo solution We assume that the solutions fo Eqn (8) and Eqn (9) in fom of: iωt u x, t) = u ( y) e (11) ( ( y, t) = θ θ ( y) e iωt Unde the modified initial and bounday conditions: u =, θ =, fo all y, t iωt dθ iωt t > : u = e, = e at = dy u =, θ as y Using Eqns (11), (1) and (13) in Eqns (8) and (9) (1) y (13) G u( y, t) = (exp( m y) exp( m1 y)) + exp( m y) R1 (14) exp( m1 y) θ ( y, t) = m (15) 1 The expession fo the skin fiction is: Whee u y y= = G R 1 [ m1 m ] m 3NP iω 1 1 m1 =, m = iω M, R1 = m1 m1 iω M 3N + P K K (16) RESULTS AND DISCUSSION 1. The influence of fequency of excitation fo size.5 on the velocity field has been illustated in Fig 1. In this case, it is noticed that as the fequency of excitation deceases the velocity deceases. Futhe, as we move away fom the bounday of the suface the velocity deceases. 18, IJMA. All Rights Reseved 5

5 1 E. Raghunandana Sai and Ch. V. Ramana Muthy* / MHD Flow Ove A Moving Infinite Vetical Poous Plate with Unifom Heat Flux in The Pesence of / IJMA- 9(5), May-18. Figue-1: Influence of fequency of excitation on velocity. Fig - illustates the influence the fequency of excitation fo size.6. The effect of the fequency of excitation and fo a constant poe size has been illustated. In this case, it is noticed that as the fequency of excitation deceases the velocity deceases. Futhe, as we move away fom the bounday of the suface the velocity deceases. Figue-: Influence of fequency of excitation on velocity 3. The influence the fequency of excitation fo poe size of.7 is shown in Fig 3. In this case, it is epoted that, as the fequency of excitation deceases the velocity also deceases. In addition to the above, as we move fa away fom the bounday of the suface the velocity deceases. Figue-3: Effect of fequency of excitation on velocity 18, IJMA. All Rights Reseved 6

6 1 E. Raghunandana Sai and Ch. V. Ramana Muthy* / MHD Flow Ove A Moving Infinite Vetical Poous Plate with Unifom Heat Flux in The Pesence of / IJMA- 9(5), May Fig 4 exhibits the consolidated effect of poosity and fequency of excitation on the velocity pofiles. In this case, it is epoted that as the poe size of the bounday size of the bounday deceases, then the velocity deceases. Futhe, as we move away fom the bounding suface the velocity deceases. Figue-4: Effect of fequency of excitation on velocity 5. The influence of poosity fo a fixed fequency of excitation ove the velocity pofiles has been depicted in Fig- 5. In this situation, it is has been epoted that, as the poe size of the bounday suface deceases, the velocity also deceases. Futhe, as we away fom the bounding suface the velocity deceases. Figue-5: Influence of fequency of excitation on velocity 6. The influence of poosity fo a fixed fequency of excitation ove the velocity pofiles is shown in Fig 6. In this case, it is epoted that as the poe size of the bounday suface size deceases, the velocity also deceases. Futhe, as we move away fom the bounding suface the velocity also deceases. Figue-6: Effect of fequency of excitation on velocity 18, IJMA. All Rights Reseved 7

7 1 E. Raghunandana Sai and Ch. V. Ramana Muthy* / MHD Flow Ove A Moving Infinite Vetical Poous Plate with Unifom Heat Flux in The Pesence of / IJMA- 9(5), May Fig 7 illustates the influence of magnetic field on the velocity field when the poe size of the bounding suface is.7. In this case, it is epoted that as magnetic intensity inceases the velocity deceases. Also as we move away fom the bounding suface, the velocity deceases. Figue-7: Influence of Magnetic field on velocity 8. The influence of magnetic field on the velocity when the poe size is.8 is illustated in Fig-8. In this case, it is epoted that as the magnetic intensity inceases, the velocity deceases and also as we move away fom the bounding suface, the velocity diminishes gadually. Figue-8: Effect of Magnetic field on velocity 9. The influence of Gashoff numbe with espect to the fequency of excitation has been illustated in fig 9. Fo diffeent values fequency of excitation, the pofiles of skin fiction ae noticed to be linea and of couse with negative slope. It is noticed that as the fequency of excitation deceases, the skin fiction on bounday suface found to be inceasing. 18, IJMA. All Rights Reseved 8

8 1 E. Raghunandana Sai and Ch. V. Ramana Muthy* / MHD Flow Ove A Moving Infinite Vetical Poous Plate with Unifom Heat Flux in The Pesence of / IJMA- 9(5), May-18. Figue-9: Influence of fequency of excitation on skin fiction 1. The consolidated effect of fequency of excitation and Gashoff numbe with espect to poosity paamete has been depicted in figue 1. It is noticed that, as the Gashoff numbe inceases, the skin fiction on the bounday suface deceases consideably. In addition to the above as the fequency of excitation inceases, The pofiles fo skin fiction ae noticed to be inceasing and ae linea in thei behavio. Figue-1: Influence of Gashoff Numbe on skin fiction 11. Fig 11 illustates that the combined effects of fequency of excitation with espect to the poosity of the bounday suface fo a constant Gashoff numbe. It is obseved. Fo a fixed poosity paamete, as the fequency of excitation inceases, the skin fiction also inceases. The pofile fo the skin fiction seems to be moe o less linea in its natue. Figue-11: Influence of poosity on skin fiction 18, IJMA. All Rights Reseved 9

9 1 E. Raghunandana Sai and Ch. V. Ramana Muthy* / MHD Flow Ove A Moving Infinite Vetical Poous Plate with Unifom Heat Flux in The Pesence of / IJMA- 9(5), May-18. REFERENCES 1. Stokes. G. G. On the effects of Intenal Fiction of Fluids on the Motion of Pendulums, Comb. Phil. Tans IX,, Pp (1851).. Binkman H.C, A calculation of viscous foce extended by flowing fluid in a dense swam of paticles. Appl.sci.Res,A(1) Pp 7-34.(1947). 3. Stewatson. K., On the impulsive Motion of a Flat plate in a Viscous Fluid. Quately Jnl. of Mechanics and Applied Mathematics, IV,, Pp , (1951). 4. Beman. A. S.Lamine flow in a channel with poous walls.jnl. Appl.Phys, 4, Pp , (1953). 5. Moi Y, on combined fee and foced convective lamina MHD flow and heat tansfe in channels with tansvese magnetic field, intenational developments in heat tansfe, ASME pape no.14,pp (1961). 6. Macey R.I., Pessue flow pattens in a cylinde with eabsobing walls, Bull Math. Bio phys, 5(1), (1963). 7. Hall. M. G.The Bounday Laye ove an impulsively Stated Flat plate, Poc. Roy. Soc. A. 31, 15, Pp , (1969). 8. Chang. I. C. Yang. K. T., Lloyd. J. R., Radiation Natual Convection Inteactions in Two Dimensional Complex Enclosues, ASME. Jnl. Heat Tansfe, 15, 1, Pp , (1983). 9. Mahajan. R. L., Gebhat. B. B., Viscous Dissipation Effects in Buoyancy-Induced Flows., Int. Jnl. of Heat Mass Tansfe, 3, 7, Pp , (1989). 1. Soundalgeka V. M. and Thake H.S.:Radiation effects onfee convection flow past a semi infinite vetical plate.modeling measuement and contol,vol.b51,pp 31-4, (1993). 11. Das. U. N., Deka. R. K., Soundalgeka. V. M., Effects of Mass Tansfe on Flow Past an Impulsively Stated Infinite Vetical Plate with Constant Heat Flux and Chemical Reaction, Foschang im Ingenieuwesen, 6, 1, Pp , (1994). 1. Hossain M. A. and Takha, Radiation effect on mixed convection along a vetical plate with unifom suface tempeatue, Heat and Mass Tansfe, 31, Pp , (1996). 13. Raptis. A and Pedikis.C, Radiation and fee convection flow past a moving plate, Int. J.App.Mech.Engg, 4, (1999). 14. Chandakala. P and Antony Raj.S, Radiation effects on MHD flow past an implusively stated vetical plate with unifom heat flux, Indian Jounal of Mathematics, 5(3), (8). Souce of suppot: Nil, Conflict of inteest: None Declaed. [Copy ight 18. This is an Open Access aticle distibuted unde the tems of the Intenational Jounal of Mathematical Achive (IJMA), which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited.] 18, IJMA. All Rights Reseved 3

The Combined Effect of Chemical reaction, Radiation, MHD on Mixed Convection Heat and Mass Transfer Along a Vertical Moving Surface

The Combined Effect of Chemical reaction, Radiation, MHD on Mixed Convection Heat and Mass Transfer Along a Vertical Moving Surface Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 93-966 Vol. 5, Issue (Decembe ), pp. 53 53 (Peviously, Vol. 5, Issue, pp. 63 6) Applications and Applied Mathematics: An Intenational Jounal (AAM)