The Effect of Physical Parameters on Flow Variables of an Electrically Conducting Viscoelastic Fluid

Transcription

1 American Jornal of Applied Mathematics 07; 5(): doi: 0.648/j.ajam ISSN: (Print); ISSN: 0-006X (Online) The Effect of Phsical Parameters on Flow Variables of an Electricall Condcting Viscoelastic Flid Binam Zigta, Prnachandra Rao Koa * School of Mathematical and Statistical Sciences, Hawassa Universit, Hawassa, Ethiopia address: tzigta@ahoo.com (B. Zigta), drkpraocecc@ahoo.co.in (P. R. Koa) * Corresponding athor To cite this article: Binam Zigta, Prnachandra Rao Koa. The Effect of Phsical Parameters on Flow Variables of an Electricall Condcting Viscoelastic Flid. American Jornal of Applied Mathematics. Vol. 5, No., 07, pp doi: 0.648/j.ajam Received: April 4, 07; Accepted: April 7, 07; Pblished: Jne, 07 Abstract: In this paper the effect phsical parameters on flow variables of nstead, incompressible, electricall condcting viscoelastic flid flowing between a pair of infinite vertical Coette poros channel walls embedded in a poros medim is analzed. A niform magnetic field is applied perpendiclar to the channel walls. The temperatre of the moving wall varies periodicall. The temperatre difference between the two walls is high enogh de to thermal radiation. The soltion of the governing eqations is obtained sing reglar pertrbation techniqes. This techniqe is applied on partial differential eqations that are difficlt to solve. These partial differential eqations are redced to a set of ordinar differential eqations in dimensionless form and ths the can be solved analticall. The effects of phsical parameters on the flow variables are stdied and the reslts have been discssed. The phsical parameters considered inclde Hartmann nmber, viscoelastic parameter, Permeabilit of poros medim, chemical reaction parameter, radiative parameter, thermal Grashof nmber for heat transfer, modified Grashof nmber for mass transfer, freqenc parameter, Prandtl nmber, mass diffsivit and Schmidt nmber. The flow variables considered inclde velocit, temperatre and concentration. The theoretical reslts have been spported b simlation std. The observations inclde: (i) velocit decreases with increasing vales of freqenc, Hartmann nmber and viscoelastic parameter (ii) velocit increases with increasing vales of temperatre, thermal Grashof nmber, modified Grashof nmber and permeabilit of poros medim, (iii) the temperatre decreases near the moving channel wall when the radiative parameter increases (iv) the temperatre approaches to zero in the region near to the bondar laer of the stationar channel wall when the radiative parameter increases (v) concentration decreases with an increment in both chemical reaction and Schmidt nmber and (vi) The velocit of flid increases as thermal Grashof nmber and modified Grashof nmber increases. Kewords: Viscoelastic Flid, MHD, Coette Channel Walls, Permeabilit of Poros Medim. Introdction The change of shear stress verss strain rate inside a flid depending on viscosit can be classified as a linear, nonlinear or plastic response. If the shear stress is linearl proportional to the strain rate then it is known as Newtonian flid, and the constant of proportionalit is called as the coefficient of dnamic viscosit. In case of non-newtonian flid, the shear stress is not onl nonlinear to strain rate bt also can be time dependent and hence coefficient of dnamic viscosit can t be a constant. The properties of non-newtonian flid flow are treated b rheolog, the science of stding deformation and flow of a sbstance. A flid is said to have Thixotropic propert if its viscosit decreases onl with time bt neither with the shear stress nor with strain rate. Also, when shear stress is independent of strain rate then the flid material possesses plastic deformation. Viscoelastic is the propert of a flid material that combines both viscos and elastic characteristic while the flid is ndergoing deformation. The behavior of viscoelastic flid is different from that of both Newtonian and inelastic non-newtonian flids. This behavior incldes the presence of normal stresses in shear flows those are sensitive to deformation. Some materials that possess viscoelastic properties inclde: shampoos, hand creams, tooth pastes,

2 American Jornal of Applied Mathematics 07; 5(): oghrts, polmers and paints. The factors that can affect the properties of viscoelastic are temperatre, time and shear rate. Unstead, incompressible, and electricall condcting viscoelastic flid embedded in a poros medim is of great importance. Sch flid has several applications in indstr inclding polmer prodction, ceramics, lbrication, electromagnetic proplsion, and magneto-chemistr. In literatre it can be observed that there have been a ver few stdies condcted on the present sbject. To fill the gap in this std it has been considered the effects of thermal radiation, chemical reaction, Permeabilit of poros medim and electricall condcting viscoelastic flid. However, second order effects in elasticit, plasticit and flid dnamics have been analzed []. The Magneto hdrodnamic free convection laminar flow of incompressible viscoelastic flid was stdied []. The MHD combined free and forced convection flow throgh two parallel poros walls were stdied []. An oscillator mixed convection flow of electricall condcting viscoelastic flid in an infinite vertical channel filled with poros medim was analzed [4]. MHD free convection flow of viscoelastic flid past an infinite poros plate was investigated [5]. The nstead flow and heat transfer throgh an elastic viscos liqid along an infinite hot vertical poros moving plate with variable free stream and sction were examined [6]. Flow and heat transfer of an electricall condcting viscoelastic flid between two horizontal sqeezing and stretching plates have been stdied [7]. Heat and mass transfer in MHD viscoelastic flid flow throgh a poros medim over a stretching sheet with chemical reaction was stdied [8]. MHD mixed convective viscoelastic flid flow in a permeable vertical channel with Dfor effect and chemical reaction parameter was stdied [9]. Viscoelastic flid flow past an infinite vertical poros plate in the presence of first order chemical reaction was investigated [0]. MHD flow throgh a poros medim past a stretched vertical permeable srface in the presence of heat sorce or sink and a chemical reaction has been stdied []. The three dimensional free convection flows throgh poros medim in a vertical channel with heat sorce and chemical reaction was stdied []. The nstead two dimensional flows and heat transfer throgh an elastic viscos liqid along an infinite hot vertical poros srface bonded b poros medim was stdied []. MHD flows in non-newtonian flids has been stdied [6]. The effects of radiation on free convection MHD flow have attracted man scholars de to its applications in space science. Sch radiations occr de to high temperatres and design of relevant eqipments. Frthermore, heat and mass transfer with thermal radiation on free convection MHD flow are significant in the plate heat transfer. Recentl, man scholars focsed on this field of std de to its applications in gas cooled nclear reactors, nclear power plants, gas trbines, space vehicles, and spersonic flights. The nstead convective flow in a moving plate with thermal radiation was stdied [7-8]. The combined effects of radiation and boanc force past a vertical plate has been stdied [9]. The inflence of thermal radiation on convective flows over a poros vertical plate was analzed [0]. The effects of thermal radiation on heat and mass transfer over a moving vertical plate were stdied []. The effects of thermal radiation on mixed convection flow over a permeable vertical plate with magnetic field have been stdied []. Stdies of energ and concentration transfer de to chemical reaction have significant applications in man branches of science and technolog. The effects of concentration transfer with the involvement of chemical reaction are important in chemical process which helps to obtain high vale prodcts from cheaper raw materials. The qalit of prodction and the flow behavior of the flid is affected b the concentration goes throgh some kind of chemical reaction with the srronding flid. The effects of mass transfer over a vertical oscillating plate with chemical reaction were stdied []. Chemical reaction and thermal radiation effects on MHD convective flow in a poros medim in the presence of sction stdied [4]. The effects of magnetic field on MHD Coette flow of a thirdgrade flid with chemical reaction stdied [5]. Even thogh, the effects of thermal radiation, chemical reaction, permeabilit of poros medim and electricall condcting viscoelastic flid in vertical Coette channel wall embedded in a poros medim are important the have not been considered in these stdies [5]. Hence the present std has been motivated to verif sch effects. The flid considered here has the properties viz., nstead, incompressible, electricall condcting viscoelastic, chemical reacting and thermal radiating. Free stream velocit of the flid is assmed to be flctating. Frthermore, it is assmed that temperatre and concentration of the flid do also flctate with time. The main objective of the present std is to analze the effect of phsical parameters on flow variables of nstead, incompressible, and electricall condcting viscoelastic flid on free convection oscillator flid flow bonded between two infinite vertical Coette channel walls embedded in a poros medim when the temperatre and concentration oscillates with time in the presence of thermal radiation, chemical reaction and permeabilit of poros medim.. Mathematical Modeling and Analsis of the Problem Note that here and in what follows flid means nstead, incompressible, electricall condcting viscoelastic flid, chemical reacting and radiating flid. Consider two dimensional free convection Coette flid flows bonded b two infinite vertical channel walls separated b a distance embedded in a poros medim. The geometrical representation of the model and coordinate sstem are shown in Figre. The axis is parallel to the gravitational acceleration vector bt in opposite direction. The axis is transverse to the channel walls. The vertical moving channel wall is located at =0 along where the temperatre is and the concentration is. The other stationar channel wall is located at = where

3 80 Binam Zigta and Prnachandra Rao Koa: The Effect of Phsical Parameters on Flow Variables of an Electricall Condcting Viscoelastic Flid the temperatre is and the concentration is. Initiall at =0, the stationar channel wall and the flid are at the same temperatre and concentration level of the flid is the same at all points. At 0, the temperatre of the moving wall and concentration of the flid are raised to and respectivel and the are maintained at constant vales. This reslt is de to poros medim i.e., poros medim is sed to inslate a heated bod b maintaining its temperatre. As time changes temperatre changes leading to boanc force acting in flid flow so the channel wall at 0 starts moving in its own plane along positive direction with velocit. And for all the later times 0 the channel wall contines to be in motion. Bt de to viscosit of the flid and acceleration de to gravit the velocit will decrease and conseqentl it starts moving in opposite direction after some time the moving channel wall being heated b sppling external heat so as to maintain its temperatre constant. However, the temperatre of the wall at rises to an pper constant vale. Figre. Geometrical representation of the model. The governing eqations of the model are derived based on the following assmptions: (i) The flid properties sch as temperatre, viscosit, pressre, specific volme, specific gravit are constant (ii) the densit varies with both temperatre and concentration in the bod force term or Bossinesq s approximation. It is a variable bt not a constant (iii) the flow of the flid is nstead, laminar, incompressible, electricall condcting viscoelastic, thermal radiating and chemical reacting (iv) the magnetic field applied is transversal between the channel walls (v) the indced magnetic field of the flid is negligible and hence the magnetic Renolds nmber is a ver small constant, the external electric field is spposed to be zero (vi) the electric field de to polarization of charges is negligible (vii) viscosit is considered with the constant permeabilit of poros medim (viii) there exists a homogeneos chemical reaction of first order with constant rate between diffsing concentration or species and the flid in the moving wall. Note that a reaction is said to be of first order if the rate of reaction is directl proportional to concentration. The expression for free stream velocit can be formed as () Here in (), is the mean constant free stream velocit, is the freqenc and is the time. We now derive the governing eqations of the present model. The momentm eqation of the model takes the form as! " # $ #% & % ' & & ()* &, / 0 + $ In (), the vector cross prodct is the Lorentz force. This term is a bod force corresponding to magneto hdrodnamics flow. The mins sign in &4"5 6 8 indicates that the flid flows along the direction from higher to lower potential. The pls sign in shear stress of viscoelastic flid 946 : ; 84<= >$ < 8? is tensile stress or plling force. Using ohm s law, the expression for the densit of crrent can be AB. Upon sbstitting A0, since electric field is assmed to be negligible, it redces B. Also, the expression for Lorentz force redces and takes the form as &@ C 5. In view of these reslt, the eqation () redces to ()! " # $ # % & % ' & & D*# E! + &, - -. / 0 + $ ()

4 American Jornal of Applied Mathematics 07; 5(): In () = >$ is the component of the shear stress of the viscoelastic flid given in [6] as = >$ =9F<5 < F G 4<= >$ < 8?. Here F is the coefficient of dnamic viscosit and G is the modls of rigidit. If G approaches to infinit or if the component of shear stress is independent of time that is at stead state, the flid behaves like a viscos flid withot elasticit. Solving for = >$ in terms of the velocit component 5 we obtain <= >$ < = < < HF<5 < I < J< KF < < HF <5 < IL 5 F<C < C FC J N <O 5 < < CP The higher order terms are neglected since their contribtions are small enogh as G approaches to infinit. In view of these the eqation () after some algebraic maniplations redces to H <5 < I=H< < I+Q+6 :"F 5 C R<C : < C+ST U+SV WC 5 : C 5 C 6 :" C F C < O 5 6" C C : J C << C The energ eqation of the model can be expressed as X =J # X $ # Y \ ] +Z [ $ The concentration eqation of the model can be expressed as Z (4) =^# Z $ # 6 _ (5) The variables and parameters sed in this std and their meanings are as follows: ST thermal Grashof nmber; SV modified Grashof nmber; `V Schmidt nmber; time; W Hartmann nmber; Pr Prandtl nmber; c _ radiative heat flx; 6 : viscoelastic parameter; 6 _ chemical reaction parameter; d radiation parameter; ^ mass diffsivit; temperatre of the flid in the bondar laer; temperatre of the moving channel wall; temperatre of the stationar channel wall; free stream velocit; dimension less free stream velocit; 5 velocit component in direction; B velocit component in direction;, Cartesian coordinates;, dimension less Cartesian coordinates; concentration at channel wall at =0; concentration at channel wall at = ; dimension less concentration; f Specific heat at constant pressre; Electric crrent densit; acceleration de to gravit; % thermal expansion coefficient; % ' concentration expansion coefficient; amplitde of free stream velocit; Freqenc of oscillation; U dimension less temperatre; g Thermal condctivit; " Kinematic viscosit; F Dnamic viscosit; J Thermal diffsivit; Electric condctivit. The set of eqations () (5) govern the present model represented in Figre. In (4), the term 94 ; f 8<c _ <? is the thermal radiation effect. The mins sign of the radiative heat flx term indicates the emission of energ awa from the objects that is heat is lost. Frthermore, in this eqation, J>0 shows heat generation and J<0 indicates heat absorption i.e., thermal energ diffses more rapidl throgh sbstances with high thermal diffsivit J and it diffses slowl throgh those with low thermal diffsivit J. In (5), the mins sign indicates that the mass diffsion represents Generative chemical reaction. Here chemical reaction is an exothermic reaction and in that case heat is generated. However, the chemical reaction is said to be generative or exothermic if the chemical reaction parameter is negative i.e. 6 _ <0.If 6 _ >0 the chemical reaction is said to be destrctive or endothermic chemical reaction. Also 6 _ is the generative chemical reaction term. Based on Figre the bondar conditions of the model 5 can be expressed as =0,5 = : + m (6) = + m (7) = + m (8) =,5 =0, =, = (9) It is assmed that the radiation heat flx is nidirectional along positive direction. Using the Roseland approximation for radiative heat transfer and the Roseland approximation for diffsion and also following the other scholarl works [7], the expression for radiative heat flx can be given as c _ = 4@ 6 o < p <. Here, the and 6 o represent the Stefan Boltzmann constant and the Roseland mean absorption coefficient respectivel. We now assme that the temperatre differences within the flid flow are sfficientl small so that p can be expressed as a linear fnction of sing Talor series expansion. The Talor series expansion of p abot, after neglecting the higher order terms, takes the form as p O p 4. Using c _ and approximation of p in (4), we obtain X =Jr # X $ #s+h Y IH YtDX v +Z [ O- w Ir # X I $ #s+hx +Z (0) [ Here in (0), the term 94z : ; f 8? denotes heat absorption of the flid.. Non Dimensionalization of the Model In order to solve the governing eqations () (5) of the model it is convenient and eas to deal with its dimensionless form. Hence, we find the dimensionless form of the model b introdcing the following non dimensional qantities.

5 8 Binam Zigta and Prnachandra Rao Koa: The Effect of Phsical Parameters on Flow Variables of an Electricall Condcting Viscoelastic Flid O =, d=4@ { g6o,5= 5 :,= :,=, = C ",}T=" J,`V=" ^, SV=~% ' C " :,ST=% C " C C ;", =, 6=6 C : " C, 6 = -.! #, U= +, # After sbstitting the above non-dimensional qantities in () (5) and after simple algebraic maniplations, the nondimensional form of the model takes the following form as =! +r+-,ƒ! #s # $ #+STU+SV WC 5! # # -, # ƒ # v -, #! # # $ # () = Y f_ r+p O s# $ #+ x. +Z [ UhC () `V Z = # Z ]Z # $ # Š () The corresponding bondar conditions (6) (9) take the form as =0,5=+,U=+, =+ (4) =,5=0,U=0, =0 (5) The sstem of eqations () () together with the bondar conditions (4) (5) constittes the non dimensional form of the present model. 4. Analtical Soltion of the Problem To find the analtical soltion of the non dimensional form of the present model we consider eqations () (5) and we assme the amplitdes of the free stream velocit, temperatre and concentration variation are ver small qantities. Hence, the amplitde of free stream velocit needs to be considered 0< Using pertrbation techniqes the soltion of the model has the following form 5,=5 : +5 Y (6) U,=U : +U Y (7),= : + Y (8) Also, the free stream velocit takes the form as =+ (9) On sbstitting eqations (6) (9) into () (), and eqating harmonic and non-harmonic terms and neglecting higher orders of we obtain 5 : ŒŒŒ 5 : ŒŒ +Ž=STU : +SV : W C 5 : (0) 5 Y ŒŒŒ Ω 5 Y ŒŒ +Γ=STU Y +SV Y W C + 5 Y Κ () U" : +U K x. # O _ L=0 () +Z [ O p U" Y +U Y K x. # +Z [ O _ L=0 () O p : " ]Z. # =0 (4) Š Y " Y ] # + `V=0 (5) Š Frther, the new bondar conditions corresponding to 4 5 obtained after non dimensionalization as =0,5 : =,5 Y =,U : =,U Y =, : =, Y = (6) =,5 : =0,5 Y =0,U : =0,U Y =0, : =0, Y =0 (7) Up on solving eqations () (5) together with the bondar conditions (6) (7), we obtain the reqired analtical soltions as U : = Y $ + C E $ (8) U Y = Y š> + C Eš> (9) : = Y œ $ + C Eœ $ (0) Y =ž Y œ #$ +ž C Eœ #$ () Varios smbols are expressed in the Appendix. 5. Simlation Std of the Problem Here simlation std of different flow parameters is condcted sing Mat lab code. We consider flid flow bonded between two infinite vertical Coette channel walls embedded in a poros medim when the temperatre and concentration oscillates with time in the presence of thermal radiation, chemical reaction, permeabilit of poros medim and electricall condcting viscoelastic flid. For simplicit we have not presented the graphs of skin friction, Nsselt nmber, Prandtl nmber, mass diffsivit and Sherwood nmber. The effects of phsical parameters sch as viscoelastic parameter, permeabilit of poros medim, thermal Grashof nmber, modified Grashof nmber, Hartmann nmber, freqenc parameter, radiation parameter, chemical reaction parameter and Schmidt nmber on flow variables sch as velocit, temperatre and concentration of the model have been stdied sing simlation.

6 American Jornal of Applied Mathematics 07; 5(): velocit profile for different vales of freqenc ω=5 ω=8 ω= Figre. Velocit profile of the model for different vales of freqenc. In Figre, the simlated reslts of the inflence of freqenc parameter on velocit have been presented. The graph in 5 plane represents respectivel the distance between the channel walls and velocit. Other parameters are held constant. It can be observed that the velocit 5 starts with a minimm vale near the moving channel wall and then it increases as it goes towards the stationar channel wall. Frthermore, the velocit decreases with increasing vales of freqenc. 5 0 velocit profiles for different vales of Hartmann nmber(m) M=5 M=0 M= Figre. Velocit profile of the model for different vales of Hartmann nmber W. In Figre, the simlated reslts of the inflence of Hartmann nmber on velocit have been presented. The graph in 5 plane represents respectivel the distance between the channel walls and velocit. Other parameters are held constant. From the figre it is observed that if the vale of Hartmann nmber W is increased more than the strength of

7 84 Binam Zigta and Prnachandra Rao Koa: The Effect of Phsical Parameters on Flow Variables of an Electricall Condcting Viscoelastic Flid resistance of the magnetic field of the flid flow will also increase more; and as a reslt the velocit also will decrease more. Frthermore, the decrement in velocit reslts in decreasing the temperatre. It can be pt in simple words that whenever the Hartmann nmber increases then the velocit as well as thermal bondar laer will decrease. In Figre 4, the simlated reslts of the inflence of viscoelastic parameter 6 : on velocit have been presented. The graph in 5 plane represents respectivel the distance between the channel walls and velocit. Other parameters are held constant. The reslt shows that for a fixed vale of viscoelastic parameter 6 :, the vale of the flid velocit denoted b 5 starts from a minimm constant vale at the moving plate, and as increases the velocit 5 increases and reaches a maximm constant vale at the stationar channel wall. Generall, it can be conclded from the simlation std that as viscoelastic parameter 6 : increases the velocit profile decreases i.e. flow of the flid retards with increasing of viscoelastic parameter. In Figre 5, the simlated reslts of the inflence of temperatre U on velocit have been presented. The graph in 5 plane represents respectivel the distance between the channel walls and velocit. Other parameters are held constant. The reslt shows that for a fixed vale of temperatre U, the vale of the flid velocit denoted b 5 starts from a minimm constant vale at the moving plate, and as increases the velocit 5 increases and reaches a maximm constant vale at the stationar plate. Generall, it can be conclded from the simlation std that as temperatre U increases the velocit increases. In Figre 6, the simlated reslts of the inflence of thermal Grashof nmber ST on velocit have been presented. The graph in 5 plane represents respectivel the distance between the channel walls and velocit. Other parameters are held constant..5 velocit profile for different vales of viscoelastic parameter ko k0=5 k0=0 k0=5.5.5 Figre 4. Velocit profile of the model for different vales of viscoelastic parameter 6. The reslt shows that for a fixed vale of thermal Grashof nmber ST, the vale of the flid velocit denoted b 5 starts from a minimm constant vale at the moving channel wall, and as increases the velocit 5 increases and reaches a maximm constant vale at the stationar channel wall. Generall, it can be conclded from the simlation std that as thermal Grashof nmber increases the velocit profile increases. Phsicall, this means that the increment of the boanc force leads to increase the vertical component of the velocit. In Figre 7, the simlated reslts of the inflence of Modified Grashof nmber SV on velocit have been presented. The graph in 5 plane represents respectivel the distance between the channel walls and velocit. Other parameters are held constant. The reslt shows that for a fixed vale of Modified Grashof nmber SV, the vale of the flid velocit denoted b 5 starts from a minimm constant vale at the moving channel wall, and as increases the velocit 5 increases and reaches a maximm constant vale at the stationar channel wall. Generall, it can be conclded from the simlation std that velocit of the flid increases as modified Grashof nmber increases.

8 American Jornal of Applied Mathematics 07; 5(): velocit profile for different vales of temperatre θ=5 θ=0 θ= Figre 5. Velocit profile of the model for different vales of temperatre U. In Figre 8, the simlated reslts of the inflence of permeabilit of poros medim 6 on velocit have been presented. The graph in 5 plane is representing respectivel the distance between the channel walls and velocit. Other parameters are held constant. The reslt shows that for a fixed vale of permeabilit of poros medim 6, the vale of the flid velocit denoted b 5 starts from a minimm constant vale at the moving channel wall, and as increases the velocit 5 increases and reaches a maximm constant vale at the stationar channel wall. Generall, it can be conclded from the simlation std that as permeabilit of poros medim increases the velocit of the flid increases. Phsicall, this means that the resistance dominated b the poros medim redces as the permeabilit of the medim increases becase of which the velocit increases. 6 5 Velocit profile for different vales of thermal Grashof nmber Gr=5 Gr=0 Gr=5 4 0 Figre 6. Velocit profile of the model for different vales of thermal Grashof nmber ST.

9 86 Binam Zigta and Prnachandra Rao Koa: The Effect of Phsical Parameters on Flow Variables of an Electricall Condcting Viscoelastic Flid In Figre 9, the simlated reslt of the inflence of Modified Grashof nmber SV on the concentration of the flid has been stdied b simlation. Natrall, the nmber SV and the concentration are inversel proportional to each other i.e. as SV increases the velocit of the flid increases and hence the concentration decreases. The same fact has been even spported b the present simlation std. In Figre 0, the simlated reslt of the inflence of radiation parameter d on transfer of the temperatre has been presented. The graph in U plane is representing respectivel the distance between the channel walls and temperatre. Other parameters are held constant. The reslt shows that for a fixed vale of thermal radiation parameter d, the temperatre U starts from a constant vale at the moving channel wall. Also as increases the temperatre decreases till it reaches a minimm vale. Nevertheless, the measre of temperatre at the moving channel wall is alwas at a higher vale than that at the stationar channel wall velocit profile for different vales of modified Grashof nmber Gc=5 Gc=0 Gc= Figre 7. Velocit profile of the model for different vales of Modified Grashof nmber SV Velocit profile for different vales of permeabilit of poros medim parameter k=5 k=0 k= Figre 8. Velocit profile of the model for different vales of permeabilit of poros medim parameter 6.

10 American Jornal of Applied Mathematics 07; 5(): As the temperatre of the moving channel wall is higher than that of the stationar channel wall, the temperatre is expected to transfer from moving channel wall towards stationar channel wall. This fact has been spported in this simlation. Also it can be observed from that simlation: (i) the vale of temperatre does not change with the change in d at the moving channel wall, (ii) the minimm vale of temperatre, occrs between the two channel walls, decreases with the increase of d i.e. minimm vale of temperatre is inversel proportional to that of d, and (iii) the vale of temperatre approaches zero with the increase of d at the stationar channel wall. In Figre, the simlated reslt of the inflence of Chemical reaction parameter ŸT on concentration has been presented. The graph in % plane is representing respectivel the distance between the channel walls and concentration. Other parameters are held constant. The reslt shows that for a fixed vale of chemical reaction parameter ŸT, the vale of flid concentration denoted b % starts from a maximm constant vale at the moving, channel wall and as increases the concentration % decreases and reaches zero constant vale at the stationar channel wall. Nevertheless, the measre of flid concentration at the moving channel wall is alwas at a higher vale than that at the stationar channel wall. As the concentration of flid at the moving channel wall is higher than that at the stationar channel wall, the concentration is expected to move from moving channel wall towards stationar channel wall. Flids diffse from the locations of higher concentrations to that of lower concentrations. This fact has been spported in this simlation. Also it can be observed from that simlation: (i) the vale of concentration of flid does not change with the change in ŸT at the moving channel wall, (ii) the vale of concentration of flid, occrring between the two, channel walls decreases with the increase of ŸT i.e. the vale of concentration of flid is inversel proportional to that of ŸT, and (iii) the vale of concentration of flid goes to zero at the stationar channel wall irrespective of the vale of ŸT..5 Concentration profile for different vales of Modified Grashof Gc=5 Gc=0 Gc=5 C Figre 9. Concentration profile of the model for different vales of Modified Grashof nmber SV. In Figre, the simlated reslt of the inflence of Schmidt nmber `V on concentration has been presented. The graph in V plane is representing respectivel the distance between the channel walls and concentration. Other parameters are held constant. The reslt shows that for a fixed vale of Schmidt nmber `V the vale of flid concentration denoted b V starts from a maximm constant vale at the moving channel wall, and as increases the concentration V decreases and reaches zero constant vale at the stationar channel wall. Nevertheless, the measre of flid concentration at the moving channel wall is alwas at a higher vale than that at the stationar channel wall. As the concentration of flid at the moving plate is higher than that at the stationar channel wall, the concentration is expected to move from moving channel wall towards stationar channel wall. Flids diffse from the locations of higher concentrations to that of lower concentrations. This fact has been spported in this simlation.

11 88 Binam Zigta and Prnachandra Rao Koa: The Effect of Phsical Parameters on Flow Variables of an Electricall Condcting Viscoelastic Flid Figre 0. Temperatre profile of the model for different vales of Radiative parameter d). Also it can be observed from that simlation: (i) the vale of concentration of flid does not change with the change in `V at the moving channel wall, (ii) the vale of concentration of flid, occrring between the two channel walls, decreases with the increase of `V i.e. the vale of concentration of flid is inversel proportional to that of `V, and (iii) the vale of concentration of flid goes to zero at the stationar channel wall irrespective of the vale of `V..5 Concentration profile for different vales of chemical reaction parameter kr=5 kr=0 kr=5 β Figre. Concentration profile of the model for different vales of chemical reaction parameter 6T. In general, the concentration decreases with an increase in `V. Phsicall, this is tre since the increase in `V implies decrease of moleclar diffsivit implies decrease in concentration bondar laer.

12 American Jornal of Applied Mathematics 07; 5(): Concentration profile for different vales of Schimdth nmber Sc=5 Sc=0 Sc=5 Co Figre. Concentration profile of the model for different vales of Schmidt nmber `V. 6. Conclsions In this paper the effect of phsical parameters on flow variables of flid flowing between a pair of infinite vertical Coette poros channel walls embedded in a poros medim is analzed. The effects of phsical parameters Viz. Hartmann nmber, Viscoelastic parameter, Permeabilit of poros medim, Chemical reaction parameter, radiative parameter, thermal Grashof nmber for heat transfer, modified Grashof nmber for mass transfer, freqenc parameter and Schmidt nmber on flow variables Viz. velocit, temperatre and concentration has been discssed. The soltion of the governing eqations is obtained sing reglar pertrbation techniqes. This techniqe is sed to transform partial differential eqations that are difficlt to solve in closed form or explicit form. Mat lab code is sed to solve and simlate the graphs of higher order ordinar differential eqations. Hence we obtain the following reslts: i. Velocit decreases with increasing vales of freqenc, Hartmann nmber, Viscoelastic parameter. ii. Velocit increases with temperatre, thermal Grashof nmber, modified Grashof nmber and permeabilit of poros medim. iii. Thermal radiation parameter increases, the temperatre decreases near the moving plate while it approaches to zero in the region near to the bondar laer of the stationar plate. iv. An increment in both chemical reaction and Schmidt nmber reslts in decreasing in concentration. Appendix = O _ p O x. # +Z [, Y= Y # EY,, C= # # EY, Y=± - ] # Š,, %= r O _ p O sx. # +Z [, C =± - ] # Š + `V Y = Y YE #, C = # # EY, ž Y = Y YE # #, ž C= # # # #EY =N JC "F P + N : C J C C 6 : " C F C P, Y= C, C= C Ž= N : p 5 : J p g" p F C P, = N : C J C 6 : 6 : " C F C P =N JC "F P+N : C J C 6 : " C F C P, Ÿ =N : C J C 6 : " C F C P References [] K. Walters, Non-Newtonian effects in some elastic-viscos liqids whose behavior at small rates of shear is characterized b a general linear eqation of state. Qant. J. Mech. Appl. Math, 5, 96, [] Bhaskara Redd N. and Bathaiah, D., Reg. J. of Energ Heat Mass Transfer, (4), 98, [] Bhaskara Redd N. and Bathaiah, D., Acta Mechanica, 4, 98, 9-5. [4] Chowdhr M. K. and Islam M. N., MHD-free convection flow of viscoelastic flid past an infinite poros plate, Int. J. Heat Mass Trans., 6, 000,

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