Hall Effcts Eon Unsteady MHD Free Convection Flow Over A Stretching Sheet With Variable Viscosity And Viscous Dissipation

Transcription

1 IOSR Journal of Matematics (IOSR-JM) e-issn: , p-issn: X. Volume, Issue 4 Ver. I (Jul - Aug. 5), PP Hall Effcts Eon Unsteady MHD Free Convection Flow Over A Stretcing Seet Wit Variable Viscosity And Viscous Dissipation Srinivas maripala, Kisan Naikoti, Department of Matematics, SNIST, yamnampet, gatkesar,hyderabad,india Dept. of Matematics, Osmania University, Hyderabad, India. Abstract: Te unsteady magnetoydrodynamics free convection flow of an incompressible viscous fluid past a stretcing surface is analyzed by taking into an account te variable viscosity and viscous dissipation under te influence of all currents effect. Te problem is governed by coupled nonlinear partial differential equations. Te coupled non-linear ordinary differential equations ave been solved numerically by te implicit finite difference metod. Te effect of magnetic parameter M, Hall parameter m, Prandtl number Pr, Eckert number Ec, on velocity and temperature fields are investigated troug graps. Keywords: MHD, viscous dissipation, implicit finite difference metod, Hall parameter. I. Introduction Te study of te flow of electrically conducting fluid in te presence of magnetic field is important from te tecnical point of view and suc types of problems ave received muc attention by many researcers. Te specific problem selected for study is te flow and eat transfer in an electrically conducting fluid adjacent to te surface. Te surface is maintained at a uniform temperature T w, wic may eiter exceed te ambient temperature T or may be less tent. Wen T w > T, an upward flow is establised along te surface due to free convection, were as for T w < T, tere is a down flow. Te interaction of te magnetic field and te moving electric carge carried by te flowing fluid induces a force, wic tends to oppose te fluid motion. Te velocity is very small so tat te magnetic force, wic is proportional to te magnitude of te longitudinal velocity and acts in te opposite direction, is also very small. Additionally, a magnetic field of strengt acts normal to te surface. Consequently, te influence of te magnetic field on te boundary layer is exerted only troug induced forces witin te boundary layer itself, wit no additional effects arising from te free stream pressure gradient. Te stress work effects in laminar flat plate natural convection flow ave been studied by Ackroyd []. However, te influence and importance of viscous stress work effects in laminar flows ave been examined by Gebart [] and Gebart and Mollendorf [3]. In bot of te investigations, special flows over semiinfinite flat surfaces parallel to te direction of body force were considered. In all te above-mentioned studies, te viscosity of te fluid was assumed to be constant. However, it is known tat te fluid pysical properties may cange significantly wit temperature canges. To accurately predict te flow beaviour, it is necessary to take into account tis variation of viscosity wit temperature. Recently, many researcers investigated te effects of variable properties for fluid viscosity and termal conductivity on flow and eat transfer over a continuously moving surface. Seddeek [4] investigated te effect of variable viscosity on ydro magnetic flow past a continuously moving porous boundary. Seddeek [5] also studied te effect of radiation and variable viscosity on an MHD free convection flow past a semi-infinite flat plate witin an aligned magnetic field in te case of unsteady flow. Dandapat et al. [6] analyzed te effects of variable viscosity, variable termal conducting, and termocapillarity on te flow and eat transfer in a laminar liquid film on a orizontal stretcing seet. Mukopadyay [7] presented solutions for unsteady boundary layer flow and eat transfer over a stretcing surface wit variable fluid viscosity and termal diffusivity in presence of wall suction. Wen te conducting fluid is an ionized gas and te strengt of te applied magnetic field is large, te normal conductivity of te magnetic field is reduced to te free spiraling of electrons and ions about te magnetic lines force before suffering collisions and a current is induced in a normal direction to bot electric and magnetic field. Tis penomenon is called Hall effect. Wen te medium is a rare field or if a strong magnetic field is present. Abo-Eldaab et al. [8] and Salem and Abd El-Aziz [9] dealt wit te effect of Hall current on a steady laminar ydromagnetic boundary layer flow of an electrically conducting and eat generating/absorbing fluid along a stretcing seet. Pal and Mondal [] investigated te effect of temperaturedependent viscosity on non-darcy MHD mixed convective eat transfer past a porous medium by taking into account Omic dissipation and non-uniform eat source/sink. Abd El-Aziz [] investigated te effect of Hall currents on te flow and eat transfer of an electrically conducting fluid over an unsteady stretcing surface in te presence of a strong magnet. Wen te strengt of magnetic field is strong, one can t neglect te effect of all current. It is of considerable importance and interest to study ow te results of te ydro dynamical DOI:.979/ Page

2 problems get modified by te effect of all curent s. Te all effect is due merely to te sideways magnetic force on te drifting free carges. Te electric field as to ave a component transverse to te direction of te current density to balance tis force. In many works on plasma pysics, te all effect is ignored. But if te strengt of magnetic field is ig and te number density of electrons is small, te all effect cannot be disregarded as it as a significant effect on te flow pattern of an ionized gas. Hall effect results in a development of an additional potential difference between opposite surfaces of a conductor for wic a current is induced perpendicular to bot te electric and magnetic field. Tis current is termed as all current. Model studies on te effect of all current on MHD convection flows ave been carried out by many autors due to application of suc studies in te problems of MHD generators and all accelerators as studied by kisan et al []. Some of tem are Pop [3], Kinyanjui et al [4], Aboeldaab [5], Datta et al [6], studied te all effects on MHD flow past an accelerated plate. Halls effect is also important wen te fluid is an ionized gas wit low density or te applied magnetic field is very strong. Because te electrical conductivity of te fluid will ten be a tensor and a current (Hall current) is induced wic is likely to be important in many engineering situations. Te effect of all current on te fluid flow wit variable concentration as many applications in MHD power generation, in several astropysical and metrological studies as well as in plasma flow troug MHD power generators. In te present investigation, it is proposed to study te effect of magnetoydrodynamics unsteady free convection flow of an incompressible viscous fluid past a stretcing surface is analysis by taking into account te all effects and viscous dissipation. Fluid viscosity is assumed to vary as an exponential function of temperature wile te fluid termal diffusivity is assumed to vary as a linear function of temperature. Te governing equations are solve by te using te implicit finite difference sceme using C-programming. Recently, S.Sateyi [7]studied te variable viscosity on magnetic ydrodynamic fluid flow and eat transfer over an unsteady stretcing surface wit all effect. II. Matematical Analysis We consider te unsteady flow and eat transfer of a viscous, incompressible, and electrically conducting fluid past a semi-infinite stretcing seet coinciding wit te plane y =, ten te fluid is occupied above te seet y. Te positive x coordinate is measured along te stretcing seet in te direction of motion, and te positive y coordinate is measured normally to te seet in te outward direction toward te fluid. Te leading edge of te stretcing seet is taken as coincident wit z-axis. Te continuous seet moves in its own plane wit velocity U w x, t and te temperature T w x, t distribution varies bot along te seet and time. A strong uniform magnetic field is applied normally to te surface causing a resistive force in te x- direction. Te stretcing surface is maintained at a constant temperature and wit significant all currents. Te magnetic Reynolds number is assumed to be small so tat te induced magnetic field can be neglected. Te effect of Hall current gives rise to a force in te z-direction, wic induces a cross flow in tat direction, and ence te flow becomes tree dimensional. To simplify te problem, we assume tat tere is no variation of flow quantities in z-direction. Tis assumption is considered to be valid if te surface is of infinite extent in te z-direction. Finally, we assume tat te fluid viscosity is to vary wit temperature wile oter fluid properties are assumed to be constant. Using boundary layer approximations, te governing equations for unsteady laminar boundary layer flows are written as follows: x + v = () t + x w t T t + u w x +v +v w = ρ (μ = ρ T T + u + v = x ρc p x ) μ w x k T σb ρ +m + (u + mw) () σb ρ +m + μ[ x (mu w) (3) + ] (4) Te boundary conditions are defined as follows; u = U w x, t, v =, T = T w x, t at y = (5) u, w, T T w as y tends to, were u and v are te velocity components along te x- and y-axis, respectively, w is te velocity component in te z direction, ρ is te fluid density, β is te coefficient of termal expansion, μ is te kinematic viscosity, g is te acceleration due to gravity, c p is te specific eat at constant pressure, and k is te temperature-dependent termal conductivity. Following Elbasbesy and Bazid, we assume tat te stretcing velocity U w x, t is to be of te following form U w = bx ( ct) (6) DOI:.979/ Page

3 were b and c are positive constants wit dimension reciprocal time. Here, b is te initial stretcing rate, wereas te effective stretcing rate b/( ct) is increasing wit time. In te context of polymer extrusion, te material properties and in particular te elasticity of te extruded seet vary wit time even toug te seet is being pulled by a constant force. Wit unsteady stretcing, owever, c becomes te representative time scale of te resulting unsteady boundary layer problem. Te surface temperature T w of te stretcing seet varies wit te distance x along te U w x, t seet and time t in te following form: T w x, t = T + T bx ( v αt) 3 (7) Were T is a (positive or negative; eating or cooling) reference temperature. Te governing differential equations ( ) (4) togeter wit te boundary conditions (5) are nondimensionalised and reduced to a system of ordinary differential equations using te following dimensionless variables: = b v (( αt) ) / y, ψ = vb αt xf(), w = bx αt (), T = T + T bx v αt 3 (),B = B ct,ec= u C w (T w T ) were ψ(x, y, ) is te pysical stream function wic automatically assures mass conservation () and B is constant. We assume te fluid viscosity to vary as an exponential function of temperature in te non dimensional form μ = μ e β, were μ is te constant value of te coefficient of viscosity far away from te seet, β is te variable viscosity parameter. Te variation of termal diffusivity wit te dimensionless temperature is written as k = k ( + β ) were β is a parameter wic depends on te nature of te fluid, k is te value of termal diffusivity at te temperature T w. Upon substituting te above transformations into () (4) we obtain te following: (8) M f β f + e β [ff (f ) S f + f +m (f + m)] = (9) M β + e β [f f S + + +m (mf )] = () + β + β + Pr f f S Ec[ f + f ] = () were te primes denote differentiation wit respect to, and te boundary conditions are reduced to f() =, f () =, ()=, ()=.(a) ( )=, f( )=, ( )=.(b) Te governing non dimensional equations (9) () along wit te boundary conditions (a)-(b) are solve by te using te implicit finite difference sceme using C-programming. III. Results and discussion In order to solve te unsteady, non-linear coupled equations 9- along wit boundary conditions an implicit finite difference sceme of cranck-nicklson type as been employed. Te finite difference scemes, te dimensionless governing equations are reduced to try-diagonal system of equations wic are solved by Tamos algoritm. Te max cosen as corresponds to after some preliminary investigation so tat te last two boundary condations (a), (b) are satisfied at wit in te tolerance limit of 5 te mes size as been fixed as.. Te numerical computations are ave been carried out for te different governing parameters suc as magnetic parameter M, Hall parameter m, Prandtal number Pr, Eckert number Ec, unsteadiness parameter S, viscosity parameter β and diffusivity parameter β only selective figures ave been sown ere for brevity. In figure (a)-(c) te influence of variable viscosity β on axial velocity, Transverse velocity and temperature profiles respectively are sown. It is observed from te figure (a) tat te effective of te variable viscosity β is to reduce te axial velocity profiles f, due to te increase of β te boundary layer tickness decreases. It can be seen from te figure (b) wit te increase of β te transverse velocity increases to a peak value near te boundary wall and ten decays rapidly to te relevant free stream velocity. From figure (c) it can be conclude tat te distribution of temperature () increases te variable viscosity β increases. Figures (a)-(c) depicts te influence of magnetic field parameter on axial velocity,transitive velocity and DOI:.979/ Page

4 temperature profiles respectively. From te figure one can find tat te axial velocity profiles decreases if te increase of magnetic parameter M, wile it can be seen tat te temperature profiles increases wit te increase of magnetic parameter M. It is obvious tat te effect of te magnetic parameter results in a decreasing velocity distribution across te boundary layer. Tis is due to act tat te effect of a transverse magnetic field gives rise to a resistive type force called te Lourntz force. Tis force as a tendency to flow down te motion of te fluid. From figure (b) it can be noticed tat te effect of magnetic flied increases te transverse velocity filed increases te transverse velocity filed increases and reaces to a peak value near te vicinity of te boundary layer and approaces to zero. It is also noticed tat more influence of magnetic filled will reac peak value and reaces to zero near te boundary layer, wereas te less magnetic field influence ave less peak value and it will reaces to zero far away from te boundary. Figure (c) it can be observed tat te effect of magnetic flied parameter increases te temperature profiles. Te effect of all current parameter m on axial velocity, transverse velocity and temperature profiles are sown in figure 3(a)-3(c). It is observed tat as te all current parameter increase te transverse velocity profiles increases up to te value of m=.5. te transverse flow in direction is to increases. However for te values of all current parameter m greater tan.5. te transverse velocity profile decreases as tese values increases. Tis is due to fact tat for large values of m te term M/(+m ) is very small ; and ence te resistive effect of te magnetic field is to diminised. Figures 4(a) to 4(c) illustrates te effect of te unsteadiness parameter S on axial velocity, transverse velocity and temperature profiles respectively. From tese figures, it can be seen tat te increase of unsteadiness parameter S is to decrease te transverse magnetic field as well as temperature profiles. it also observe from te figure 4(b) tat te transverse velocity profiles is to decreases te transverse velocity greatly near te plate and te reverse appen far away from te plate. Te effects due to viscous dissipation Ec on axial velocity, transverse velocity and te temperature profiles are sown in plated figures 5(a)-5(c). From tese figures revels tat te influence of viscous dissipation effects is to increases te axial velocity profile and temperature profiles. It can be observed from figure 5(b) tat te transverse velocity profiles decreases wit te increases of Eckert number nearest vicinity of te wall and te reverse penomenon is observed away from te wall and te effect is ig away from te plate. Tis is due to te fact tat te eat energy is stored in te fluid due to te frictional eating. So we can say tat te strong frictional eating slow down te cooling processes and in tis case te study suggest tat te rapidly cooling of te surface can be made possible if te viscous dissipation can be made as small as possible. Figure 6(a) presents typical profile of transverse velocity profiles for te different values of termal diffusivity parameter β. It can be noticed figure is decreases te transverse velocity profiles. Te effect of termal diffusivity parameter β is to increases te temperature distribution is noticed from figure 6(b). Tis is due to te ticking of te termal boundary layer as results of increasing of termal diffusivity. Figure.7 presents te effects of prandtl number Pr effect on temperature profiles increasing te value of Pr as te tendenstive decreases te fluid temperature in te boundary layer as were as te termal boundary layer tickness. β=.,s=.,pr=.7,m=.,m=.,ec=..5 β =. β =.5 β =. β = Fig.(a).Te varation axial velocity profiles wit te values of β β=.,s=.,pr=.7,m=.,m=.,ec=..5. β =. β =. β =.5 β = Fig.(b).Transverse velocity profiles for various values of β DOI:.979/ Page

5 β=.,s=.,pr=.7,m=., m=.,ec=..5 β =. β =.5 β = Fig.(c). Temperature profiles for various values of β β=.,β=.,s=.,pr=.7,m=5.,ec=. M =. M =. M = Fig.(a).Te varation axial velocity profiles wit values of M β=.,β=.,s=.,pr=.7,m=.,ec=. M =. M =. M = 3. M = 5. M = Fig.(b).Transverse velocity profiles for various values of M.5 β=.,β=.,s=.,pr=.7,m=.,ec=. M =. M =. M = Fig.(c). Temperature profiles for various values of M.5 β=.,β=.,s=.,pr=.7,m=.,ec=. m =. m =. m = Fig.3(a).Te varation axial velocity profiles wit increasing values of m DOI:.979/ Page

6 β=.,β=.,s=.,pr=.7,m=.,ec=. m =. m =. m = 3. m = Fig.3(b).Transverse velocity profiles for various values of m.5 β=.,β=.,s=.,pr=.7,m=.,ec=. m =. m =. m = Fig.3(c). Temperature profiles for various values of m.5 β=.,β=.,pr=.7,m=.,m=.,ec=. S =. S =.5 S = Fig.4(a).Te varation axial velocity profiles wit increasing values of S.4 β=.,β=.,pr=.7,m=.,m=.,ec=..7 S =. S =.5 S = Fig.4(b).Transverse velocity profiles for various values of S DOI:.979/ Page

7 β=.,β=.,pr=.7,m=.,m=.,ec=..5 S =. S =. S = Fig.4(c). Temperature profiles for various values of S β=.,β=.,s=.,pr=.7,m=.,m=. Ec =..5 Ec =. Ec =.5 Ec = Fig.5(a).Te varation axial velocity profiles wit increasing values of Ec..6 β=.,β=.,s=.,pr=.7,m=.,m=. Ec =. Ec =.5 Ec = Fig.5(b).Transverse velocity profiles for various values of Ec β=.,β=.,s=.,pr=.7,m=3.,m=. Ec =..5 Ec =.5 Ec =. Ec = Fig.5(c). Temperature profiles for various values of Ec DOI:.979/ Page

8 .5 β=.,s=.,pr=.7,m=.,m=.,ec=...5 β =. β =. β = Fig.6(a).Transverse velocity profiles for various values of β β=.,s=.,pr=.7,m=.,m=.,ec=..5 β =. β =.5 β = Fig.6(b). Temperature profiles for various values of β β=.,β=.,s=.,m=.,m=.,ec=...5 Pr =. Pr =. Pr = Fig.7. Temperature profiles for various values of Pr References []. J.A.D. Ackroyd, Stress work effects in laminar flat-plate natural convection, J. Fluid Mec., 6(4), pp , 974. []. B. Gebart, Effects of viscous dissipation in natural convection, J. Fluid Mec., 4(), pp. 5 3, 96. [3]. B. Gebart, J. Mollendorf, Viscous dissipation in external natural convection flows, J. Fluid Mec., 38(), pp. 97 7,969. [4]. M.A. Seddeek, Te effect of variable viscosity on yromagnetic flow and eat transfer past a continuously moving porous boundary wit radiation, International Communications in Heat and Mass Transfer, vol. 7, no. 7, pp ,. [5]. M.A. Seddeek, Effects of radiation and variable viscosity on amhdfree convection flow past a semi infinite flat plate wit an aligned magnetic field in te case of unsteady flow, International Journal of of Heat and Mass Transfer, vol. 5, no. 5-6, pp , 7. [6]. B.s Dandapat, B. Santra, and K. Vajravelu, Te effects of variable fluid properties and termocapillarity on te flow of a tin film on an unsteady stretcing seet,., International Journal of Heat and Mass Transfer, vol. 5, no. 5-6, pp , 7. [7]. S. Mukopadyay, Unsteady boundary layer flow and eat transfer past a porous stretcing seet in presence of variable viscosity and termal diffusivity, International Journal of Heat and Mass Transfer, vol. 5, no. -, pp , 9. [8]. E.M. Abo-Eldaab, M. A. El-Aziz, A. M. Salem, and K. K. Jaber, Hall current effect on MHDmixed convection flow from an inclined continuously stretcing surface wit blowing/suction and internal eat generation/absorption, Applied Matematical Modelling, vol. 3, no. 9, pp , 7. [9]. A.M. Salem and M. Abd El-Aziz, Effect of Hall currents and cemical reaction on ydro magnetic flow of a stretcing vertical surface wit wit internal eat generation/absorption, Applied Matematical Modelling, vol. 3, no. 7, pp , 8. []. D.. Pal and H. Mondal, Effect of variable viscosity on MHD non-darcy mixed convective eat transfer over a stretcing seet embedded in a porous medium wit non-uniform eat source/sink. Communications in Nonlinear Science and Numerical Simulation, vol. 5, no. 6, pp ,. []. M. Abd El-Aziz, Flow and eat transfer over an unsteady stretcing surface wit Hall effect, Meccanica, vol. 45, no., pp. 97 9,. []. Bala Siddulu Malga, Govardan Kamatam and Naikoti Kisan. Finite Element Analysis of Hall Effects on MHD Flow past an Accelerated Plate., International Journal of Matematics Researc, Volume 4, Number 3 (), pp [3]. Pop I. Te effect of Hall currents on ydro magnetic flow near, an accelerated plate. DOI:.979/ Page

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