An Application of HAM for MHD Heat Source Problem. with Variable Fluid Properties

Transcription

1 Advances in Theoretical and Applied Mechanics, Vol. 7, 14, no., HIKARI Ltd, An Application of HAM for MHD Heat Source Proble with Variable Fluid Properties T. Raja Rani 1, T S L Radhia and Rajendar Palli 3 Military Technological College, Muscat, Oan 1,3 BITS Pilani, Hyderabad, India Copyright 14 T. Raja Rani et al. This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original wor is properly cited. Abstract The present study deals with the free convective flow of a viscous fluid past an isotheral vertical plate ebedded in a hoogeneous porous ediu. It is assued that both the viscosity and theral conductivity vary linearly with teperature. The solution to the proble is obtained using Hootopy Analysis ethod (HAM) and the slip velocity values thus obtained are copared with the exact values obtained fro the expression derived fro the governing equations. It has been found that these values are in good agreeent. Thus, the authors could present an approxiate analytical solution to the class of MHD heat source probles which are ore often solved using nuerical ethods. Keywords: HAM, Heat source, Magnetic field, Variable fluid properties I. Introduction Heat transfer probles in porous edia, due to their enorous engineering applications, have attracted researchers fro diversified fields for the past few decades. Attepts have been ade to understand the effect of fluid properties, properties of the porous edia on the flow and the heat transfer. Classical boos on heat transfer [4,14] provide a great insight into this, where solutions (analytical or ostly nuerical) to the probles have been found under soe valid assuptions such as: the fluid viscosity is a constant, theral conductivity of ediu is a constant and so on. This is because the governing heat flow equations are highly nonlinear,

2 8 T. Raja Rani et al. that, in general cannot be solved analytically or even nuerically. Later, attepts were ade to find ore realistic solutions by assuing variable fluid properties etc. However, ost of the solutions were sought out using nuerical ethods as can be seen in for instance in references [5,9]. Further, few other engineering applications require the study of the effect of agnetic field and the presence of heat source in the flow doain. Here again, researchers found ostly nuerical solutions only [3, 1, 1, 15, 16]. But, with the advent of approxiate analytical ethods lie Adoian decoposition ethod, Hootopy perturbation ethod, Hootopy analysis ethod, it is now possible to find analytical solutions to nonlinear equations as well [17]. Thus, the present study considers the heat transfer proble with variable viscosity and theral conductivity under the presence of agnetic field and heat source. Though, in general, all the properties of the fluids change with teperature, in fact, changes in viscosity and theral conductivity are quite significant for any fluids and hence their effect on the heat flow and fluid flow are studied in the present paper. An approxiate analytical solution using the Hootopy analysis ethod (HAM) introduced by Liao in 199 [18, ] is found for the proble considered. Although several other approxiate analytical ethods (listed above) are available, the authors chose to wor with this ethod as it has been found to be a very successful ethod in finding solutions to highly nonlinear equations and coupled equations [,1,6]. References [1, 7, 13, 8] ephasize the effectiveness of this ethod in solving a few heat transfer probles as well. II. Matheatical forulation of the proble Consider an isotheral vertical flat plate ebedded in a saturated porous ediu with viscous incopressible fluid. The porous ediu is assued to be hoogeneous and is in the theral equilibriu with the surrounding fluid in the presence of heat source Q and a agnetic field of unifor strength is assued to be acting in a direction noral to the plate and transverse to the vertical plate. Let x-axis be taen vertically along the plate and y-axis perpendicular to it, T w is assued as teperature of the plate and T is an abient teperature of the fluid. The orientation of the plate for both hot and cold plate for free convection is presented in the Fig-1. The equations governing the free convection boundary-layer flow are: u v (1) x y p x p y () K g B u u v K (3)

3 Application of HAM for MHD heat source proble 81 T T T cpu v Q T T (4) x y y y where u, v are fluid velocity coponents, T is fluid teperature, K is Pereability, is effective theral conductivity of the porous ediu, B is the agnetic flux. Boussinesq approxiation [ 1 ( T T )] is taen as the body force ter and Q is the heat source. Now expressing the velocity coponents in ters of the strea function, we have: u y and v. x After introducing the strea function and eliinating fluid pressure fro equations () and (3), the governing equations siplify to T K B K g (5) y y y y T T 1 T T Q T T y x x y c p y y y cp The boundary conditions on T and are at y, T T w,, x (7) as y, T T, y Introducing Rayleigh nuber ( Ra x ), Hartan nuber ( M ), a agnetic interaction paraeter C, a heat source paraeter Q and the non-diensional functions f, together with a siilarity variable through the relations (6) * K g Tw T x BL * L K Rax ; M ; K ; C ; * f f K K M 1 Qx T T y Q ; f ( ) ; ( ) ; Ra 1 x Tw T Rax T w T x Rax (8) Equations (5), (6) are rewritten as

4 8 T. Raja Rani et al. 1 1 C f C f C (9) f Q (1) where is the viscosity variation coefficient and is the theral conductivity variation coefficient. 1 1 Taing f 1 and f 1 (Ref [5],[1][15]). The boundary condition (7) becoes at, 1, f, (11) as,, f Equation (9) can be integrated once using the condition on f at infinity to get C f (1) 1 C 1 Evaluating this expression at which gives the analytical expression for the slip velocity f () as C f () (13) C III. Paraeters of the Proble In the present proble, the flow and heat transfer depend on the paraeters: C ( the agnetic field paraeter), ( the viscosity variation coefficient), (theral conductivity variation coefficient) and Q (the heat source paraeter ). The agnetic paraeter C, taes on saller values (less than unity) when either the porous paraeter taes saller values or the Hartann nuber taes larger values. It is the case when porosity of the ediu or the intensity of agnetic field is high. Further, when there is no applied agnetic field, M taes a zero value as a result of which C taes the value unity as can be seen fro equation (8). So, in the present study, solutions are found for the values of C=.1,.5 and 1. As in [1] and [15], solutions has been found for the two cases (i) Tw T (Hot Plate) for which, as the teperature increases, increases ( w ) whereas decreases ( w ) as a result of which,

5 Application of HAM for MHD heat source proble 83 (ii) Tw T (Cold Plate) for which as the teperature increases, decreases ( w ) and hence we have that,. increases ( w ) while In this paper solutions are found for the values of both and as.. The heat source paraeter Q taes on positive values and the present study taes up the values as Q =.1 and.. IV. HAM Consider a nonlinear differential equation of the for: N u( x) (14) where N is a nonlinear operator, x is the independent variable and ux ( ) is the unnown function. Let u ( ) x be the initial approxiation of the exact solution ux ( ) and L be an auxiliary linear operator with the property that L( f) when f. (15) In this ethod, we construct the Hootopy which is a continuous apping H : u( x) x; q defined as ; ; 1 ; ( ) ; H x q q q L x q u x hh x qn x q (16) Here H( x) is an auxiliary function and h is an auxiliary paraeter called the convergence control paraeter,,1 q is an ebedding paraeter and xq ; is the approxiate solution to the given proble. We notice fro equation (16) that the solution obtained using this ethod, depends on the four iportant factors naely the initial approxiation u x, the linear operator L, the auxiliary function H( x ) and the auxiliary paraeter h. Taing q and the Hootopy defined by equation (16) to be zero, the so called zeroth order deforation equation is given by L x; u x (17) In view of the linearity of the operator L, the zeroth deforation equation is given by x; u x (18) Now, when q 1, equation (16) taes the for N x;1 This equation is sae as the given equation provided

6 84 T. Raja Rani et al. x;1 ux This shows that as the ebedded paraeter q varies fro to 1, xq ; (19) varies fro the initial guess u ( ) x (as is seen in equation (18)) to the exact solution ux(as ( ) seen equation (19)). Define the th order deforation derivatives as u x x q q ; q Then, using the Taylor s theore, xq ; u x x; q x; q Writing u x 1 can be expanded as a power series of q as () (1)! x u and using equation (18), the above taes the for! () 1 ; x q u x u x q With suitable choice of the initial guess, the auxiliary linear operator, the convergence control paraeter and the auxiliary function, Liao proved that the above power series solution converges for q 1 [17]. Now, to find the solution using equation (), the functions u x are to be found for =1,,3 Liao [17] has derived that these functions are given by the th order deforation equation defined as follows: where and 1 1 L u x u x hh x R u x (3) 1 1 Ru 1 x 1 N x; q ( 1)! q, 1 1, otherwise After deterining is q (4) (5) u x for =1,, an approxiate solution to the proble given in (14) u x u x u x (6) 1

7 Application of HAM for MHD heat source proble 85 Solution of equations (9) and (1): Let L f C Here f L f C e 1 e 1 N f 1 C f C f C 1 1 N 1 f Q (9) Construct the hootopy for the above nonlinear coupled equations as: H f ; q ; q 1 q L f ; q f hn f ; q f ; ; 1 ; ; H q q q L q hn q Assue the solutions as where 1 ; f q f f q ; q q f and 1 are obtained using equations (3)- (5). Using MATHEMATICA, the first four approxiations for the plots are presented. (7) (8) (3) (31) f and are calculated and Results and Discussions COLD PLATE: For and To find the values of the convergence control paraeter h in equations (3), the h-graphs for f i.e f h f h are plotted at in Fig. It can be seen that for C=.1 and Q=.1 and the control paraeter h=-.75. Siilarly, to find the convergence control paraeter h for finding, the h-graphs for are plotted as in Fig 3. The values of the slip velocity f () are found for a particular set of values of C and Q using the HAM solutions and also using the equation (1). The values are found to be in good agreeent with each other.

8 86 T. Raja Rani et al. A point to be noted here: Equation (1) shows that the slip velocity i.e., the velocity at the vicinity of the plate is independent of Q and the sae can be observed fro tables (1) and R u x as shown in (). Though HAM uses the value of Q in the evaluation of the equations (4) and (9), but still the values continued to be a constant for different values of Q. This ephasizes the fact that HAM is an efficient tool for solving coupled nonlinear equations. HOT PLATE: For and : As discussed above for the hot plate, the h curves are drawn for calculating the convergence control paraeter h for f and, the slip velocity values calculated using equation (1) and the HAM solution are tabulated in table. The values of slip velocities with distinct paraeters are shown in the table 1 and table for cold as well as hot plate. (i) (ii) (iii) (iv) It is apparent that the agnetic field lines obstruct the flow of fluid and hence reduced flow can be expected with the increase in the intensity of the agnetic field. The sae has been observed here: slip velocity decreased with the increase in the intensity of the agnetic field (i.e., C changes fro 1 to.1). It can also be seen that the values of the slip velocity will not vary with the strength of the heat source. Further, tables (1) and () show that the value of slip velocity f () is larger for hot plate ( ie..,.,.) than cold plate ( ie..,.,.). The teperature profiles are presented in fig 5 and fig 9, it can be observed that theral boundary layer thicness for cold plate is ore than hot plate. 1 Conclusion In the present study, the approxiate analytical solution using HAM is found for the hot as well as the cold plate in presence of heat source and agnetic field. The slip velocity values calculated using the approxiate ethod are copared with the exact values and are found to be in good agreeent with each other. The slip velocity for hot plate is ore than cold plate and the theral boundary layer thicness for the cold plate is ore than the hot plate. References 1. Abbasbandy S (7), The application of hootopy analysis ethod to nonlinear heat equations arising in heat transfer, Physics Letters (A),

9 Application of HAM for MHD heat source proble 87. Aloari A, Noorani MSM, Nazar R (8), The hootopy analysis ethod for the exact solutions of the K(,) Burgers and coupled Burgers equations, Applied Matheatical Sciences,, Bala Ani Reddy, P., Bhasar Reddy, N (13), MHD Free convection flow with variable viscosity and theral diffusivity along a oving vertical plate ebedded in a porous ediu, Acta Technica corviniensis- Bulliten of Engineering Toe-VI, vol 3, ISSN , Bejan A (4), Convection Heat Transfer, Third ed., Wiley. 5. Carey, V.P, and Mollendorf, J (198), Variable Viscosity effects in several Natural Convection Flows, Int. J. Heat Mass Transfer, 3, E.O Ifidon (9) An application of Hootopy Analysis to the viscous flow past a circular cylinder, Journal of Applied Matheatics, Hayat, T., M. Khan and S. Asghar (4) Hootopy analysis of MHD flows of an Oldroyd 8-constant fluid. Acta Mech., Vol 168: I. A. Abdallah (9), Hootopy Analytical Solution of MHD Fluid Flow and Heat Transfer Proble, Applied Matheatics & Inforation Sciences 3(), Lai, F.C, and Kulaci, F.A (199), The Effect of Variable Viscosity on Convective Heat Transfer along a Vertical Surface in a Saturated Porous ediu, Int.J. Heat Mass Transfer, 33, Lashi Prasanna, V, Raja Rani, T, C.N.B.Rao (9), Free Convection in a Porous ediu with Magnetic field, Variable Physical Properties and Varying Wall Teperature, IJCMI, 1(3), Liao S.J (4), On the hootopy analysis ethod for nonlinear probles, Applied Matheatics and Coputation, 147, Mahanti, N.C, Gaur,P (9), Effects of varying viscosity and theral conductivity on steady free convective flow and heat transfer along an isotheral vertical plate in the presence of heat sin, Journal of applied fluid echanics, Vol (1), Masoue Rahii, Mohaad Javad Hosseini, Ain Barari, Ganji Doairry, Mohaad Ebrahipour (11), Analytical Evaluation Of Heat Transfer Conductivity With Variable Properties, Technical Gazette 18, Nield,D.A, Bejan,A (6), Convection in Porous Media, Third ed., Springer, New Yor. 15. Rao, C.N.B, Raja Rani, T (9), Effect of variable viscosity on convection flows at an isotheral plate ebedded in a porous ediu in the presence of agnetic field, Int.J.of Logic Based Intelligent Systes, 3(1), Singh, G.,Shara, R.P, Chapa, A,J (1), Effect of voluetric heat generation/ Absorption on Mixed convection stagnation point flow on an Isotheral vertical plate in porous edia, Int. J. Industrial atheatics, Vol, (), S.J. Liao (3), Beyond Perturbation: Introduction to the Hootopy Analysis Method. Chapan & Hall/CRC Press, Boca Raton.

10 88 T. Raja Rani et al. 18. S.J. Liao (9), Notes on the hootopy analysis ethod: Soe definitions and theores. Co. Nonlinear Sci. Nuer. Siul Figures

11 Application of HAM for MHD heat source proble 89 Received: August 1, 14