NEW ANALYTICAL SOLUTION FOR NATURAL CONVECTION OF DARCIAN FLUID IN POROUS MEDIA PRESCRIBED SURFACE HEAT FLUX

Transcription

1 NEW ANALYICAL SOLUION FOR NAURAL CONVECION OF DARCIAN FLUID IN POROUS MEDIA PRESCRIBED SURFACE HEA FLUX Yousef VAZIFESHENAS, Davood DOMIRY GANJI*, Hasan SAJJADI Department of Mechanical Enineerin, Babol Universit of echnolo, P. O. Bo 484, Babol, Iran Abstract A new analtical method called He's Variational Iteration Method is introduced to be applied to solve nonlinear equations. In this method, eneral Larane multipliers are introduced to construct correction functional for the problems. It is stronl and simpl capable of solvin a lare class of linear or nonlinear differential equations without the tanible restriction of sensitivit to the deree of the nonlinear term and also is ver user friend because it reduces the size of calculations besides; its iterations are direct and straihtforward. In this paper the powerful method called Variational Iteration Method is used to obtain the solution for a nonlinear Ordinar Differential Equations that often appear in boundar laers problems arisin in heat and mass transfer which these kinds of the equations contain infinit boundar condition. he boundar laer approimations of fluid flow and heat transfer of vertical full cone embedded in porous media ive us the similarit solution for full cone subjected to surface heat flu boundar conditions. he obtained Variational Iteration Method solution in comparison with the numerical ones represents a remarkable accurac. Kewords: Porous Media, Variational Iteration Method (VIM), Ordinar Differential Equations (ODE) 1. Introduction Most scientific problems and phsical phenomena occur nonlinearl. Ecept in a limited number of these problems, we have difficult in findin their eact analtical solutions. herefore, there have been attempts to develop new techniques for obtainin analtical solutions which reasonabl approimate the eact solutions [1]. In recent decades, numerical calculation methods were ood means of analzin the nonlinear equations; but as the numerical calculation methods improved, semi-eact analtical methods did, too. Most scientists believe that the combination of numerical and semi-eact analtical methods can

2 also end with useful results. he investiation of convective heat transfer in fluid-saturated porous media has man important applications in technolo eothermal ener recover such as oil recover, food processin, fiber and ranular insulation, porous burner and heater, combustion of low-calorific fuels to diesel enines and desin of packed bed reactors. Man problems have been investiated about eternal natural convection in a porous medium adjacent to heated bodies [-3]. In these analses, the boundar laer approimations are applied to the problem of eternal natural convection and also it is assumed that Darc law approimations are applicable. he coupled set of overnin equations was solved b numerical methods or analtical solution. A reat deal of information is available on heat and fluid flow about such cones as reviewed b Ref. [-4]. In this paper the same approimations are applied to the problem of natural convection about an inverted heated cone embedded in a porous medium of infinite etent. No similarit solution eists for the truncated cone, but for the case of full cone similarit solutions eist if the prescribed surface heat flu or surface heat flu is a power function of distance from the verte of the inverted cone [, 5].As we know Perturbation method is one of the well-known methods to solve nonlinear problems; it is based on the eistence of small/lare parameters, the so-called perturbation quantit [6, 7]. Man nonlinear problems do not contain such kind of perturbation quantit, and we cannot use perturbation methods, such as the artificial small parameter method [8], the δ-epansion method [9], the Adomian s decomposition method [1], the homotop perturbation method (HPM) [11-9], and the variational iteration method (VIM) [3-33]. In this stud, we have applied VIM to find the analtical solutions of nonlinear ordinar differential equations arisin from similarit solution of natural convection of Darcian fluid about a vertical full cone embedded in porous media, and have made a comparison with numerical solution, which is solved b MALAB software [34] command ODE45 with the shootin procedures for the sstematicall uessin of the missin initial conditions.

3 Nomenclature acceleration due to ravit A prescribed constant ambient temperature λ prescribed constants, Cartesian coordinate sstem f similarit function for stream function distance of start point of cone from the verte θ similarit function for temperature β epansion coefficient of the fluid Ψ stream function η Independent dimensionless parameter q w surface heat flu υ kinematic viscosit of the fluid temperature Nu local Nusselt number α m hermal diffusivit the fluid-saturated porous medium r local radius of the cone u,v velocit vector alon, ais Ra local Raleih number w wall temperature K permeabilit of the fluid-saturated porous medium. Mathematical Formulation: Consider an inverted cone with semi anle γ and take aes in the manner indicated in Fi. 1. he boundar laer develops over the heated frustum. In terms of the stream function ψ defined b: 1 ψ 1 ψ u, v r r (1) he boundar laer equations are:

4 1 1 r K r m α ψ ψ ν β ψ () For a thin boundar laer we have approimatel r sin (γ). We suppose that either a power law of temperature or a power law of heat flu is prescribed on the frustum. Accordinl, the boundar conditions are: < < A k q u u m m, ) (,, λ (3) For the case of a full cone (, fi. 1(b)) a similarit solution eists. In the case of prescribed wall temperature, we let:, ), ( ), ( 1/ 3 1/ 3 1/ 3 m w m Ra Ra k q f rra θ α ψ (4) Where the Raleih number is based on heat flu, m m w k q K Ra υα γ β ) cos( (5) he overnin equations become h f h f h h f λ λ (6) Subject to

5 f ( ), h'() 1, h( ) (7) Finall from eqs. (6) and (7) we have: λ + f + f (), f 5 λ + 1 f f ( f ) 3 () 1, f ( ),, (8) It is of interest to obtain the value of the local Nusselt number which is defined as: Nu 1/ 3 Ra h () 1 (9) From eqs. (4), (5) and (9) it follows that the local Nusselt number is iven b: Nu 1/ 3 [ h() ] Ra (1) 3. Applications of VIM Consider overnin equation of fluid flow and heat transfer of full cone embedded in porous medium that is epressed b eq. (8) for wall temperature boundar condition. Consider equation that prescribed wall temperature case that is epressed b eq. (8). Fi.1: Coordinate sstem for a full cone ( ).

6 o clarif the basic ideas of He's VIM, we consider the followin differential equation: L u + Nu ( ) (11) Where L is a linear operator, N a nonlinear operator and (t) an inhomoeneous term. Accordin to VIM, we can write down a correction functional as follows: u + ( ) u ( ) + λ{ Lu ( τ) + Nu ( τ) ( τ)} dτ (1) n 1 n n n o solve eq. (8), usin the VIM, we have the correction functional as 1 1 λ λ λ ds (13) 3 f ( ) 1 e + [ f ( S) + ( + 5) f( s) f ( s) ( + 1)( f ( s)) hen 1 1 λ λ ds (14) 3 f ( ) 1 e + ( s )[ f ( s) + ( + 5) f( s) f ( s) ( + 1)( f ( s)) So finall we have f ( ) + e λ+ + λ+ λe + λe e (15) And also f ( ) e + + λ λe λe + e (16) So now for different amounts of λ and, we can compare the answers.

7 Fi : f () for different amounts of λ. able 1: f () for different amounts of λ. λ λ 1/4 λ 1/

8 able - f values obtained b Numerical and VIM solution for λ 1/4. λ 1/4-VIM λ 1/4-NUM Fi 3: Comparison between VIM & Numerical solution for λ 1/4. 4. Results and discussion he results for f'() have been shown in fi. with selected λ as, 1/4, 1/3, and it is seen that increasin these amounts, makes more inclination from the main answer, and also the trend confirms the boundar

9 condition which is applied b bein near zero in infinit. Additionall there is a comparison between the VIM solution and numerical solution in tab. and fi. 3. Here nonlinear differential equation arisin from similarit solution of inverted cone embedded in porous medium has been studied usin variational iteration method. he comparison with numerical results and converence stud shows that b increasin the number of order, the accurac of the solution increases. Some of the advantae of VIM are that reduces the volume of calculations with the fewest number of iterations or even in some cases, once, it can convere to correct results. he proposed method is ver simple and straihtforward. In our work, we use the Maple Packae to calculate the functions obtained from the variational iteration method. References [1] A.H. Nafeh, Introduction to Perturbation echniques, John Wile and Sons, New York (1981). [] D.A. Nield, A. Bejan, Convection in Porous Media, third ed., Spriner-Verla, New York, (6). [3] I. Pop,. Y. Na, Natural convection over a frustum of a wav cone in a porous medium, Mech. Res. Comm. (1995), pp [4] I. Pop, D. B. Inham: Convective Heat ransfer: Mathematical and Computational Modelin of Viscous Fluids and Porous Media, Peramon, Oford (1). [5] P. Chen,.. Le, I. Pop, Natural convection of a Darcian fluid about a cone, International Communications Heat Mass ransfer 1 (1985). pp [6] J.D. Cole, Perturbation Methods in Applied Mathematics, Blaisdell Waltham, MA, (1968). [7] A.H. Nafeh, Perturbation Methods, Wile, New York, (). [8] A.M. Lapunov, General Problem on Stabilit of Motion, alor & Francis, London, (Enlish translation). (199). [9] A.V. Karmishin, A.I. Zhukov, V.G. Kolosov, Methods of Dnamics Calculation and estin for hin- Walled Structures, Mashinostroenie, Moscow, (199). [1] G. Adomian, Solvin Frontier Problems on Phsics: he Decomposition Method, Kluwer Academic, Dordrecht, (1994). [11] J.H. He, Homotop Perturbation Method for Bifurcation of Nonlinear Problems, International Journal of Nonlinear Sciences and Numerical Simulation. 6 (5). pp [1] J.H. He, he homotop perturbation method for nonlinear oscillators with discontinuities Applied Mathematics and Computation, 151 (4). pp [13] D.D. Ganji, A. Rajabi, Assessment of homotop perturbation and perturbation methods in heat radiation equations International Communications in Heat and Mass ransfer, 33 (6), pp

10 [14] J.H. He, Homotop perturbation method for solvin boundar value problems, Phsics Letters A, 35 (6), pp [15] J.H. He, Some asmptotic methods for stronl nonlinear equations, International Journal of Modern Phsics B (6), pp [16] Ganji, D.D., H. Mirolbabaei, Me.miansari, Mo. Miansari, "Application of homotop perturbation method to Solve linear and non-linear sstems of ordinar differential equations and differential equation of order three", Journal of Applied Sciences 8 (8), pp [17] H. Mirolbabaei, D.D. Ganji, H. aherian, Soliton sulotion of the Kadomtse-Petviashvili equation b homotop perturbation method, World Journal of Modelin and Simulation 5 (9), pp [18] H. Mirolbabaei, D.D. Ganji, Application of homotop perturbation method to the combined KdV MKdV equation, Journal of Applied Sciences, 9 (9), pp [19] M. Omidvar, A. Barari, H. Mirolbabaei, D.D. Ganji, Solution of Diffusion Equations usin Homotop Perturbation and Variational Iteration Methods, MAEJO International Journal of Science and echnolo(in press). [] A. Barari, H. Mirolbabaei, D.D. Ganji, Application of He s Variational Iteration Method to Nonlinear Helmholtz and Fifth-order KDV Equations, SDU Journal of Science(in press). [1] He JH. A couplin method of homotop and perturbation technique for nonlinear problems. International Journal of Nonlinear Mech. 35 (), pp [] He JH. Homotop perturbation method, a new nonlinear analtic technique. Applied Mathematics and computations. 135 (3), pp [3] He JH. A new approach to nonlinear partial differential equations. Communication in Nonlinear Sciences and Numerical Simulation. (1997), pp [4] He JH. Comparison of homotop6 perturbation and homotop analsis method. Applied Mathematics and Computations, 156 (4), pp [5] He JH. An approimation solution technique dependin upon an artificial parameter Communication in Nonlinear Sciences and Numerical Simulation. 3(1998), pp [6] He JH. Application of homotop perturbation method to nonlinear wave equations. Chaos Solutions and Fractals. 6 (5), pp [7] He JH. he homotop perturbation method for nonlinear oscillators with discontinuities. Applied Mathematics and Computations. 151(4), pp [8] He JH. A simple perturbation approach to Blasius equation. Applied Mathematics and Computations. 14 (3), pp [9] Öziş., A. Yildrim., Comparison between Adomian s method and He s homotop perturbation method. Computers and Mathematics with Applications. 56 (8), pp

11 [3] Konuralp, A., the stead temperature distributions with different tpes of nonlinearities. ( in press) (9). [31] J.H. He, Variational iteration method for autonomous ordinar differential sstems, Applied Mathematics and Computation, 114 (), pp [3] J.H. He, Approimate solution of nonlinear differential equations with convolution product nonlinearities, Computer Methods in Applied Mechanics and Enineerin, 167 (1998), pp [33] D.D. Ganji, A. Sadihi, Application of homotop-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations, Journal of Computational and Applied Mathematics. 7 (7), pp [34] MALAB 6 the lanuae of technical computin; 6.