The Spectral Homotopy Analysis Method Extended to Systems of Partial Differential Equations

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 241594, 11 pages http://dx.doi.org/10.1155/2014/241594 Research Article The Spectral Homotopy Analysis Method Extended to Systems of Partial Differential Equations S. S. Motsa, F. G. Awad, Z. G. Makukula, and P. Sibanda School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg 3209, South Africa Correspondence should be addressed to S. S. Motsa; Received 20 January 2014; Accepted 9 April 2014; Published 29 April 2014 Academic Editor: Dianchen Lu Copyright © 2014 S. S. Motsa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The spectral homotopy analysis method is extended to solutions of systems of nonlinear partial differential equations. The SHAM has previously been successfully used to find solutions of nonlinear ordinary differential equations. We solve the nonlinear system of partial differential equations that model the unsteady nonlinear convective flow caused by an impulsively stretching sheet. The numerical results generated using the spectral homotopy analysis method were compared with those found using the spectral quasilinearisation method (SQLM) and the two results were in good agreement. 1. Introduction The current study serves to demonstrate the extension of the spectral homotopy analysis method (SHAM) to systems of nonlinear partial differential equations. The method, originally introduced by Motsa et al. [1, 2], was used to solve nonlinear ordinary differential equations. A recent modification (see Motsa [3]) allows the method to be used to find solutions of nonlinear PDEs. A nonlinear system of partial differential equations that describe unsteady nonlinear convective flow caused by an impulsively stretched plate is used as the test problem in the study. The spectral quasilinearisation method (SQLM), an adaptation of the quasilinearisation method (Bellman and Kalaba [4]), is used as a benchmark to prove the accuracy of the spectral homotopy analysis method. The test equations for this study are obtained by considering unsteady nonlinear convection flow over an impulsively stretched flat surface. Fluid flow over stretching surfaces is important in many practical applications such as extrusion of plastic sheets, paper production, glass blowing, metal spinning, and drawing plastic films. The quality of the final product depends on the rate of heat transfer at the stretching surface. In a previous study in this field, Kumari et al. [5] used the Keller box method and the Nakamura method to investigate the problem of heat transfer in the unsteady free convection flow over a continuous moving vertical sheet in an ambient fluid. Ishak et al. [6] investigated theoretically the unsteady mixed convection boundary layer flow and heat transfer due to a stretching vertical surface in a quiescent viscous and incompressible fluid. Further, Pop and Na [7] and Wang et al. [8] dealt with the unsteady boundary layer flow due to impulsive flow starting from rest of a stretching sheet in a viscous fluid. A numerical solution for unsteady mixed convection boundary layer nanofluid flow and heat transfer due to a stretching vertical sheet was presented by Mahdy [9]. The equations were solved using the implicit finite difference method. Bachok et al. [10] studied the flow and heat transfer problem due to the unsteady, twodimensional laminar flow of a viscous nanofluid caused by a permeable stretching/shrinking sheet in a quiescent fluid. Numerical solutions of the transformed governing equations were obtained using a shooting method. Sharma et al. [11] used the finite element method to find numerical solutions of the flow and heat transfer problem over a stretching sheet immersed in a nanofluid, with velocity slip at the boundary. In fluid flow problems involving heat transfer, it may be essential to consider a nonlinear relationship between density and temperature. Thermal stratification and heat released by the viscous dissipation triggers some changes in density gradients. To the best of our knowledge there 2 Abstract and Applied Analysis is no study that has been conducted to discuss the effects of nonlinear density on unsteady nonlinear convection flow over an impulsively stretched flat surface. Accordingly, the main aim of the study is to investigate the nonlinear convective flow over an impulsively stretched flat surface under the nonlinear density-temperature relationship. Using the boundary layer approximation, the unsteady momentum, heat, and mass transfer equations are transferred to nonlinear partial differential equations form and solved using the spectral homotopy analysis method (SHAM). This work is the first attempt at applying the SHAM to system of coupled nonlinear PDEs. The SHAM results presented in this study are validated against results from a quasilinearisation based spectral collocation method. With the usual Boussinesq and the boundary layer approximations, the governing equations are in the form 𝜕𝑢 𝜕V + = 0, 𝜕𝑥 𝜕𝑦 𝜕𝑢 𝜕𝑢 𝜕𝑢 𝜕2 𝑢 2 +𝑢 +V = ] 2 + [𝛽0 (𝑇 − 𝑇∞ ) + 𝛽1 (𝑇 − 𝑇∞ ) ] 𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑦 2 + [𝛽2 (𝐶 − 𝐶∞ ) + 𝛽3 (𝐶 − 𝐶∞ ) ] g, (6) 𝑢 𝜕2 𝑇 𝜕𝑇 𝜕𝑇 +V = 𝛼𝑚 2 , 𝜕𝑥 𝜕𝑦 𝜕𝑦 (7) 𝜕𝐶 𝜕𝐶 𝜕2 𝐶 +V =𝐷 2, 𝜕𝑥 𝜕𝑦 𝜕𝑦 (8) 𝑢 2. Governing Equations Consider the problem of unsteady nonlinear convection of a fluid over a stretching flat plate. Initially (𝑡 = 0), both the fluid and stretching plate are kept at a constant temperature 𝑇𝑤 and concentration 𝐶∞ where 𝑇𝑤 > 𝑇∞ is for a heated plate and 𝑇𝑤 < 𝑇∞ corresponding to a cooled plate. We assume that at 𝑡 = 0 the velocity of the stretching plate is 𝑢𝑤 = 𝑎𝑥, where 𝑎 is a positive constant. From the Boussinesq approximation, density is related to temperature and the concentration by the equation 𝜌 = 𝜌0 [1 − 𝛽𝑇 (𝑇 − 𝑇∞ ) + 𝛽𝐶 (𝐶 − 𝐶∞ )] . (1) In the case of thermal stratification and heat released by viscous dissipation, wall jet like profiles induce significant changes in density gradients, and the density depends on the temperature or temperature and concentration in a nonlinear form (see [12–17]): 2 𝜌 = 𝜌∞ [1 − 𝛽𝑇 (𝑇 − 𝑇∞ ) + 𝛽1 (𝑇 − 𝑇∞ ) ] . (2) Karcher and Müller [18] used the formulation below to define the nonlinearity of the relationship between the density, the temperature, and the concentration: subject to the boundary conditions 𝑢 = 𝑎𝑥, where 𝜌∞ is the constant fluid density, 𝑇∞ and 𝐶∞ are the fluid temperature and solutal concentration, respectively, 𝛽0 and 𝛽2 are the coefficients of thermal and solutal expansion, and 𝛽1 denotes the nonlinear coefficient of thermal expansion. Partha [16] investigated the natural nonlinear convection in a non-Darcy porous medium using a temperatureconcentration-dependent density relation in the form 2 𝜌 = 𝜌∞ [𝛽0 (𝑇 − 𝑇∞ ) + 𝛽1 (𝑇 − 𝑇∞ ) 2 (...truncated)