A spectral‐homotopy analysis method for heat transfer flow of a third grade fluid between parallel plates

Abstract

PurposeThe purpose of this paper is to study the steady laminar flow of a pressure driven third‐grade fluid with heat transfer in a horizontal channel. The study serves two purposes: to correct the inaccurate results presented in Siddiqui et al., where the homotopy perturbation method was used, and to demonstrate the computational efficiency and accuracy of the spectral‐homotopy analysis methods (SHAM and MSHAM) in solving problems that arise in fluid mechanics.Design/methodology/approachExact and approximate analytical series solutions of the non‐linear equations that govern the flow of a steady laminar flow of a third grade fluid through a horizontal channel are constructed using the homotopy analysis method and two new modifications of this method. These solutions are compared to the full numerical results. A new method for calculating the optimum value of the embedded auxiliary parameter ∼ is proposed.FindingsThe “standard” HAM and the two modifications of the HAM (the SHAM and the MSHAM) lead to faster convergence when compared to the homotopy perturbation method. The paper shows that when the same initial approximation is used, the HAM and the SHAM give identical results. Nonetheless, the advantage of the SHAM is that it eliminates the restriction of searching for solutions to the nonlinear equations in terms of prescribed solution forms that conform to the rule of solution expression and the rule of coefficient ergodicity. In addition, an alternative and more efficient implementation of the SHAM (referred to as the MSHAM) converges much faster, and for all parameter values.Research limitations/implicationsThe spectral modification of the homotopy analysis method is a new procedure that has been shown to work efficiently for fluid flow problems in bounded domains. It however remains to be generalized and verified for more complicated nonlinear problems.Originality/valueThe spectral‐HAM has already been proposed and implemented by the authors in a recent paper. This paper serves the purpose of verifying and demonstrating the utility of the new spectral modification of the HAM in solving problems that arise in fluid mechanics. The MSHAM is a further modification of the SHAM to speed up converge and to allow for convergence for a much wider range of system parameter values. The utility of these methods has not been tested and verified for systems of nonlinear equations. For this reason as much emphasis has been placed on proving the reliability and validity of the solution techniques as on the physics of the problem.

Two - layers Couette flow of a third grade fluid

“…In [15] authors suggest instead of choosing a "random" h-value from the horizontal segment of the h-curve, to choose the h-value that corresponds to the extreme of the second-order approximation. In our case the highest approximation available is the secondorder approximation and we proceed using the extreme value from the h-curves constructed for the first derivative of each of the two layers at the boundary, 0 y 0 .…”

“…(1) 7 7 7 7 (15) As a result the analytical solution is 1.a) x-axes -h value; y-axes -the derivative of the velocity of the two layers; b) Velocity profile in case ; ;…”

Numerical investigation of the flow of a micropolar fluid through a porous channel with expanding or contracting walls

“…The standard way of choosing admissible values of ℏ that ensure convergence of the approximate series solution is to select a value of ℏ on the horizontal segment of the so-called ℏ-curves. Sibanda and Motsa et al [6] have suggested that the optimal value of ℏ to use corresponds to the turning point of the second order ℏ-curve. In Figure 2, we show the ℏ curves for different orders of the SHAM approximation.…”

“…However, the HAM suffers from a number of restrictive measures, such as the requirement that the solution sought ought to conform to the so-called rule of solution expression and the rule of coefficient ergodicity. In a recent study, Motsa and Sibanda et al [5,6] proposed a spectral modification of the homotopy analysis method, the spectral-homotopy analysis method (SHAM) that seeks to remove some restrictive assumptions associated with the implementation of the standard homotopy analysis method.…”

On the practical use of the spectral homotopy analysis method and local linearisation method for unsteady boundary-layer flows caused by an impulsively stretching plate

“…Spectral methods are now becoming the preferred tools for solving ordinary and partial differential equations because of their elegance and high accuracy in resolving problems with smooth functions [6,33]. The use of the SHAM has largely been restricted to the solution of nonlinear boundary value problems [1,8,13,18,[21][22][23][24][25]31] However, Atabakan et al [3] recently used the method to solve Volterra and Fredholm integro-differential equations. A slightly different version of the SHAM that uses Chebyshev-Tau method to convert a BVP to algebraic equations is proposed in Kazem and Shaban [12].…”