On a linearisation method for Reiner-Rivlin swirling flow

Abstract

Abstract.The steady flow of a Reiner-Rivlin fluid with Joule heating and viscous dissipation is studied. We present a novel technique for accelerating the convergence of the spectral-homotopy analysis method. Solutions of the nonlinear momentum and energy equations are obtained using the improved spectral homotopy analysis method. Solutions were also generated using the spectral-homotopy analysis method and benchmarked against results in the literature.Mathematical subject classification: Primary: 76A05, 76N05; Secondary: 76M25.

On spectral relaxation method approach for steady von Kármán flow of a Reiner-Rivlin fluid with Joule heating, viscous dissipation and suction/injection

“…1 and 2 we give a comparison of the present SRM solutions against the results reported in [32] and [29] Basic SRM SRM with SOR (ω)…”

“…The SRM results were validated against numerical results generated using Matlab bvp4c routine and against available literature results of [22,29,32]. The SRM gives very accurate results which rapidly converge to the benchmark bvp4c numerical solutions.…”

“…In order to assess the accuracy and convergence of the SRM method, the present results were verified by comparing with other results from literature which have been reported to be accurate to within a certain number of decimal digits. In particular comparison was done against the results reported in [29] and [32] who worked on the same governing equations using the spectral homotopy analysis method and finite differences, respectively. Further validation of the solution was established by comparing the present results with numerical solutions obtained using the MATLAB in-built routine bvp4c which is a finite difference code that implements the three-stage Lobatto III formula 1 .…”

“…Numerical studies were carried out using the Crank-Nicolson method by Attia [18][19][20][21][22], the shooting method by Sibanda et al [23], Osalusi [24] and Takhar [25], the finite difference method by Attia [26], Aboul-Hassan [27] and Hatzikonstantinou [28]. Other recent numerical approaches to the von Kármán equations include a combination of the spectral homotopy analysis method and the successive linearisation method [29][30][31].…”

“…Akbar and Nadeem [35] used a perturbation series to analyse the effects of heat and mass transfer in a blood model of a Reiner-Rivlin fluid. Makukula et al [29] recently used the same problem as a benchmark for testing the performance of their improved spectral homotopy analysis method. In this work, we revisit the von Kármán swirling flow problem of [32] and consider suction/injection and solve the underlying highly nonlinear differential equations using a new method called the spectral relaxation method (SRM).…”