On a bivariate spectral relaxation method for unsteady magneto-hydrodynamic flow in porous media

Magagula, Vusi Mpendulo; Motsa, S. S.; Sibanda, Precious; Dlamini, Phumlani

doi:10.1186/s40064-016-2053-4

Cited by 17 publications

(6 citation statements)

References 19 publications

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“…Most of the standard methods of solving the boundary layer problems are the numerical approach based on the shooting algorithm with the Runge-Kutta scheme, finite difference method, spectral homotopy analysis method, and Newton-Raphson based methods such as the quasilinearization method and the successive linearization method. Recently, spectral based numerical techniques such as the Spectral Quasilinearization Method and Spectral Relaxation Method have been developed (see Motsa et al [23], Motsa [24], and Magagula et al [25]). As indicated by Motsa et al [26] and Zhou [27] Chebyshev spectral collocation methods are easy to implement and adaptable to various problems and provide more accurate approximations with a relatively small number of unknowns.…”

Section: Introductionmentioning

confidence: 99%

Magnetohydrodynamic (MHD) Boundary Layer Flow Past a Wedge with Heat Transfer and Viscous Effects of Nanofluid Embedded in Porous Media

Ibrahim

Tulu

2019

Mathematical Problems in Engineering

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The problem of two-dimensional steady laminar MHD boundary layer flow past a wedge with heat and mass transfer of nanofluid embedded in porous media with viscous dissipation, Brownian motion, and thermophoresis effect is considered. Using suitable similarity transformations, the governing partial differential equations have been transformed to nonlinear higher-order ordinary differential equations. The transmuted model is shown to be controlled by a number of thermophysical parameters, viz. the pressure gradient, magnetic, permeability, Prandtl number, Lewis number, Brownian motion, thermophoresis, and Eckert number. The problem is then solved numerically using spectral quasilinearization method (SQLM). The accuracy of the method is checked against the previously published results and an excellent agreement has been obtained. The velocity boundary layer thickness reduces with an increase in pressure gradient, permeability, and magnetic parameters, whereas thermal boundary layer thickness increases with an increase in Eckert number, Brownian motion, and thermophoresis parameters. Greater values of Prandtl number, Lewis number, Brownian motion, and magnetic parameter reduce the nanoparticles concentration boundary layer.

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Section: Introductionmentioning

confidence: 99%

Magnetohydrodynamic (MHD) Boundary Layer Flow Past a Wedge with Heat Transfer and Viscous Effects of Nanofluid Embedded in Porous Media

Ibrahim

Tulu

2019

Mathematical Problems in Engineering

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“…We should remark that the BSRM method uses a similar approach to the BSLLM method except that the BSRM method does not use the quasilinearization approach but rather the Gauss‐Seidel approach to decouple the equations and hence resulting in different variable coefficients from the BSLLM. The BSRM for systems of three equations or more have been applied to various problems by many researchers including References 19,29. Rearranging terms in Equation (), we obtain

\begin{align} true \sum_{s = 0}^{p} α_{s, r}^{false(1 false)} false(η, ζ false) f_{1, r + 1}^{false(s false)} + β_{r}^{false(1 false)} false(η, ζ false) \frac{\partial f_{1, r + 1}^{false(0 false)}}{\partial ζ} + γ_{r}^{false(1 false)} false(η, ζ false) \frac{\partial f_{1, r + 1}^{false(1 false)}}{\partial ζ} & = R_{1} false(η, ζ false), \\ true \sum_{s = 0}^{p} α_{s, r}^{false(2 false)} false(η, ζ false) f_{2, r + 1}^{false(s false)} + β_{r}^{false(2 false)} false(η, ζ false) \frac{\partial f_{2, r + 1}^{false(0 false)}}{\partial ζ} + γ_{r}^{false(2 false)} false(η, ζ false) \frac{\partial f_{2, r + 1}^{false(1 false)}}{\partial ζ} & = R_{2} false(η, ζ false), \\ ⋮ \\ true \sum_{s = 0}^{p} \end{align}

…”

Section: Pseudospectral Numerical Methodsmentioning

confidence: 99%

“…Thus combining finite differences and spectral methods compromises the computational accuracy of the spectral method. This method was extended to solve system of nonlinear partial differential equations 18,19 with spectral methods applied independently on both space and time derivatives. This method was named the bivariate spectral relaxation method (BSRM).…”

Section: Introductionmentioning

confidence: 99%

A comparison of bivariate pseudospectral methods for nonlinear systems of steady nonsimilar boundary layer partial differential equations

Magagula

Motsa

Sibanda

2020

Comp and Math Methods

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In this work, three pseudospectral methods, namely bivariate spectral relaxation method, spectral local linearization, and bivariate spectral quasilinearization method are analyzed for steady nonlinear nonsimilar boundary layer partial differential equations. Their accuracy and general performance are discussed. A system of three nonlinear coupled partial differential equations that models an unsteady three‐dimensional MHD‐boundary‐layer flow due to an impulsive motion of a stretching surface is used to demonstrate the accuracy, convergence, and general performance of the three pseudospectral methods. The general comparison of the three pseudospectral methods is presented in graphical and tabular forms. Computational times of each method is presented in tabular form, showing the skin friction and Nusselt number.

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“…These useful features of the MD-BSQLM enable the approach to yield a very accurate solution. The method has a much better region of convergence for the approximate solution when compared to other Chebyshev spectral collocationbased methods such as bivariate spectral homotopy analysis method [47], bivariate spectral quasilinearization [48], and bivariate spectral relaxation method [49], among others. This study sought, among other things, to check the accuracy and robustness of the MD-BSQLM scheme in finding solutions to this class of problems with significant complexities.…”

Section: Introductionmentioning

confidence: 99%

Numerical Investigation of Natural Convection Viscoelastic Jeffrey’s Nanofluid Flow from a Vertical Permeable Flat Plate with Heat Generation, Thermal Radiation, and Chemical Reaction

Agbaje

2020

Abstract and Applied Analysis

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The boundary layer flow of an incompressible viscoelastic Jeffrey’s nanofluid from a vertical permeable flat plate is investigated. We consider the effects of heat generation, thermal radiation, and chemical reaction on the fluid flow. The nonlinear transformed coupled differential equations that describe the transport processes are solved numerically using a multidomain bivariate spectral quasilinearization method (MD-BSQLM). This innovative method involves blending the quasilinearization idea with the bivariate Lagrange interpolation. The solutions of the resulting system of equations are then obtained sequentially on multiple intervals using the Chebyshev spectral collocation method. The method is shown to give accurate solutions for boundary layer-type equations. The influence of various physical parameters on velocity, temperature, and nanoparticle concentration fields, as well as on the skin friction and heat and mass transfer coefficients, is shown and discussed in detail. The range of the values of the governing parameters considered in this study is between 0 , 4 . For qualitative validation of the results and the numerical method used, calculations were carried out to graphically obtain the velocity, temperature, and nanoparticle concentration fields for selected physical parameter values. The results obtained were found to correlate with the results from published literature. For quantitative verification of our findings, the MD-BSQLM numerical solutions were again confirmed against published results reported in the literature, and the results were observed to be in perfect agreement. This study’s findings indicate that the Deborah number and suction parameter have related effects on the velocity profile, which is to suppress both the flow velocity and the momentum boundary layer thickness. Increasing the heat generation and thermal radiation parameters enhances both the temperature and thermal boundary layer depths. In contrast, an increase in the chemical reaction parameter causes a decrease in the fluid concentration.

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On a bivariate spectral relaxation method for unsteady magneto-hydrodynamic flow in porous media

Cited by 17 publications

References 19 publications

Magnetohydrodynamic (MHD) Boundary Layer Flow Past a Wedge with Heat Transfer and Viscous Effects of Nanofluid Embedded in Porous Media

Magnetohydrodynamic (MHD) Boundary Layer Flow Past a Wedge with Heat Transfer and Viscous Effects of Nanofluid Embedded in Porous Media

A comparison of bivariate pseudospectral methods for nonlinear systems of steady nonsimilar boundary layer partial differential equations

Numerical Investigation of Natural Convection Viscoelastic Jeffrey’s Nanofluid Flow from a Vertical Permeable Flat Plate with Heat Generation, Thermal Radiation, and Chemical Reaction

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