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The 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.

Exact 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.

The “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.

The 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.

The 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.

The purpose of this paper is to consider heat and mass transfer on magnetohydrodynamics (MHD) Williamson fluid flow over a slendering stretching sheet with variable thickness in the presence of radiation and chemical reaction. All pertinent flow parameters are discussed and their influence on the hydrodynamics, thermal and concentration boundary layer are presented with the aid of the diagram.

The governing partial differential equations are reduced into a system of ordinary differential equations with the help of suitable similarity variables. A discrete version of the homotopy analysis method (HAM) called the spectral homotopy analysis method (SHAM) was used to solve the transformed equations. SHAM is efficient, and it converges faster than the HAM. The SHAM provides flexibility when solving linear ordinary differential equations with the use of the Chebyshev spectral collocation method.

The findings revealed that an increase in the variable thermal conductivity hike the temperature and the thermal boundary layer thickness, whereas the reverse is the case for velocity close to the wall.

The uniqueness of this paper is the exploration of combined effects of heat and mass transfer on MHD Williamson fluid flow over a slendering stretching sheet. The Williamson fluid term in the momentum equation is expressed as a linear function and the viscosity and thermal conductivity are considered to vary in the boundary layer.

This paper aims to address the problem of double-diffusive magnetohydrodynamics (MHD) non-Darcy convective flow of heat and mass transfer over a stretching sheet embedded in a thermally-stratified porous medium. The controlling parameters such as chemical reaction parameter, permeability parameter, etc., are extensively discussed and illustrated in this paper.

With the help of appropriate similarity variables, the governing partial differential equations are converted into ordinary differential equations. The transformed equations are solved using the spectral homotopy analysis method (SHAM). SHAM is a numerical method, which uses Chebyshev pseudospectral and homotopy analysis method in solving science and engineering problems.

The effects of all controlling parameters are presented using graphical representations. The results revealed that the applied magnetic field in the transverse direction to the flow gives rise to a resistive force called Lorentz. This force tends to reduce the flow of an electrically conducting fluid in the problem of heat and mass transfer. As a result, the fluid velocity reduces in the boundary layer. Also, the suction increases the velocity, temperature, and concentration of the fluid, respectively. The present results can be used in complex problems dealing with double-diffusive MHD non-Darcy convective flow of heat and mass transfer.

The uniqueness of this paper is the examination of double-diffusive MHD non-Darcy convective flow of heat and mass transfer. It is considered over a stretching sheet embedded in a thermally-stratified porous medium. To the best of the knowledge, a problem of this type has not been considered in the past. A novel method called SHAM is used to solve this modelled problem. The novelty of this method is its accuracy and fastness in computation.

This paper aims to analyze heat and mass transfer of magnetohydrodynamic (MHD) non-Newtonian fluids flow past an inclined thermally stratified porous plate using a numerical approach.

The flow equations are set up with the non-linear free convective term, thermal radiation, nanofluids and Soret–Dufour effects. Thus, the non-linear partial differential equations of the flow analysis were simplified by using similarity transformation to obtain non-linear coupled equations. The set of simplified equations are solved by using the spectral homotopy analysis method (SHAM) and the spectral relaxation method (SRM). SHAM uses the approach of Chebyshev pseudospectral alongside the homotopy analysis. The SRM uses the concept of Gauss-Seidel techniques to the linear system of equations.

Findings revealed that a large value of the non-linear convective parameters for both temperature and concentration increases the velocity profile. A large value of the Williamson term is detected to elevate the velocity plot, whereas the Casson parameter degenerates the velocity profile. The thermal radiation was found to elevate both velocity and temperature as its value increases. The imposed magnetic field was found to slow down the fluid velocity by originating the Lorentz force.

The novelty of this paper is to explore the heat and mass transfer effects on MHD non-Newtonian fluids flow through an inclined thermally-stratified porous medium. The model is formulated in an inclined plate and embedded in a thermally-stratified porous medium which to the best of the knowledge has not been explored before in literature. Two elegance spectral numerical techniques have been used in solving the modeled equations. Both SRM and SHAM were found to be accurate.

The purpose of this study is to explore numerical scrutinization of micropolar and Walters-B non-Newtonian fluids motion under the influence of thermal radiation and chemical reaction.

The two fluids micropolar and Walters-B liquid are considered to start flowing from the slot to the stretching sheet. A magnetic field of constant strength is imposed on their flow transversely. The problems on heat and mass transport are set up with thermal, chemical reaction, heat generation, etc. to form partial differential equations. These equations were simplified into a dimensionless form and solved using spectral homotopy analysis method (SHAM). SHAM uses the basic concept of both Chebyshev pseudospectral method and homotopy analysis method to obtain numerical computations of the problem.

The outcomes for encountered flow parameters for temperature, velocity and concentration are presented with the aid of figures. It is observed that both the velocity and angular velocity of micropolar and Walters-B and thermal boundary layers increase with increase in the thermal radiation parameter. The decrease in velocity and decrease in angular velocity occurred are a result of increase in chemical reaction. It is hoped that the present study will enhance the understanding of boundary layer flow of micropolar and Walters-B non-Newtonian fluid under the influences of thermal radiation, thermal conductivity and chemical reaction as applied in various engineering processes.

All results are presented graphically and all physical quantities are computed and tabulated.

The purpose of this study is to investigate the Dynamics of micropolar – water B Fluids flow simultaneously under the influence of thermal radiation and Soret–Dufour Mechanisms.

The thermal radiation contribution, the chemical change and heat generation take fluidity into account. The flow equations are used to produce a series of dimensionless equations with appropriate nondimensional quantities. By using the spectral homotopy analysis method (SHAM), simplified dimensionless equations have been quantitatively solved. With Chebyshev pseudospectral technique, SHAM integrates the approach of the well-known method of homotopical analysis to the set of altered equations. In terms of velocity, concentration and temperature profiles, the impacts of Prandtl number, chemical reaction and thermal radiation are studied. All findings are visually shown and all physical values are calculated and tabulated.

The results indicate that an increase in the variable viscosity leads to speed and temperature increases. Based on the transport nature of micropolar Walters B fluids, the thermal conductivity has great impact on the Prandtl number and decrease the velocity and temperature. The current research was very well supported by prior literature works. The results in this paper are anticipated to be helpful for biotechnology, food processing and boiling. It is used primarily in refrigerating systems, tensile heating to large-scale heating and oil pipeline reduction.

All results are presented graphically and all physical quantities are computed and tabulated.

This paper aims to suggest a novel modified Laplace decomposition method (MLDM) for MHD flow over a non-linear stretching sheet with slip condition by suitable choice of an initial solution.

The governing partial differential equations are converted into dimensionless non-linear ordinary differential equation by similarity transformation, which is solved by MLDM. The method is based on the application of Laplace transform to boundary layers in fluid mechanics. The non-linear term can be easily handled by the use of He's polynomials.

The series solution of the MHD flow of an incompressible viscous fluid over a non-linear stretching sheet subject to slip condition is obtained. An excellent agreement between the MLDM and HPM is achieved. Convergence of the obtained series solution is properly checked by using the ratio test.

Stretching surface is an important type of flow occurring in a number of engineering processes such as heat-treated materials travelling between a feed roll and a wind up roll, aerodynamic extrusion of plastic sheets, glass fiber and paper production, cooling of an infinite metallic plate in a cooling path, manufacturing of polymeric sheets are few examples of flow due to stretching surfaces. This work provides a very useful source of information for researchers on this subject.

Such flow analysis is even not available yet for the hydrodynamic fluid. The series solution for MHD boundary layer problem with slip condition by means of MLDM is yet not available in the literature.

The purpose of this paper is to demonstrate the eligibility of the weighted residual methods (WRMs) applied to Jeffery-Hamel Flow. Selecting the most appropriate method among the WRMs and discussing about Jeffery-Hamel flow's treatment in divergent and convergent channels are the other important purposes of the present research.

Three analytical methods (collocation, Galerkin and least square method) have been applied to solve the governing equations. The reliability of the methods is also approved by a comparison made between the forth order Runge-Kutta numerical method.

The obtained solutions revealed that WRMs can be simple, powerful and efficient techniques for finding analytical solutions in science and engineering non-linear differential equations.

It could be considered as a first endeavor to use the solution of the Jeffery-Hamel flow using these kind of analytical methods along with the numerical approach.

The purpose of this paper is to find analytic approximate solutions for time-dependent flow and heat transfer of a Sisko fluid.

The homotopy analysis method is used to find a family of travelling wave solutions of the governing non-linear problem.

The effects of different parameters on the velocity and temperature profiles are shown graphically.

The analytic solutions of the system of non-linear ordinary differential equations are constructed in the series form for various values of the power index.

The aim of this work is to analyze the combined effects of melting, thermal-diffusion and diffusion-thermo on the flow of non-Newtonian fluid.

An efficient approach namely homotopy analysis method is applied to compute the solution of the non-linear problem. Moreover, numerical results using MATLAB function bvp4c are also computed.

Main findings are an increase in the melting process corresponding to increase in the velocity and the boundary layer thickness. However, surface heat and mass transfer decrease by increasing the values of melting parameter M.

Combined effects of thermal-diffusion and diffusion-thermo are analyzed and the solutions are computed both numerically and analytically. Some deduced results can be obtained in a limiting sense.