In this paper, analytic approximation to the heat and mass transfer characteristics of a two-dimensional time-dependent flow of Williamson nanofluids over a permeable stretching sheet embedded in a porous medium has been presented by considering the effects of magnetic field, thermal radiation, and chemical reaction. The governing partial differential equations along with the boundary conditions were reduced to dimensionless forms by using suitable similarity transformation. The resulting system of ordinary differential equations with the corresponding boundary conditions was solved via the homotopy analysis method. The results of the study show that velocity, temperature, and concentration boundary layer thicknesses generally decrease as we move away from the surface of the stretching sheet and the Williamson parameter was found to retard the velocity but it enhances the temperature and concentration profiles near the surface. It was also found that increasing magnetic field strength, thermal radiation, or rate of chemical reaction speeds up the mass transfer but slows down the heat transfer rates in the boundary layer. The results of this study were compared with some previously published works under some restrictions, and they are found in excellent agreement.
In this study, a convective heat and mass transfer phenomena in a time-dependent boundary layer flow of tangent hyperbolic nanofluid over a permeable stretching wedge has been examined with respect to some pertinent thermo-physical parameters. Convenient similarity transformation is used to reformulate the dimensional partial differential equations into dimensionless system of ordinary differential equations. The reduced set of equations is solved by the homotopy analysis method implemented in Mathematica environment. The effects of the relevant parameters on velocity, temperature and concentration profiles were examined in detail. The impacts of the parameters on the rates of momentum, heat and mass transfer are also analyzed quantitatively in terms of the wall friction coefficient, local Nusselt number and Sherwood number, respectively. Analysis of the results reveals that the increase in the buoyancy ratio parameter facilitates the flow velocity and the increase in the dissipation parameter maximizes the temperature distribution and nanoparticle concentration near the surface of the wedge. Moreover, the analytic approximations obtained by implementing the homotopy analysis method are found in excellent agreement with some previously published results.
Although large and sparse linear systems can be solved using iterative methods, its number of iterations is relatively large. In this case, we need to modify the existing methods in order to get approximate solutions in a small number of iterations. In this paper, the modified method called second-refinement of Gauss-Seidel method for solving linear system of equations is proposed. The main aim of this study was to minimize the number of iterations, spectral radius and to increase rate of convergence. The method can also be used to solve differential equations where the problem is transformed to system of linear equations with coefficient matrices that are strictly diagonally dominant matrices, symmetric positive definite matrices or M-matrices by using finite difference method. As we have seen in theorem 1and we assured that, if A is strictly diagonally dominant matrix, then the modified method converges to the exact solution. Similarly, in theorem 2 and 3 we proved that, if the coefficient matrices are symmetric positive definite or M-matrices, then the modified method converges. And moreover in theorem 4 we observed that, the convergence of second-refinement of Gauss-Seidel method is faster than Gauss-Seidel and refinement of Gauss-Seidel methods. As indicated in the examples, we demonstrated the efficiency of second-refinement of Gauss-Seidel method better than Gauss-Seidel and refinement of Gauss-Seidel methods.
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