On the Expedient Solution of the Boltzmann Equation by Modified Time Relaxed Monte Carlo (MTRMC) Method

In the present study, a modified time relaxed Monte Carlo (MTRMC) method is developed for numerical solution of the Boltzmann equation in rarefied regimes. Taylor series expansion is employed to obtain a generalized form of the Wild sum expansion and consequently the modified collision functions with fewer inter-molecular interactions are obtained. The proposed algorithm is applied on the lid-driven micro cavity flow with different lid velocities and the results for velocity and shear stress distributions are compared with those from the standard DSMC and TRMC methods. The comparisons show excellent agreement between the results of the MTRMC method with their counterparts from TRMC and DSMC methods. The present study illustrates appreciable improvement in the computational expense of the MTRMC method compared to those from standard TRMC and DSMC methods. The improvement is more pronounced compared to the standard DSMC method. It is observed that up to 56% reduction in CPU time is obtained in the studied cases

On the Instability of Two Dimensional Backward-Facing Step Flow using Energy Gradient Method

In the present paper, the energy gradient method is implemented to study the instability of 2-D laminar backward-facing step (BFS) flow under different Reynolds numbers and expansion ratios. For this purpose, six different Reynolds numbers (50 ≤ Re ≤ 1000) and two various expansion ratios of 1.9423 and 3 are considered. We compared our results of the present study with existing experimental and numerical data and good agreement is achieved. To study of fluid flow instability, we evaluated the distributions of velocity, vorticity and energy gradient function K. The results of our study show that as the expansion ratio decreases the flow becomes more stable. We also found that the origin of instability in the entire flow field is located on the separated shear layer nearby the step edge. In addition, we approved that the inflection point on the profile of velocity corresponds to the maximum of vorticity resulted to the instability

Two Semi-Analytical Methods Applied to Hydrodynamic Stability of Dean Flow

Hydrodynamic stability of Dean flow is studied using two semi-analytical methods of differential transform method (DTM) and Homotopy perturbation method (HPM). These two methods are evaluated to examine the effectiveness and accuracy of the solution of considered eigenvalue problem. Very good accordance is achieved between our semi-analytical results compared to existing numerical data. Based on our analysis, in the similar number of truncated terms, HPM is more accurate in comparison with DTM. We also concluded that for the higher wave numbers, HPM provide more accurate results with less truncated terms compared to the DTM. Finally, we found the critical Dean number 35.927 corresponding to wave number of 3.952 for onset of instability of Dean flow

On the hybrid of fourier transform and adomian decomposition method for the solution of nonlinear cauchy problems of the Reaction-Diffusion equation

The physical science importance of the Cauchy problem of the reaction-diffusion equation appears in the modelling of a wide variety of nonlinear systems in physics, chemistry, ecology, biology, and engineering. A hybrid of Fourier transform and Adomian decomposition method (FTADM) is developed for solving the nonlinear non-homogeneous partial differential equations of the Cauchy problem of reaction-diffusion. The results of the FTADM and the ADM are compared with the exact solution. The comparison reveals that for the same components of the recursive sequences, the errors associated with the FTADM are much lesser than those of the ADM. We show that as time increases the results of the FTADM approaches 1 with only six recursive terms. This is in agreement with the physical property of the density-dependent nonlinear diffusion of the Cauchy problem which is also in agreement with the exact solution. The monotonic and very rapid convergence of the results of the FTADM towards the exact solution is shown to be much faster than that of the ADM. © 2012 Verlag der Zeitschrift für Naturforschung, Tübingen

The HPM Applied to MHD Nanofluid Flow over a Horizontal Stretching Plate

The nonlinear two-dimensional forced-convection boundary-layer magneto hydrodynamic (MHD) incompressible flow of nanofluid over a horizontal stretching flat plate with variable magnetic field including the viscous dissipation effect is solved using the homotopy perturbation method (HPM). In the present work, our results of the HPM are compared with the results of simulation using the finite difference method, Keller's box-scheme. The comparisons of the results show that the HPM has the capability of solving the nonlinear boundary layer MHD flow of nanofluid with sufficient accuracy