Combined effect of magnetic field and heat absorption on unsteady free convection and heat transfer flow in a micropolar fluid past a semi-infinite moving plate with viscous dissipation using element free Galerkin method

The fully developed electrically conducting micropolar fluid flow and heat transfer along a semi-infinite vertical porous moving plate is studied including the effect of viscous heating and in the presence of a magnetic field applied transversely to the direction of the flow. The Darcy-Brinkman-Forchheimer model which includes the effects of boundary and inertia forces is employed. The differential equations governing the problem have been transformed by a similarity transformation into a system of non-dimensional differential equations which are solved numerically by element free Galerkin method. Profiles for velocity, microrotation and temperature are presented for a wide range of plate velocity, viscosity ratio, Darcy number, Forchhimer number, magnetic field parameter, heat absorption parameter and the micropolar parameter. The skin friction and Nusselt numbers at the plates are also shown graphically. The present problem has significant applications in chemical engineering, materials processing, solar porous wafer absorber systems and metallurgy.     

Numerical simulation of a flexible fiber deformation in a viscous flow by the immersed boundary-lattice Boltzmann method

In this paper, deformation of a mass-less elastic fiber with a fixed end, immersed in a two-dimensional viscous channel flow, is simulated numerically. The lattice-Boltzmann method (LBM) is used to solve the Newtonian flow field and the immersed-boundary method (IBM) is employed to simulate the deformation of the flexible fiber interacting with the flow. The results of this unsteady simulation including fiber deformation, fluid velocity field, and variations of the fiber length are depicted in different time-steps through the simulation time. Similar trends are observed in plots representing length change of fibers with different values of stretching constant. Also, the numerical solution reaches a steady state equivalent to the fluid channel flow over a flat plate.     

The element‐free Galerkin method for a quasistatic contact problem with the Tresca friction in elastic materials

This paper is proposed for the error estimates of the element‐free Galerkin method for a quasistatic contact problem with the Tresca friction. The penalty method is used to impose the clamped boundary conditions. The duality algorithm is also given to deal with the non‐differentiable term in the quasistatic contact problem with the Tresca friction. The error estimates indicate that the convergence order is dependent on the nodal spacing, the time step, the largest degree of basis functions in the moving least‐squares approximation, and the penalty factor. Numerical examples demonstrate the effectiveness of the element‐free Galerkin method and verify the theoretical analysis.     

Analysis of a combined mixed finite element and discontinuous Galerkin method for incompressible two‐phase flow in porous media

We analyze a combined method consisting of the mixed finite element method for pressure equation and the discontinuous Galerkin method for saturation equation for the coupled system of incompressible two‐phase flow in porous media. The existence and uniqueness of numerical solutions are established under proper conditions by using a constructive approach. Optimal error estimates in L²(H¹) for saturation and in L ^∞ (H(div)) for velocity are derived. Copyright © 2013 John Wiley & Sons, Ltd.     

A shift‐adaptive meshfree method for solving a class of initial‐boundary value problems with moving boundaries in one‐dimensional domain

A new shift‐adaptive meshfree method for solving a class of time‐dependent partial differential equations (PDEs) in a bounded domain (one‐dimensional domain) with moving boundaries and nonhomogeneous boundary conditions is introduced. The radial basis function (RBF) collocation method is combined with the finite difference scheme, because, unlike with Kansa's method, nonlinear PDEs can be converted to a system of linear equations. The grid‐free property of the RBF method is exploited, and a new adaptive algorithm is used to choose the location of the collocation points in the first time step only. In fact, instead of applying the adaptive algorithm on the entire domain of the problem (like with other existing adaptive algorithms), the new adaptive algorithm can be applied only on time steps. Furthermore, because of the radial property of the RBFs, the new adaptive strategy is applied only on the first time step; in the other time steps, the adaptive nodes (obtained in the first time step) are shifted. Thus, only one small system of linear equations must be solved (by LU decomposition method) rather than a large linear or nonlinear system of equations as in Kansa's method (adaptive strategy applied to entire domain), or a large number of small linear systems of equations in the adaptive strategy on each time step. This saves a lot in time and memory usage. Also, Stability analysis is obtained for our scheme, using Von Neumann stability analysis method. Results show that the new method is capable of reducing the number of nodes in the grid without compromising the accuracy of the solution, and the adaptive grading scheme is effective in localizing oscillations due to sharp gradients or discontinuities in the solution. The efficiency and effectiveness of the proposed procedure is examined by adaptively solving two difficult benchmark problems, including a regularized long‐wave equation and a Korteweg‐de Vries problem. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1622–1646, 2016     

Existence,blow‐up,and exponential decay estimates for a system of semilinear wave equations associated with the helicalflows of Maxwell fluid

The paper is devoted to the study of a system of semilinear wave equations associated with the helical flows of Maxwell fluid. First, based on Faedo–Galerkin method and standard arguments of density corresponding to the regularity of initial conditions, we establish two local existence theorems of weak solutions. Next, we prove that any weak solutions with negative initial energy will blow up in finite time. Finally, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions via the construction of a suitable Lyapunov functional. Copyright © 2015 John Wiley & Sons, Ltd.     

The P1 and P2 approximations of the spherical harmonic method as applied to radiative heat transfer computations with allowance for strong blowing from the surface of a vehicle moving in the Jovian atmosphere

The spherical harmonics method in the P₁ and P₂ approximations is used to analyze radiative heat transfer for a space vehicle entering a planet’s atmosphere. Strong blowing of ablated materials from the vehicle surface is taken into account by using a two-layer flow model without allowance for the viscosity and thermal conduction of the gas. The heating and ablation of a multilayered thermal protection system are determined simultaneously with the vehicle’s flight trajectory, which is calculated taking into account the mass loss due to the ablation. The approach is illustrated by computing the flight of a space vehicle shaped as a spherical segment or a spherically blunted cone entering the Jovian atmosphere at a speed of 60 km/s and an entry angle of ?5°.