Elsayed M. E. Elbarbary

Hall current effect on magnetohydrodynamic free-convection flow past a semi-infinite vertical plate with mass transfer

Abstract Heat and mass transfer along a vertical plate under the combined buoyancy force effects of thermal and species diffusion is investigated in the presence of a transversely applied uniform magnetic field and the Hall currents are taken into account. The governing fundamental equations on the assumption of small magnetic Reynolds number are approximated by a system of nonlinear ordinary differential equations, which are integrated by fourth-order Runge–Kutta method. Velocity, temperature and concentration are shown on graphs. The numerical values of the local shear stress, the local Nusselt number Nu and the local Sherwood number Sh are entered in Table 1 . The effects of magnetic parameter, Hall parameter and the relative buoyancy force effect between species and thermal diffusion on the velocity, temperature and concentration are discussed. The results are compared with those known from the literature.

Chebyshev finite difference approximation for the boundary value problems

This paper presents a numerical technique for solving linear and non-linear boundary value problems for ordinary differential equations. This technique is based on using matrix operator expressions which applies to the differential terms. It can be regarded as a non-uniform finite difference scheme. The values of the dependent variable at the Gauss-Lobatto points are the unknown one solves for. The application of the method to boundary value problems leads to algebraic systems. The method permits the application of iterative method in order to solve the algebraic systems. The effective application of the method is demonstrated by four examples.

Chebyshev finite difference method for the solution of boundary-layer equations

A new Chebyshev finite difference method is proposed for solving the governing equations of the boundary-layer flow. The Falkner-Skan equation has been solved as a model problem. The more general problem of the equations governing magnetohydrodynamic three dimensional free convection on a vertical stretching surface is solved. The comparisons between the data resulting from the present method and those obtained by others are made. The results indicate that the suggested method yields more accurate results than those computed by others.

A Chebyshev expansion method for solving nonlinear optimal control problems

This paper presents a numerical technique for solving the controlled Duffing oscillator. The control and state variables are approximated by Chebyshev series. A technique is provided for approximating the system dynamics, boundary conditions, and performance index. This technique is based on using an explicit formula for the Chebyshev polynomials in terms of arbitrary order of their derivatives. The system dynamics and the performance index are converted into some algebraic equations. Then the optimal control problem is reduced to constrained optimization problem. Results and comparisons are given at the end of this paper.

Finite difference solution of radiation effects on MHD unsteady free-convection flow over vertical porous plate

The effect of radiation on magnetohydrodynamic (MHD) unsteady free-convection flow past a semi-infinite vertical porous plate in the presence of transversal uniform magnetic field has been analyzed. By means of an implicit finite-difference method the boundary layer equations are transformed into a linear algebraic system. The effect of radiation, magnetic, Prandtl number and suction (or injection) parameters have been studied. The numerical results reveal that the radiation has significant influences on the velocity and temperature profiles, skin friction and Nusselt number. The results indicate that the velocity, temperature, local and average skin friction increase as the radiation parameter increases. While the local and average Nusselt numbers decrease as the radiation parameter increases.

Restrictive taylor's approximation and parabolic partial differential equations

In this paper, we use the restrictive Pade approximation to approximate the exponential matrix exp(rA). The advantage is that it has the exact value at certain r. We define a new accurate, fast implicit method for the finite difference solution of a parabolic partial differential equations. The stability region is discussed, the obtained results are compared with the exact solution and some famous finite difference methods. The numerical results are in satisfactory agreement with the exact solution.

Chebyshev expansion method for solving second and fourth-order elliptic equations

In this paper we introduce a new spectral method based on Chebyshev polynomials for solving second and fourth-order elliptic equations. Moreover the suggested method is applicable for a wide area of differential equations. An explicit formula for the Chebyshev polynomials in terms of arbitrary order of their derivatives is presented. Also a formula for the successive integration of Chebyshev polynomials in terms of Chebyshev polynomials is proved. Numerical results indicate that the suggested method is significantly more accurate than that based on the Chebyshev-Tau method. The present results are in satisfactory agreement with the exact solutions.

Chebyshev finite difference method for MHD flow of a micropolar fluid past a stretching sheet with heat transfer

In this paper, the problem of heat transfer to MHD flow of a micropolar fluid from a stretching sheet with suction and blowing through a porous medium is studied numerically by using Chebyshev finite difference method (ChFD). A similarity solution to governing momentum, angular momentum and energy equations is derived. The effects of surface mass transfer, Prandtl number, magnetic field and porous medium on the velocities and temperature profiles have been studied. The numerical results indicate that, the velocity and the angular velocity increase as the permeability parameter increases but they decrease as the magnetic field increases. Also, the temperature decreases as the permeability parameter increases but it increases as the magnetic field increases.

MHD flow of an elastico-viscous fluid under periodic body acceleration.

Magnetohydrodynamic (MHD) flow of blood has been studied under the influ- ence of body acceleration. With the help of Laplace and finite Hankel transforms, an exact solution is obtained for the unsteady flow of blood as an electrical conducting, incompress- ible and elastico-viscous fluid in the presence of a magnetic field acting along the radius of the pipe. Analytical expressions for axial velocity, fluid acceleration and flow rate has been obtained.

Finite difference solution of radiation effects on MHD unsteady free-convection flow over vertical plate with variable surface temperature

An analysis is performed to study radiation effects on magnetohydrodynamic (MHD) unsteady free-convection flow past a semi-infinite vertical plate with variable surface temperature in the presence of transversal uniform magnetic field. The boundary layer equations are transformed into a linear algebraic system by an implicit finite-difference method. A parametric study is performed to illustrate the influence of radiation parameter, magnetic parameter, and Prandtl number on the velocity and temperature profiles. The numerical results reveal that the radiation has significant influences on the velocity and temperature profiles, skin friction, and Nusselt number. The results indicate that the velocity, temperature, and local and average skin friction increase as the radiation parameter increases, while the local and average Nusselt numbers decrease as the radiation parameter increases.