Mina B. Abd-el-Malek

Solution of the Rayleigh problem for a power law non-Newtonian conducting fluid via group method

An investigation is made of the magnetic Rayleigh problem where a semi-infinite plate is given an impulsive motion and thereafter moves with constant velocity in a non-Newtonian power law fluid of infinite extent. The solution of this highly non-linear problem is obtained by means of the transformation group theoretic approach. The one-parameter group transformation reduces the number of independent variables by one and the governing partial differential equation with the boundary conditions reduce to an ordinary differential equation with the appropriate boundary conditions. Effect of the some parameters on the velocity u (y, t) has been studied and the results are plotted.

Group method analysis of unsteady free-convective laminar boundary-layer flow on a nonisothermal vertical circular cylinder

The transformation group theoretic approach is applied to present an analysis of the problem of unsteady laminar free convection from a non-isothermal vertical flat plate. The application of two-parameter groups reduces the number of independent variables by two, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The possible forms of surface-temperature variations with position and time are derived. The ordinary differential equations are solved numerically using a fourth-order Runge-Kutta scheme and the gradient method. The heat-transfer characteristics for finite values of the Prandtl number Pr are presented, as temperature and velocity distributions.

Group theoretic methods applied to Burgers' equation

Abstract In this study, the group-theoretic methods for calculating the solution of Burgers’ equation with appropriate boundary- and initial-conditions is presented. The application of a one-parameter group reduces the number of independent variables by one, and consequently the governing partial differential equation with the boundary- and initial-conditions to an ordinary differential equation with the appropriate corresponding conditions. The obtained differential equation is solved analytically and the solution obtained in closed form, for a specific choice of boundary condition.

Similarity solutions for unsteady free-convection flow from a continuous moving vertical surface

The transformation group theoretic approach is applied to present an analysis of the problem of unsteady free convection flow over a continuous moving vertical sheet in an ambient fluid. The thermal boundary layer induced within a vertical semi-infinite layer of Boussinseq fluid by a constant heated bounding plate. The application of two-parameter groups reduces the number of independent variables by two, and consequently the system of governing partial differential equations with the boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The obtained ordinary differential equations are solved analytically for the temperature and numerically for the velocity using the shooting method. Effect of Prandtl number on the thermal boundary-layer and velocity boundary-layer are studied and plotted in curves.

Group method analysis of steady free-convective laminar boundary-layer flow on a nonisothermal vertical circular cylinder

The transformation group theoretic approach is applied to present an analysis of the problem of unsteady free convection from the outer surface of a vertical circular cylinder. The application of two-parameter group reduces the number of independent variables by two, and consequently the system of the governing partial differential equations with the boundary conditions reduces to a system of ordinary differential equations with the appropriate boundary conditions. The ordinary differential equations are solved numerically using a fourth-order Runge-Kutta scheme and the gradient method. Numerical results are obtained for the study of the boundary-layer characteristics. The general analysis developed in this study corresponds to the case of surface temperature that varies exponentially with time and uniform with respect to the axial coordinate, i.e., in the formT w =ae bt , wherea andb are constants. The effect of Prandtl number,Pr, andb on the boundary layer characteristics and the maximum value of the vertical component of the velocity are studied.

Approximate solution of a flow over a ramp for large Froude number

Abstract An approximate method is presented to solve the problem of steady free-surface flow of an ideal fluid over a semi-infinite ramp in the bottom. Schwartz-Christoffel transformation is used to map the region of flow, in the complex potential-plane, onto the upper half-plane. The Hilbert transformation as well as the perturbation technique are used as a basis for the approximate solution of the problem for large Froude number and small inclination angle of the ramp. General equations, in integral form, for any order of approximation are obtained. Solution up to first-order approximation is discussed and illustrated. Elevation of the free-surface for different ramp heights, different inclination angles of the ramp and different Froude numbers are plotted. An approximate formula of maximum elevation of the free-surface in terms of the ramp heights and its inclination angle is found.

Hilbert's method for numerical solution of flow from a uniform channel over irregular bottom topographics

Abstract A nonlinear problem is presented for determining the shape of the free surface of a stream which is disturbed by some irregularities lying on the bottom. Using Schwarz-Christoffel transformation and the Hilbert solution of a mixed boundary value problem in the upper half-plane, then, solving the obtained integral equations, we obtain the nonlinear free-surface profile. The nongravity solution was used as an initial approximation to find the gravity-affected solution. The effect of the bottom shape on the free surface has been studied as well as the distribution of the pressure coefficient along the bottom boundaries. Some difficulties of computation and some checking of the numerical solution are discussed.

Group theoretic approach for solving the problem of diffusion of a drug through a thin membrane

The transformation group theoretic approach is applied to study the diffusion process of a drug through a skinlike membrane which tends to partially absorb the drug. Two cases are considered for the diffusion coefficient. The application of one parameter group reduces the number of independent variables by one, and consequently the partial differential equation governing the diffusion process with the boundary and initial conditions is transformed into an ordinary differential equation with the corresponding conditions. The obtained differential equation is solved numerically using the shooting method, and the results are illustrated graphically and in tables.

Approximate solution of gravity-affected flow from planar sluice gate at high Froude number

Abstract An approximate method is developed for solution of the full nonlinear equations governing irrotational flow with a free surface under a sluice gate, of small slope, at high Froude number. Schwarz-Christoffel transformation is used to map the region, in the complex potential-plane, onto the upper half-plane. The Hilbert transformation as well as the perturbation technique, for small inclination angle, are used as a basis for the approximate solution of the problem. General equations, in integral form, for any order of approximation are obtained. A complete solution, up to first-order approximation, for the downstream free surface profile for different values of inclination angle, depth ratio and Froude number is discussed and illustrated. The contraction coefficient for different values of inclination angle has been found. The curvature of the downstream free surface has been obtained in general form and plotted as a function of the spatial variable for different values of the inclination angle. The results obtained are sufficiently accurate for practical purposes in case of high Froude number and small slope of the gate.

Lie group analysis of nonlinear inviscid flows with a free surface under gravity

We applied the Lie group method in studying nonlinear inviscid flows with a free surface under gravity. This method reduces the number of independent variables by one. Therefore, for a system of partial differential equations with three independent variables we applied the method twice to yield a system represented by ordinary differential equations with appropriate corresponding conditions. We obtained analytical solutions for this system. Solutions for the free surface and the velocity components are obtained in closed form. The results are illustrated graphically for different parameters.