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FLUID FLOW AT SMALL REYNOLDS NUMBER: Numerical Applications

by Tayfour El-Bashir


Preface

This book is concerned with the numerical solution of the Navier-Stokes equations for some steady, two-dimensional, incompressible viscous fluid flow problems at small and moderate values of the Reynolds number.

The first problem relates to a two-dimensional, incompressible flow both with and without a line rotlet at various small values of the Reynolds number. A circular cylinder is rotated with a constant angular velocity in the presence of a uniform stream of magnitude U. Two techniques are introduced, one in order to avoid the difficulties in satisfying the boundary conditions at large distances from the cylinder, the other to achieve convergence of the solution at zero Reynolds number. Transformations are applied to both the coordinate system and the stream function.

The second problem considers the solution of the biharmonic equation for the slow viscous flow generated by a line rotlet in the presence of a circular cylinder. On identifying the coefficients of some of the terms in the asymptotic expansion of the stream function, in terms of the force components and the torque on the body, and using an integral constraint, the Boundary Element Method provides a closed system of equations. Excellent agreement is obtained between the numerical results and the analytical expressions and some new results relating to forces and torques on the cylinder are presented.

In the third problem the solution of the biharmonic equation, which represents the Stokes flow created by two rotating circular cylinder in which the force system acting on the two cylinders is in a state of overall equilibrium is examined. In this situation the total force and the total torque are both assumed to be zero and the Boundary Element Method, together with the relationships between the forces and the torque on the combined system and the coefficients in the asymptotic expansion for the stream function, is applied.

The final problem relates to a line rotlet outside an elliptical cylinder and the solution shows that it is possible to generate a flow at infinity which corresponds to that of rigid body rotation. This contrary to the situation for a line rotlet outside a circular cylinder, where the fluid flow at infinity corresponds to that of a uniform stream which is the direction perpendicular to the line joining the rotlet to the center of the cylinder.


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