The electric current generated by the laminar flow of an insulating liquid through a cylindrical pipe has been calculated using a model that incorporates nonequilibrium boundary conditions at the wall. Analytical solutions to the model equations are derived in the limits of high and low electrical conductivity and numerical solutions are given at intermediate conductivities. The solutions depend on four parameters that are dimensionless representations of the electrical conductivity, the rate of adsorption of positive ions, the rate of adsorption of negative ions and the ratio of the diffusivities of positive and negative ions. The results are presented as graphs of current versus the electrical conductivity parameter for various values of the other three parameters. Physical reasons for the shapes of the current/conductivity curves are discussed. According to the model, the currents are generated by differences in the ionic diffusivities and/or by differences in the ionic adsorption rates at the wall.
SUMMARYThe general nine point isoparametric transformation is analysed with respect to a triangle with two straight sides and one curved side. Point placements are determined on the curved side which ensure that the Jacobian of the transformation remains positive inside and on the boundary of the triangle. Special cases of the transformation are considered which lead t o implied curves of the form of symmetric cubics, parabolas, and explicit cubics respectively, thus enabling isoparametric transformations to match a variety of boundary shapes.
The choice of suitable graded meshes and the use of extrapolation techniques as applied to finite difference methods of solution for firstly, linear two-point boundary value problems and secondly, the prototype cavity problem in fluid dynamics are examined. Consider two-point boundary value problems of the form [x) %+ b{x)y = 0 lx-ŝ ubject to the conditions y(0) = y(l) = 1 .It is shown how to construct systematically graded meshes that ensure that the appropriate finite difference method has optimal numerical properties.This scheme for choosing graded meshes is designed to allow the use of extrapolation processes as well. This is an attribute not usually available when using graded meshes. Both first order and second order formulations of the above two-point boundary value are examined.The prototype cavity problem of fluid dynamics is used as an example of the application of some of the above-mentioned ideas. The cavity problem consists of finding the flow pattern for an inviscid fluid in a square two-dimensional box with a sliding roof. A similar method to the above is not known as yet for choosing an appropriate graded mesh for the non-linear Navier-Stokes equation but the principle for choosing a graded mesh is applicable and does still allow extrapolation to be used with that dx 2
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