Mathematics & Statistics Theses & Dissertations

Spring 1996

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Statistics

Program/Concentration

Computational and Applied Mathematics

Committee Director

John Mark Dorrepaal

Committee Member

John H. Heinbockel

Fang Q. Hu

Abstract

The viscous plane flow of an electrically conducting fluid towards an infinite wall is solved in the presence of a magnetic field which is aligned with the flow far from the wall. The problem has two dimensionless parameters-- ε, the magnetic Prandtl number, and β, the square of the ratio of Alfven velocity to fluid velocity far from the wall. The problem has a similarity solution which reduces the governing equations to a system of coupled ordinary differential equations which can be solved numerically. For extreme values of ε, both large and small, singular perturbation techniques are used to derive asymptotic expansions for the physically relevant quantities in the flow--the skin friction at the wall and the tangential component of magnetic intensity at the wall. Extensive comparisons are made between the asymptotic predictions and the numerical results with remarkably good agreement being obtained when ε ≥100 and ε ≤ 0.001.

The flow is next combined with a shear flow to yield a flow impinging on the wall at some angle of incidence. The problem has a similarity solution and the resulting system of coupled differential equations is solved numerically. A series solution for the shear component of tangential stress at the wall for small and large values of $\epsilon$ is derived using singular perturbation techniques. The asymptotic expansions obtained are shown to be in excellent agreement with the numerical results when ε ≤ 0.001 and ε ≥100. The behavior of the flow near the wall is analyzed and the slope-ratio constant is evaluated for a variety of (ε,β) cases.

DOI

10.25777/hqxg-n236

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