# Finite element analysis of the axially symmetric motion of an incompressible viscous fluid in a spherical annulus

#### Abstract

The primary objective of this research is to employ a modified Galerkin finite element procedure to analyze the steady state flow of a fluid contained between two concentric rotating spheres. The spheres are assumed to be rigid and the region in the cavity between the spheres is constrained to rotate about a vertical axis due to a prescribed angular velocity while the outer sphere is fixed. The fluid problem, which is originally formulated in terms of the stream function and circumferential function variables, is reduced to second order form by introducing the vorticity variable. The system region is discretized with the use of finite elements. The fluid equations are reduced to a system of non-linear algebraic equations by employing a modification of the conventional Galerkin finite element method. The field variables are expanded (over each element) in terms of products of cubic interpolation functions and element nodal coordinates. The nodal coordinates are selected so as to satisfy all of the boundary conditions. The finite element equations are linearized, assembled and then solved by employing a Newton-Raphson algorithm. Results for the circumferential function ($\Omega$), stream function ($\Psi$), vorticity function ($\zeta$) and the torques ($T\sb{i}$) at the fluid boundaries are presented for Reynolds number ($R\sb{e})\le 2000$ and radius ratio $0.2\le (\alpha)\le 0.9$. Previous investigators who studied this problem experienced difficulties in obtaining results for wide gap cases ($\alpha\le 0.2$), except for small Reynolds numbers ($R\sb{e}\le 100$). The research indicates that the modified Galerkin procedure employed in this study can be effective for analyzing those non-linear fluid flow problems which are not amenable to the Ritz or the conventional Galerkin finite element methods. To the best knowledge of the writer, this procedure has not been employed by any other investigators.

#### Recommended Citation

Ni, Weishun, "Finite element analysis of the axially symmetric motion of an incompressible viscous fluid in a spherical annulus" (1993). *Dissertations (1962 - 2010) Access via Proquest Digital Dissertations*. AAI9325690.

https://epublications.marquette.edu/dissertations/AAI9325690