Magnetically confined fusion plasmas are known to develop a variety of instabilities. Some of these instabilities can be understood with the ideal magnetohydrodynamic (MHD) model. A plasma in MHD equilibrium can be unstable to small perturbations, which are always present in experiments. One particular instability is the external kink mode. While this mode might be stabilized by a perfectly conducting wall, actual walls have some finite resistivity such that the kink still grows on the L/R time of the wall---it is a resistive wall mode (RWM).;In this dissertation, the RWM is studied with the ideal MHD equilibrium and stability equations that include equilibrium flow. The stability equation is a nonlinear eigenvalue problem, which is transformed by the use of an auxiliary variable into a set of linear eigenvalue equations. For a flowing cylindrical plasma, these equations are formulated as a matrix eigenvalue problem by expanding the radial dependence of the perturbations as finite elements. The perturbations at the edge of the plasma are coupled to the surrounding resistive wall by the use of a Green's function for the vacuum fields and by the introduction of an additional unknown that represents the induced current in the wall.;The matrix eigenvalue formulation of the RWM problem was solved numerically with a new finite element eigenvalue code. The code is benchmarked both analytically and numerically for the boundary conditions of a close fitting conducting wall, no wall, a perfectly conducting wall, and a resistive wall. The RWM is shown to be stabilized by flow for a window of wall positions, both with and without parallel viscosity, the latter requiring an extrapolation to a grid step size of zero in the region of resonance between the RWM and the sound continuum. Finally, flow shear is introduced, which reveals that the RWM does not move with the plasma at the resonance location, but rather with the plasma at a radial location which is independent of the value of the poloidal mode number.
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