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An MHD Stokes eigenvalue problem and its approximation by a spectral collocation method

An eigenvalue problem is introduced for the magnetohydrodynamic (MHD) Stokes equations describing the flow of a viscous and electrically conducting fluid in a duct under the influence of a uniform magnetic field. The solution of the eigenproblem is approximated by using a spectral collocation method that is based on vanishing the residual equation at the collocation points on the physical domain which are chosen to be the Chebyshev–Gauss–Lobatto points. As the solutions are sought in the physical space, the approximations to the derivatives of the unknowns are directly evaluated. The equations are formulated in the primitive variables, and hence with inclusion of the continuity equation, the discretization of the operator results in a generalized eigenproblem with zero diagonal entries. Therefore, a penalty method is applied to circumvent the degeneracy where a perturbed form of the problem is considered, and a zero mean pressure value is introduced. The numerical prospects of the algorithm are investigated and demonstrated by a number of characteristic tests. The key features of interest are the effects of introducing a magnetic field on the eigenspectrum focusing mainly on the change of the fundamental eigenpairs, and the consequential variation of the eigenstructure with the magnetic field. The mechanisms that underlie these effects are examined by the numerical model proposed, the implications of these effects are presented, and it is shown that the flow field is considerably affected with the introduction of a magnetic field into the physical model.