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Analysis of superposed fluids by the finite element method: Linear stability and flow development. (English) Zbl 0644.76042

A Galerkin finite element method is described for studying the stability of two superposed immiscible Newtonian fluids in plane Poiseuille flow. The formulation results in an algebraic eigenvalue problem of the form \(A\lambda^ 2+B\lambda +C=0\) which, after transforming to a standard generalized eigenvalue problem, is solved by the QR algorithm. The numerical results are in good agreement with previous asymptotic results. Additional results show that the finite element method is ideally suited linear stability of superposed fluids when parameters characterizing the flow fall outside the range amenable to perturbation methods. The applicability of the finite element method to similar eigenvalue problems is demonstrated by analysing the steady-state spatial development of two superposed fluids in a channel.

MSC:

76E05 Parallel shear flows in hydrodynamic stability
76T99 Multiphase and multicomponent flows
76M99 Basic methods in fluid mechanics

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