Yiantsios, Stergios; Higgins, Brian G. Analysis of superposed fluids by the finite element method: Linear stability and flow development. (English) Zbl 0644.76042 Int. J. Numer. Methods Fluids 7, 247-261 (1987). 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. Cited in 1 Document MSC: 76E05 Parallel shear flows in hydrodynamic stability 76T99 Multiphase and multicomponent flows 76M99 Basic methods in fluid mechanics Keywords:Galerkin finite element method; stability of two superposed immiscible Newtonian fluids; plane Poiseuille flow; algebraic eigenvalue problem; standard generalized eigenvalue problem; QR algorithm; perturbation methods Software:EISPACK PDFBibTeX XMLCite \textit{S. Yiantsios} and \textit{B. G. Higgins}, Int. J. Numer. Methods Fluids 7, 247--261 (1987; Zbl 0644.76042) Full Text: DOI References: [1] Yih, J. Fluid Mech. 27 pp 337– (1967) [2] Nakaya, J. Phys. Soc. Japan 37 pp 214– (1974) [3] Hooper, Phys. Fluids 28 pp 1613– (1985) · Zbl 0586.76062 [4] Li, Int. j. numer. methods eng. 17 pp 853– (1981) [5] Orszag, J. Fluid Mech. 50 pp 689– (1971) [6] Wilson, J. Fluid Mech. 38 pp 793– (1969) [7] Bramley, J. Comp. Phys. 47 pp 179– (1982) [8] , , and , Matrix eigensystem routines: EISPACK Guide, 2nd edn, Springer-Verlag, New York, 1970. · Zbl 0289.65017 [9] Kao, J. Fluid Mech. 52 pp 401– (1972) [10] and , submitted to Phys. Fluids (1986). [11] and , Matrix Computations, Johns Hopkins University Press, Baltimore, 1983. [12] Thomas, Phys. Rev. 91 pp 780– (1953) [13] Ng, J. Comp. Phys. 30 pp 125– (1979) [14] Davey, J. Comp. Phys. 35 pp 36– (1980) [15] and , Convection in Liquids, Springer-Verlag, Berlin, 1984. · Zbl 0545.76048 · doi:10.1007/978-3-642-82095-3 [16] Higgins, I&EC Fundam. 21 pp 168– (1982) [17] The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965. · Zbl 0258.65037 [18] Stewart, Numer. Math. 25 pp 123– (1976) [19] Wang, Phys. Fluids. 21 pp 1669– (1978) [20] and , ’Stability of stratified film flow’, paper 46d, AIChE Winter National Meeting, Orlando (1982). [21] Kistler, Int. j. numer. methods fluids 4 pp 207– (1984) [22] Ph. D. Thesis, University of Minnesota, Minneapolis, 1982. [23] Ruschak, Comp. Fluids 11 pp 391– (1983) [24] and , ’Liquid flow in a forward roll coater’, TAPPI Proceedings 1985 Coating Conference, 161-169 (1985). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.