Journal of Mathematical Fluid Mechanics

, Volume 19, Issue 4, pp 645–657 | Cite as

Mathematical Study of the Small Oscillations of Two Nonmixing Fluids, the Lower Inviscid, the Upper Viscoelastic, in an Open Container

  • H. EssaouiniEmail author
  • L. El Bakkali
  • P. Capodanno


The authors study the small oscillations of a system of two nonmixing fluids, the lower inviscid, the upper viscoelastic, in an open container, restricting themselves for the second to the more simple Oldroyd model. From the equations of motion, they obtain a variational formulation of the problem, from which they deduce a variational equation for the viscoelastic fluid only, then a system of operatorial equations in suitable Hilbert space. They show the existence and the symmetry of the spectrum, prove the stability of the system and specify the location of the eigenvalues. They prove the existence of two sets of positive real eigenvalues having, the first l’infinity, the second a point of the real axis, as point of accumulation. Finally, after a suitable transformation of the operatorial equations of motion, they obtain an existence and unicity theorem of the solution of the associated evolution problem by means of the semigroups theory.


Viscoelastic fluid small oscillations variational operatorial and spectral methods semi-groups 

Mathematics Subject Classification

76A10 76M22 76M30 49R50 47A75 


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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Faculty of Sciences, M2SM ER28/FS/05Abdelmalek Essaâdi UniversityTetuanMorocco
  2. 2.Université de Franche-ComtéBesançonFrance

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