A quadratic eigenvalue problem involving Stokes equations

• Published in: Computer methods in applied mechanics and engineering Vol. 100; no. 3; pp. 295 - 313
• Main Authors: Conca, C., Duran, M., Planchard, J.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.11.1992; Elsevier
• Subjects: Exact sciences and technology; Fundamental areas of phenomenology (including applications); Physics; Solid mechanics; Structural and continuum mechanics; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Hydroelasticity; Viscous fluid; Free vibration; Eigenvalue problem; Incompressible fluid; Fluid structure interaction; Stokes flow
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/0045-7825(92)90086-Y

An existence theorem for the eigenvalues of a spectral problem is studied in this paper. The physical situation behind this mathematical problem is the determination of the eigenfrequencies and eigenmotions of a fluid-solid structure. The liquid part in this structure is represented by a viscous incompressible fluid, while the solid part is a set of parallel rigid tubes. The spectral problem governing this system is a quadratic eigenvalue problem which involves Stokes equations with a non-local boundary condition. The strategy for tackling the question of existence of eigenvalues consists of proving that the original problem is equivalent to that of determining the characteristic values of a linear (non-selfadjoint) compact operator. Sharp estimates for the eigenvalues give precise information about the region of ω where the eigenvalues are located. In particular, we prove that this problem admits a countable set of eigenvalues in which only a finite number of them have a non-zero imaginary part.