Abstract
We derive statements of linearized boundary value problems in small perturbations arising in continuum mechanics for incompressible viscous media and inviscid media. The known main three-dimensional flow is assumed to be steady-state; along with this flow, a perturbed flow of the same medium induced by the same bulk and surface forces is considered in a domain with unknown moving boundary. The arising linearized statements are reduced to a system of four equations for the perturbations of pressure and velocity components in the unperturbed domain and to a system of homogeneous boundary conditions carried over to the unperturbed boundaries. It turns out that in such statements, the spectral parameter α—the complex vibration frequency—occurs linearly in three equations of motion and one boundary condition. In special cases of the perturbation pattern, reduction is possible to one equation for the stream function amplitude linearly containing the parameter α and four boundary conditions, two of which contain the parameter α (occurring linearly in one condition and quadratically in the other). Examples include a layer of a heavy Newtonian fluid flowing down a sloping plane and vibrations in a two-layer system of heavy perfect fluids.
Similar content being viewed by others
References
Il’yushin, A.A., Deformation of viscoplastic bodies, Uch. Zap. Mosk. Gos. Univ. Mekh., 1940, vol. 39, pp. 3–81.
Georgievskii, D.V., Linearization of tensor nonlinear constitutive relations in the problems on stability of flows, Chebyshevskii Sb., 2017, vol. 18, no. 3, pp. 202–209.
Lidskii, V.B. and Sadovnichii, V.A., Trace formulas in the case of the Orr-Sommerfeld equation, Math. USSR Izv., 1968, vol. 2, no. 3, pp. 585–600.
Betchov, R. and Criminale, W.O., Stability of Parallel Flows, New York: Academic, 1967.
Georgievskii, D.V., Izbrannye zadachi mekhaniki sploshnoi sredy (Selected Problems of Continuum Me chanics), Moscow: URSS, 2018.
Il'in, V.A. and Filippov, A.F., The nature of the spectrum of a selfadjoint extension of the Laplace oper ator in a bounded region (fundamental systems of functions with an arbitrary preassigned subsequence of fundamental numbers), Sov. Math. Dokl., 1970, vol. 11, pp. 339–342.
Gekhtman, M.M., Spectrum of some nonclassical self-adjoint extensions of the Laplace operator, Funct. Anal. Appl., 1970, vol. 4, no. 4, pp. 325–326.
Yakubov, S.Ya., A boundary value problem for the Laplace equation with nonclassical spectral asymptotics, Sov. Math. Dokl., 1982, vol. 26, pp. 276–279.
Gorbachuk, V.I. and Rybak, M.A., On boundary value problems for a Sturm-Liouville operator equation with a spectral parameter in the equation and in the boundary condition, in Pryamye i obratnye zadachi teorii rasseyaniya (Direct and Inverse Problems of Scattering Theory), Kiev, 1981, pp. 3–16.
Aliev, B.A., Asymptotic behavior of the eigenvalues of a boundary-value problem for a second-order elliptic operator-differential equation, Ukr. Math. J., 2006, vol. 58, no. 8, pp. 1298–1306.
Kapustin, N.Yu., On a spectral problem in the theory of the heat operator, Differ. Equations, 2009, vol. 45, no. 10, pp. 1544–1546.
Kapustin, N.Yu., Basis property of the system of root functions of a classical spectral problem with a multiple eigenvalue, Differ. Equations, 2018, vol. 54, no. 10, pp. 1399–1402.
Kapustin, N.Yu., On the uniform convergence in C 1 of Fourier series for a spectral problem with squared spectral parameter in a boundary condition, Differ. Equations, 2011, vol. 47, no. 10, pp. 1408–1413.
Aliev, B.A., Asymptotic behavior of eigenvalues of a boundary value problem for a second-order elliptic differential-operator equation with spectral parameter quadratically occurring in the boundary condition, Differ. Equations, 2018, vol. 54, no. 9, pp. 1256–1260.
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2019, published in Differentsial’nye Uravneniya, 2019, Vol. 55, No. 5, pp. 683–690.
Rights and permissions
About this article
Cite this article
Georgievskii, D.V. Statements of Linearized Boundary Value Problems of Continuum Mechanics with a Spectral Parameter in the Boundary Conditions. Diff Equat 55, 669–676 (2019). https://doi.org/10.1134/S0012266119050082
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0012266119050082