2018
DOI: 10.22363/2413-3639-2018-64-3-547-572
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To the Problem on Small Motions of the System of Two Viscoelastic Fluids in a Fixed Vessel

Abstract: In this paper, we study the problem of small motions of two Oldroyd viscoelastic incompressible fluids contained in a fixed vessel. By means of the operator approach, we reduce the original initialboundary value problem to the Cauchy problem for a differential operator equation in a Hilbert space and prove the well-posed solvability of the problem on an arbitrary interval of time. We obtain the equation for normal oscillations of the hydraulic system under consideration (Krein generalized operator pencil).

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Cited by 5 publications
(6 citation statements)
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“…In [9], relations for the orthoprojectors 0 and 1 (see Sec. 4) are obtained for the case where an immovable vessel is filled by three immiscible viscoelastic liquids (instead of two ones).…”
Section: Discussionmentioning
confidence: 99%
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“…In [9], relations for the orthoprojectors 0 and 1 (see Sec. 4) are obtained for the case where an immovable vessel is filled by three immiscible viscoelastic liquids (instead of two ones).…”
Section: Discussionmentioning
confidence: 99%
“…Finally, once relations for the orthoprojectors 0 and 1 for the problem on small motions of a system an arbitrary amount of immiscible viscoelastic liquids completely filling an immovable vessel are obtained, the approach applied in this paper can be used to prove strong solvability theorem for the problem on small motions of a system on an interval of time. Scalar and vector transmission problems described in [9] arise when one describes the action of orthoprojectors 0 and 1 in the case of three liquids.…”
Section: Discussionmentioning
confidence: 99%
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“…This approach develops the constructions conducted earlier by Krein and his disciples (see [14,16] and also papers [12,13]) applicable to the problems of small oscillations of a viscoelastic fluid in a partially filled vessel of a system of immiscible fluids. The case of a fully filled cavity with a viscoelastic fluid was considered in [2], and also in [11]. An instance of initial-boundary value problem for the vessel filled with two immiscible viscoelastic fluids was studied in [5].…”
Section: Statement Of the Scalar Model Problemmentioning
confidence: 99%