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).

Problems on Small Motions of Systems of Two Viscoelastic Fluids in Fixed Vessels

“…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).…”

“…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.…”

“…In this paper, the investigation from [7] is continued as follows: we study the problem of small motions for a system of two viscoelastic immiscible liquids completely filling an immovable vessel. For simplicity, the Oldroyd model ( = 1) is taken though the same scheme fits for the generalized Oldroyd model ( 1) as well.…”

“…Further, are the operators of the trace on the boundary Γ Ω of scalar functions defined in Ω By the Galiardo theorem (see [5]), we have 1 2 (Γ) and these traces satisfy the normalizing conditions imposed by (4.7). It is known from [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] that the set…”

To the Problem of Small Oscillations of a System of Two Viscoelastic Fluids Filling Immovable Vessel: Model Problem

“…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].…”