Abstract
This study presents mathematical model of the internal waves and examines wave propagation in a two-layer fluid flow. Elements of the functional-analytical approach are used to develop the model. A flat unsteady motion of a two-layer liquid under a cover over a flat bottom is considered. The fluid is assumed to be ideal and incompressible. Internal waves are caused by external pressure application to the interface between the layers, oscillation of the flat bottom and disturbances in the flow. The power of the wave source is characterized by dimensionless parameter ε. The problem is formulated, and its solution is based on asymptotic analysis for 0<ε<1.
Highlights
The mathematical formulation of internal waves is in particular interest for researchers as it has wide range of application in the industry, from oil transportation to deep sea constructions
More convenient and easier methods of describing the internal waves can be achieved by the use of asymptotic models with introduction of the dimensionless parameters
There are several works on a two-fluid systems, including studies based on the rigid-lid assumption [2, 3], weakly nonlinear models for the free-surface case [4], stronly nonlinear models [5, 6, 7, 8, 9]
Summary
The mathematical formulation of internal waves is in particular interest for researchers as it has wide range of application in the industry, from oil transportation to deep sea constructions. The following assumptions are adopted: 1) The fluid is ideal and incompressible ; 2) Surface tension is neglected ; 3) The fluid motion is uniform with a constant velocity of Vuntil some initial moment t = 0 ; 4) The source of internal waves begins to act at t = 0 and oscillates with a frequency of ω ; 5) The source power is characterized by a dimensionless parameter ε> 0 ; 6) We consider the case where ε < 1 ; 7) There are three sources of internal waves: a) pressure εεεεis applied to the interface between the layers; b) some section of the bottom (or lid) oscillates according to a given law; c) a regular or singular potential of disturbances is superimposed on the flow. The problem is formulated as determination of the fluid flow and shape of the interface between the layers at t '> 0 according to the given mechanism of wave generation
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