On a Long-Wave Instability of Steady-State Plane-Parallel Flows of an Ideal Fluid with Free Boundary in the Gravity Field

Yu G Gubarev

doi:10.5923/j.ijtmp.20120201.02

Abstract

We consider the problem of linear stability of steady-state plane-parallel flows of inviscid incompressible fluid of uniform density with free surface in the gravity field. Using the method of coupling the integrals of motion, we prove that sufficient conditions for the stability of these flows to small plane long-wave perturbations are absent. We construct an analytic example of a steady-state plane-parallel flow with small plane long-wave perturbations superimposed as normal modes

Highlights

The wave motions in fluids undoubtedly belong to those natural phenomena which the human civilization has been dealing with essentially throughout its entire history
It is quite natural that people have constantly tried, and keep trying, to learn how to use waves in fluids beneficially. This requires a thorough understanding of wave motions in fluids and their properties
In this article we study precisely the problem of adequacy of mathematical modelling of waves in fluids on an example of a basic mathematical model of long-wave fluid motions: the model of propagation of long waves on the free boundary

Summary

Statement of the Exact Problem

We study plane long-wave flows of inviscid incompressible fluid of uniform density in an infinitely long thin layer above horizontal bottom in the gravity field. In the initial-boundary value problem (1)-(4) it is exact stationary solutions (10) are stable to small plane convenient to pass from the Euler independent variables long-wave perturbations u′(t, x, λ), ρ ′(t, x, λ), and H ′(t, x). In accordance with the method of coupling the integrals of motion, the exact stationary solutions (10) to the initialboundary value problem (7) are stable to small plane longwave perturbations (11) if and only if the functional E1 of (12) is sign definite or at least semidefinite. ∆2 = − u 02 4 < 0, which prevents the integral E1 from being sign (semi)definite This implies that no sufficient conditions exist for the stability of the exact stationary solutions (10) to the mixed problem (7) to small plane long-wave perturbations u′(t, x, λ), ρ ′(t, x, λ), and H ′(t, x). An analytic example of an exact stationary solution (10) to the mixed problem (7) with small plane long-wave perturbations (11) superimposed as normal modes

Example

Conclusions