Abstract
A new model for Korteweg and de-Vries equation (KdV) is derived. The system under study is an open channel consisting of two concentric cylinders, rotating about their vertical axis, which is tilted by slope tau from the inertial vertical z, in uniform rate {Omega }_{1}=tau Omega, and the whole tank is elevated over other table rotating at rate Omega. Under these conditions, a set of Kelvin waves is formed on the free surface depending on the angle of tilt, characterized by the slope tau, volume of water, and rotation rate. The resonant mode in the system appears in the form of a single Kelvin solitary wave, whose amplitude satisfies the Korteweg-de Vries equation with forced term. The equation was derived following classical perturbation methods, the additional term made the equation a non-integrable one, that cannot be solved without the help of numerical methods. Invoking the simple finite difference scheme method, it was found that the numerical results are in a good agreement with the experiment.
Highlights
Most wave phenomena in modern science are decribed by nonlinear equations
The most beautiful model was the one derived in 1895 by two Dutch mathematicians, which the equation takes their names: Korteweg and de-Vries (KdV) equation2. It was first dedicated for water waves in open channel, after the discovery of solitary wave by Russell (1834), later in the sixties the equation was noticed in other applications like: internal gravity waves in stratified fluid, ion-acoustic waves in plasma, axisymmetric waves in a nonuniformly rotating fluid, nonlinear waves in cold plasmas, axisymmetric magnetohydrodynamic waves and several other physical applications. (Debnath 2012)
In order to track the different oscillations in the channel under precession conditions, a real flume was constructed for this purpose
Summary
Most wave phenomena in modern science are decribed by nonlinear equations. The most beautiful model was the one derived in 1895 by two Dutch mathematicians, which the equation takes their names: Korteweg and de-Vries (KdV) equation. In this paper I introduce a new model of this KdV equation with azimuthal dependence where the cylindrical geometry, incorporates both tilt and rotation effects This is a new case, as all the previous cases that discussed the solitary wave invoked the rotation effect only. The earliest experiment was carried out by Maxworthy (1983) to study the rotation effect on the internal solitary wave in a rectangular tank mounted over a rotating table, he noticed that the rotation affected the wave form as it was noticed to be curved backward while moving. Their observations about the wave showed that the crest of the wave is neither horizontal nor contained in a vertical plane perpendicular to the side due to rotation effect, but is curved backward, there is a spatial phase shift which increases by increasing the distance from the wall, and at a given distance from the wall increases by increasing Coriolis parameter
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