Abstract
The Korteweg–de Vries partial differential equation that has nonlinear-dispersion balance was derived under potential conditions to match the case of a single Kelvin mode that was noticed revolving on the outer periphery of an open cylindrical channel under precession conditions, which is assumed the solitary wave case in the channel. This led to a new version of the equation with a forcing term that includes the tilt effect with coefficients include the rotation effect. It was solved numerically using Fourier transformation methods for space discretization and the fourth order Runge–Kutta method for time discretization; the results were in a good match with the experiment. The rotational case led to a new Benjamin–Bona–Mahony equation that has variable coefficients with time and space mainly coming from the Coriolis effect in the axial direction of motion, with a forcing term comes from the gravity force. It was also solved numerically using a simple implicit finite difference scheme. This equation has two versions, one in terms of the velocity and one in terms of the amplitude. The first was compared with the bore velocity signal, which reflected the cnoidal type of waves, and the results were in a satisfactory match with the extracted signals; the second one was tracked with time to see the role Coriolis and gravity forces play on the single Kelvin wave form.
Highlights
The models of nonlinear partial differential equations are varied, and all those models can be found in many physical applications; in hydrodynamics, for instance, there is the physics of waves and their evolution and propagation, which are of high importance due to a lot of real evidences and cases; the famous solitary wave, for instance, was first reported by Russell1 in a real channel in Scotland; it has a mathematical model that is described by the Korteweg–de Vries partial differential equation, ut + uux + uxxx = 0
The effect of the Coriolis constant on the solution of the Geophysical Korteweg–de Vries equation has been carried out by Karunakar and Chakraverty11 by using the Homotopy Perturbation Method (HPM); they concluded that the Coriolis constant is directly proportional to wave height and inversely proportional to wavelength and that the presence of the Coriolis term in the gKdV equation has a remarkable change in the shape of the solution
We have presented two different nonlinear models: one is for the KdV equation, and the other is the BBM equation; the key in both equations is the simplest balance between the nonlinearity and dispersion
Summary
In the course of nanotechnology, another equation that incorporates the dispersion and damping effects was derived by another team of Rashid to investigate the shape effects of nanoparticles on the Marangoni boundary layer of graphene–water nanofluid flow and the heat transfer over a porous medium under the influence of the suction parameter, thermal radiation, and a magnetic field This time, the problem was solved numerically using the NDSolve technique of Mathematic 10.3 software, and the results were compared with the analytical solution using the HAM method. The HAM method showed its effective results in reducing the errors fast, and again, it was used by other team of Rashid, this time to study the effect of gold nanoparticles’ shape on squeezing nanofluid flow and heat transfer between parallel plates This time, they incorporated the hexahedron, tetrahedron, and lamina shapes as well.
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