scholarly journals Expansion shock waves in regularized shallow-water theory

2016 ◽  
Vol 472 (2189) ◽  
pp. 20160141 ◽  
Author(s):  
Gennady A. El ◽  
Mark A. Hoefer ◽  
Michael Shearer
Keyword(s):  
Shallow Water ◽  
Initial Conditions ◽  
Simple Wave ◽  
Boussinesq Equations ◽  
Shallow Water Theory ◽  
Initial Value ◽  
Algebraic Decay ◽  
New Type ◽  
Non Local ◽  
Shock Solution

We identify a new type of shock wave by constructing a stationary expansion shock solution of a class of regularized shallow-water equations that include the Benjamin–Bona–Mahony and Boussinesq equations. An expansion shock exhibits divergent characteristics, thereby contravening the classical Lax entropy condition. The persistence of the expansion shock in initial value problems is analysed and justified using matched asymptotic expansions and numerical simulations. The expansion shock's existence is traced to the presence of a non-local dispersive term in the governing equation. We establish the algebraic decay of the shock as it is gradually eroded by a simple wave on either side. More generally, we observe a robustness of the expansion shock in the presence of weak dissipation and in simulations of asymmetric initial conditions where a train of solitary waves is shed from one side of the shock.

scholarly journals Blow-Up of Solutions to a Novel Two-Component Rod System

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Shengrong Liu ◽  
Zhengguang Guo ◽  
Weiming Wang
Keyword(s):  
Shallow Water ◽  
Finite Time ◽  
Blow Up ◽  
Strong Solutions ◽  
Shallow Water Theory ◽  
Initial Value ◽  
Rod System ◽  
Two Component ◽  
Special Classes

We consider a novel two-component rod system which is closely connected to the shallow water theory. The present work is mainly concerned with the blow-up mechanism of strong solutions; we establish new conditions in view of some special classes of initial value to guarantee finite time blow-up of solutions.

scholarly journals Asymptotic Stability of Nonlinear Discrete Fractional Pantograph Equations with Non-Local Initial Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 473
Author(s):  
Jehad Alzabut ◽  
A. George Maria Selvam ◽  
Rami Ahmad El-Nabulsi ◽  
D. Vignesh ◽  
Mohammad Esmael Samei
Keyword(s):  
Fixed Point ◽  
Asymptotic Stability ◽  
Fixed Point Theorems ◽  
Initial Conditions ◽  
Current Collector ◽  
Stability Of Solutions ◽  
Electric Bus ◽  
Pantograph Equation ◽  
Non Local

Pantograph, the technological successor of trolley poles, is an overhead current collector of electric bus, electric trains, and trams. In this work, we consider the discrete fractional pantograph equation of the form Δ*β[k](t)=wt+β,k(t+β),k(λ(t+β)), with condition k(0)=p[k] for t∈N1−β, 0<β≤1, λ∈(0,1) and investigate the properties of asymptotic stability of solutions. We will prove the main results by the aid of Krasnoselskii’s and generalized Banach fixed point theorems. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.

Study on the working state of the five hundred-meter aperture spherical telescope using step-wise assignment method

2014 ◽  
Vol 229 (2) ◽  
pp. 316-324 ◽  
Author(s):  
Yu Liu ◽  
Feng Gao
Keyword(s):  
Nonlinear Equations ◽  
Analytical Solutions ◽  
Driving Mechanism ◽  
Initial Value ◽  
Unknown Parameters ◽  
Support Structure ◽  
Assignment Method ◽  
New Type ◽  
Set Up ◽  
The Mathematical Model

The working state of the five hundred-meter aperture spherical telescope (FAST) is solved using the step-wise assignment method. In this paper, the mathematical model of the cable-net support structure of the FAST is set up by the catenary equation. There are a large number of nonlinear equations and unknown parameters of the model. The nonlinear equations are solved by using the step-wise assignment method. The method is using the analytical solutions of the cable-net equations of one working state as the initial value for the next working state, from which the analytical solutions of the nonlinear equations of the cable-net for each working state of the FAST and the tension and length of each driving cable can be obtained. The suggested algorithm is quite practically well suited to study the working state of the cable-net structures of the FAST. Also, the working state analysis result of the cable-net support structure of a reduced model of the cable-net structure reflector for the FAST is given to verify the reliability of the method. In order to show the validity of the method, comparisons with another algorithm to set the initial value are presented. This method has an important guiding significance to the further study on the control of the new type of flexible cable driving mechanism, especially the FAST.

scholarly journals A Shallow-Water Theory for the Sun's Active Longitudes

2005 ◽  
Vol 635 (2) ◽  
pp. L193-L196 ◽  
Author(s):  
Mausumi Dikpati ◽  
Peter A. Gilman
Keyword(s):  
Shallow Water ◽  
Shallow Water Theory

Natural Convection in a Porous, Horizontal Cylindrical Annulus

1994 ◽  
Vol 116 (3) ◽  
pp. 621-626 ◽  
Author(s):  
J. P. Barbosa Mota ◽  
E. Saatdjian
Keyword(s):  
Natural Convection ◽  
Initial Conditions ◽  
Hysteresis Loops ◽  
Boussinesq Equations ◽  
Radius Ratio ◽  
Fine Grid ◽  
Cylindrical Annulus ◽  
Alternating Direction ◽  
Fluid Layers ◽  
Horizontal Cylinders

Natural convection in a porous medium bounded by two horizontal cylinders is studied by solving the two-dimensional Boussinesq equations numerically. An accurate second-order finite difference scheme using an alternating direction method and successive underrelaxation is applied to a very fine grid. For a radius ratio above 1.7 and for Rayleigh numbers above a critical value, a closed hysteresis loop (indicating two possible solutions depending on initial conditions) is observed. For a radius ratio below 1.7 and as the Rayleigh number is increased, the number of cells in the annulus increases without bifurcation, and no hysteresis behavior is observed. Multicellular regimes and hysteresis loops have also been reported for fluid layers of same geometry but several differences between these two cases exist.

A shock solution for the nonlinear shallow water equations

2010 ◽  
Vol 658 ◽  
pp. 166-187 ◽  
Author(s):  
MATTEO ANTUONO
Keyword(s):  
Shock Wave ◽  
General Solution ◽  
Theoretical Analysis ◽  
Asymptotic Behaviour ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Boundary Data ◽  
Depth Analysis ◽  
Nonlinear Shallow Water Equations ◽  
Shock Solution

A global shock solution for the nonlinear shallow water equations (NSWEs) is found by assigning proper seaward boundary data that preserve a constant incoming Riemann invariant during the shock wave evolution. The correct shock relations, entropy conditions and asymptotic behaviour near the shoreline are provided along with an in-depth analysis of the main quantities along and behind the bore. The theoretical analysis is then applied to the specific case in which the water at the front of the shock wave is still. A comparison with the Shen & Meyer (J. Fluid Mech., vol. 16, 1963, p. 113) solution reveals that such a solution can be regarded as a specific case of the more general solution proposed here. The results obtained can be regarded as a useful benchmark for numerical solvers based on the NSWEs.

A Quasi-three-dimensional Finite-volume Shallow Water Model for Green Water on Deck

2013 ◽  
Vol 57 (03) ◽  
pp. 125-140
Author(s):  
Daniel A. Liut ◽  
Kenneth M. Weems ◽  
Tin-Guen Yen
Keyword(s):  
Shallow Water ◽  
Finite Volume ◽  
Time Domain ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Green Water ◽  
Water Model ◽  
Ship Motions ◽  
Shallow Water Model

A quasi-three-dimensional hydrodynamic model is presented to simulate shallow water phenomena. The method is based on a finite-volume approach designed to solve shallow water equations in the time domain. The nonlinearities of the governing equations are considered. The methodology can be used to compute green water effects on a variety of platforms with six-degrees-of-freedom motions. Different boundary and initial conditions can be applied for multiple types of moving platforms, like a ship's deck, tanks, etc. Comparisons with experimental data are discussed. The shallow water model has been integrated with the Large Amplitude Motions Program to compute the effects of green water flow over decks within a time-domain simulation of ship motions in waves. Results associated to this implementation are presented.

Modified Shallow Water Equations With Application for Horizontal Centrifugal Casting of Rolls

2015 ◽  
Vol 137 (11) ◽  
Author(s):  
Abdellah Kharicha ◽  
Jan Bohacek ◽  
Andreas Ludwig ◽  
Menghuai Wu
Keyword(s):  
Liquid Metal ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Initial Conditions ◽  
Centrifugal Casting ◽  
Axial Direction ◽  
Mean Amplitude ◽  
Angular Frequency ◽  
Average Height ◽  
Mold Deformation

A numerical model based on the shallow water equations (SWE) was proposed to simulate the two-dimensional (2D) average flow dynamics of the liquid metal spreading inside a horizontally rotating mold. The SWE were modified to account for the forces, such as the centrifugal force, Coriolis force, shear force with the mold wall, and gravity. In addition, inherent vibrations caused by a poor roundness of the mold and the mold deformation due to temperature gradients were applied explicitly by perturbing the gravity and the axis bending, respectively. Several cases were studied with the following initial conditions: a constant average height of the liquid metal (5, 10, 20, 30, and 40 mm) patched as a flat or a perturbed surface. The angular frequency Ω of the mold (∅1150–3200) was 71.2 (or 30) rad/s. Results showed various wave patterns propagating on the free surface. In early stages, a single longitudinal wave moved around the circumference. As the time proceeded, it slowly diminished and waves traveled mainly in the axial direction. It was found that the mean amplitude of the oscillations grows with the increasing liquid height.

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