The domain of existence of solitary waves in fluid-filled thin elastic tubes

2021 ◽  
pp. 108128652098481
Author(s):  
R. Ivan Defaz ◽  
Marcelo Epstein ◽  
Salvatore Federico
Keyword(s):  
Differential Equations ◽  
Solitary Waves ◽  
Wave Speed ◽  
Fluid Velocity ◽  
Wave Profile ◽  
The Self ◽  
System Of Differential Equations ◽  
Elastic Tubes ◽  
Governing System ◽  
Domain Of Existence

Under given prestress conditions, solitary waves in fluid-filled elastic tubes are confined to a rather narrow set of combinations of the background fluid velocity and the wave speed. This set, which we call the domain of existence, is bounded by several curves that represent various physical and mathematical restrictions. Remarkably, these restrictions can be cast as purely algebraic conditions to be imposed upon the governing system of differential equations. Paramount among the physical restrictions are the avoidance of wrinkles and the self-impenetrability of the wave profile. In particular, the existence of a critical wave speed of impending wrinkling, independent of the background fluid velocity, is established rigorously.

scholarly journals Heat Transport Phenomena for the Darcy–Forchheimer Flow of Casson Fluid over Stretching Sheets with Electro-Osmosis Forces and Newtonian Heating

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2525
Author(s):  
Xianqin Zhang ◽  
Dezhi Yang ◽  
Muhammad Israr Ur Rehman ◽  
Aamir Hamid
Keyword(s):  
Differential Equations ◽  
Fluid Velocity ◽  
Casson Fluid ◽  
Similarity Transformations ◽  
Governing System ◽  
Electro Osmosis ◽  
The Impact ◽  
Osmotic Effects ◽  
Forchheimer Flow ◽  
Good Agreement

In this study, an investigation has been carried out to analyze the impact of electro-osmotic effects on the Darcy–Forchheimer flow of Casson nanofluid past a stretching sheet. The energy equation was modelled with the inclusion of electro-osmotic effects with viscous and Joule dissipations. The governing system of partial differential equations were transformed by using the suitable similarity transformations to a system of ordinary differential equations and then numerically solved by using the Runge–Kutta–Fehlberg method with a shooting scheme. The effects of various parameters of interest on dimensionless velocity and temperature distributions, as well as skin friction and heat transfer coefficient, have been adequately delineated via graphs and tables. A comparison with previous published results was performed, and good agreement was found. The results suggested that the electric and Forchheimer parameters have the tendency to enhance the fluid velocity as well as momentum boundary layer thickness. Enhancements in temperature distribution were observed for growing values of Eckert number. It was also observed that higher values of electric field parameter diminished the wall shear stress and local Nusselt number.

scholarly journals A Dynamic Model of a Belt Driven Electromechanical XY Plotter Cutter

2012 ◽  
Author(s):  
Joseph V. Prisco ◽  
Philip A. Voglewede
Keyword(s):  
Differential Equations ◽  
Dynamic Model ◽  
Motion Control ◽  
Development Time ◽  
Arts And Crafts ◽  
Performance Parameters ◽  
System Of Differential Equations ◽  
Electromechanical Model ◽  
Governing System

Currently, models for XY plotter cutters specific to industrial and arts and crafts applications are not publicly available. This paper mathematically models the XY motion control for a commercial plotter cutter. In this particular application, the Y motion is controlled by media feed and the X motion is controlled by a gantry arm. A dynamic, electromechanical model consisting of a governing system of differential equations for the gantry arm is developed and simulated using Matlab. Once the model is developed, it will be used to decrease development time and optimize performance parameters.

Breaking and broadening of internal solitary waves

2000 ◽  
Vol 413 ◽  
pp. 181-217 ◽  
Author(s):  
JOHN GRUE ◽  
ATLE JENSEN ◽  
PER-OLAV RUSÅS ◽  
J. KRISTIAN SVEEN
Keyword(s):  
Solitary Waves ◽  
Wave Breaking ◽  
Wave Speed ◽  
Wave Amplitude ◽  
Fluid Velocity ◽  
Fluid Motion ◽  
Constant Density ◽  
Image Velocimetry ◽  
The Waves

Solitary waves propagating horizontally in a stratified fluid are investigated. The fluid has a shallow layer with linear stratification and a deep layer with constant density. The investigation is both experimental and theoretical. Detailed measurements of the velocities induced by the waves are facilitated by particle tracking velocimetry (PTV) and particle image velocimetry (PIV). Particular attention is paid to the role of wave breaking which is observed in the experiments. Incipient breaking is found to take place for moderately large waves in the form of the generation of vortices in the leading part of the waves. The maximal induced fluid velocity close to the free surface is then about 80% of the wave speed, and the wave amplitude is about half of the depth of the stratified layer. Wave amplitude is defined as the maximal excursion of the stratified layer. The breaking increases in power with increasing wave amplitude. The magnitude of the induced fluid velocity in the large waves is found to be approximately bounded by the wave speed. The breaking introduces a broadening of the waves. In the experiments a maximal amplitude and speed of the waves are obtained. A theoretical fully nonlinear two-layer model is developed in parallel with the experiments. In this model the fluid motion is assumed to be steady in a frame of reference moving with the wave. The Brunt-Väisälä frequency is constant in the layer with linear stratification and zero in the other. A mathematical solution is obtained by means of integral equations. Experiments and theory show good agreement up to breaking. An approximately linear relationship between the wave speed and amplitude is found both in the theory and the experiments and also when wave breaking is observed in the latter. The upper bound of the fluid velocity and the broadening of the waves, observed in the experiments, are not predicted by the theory, however. There was always found to be excursion of the solitary waves into the layer with constant density, irrespective of the ratio between the depths of the layers.

Boussinesq solution for solitary waves in uniform channels with sloping side walls

2000 ◽  
Vol 214 (6) ◽  
pp. 781-787 ◽  
Author(s):  
M H Teng
Keyword(s):  
Solitary Waves ◽  
Small Amplitude ◽  
Wave Speed ◽  
Side Wall ◽  
Boussinesq Equations ◽  
Wave Profile ◽  
Cross Sectional ◽  
Side Walls ◽  
Boussinesq Solution ◽  
Wall Slope

The analytical solution to the Boussinesq equations for solitary waves travelling in uniform water channels with sloping side walls is presented. Quantitative effects of channel cross-sectional geometry and channel side-wall slope at the waterline on the wave profile and wave speed, as well as the criteria for positive solitary waves to exist, are discussed. The new Boussinesq solution is also compared with the existing Korteweg-de Vries solution obtained by Peregrine. It is found that the two solutions are consistent for small amplitude waves while, for relatively large waves, the Boussinesq solution gives different predictions for wave speed and for the criteria for solitary waves to exist.

On the speed and profile of steep solitary waves

1976 ◽  
Vol 350 (1661) ◽  
pp. 175-189 ◽  
Keyword(s):  
Integral Equation ◽  
Solitary Waves ◽  
Wave Speed ◽  
Wave Amplitude ◽  
Previous Method ◽  
Wave Profile ◽  
Maximum Speed ◽  
Surface Slope ◽  
Point Of Intersection ◽  
Good Agreement

Previous estimates of the speed of solitary waves in shallow water unexpectedly showed that the speed and energy were greatest for waves of less than the maximum possible height. These calculations were based on Pade approximants. In the present paper we present some quite independent calculations based on an integral equation for the wave profile (Byatt-Smith 1970), now modified so that the wave speed appears as a dependent variable. There is remarkably good agreement with the previous method. In particular the existence of a maximum speed and energy are verified. The method also yields a more accurate profile for the free surface of steep solitary waves. As the wave amplitude increases, it is found that the point of intersection of neighbouring profiles moves up towards the crest. Hence the highest wave lies mostly beneath its neighbours, which helps to explain why its speed is less. Tables are given not only of the wave speed but also of the maximum surface slope as a function of wave amplitude. In no case does the slope exceed 30°, but for still higher waves this possibility is not excluded.

scholarly journals Development of Differential Equations and their Solution Using the Simulink Matlab Program, which Calculate the Self-Swinging of Synchronous Machines with Traditional and Longitudinal-Transverse Excitation

2020 ◽  
Vol 216 ◽  
pp. 01116
Author(s):  
Olimjon Toirov ◽  
Аllabergan Bekishev ◽  
Sardor Urakov ◽  
Utkir Mirkhonov
Keyword(s):  
Differential Equations ◽  
Block Diagram ◽  
Operator Method ◽  
Synchronous Generator ◽  
The Self ◽  
Synchronous Machines ◽  
System Of Differential Equations ◽  
Synchronous Generators ◽  
Matlab Program ◽  
Transverse Excitation

The article presents the differential equations of a synchronous generator in phase coordinates and in the coordinate system (d, q). In addition, differential equations of synchronous machines with longitudinal-transverse excitation and a block diagram based on these equations are given. The system of differential equations is solved by the operator method. On the basis of a system of differential equations using the Simulink Matlab program, a structural diagram was created and a graph of the self-swinging processes taking place in synchronous machines with conventional and longitudinal-transverse excitation was obtained. On the basis of the obtained graph, the processes of self-swinging of synchronous generators with traditional and longitudinal and transverse excitation are compared.

scholarly journals LONGITUDINAL-FLEXURAL SELF-SUSTAINED VIBRATIONS OF NANOTUBE CONVEYING FLUID

2018 ◽  
Keyword(s):  
Differential Equations ◽  
Fluid Velocity ◽  
Period Doubling ◽  
The Self ◽  
Beam Model ◽  
Algebraic Equations ◽  
Chaotic Motions ◽  
Geometrical Nonlinear ◽  
Nonlinear Algebraic Equations ◽  
Conveying Fluid

Beam model of geometrical nonlinear longitudinal-flexural self-sustained vibrations of nanotube conveying fluid is obtained with account of nonlocal elasticity. The Euler-Bernoulli hypotheses are the basis of this model. The geometrical nonlinear deformations are described by nonlinear relations between strains and displacements. It is assumed, that the amplitudes of the longitudinal and bending vibrations are commensurable. Using variational methods of mechanics, the system of two nonlinear partial differential equations is derived to describe the nanotube self-sustained vibrations. The Galerkin method is applied to obtain the system of nonlinear ordinary differential equations. The harmonic balanced method is used to analyze the monoharmonic vibrations. Then the analysis of the self-sustained vibrations is reduced to the system of the nonlinear algebraic equations with respect to the vibrations amplitudes. The Newton method is used to solve this system of nonlinear algebraic equations. As a result of the simulations, it is determined that the stable self-sustained vibrations originate in the Hopf bifurcation due to stability loss of the trivial equilibrium. These stable vibrations are analyzed, when the fluid velocity is changed. The results of the self- sustained vibrations analysis are shown on the bifurcation diagram. The infinite sequence of the period-doubling bifurcations of the monoharmonic vibrations is observed. The chaotic motions take place after these bifurcations. As a result of the numerical simulations it is determined, that the amplitudes of the longitudinal and flexural vibrations are commensurable.

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