scholarly journals Nonlinear evolution equation for modeling waves at the boundary of a stratified flow of viscous liquids in an inclined channel

2021 ◽  
Vol 2119 (1) ◽  
pp. 012058
Author(s):  
D G Arkhipov ◽  
G A Khabakhpashov
Keyword(s):  
Differential Equation ◽  
Nonlinear Evolution Equation ◽  
Wave Motion ◽  
Nonlinear Evolution ◽  
Stationary Flow ◽  
The Other ◽  
Two Dimensional ◽  
Viscous Liquids ◽  
Inclined Channel ◽  
Small Flow

Abstract The dynamics of perturbations of the interface of a two-layer Poiseuille flow in a flat closed inclined channel is studied. The velocity profiles of wave motion are analytically found neglecting dissipation, dispersion and pumping of perturbations. On the basis of the found solution, a nonlinear evolution integro-differential equation for plane moderately long perturbations of the interface of the liquids is derived. The coefficients of the equation are represented by integrals over the layer thicknesses from functions depending on the stationary flow and perturbation profiles. The equation takes into account viscous dissipation: one of the integrals in this equation corresponds to dissipation in lion-stationary boundary layers, and the other corresponds to the transfer of energy from the flow to the wave. For the case of small flow velocities, the coefficients of the equation are analytically calculated. The equation has also been generalized to the quasi-two-dimensional case when the gradients along the transversal coordinate are small.

A nonlinear system for irreversible phase changes

1990 ◽  
Vol 1 (1) ◽  
pp. 91-100 ◽  
Author(s):  
Dominique Blanchard ◽  
Hamid Ghidouche
Keyword(s):  
Nonlinear System ◽  
Existence And Uniqueness ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Phase Changes ◽  
The Other ◽  
System Modelling ◽  
Mathematical Study ◽  
Uniqueness Of The Solution ◽  
Change Problem

This paper is concerned with the mathematical study of a nonlinear system modelling an irreversible phase change problem. Uniqueness of the solution is proved using the accretivity of the system in (L1)2. Expressing one of the two unknowns as an explicit functional of the other reduces the system to a single nonlinear evolution equation and ultimately leads to an existence theorem.In this paper the existence and uniqueness of the solution of a nonlinear system modelling some irreversible phase changes is established.

scholarly journals Bäcklund Transformations between the KdV Equation and a New Nonlinear Evolution Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Xifang Cao
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
The Other ◽  
Bäcklund Transformations ◽  
Backlund Transformations ◽  
The Kdv Equation ◽  
New Equation

We first give a Bäcklund transformation from the KdV equation to a new nonlinear evolution equation. We then derive two Bäcklund transformations with two pseudopotentials, one of which is from the KdV equation to the new equation and the other from the new equation to itself. As applications, by applying our Bäcklund transformations to known solutions, we construct some novel solutions to the new equation.

Application of Improved (G′/G)–Expansion Method to Traveling Wave Solutions of Two Nonlinear Evolution Equations

2012 ◽  
Vol 4 (1) ◽  
pp. 122-130 ◽  
Author(s):  
Xiaohua Liu ◽  
Weiguo Zhang ◽  
Zhengming Li
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Nonlinear Evolution Equation ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
The Other ◽  
Wave Solutions

AbstractIn this work, the improved (G′/G)-expansion method is proposed for constructing more general exact solutions of nonlinear evolution equation with the aid of symbolic computation. In order to illustrate the validity of the method we choose the RLW equation and SRLW equation. As a result, many new and more general exact solutions have been obtained for the equations. We will compare our solutions with those gained by the other authors.

BIFURCATION STRUCTURE IN A MODEL OF A FREQUENCY MODULATED CO2 LASER

1993 ◽  
Vol 03 (01) ◽  
pp. 97-111 ◽  
Author(s):  
C. MIRA ◽  
I. DJELLIT
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Organization Structure ◽  
The Other ◽  
Two Dimensional ◽  
Periodic Excitation ◽  
Bifurcation Structure ◽  
Higher Harmonic ◽  
Bifurcation Structures ◽  
Qualitative Changes

This paper concerns the bifurcation properties of a model of a frequency modulated CO 2 laser in the form of a two-dimensional ordinary differential equation with a parametric periodic excitation. These properties are related to the bifurcation curves organization (structure) in a parameter plane (amplitude, frequency of the modulation). Two basic bifurcation structures appear, one concerning the higher harmonic solutions, the other the subharmonic solutions. Qualitative changes of these structures are considered when a third parameter (pump parameter) is varied.

Two-dimensional patterns in Rayleigh-Taylor instability of a thin layer

1992 ◽  
Vol 236 ◽  
pp. 349-383 ◽  
Author(s):  
M. Fermigier ◽  
L. Limat ◽  
J. E. Wesfreid ◽  
P. Boudinet ◽  
C. Quilliet
Keyword(s):  
Evolution Equation ◽  
Thin Layer ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Amplitude Equation ◽  
Taylor Instability ◽  
Two Dimensional ◽  
Rayleigh Taylor Instability ◽  
Evolution Laws ◽  
Amplitude Reflection

We study experimentally and theoretically the evolution of two-dimensional patterns in the Rayleigh—Taylor instability of a thin layer of viscous fluid spread on a solid surface. Various kinds of patterns of different symmetries are observed, with possible transition between patterns, the preferred symmetries being the axial and hexagonal ones. Starting from the lubrication hypothesis, we derive the nonlinear evolution equation of the interface, and the amplitude equation of its Fourier components. The evolution laws of the different patterns are calculated at order two or three, the preferred symmetries being related to the non-invariance of the system by amplitude reflection. We also discuss qualitatively the dripping at final stage of the instability.

scholarly journals Progression of Aortic Dissection: An Application of Soliton Solutions of Nonlinear Evolution Equation in (3+1)-Dimensions

2021 ◽  
Author(s):  
VISHAKHA JADAUN ◽  
Nitin Singh
Keyword(s):  
Differential Equation ◽  
Aortic Dissection ◽  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Early Stage ◽  
Nonlinear Partial Differential Equation ◽  
Aortic Tissue ◽  
The Impact

Abstract Aortic dissection is a serious pathology involving the vessel wall of the aorta with significant societal impact. To understand aortic dissection we explain the role of the dynamic pathology in the absence or presence of structural and/or functional abnormalities. We frame a differential equation to evaluate the impact of mean blood pressure on the aortic wall and prove the existence and uniqueness of its solution for homeostatic recoil and relaxation for infinitesimal aortic tissue. We model and analyze generalized (3+1)-dimensional nonlinear partial differential equation for aortic wave dynamics. We use the Lie group of transformations on this nonlinear evolution equation to obtain invariant solutions, traveling wave solutions including solitons. We find that abnormalities in the dynamic pathology of aortic dissection act as triggers for the progression of disease in early-stage through the formation of soliton-like pulses and their interaction. We address the role of unstable wavefields in waveform dynamics when waves are unidirectional. Moreover, the notion of dynamic pathology within the domain of vascular geometry may explain the evolution of aneurysms in cerebral arteries and cardiomyopathies even in the absence of anatomical and physiological abnormalities.

scholarly journals On nonlinear evolution equation of second order in Banach spaces

2018 ◽  
Vol 16 (1) ◽  
pp. 268-275
Author(s):  
Kamal N. Soltanov
Keyword(s):  
Differential Equation ◽  
Evolution Equation ◽  
Banach Spaces ◽  
Traffic Flow ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Second Order ◽  
Nonlinear Hyperbolic Equation ◽  
Existence Of A Solution ◽  
Special Case

AbstractHere we study the existence of a solution and also the behavior of the existing solution of the abstract nonlinear differential equation of second order that, in particular, is the nonlinear hyperbolic equation with nonlinear main parts, and in the special case, is the equation of the type of equation of traffic flow.

The bursting of two-dimensional drops in slow viscous flow

1973 ◽  
Vol 60 (4) ◽  
pp. 625-639 ◽  
Author(s):  
J. D. Buckmaster ◽  
J. E. Flaherty
Keyword(s):  
Differential Equation ◽  
Surface Tension ◽  
Analytic Function ◽  
Function Theory ◽  
Equilibrium Shape ◽  
The Other ◽  
Two Dimensional ◽  
Approximate Analysis ◽  
Slow Viscous Flow ◽  
Corner Flow

We consider the deformation of two-dimensional drops when immersed in a slow viscous corner flow. The problem is formulated as one of analytic function theory and simplified by assuming that both the drop and the exterior fluid have the same viscosity. An approximate analysis is carried out, in which the conditions at the interface are satisfied in an average sense, and this reveals the following features of the solution. A drop of given physical properties (volume, surface tension and viscosity), when immersed in a corner flow, has no steady equilibrium shape if the rate of strain of the applied flow is too large. On the other hand, if the rate of strain is small enough for a steady solution to exist, then in general there are two possible solutions. These features are confirmed by formulating the exact problem in terms of a nonlinear integro-differential equation, which is solved numerically.

Long–Wave Morphologies in Directional Solidification

1990 ◽  
Vol 43 (5S) ◽  
pp. S85-S88
Author(s):  
D. S. Riley
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Stability Limit ◽  
Two Dimensional ◽  
Segregation Coefficient ◽  
Directionally Solidified ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
Long Wave

Long–wave instabilities in a directionally–solidified binary mixture may occur in several limits. Sivashinsky identified a small–segregation–coefficient limit and obtained a weakly–nonlinear evolution equation governing subcritical two–dimensional bifurcation. Brattkus and Davis identified a near–absolute–stability limit and obtained a strongly–nonlinear evolution equation governing supercritical two–dimensional bifurcation. In this presentation these previous analyses are set into a logical framework, and a third distinguished (small–segregation–coefficient, large–surface–energy) limit identified. The corresponding strongly–nonlinear, evolution equation equation links both of the previous and describes the change from sub– to super–critical bifurcations.

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