scholarly journals On instability of Rayleigh–Taylor problem for incompressible liquid crystals under $L^{1}$-norm

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mengmeng Liu ◽  
Xueyun Lin
Keyword(s):  
Gravitational Field ◽  
Incompressible Liquid ◽  
Steady State Solution ◽  
Nonlinear Instability ◽  
Linearized Problem ◽  
Unstable Solutions ◽  
Taylor Problem ◽  
Linear And Nonlinear ◽  
Nonlinear Solutions ◽  
Rt Instability

AbstractWe investigate the nonlinear Rayleigh–Taylor (RT) instability of a nonhomogeneous incompressible nematic liquid crystal in the presence of a uniform gravitational field. We first analyze the linearized equations around the steady state solution. Thus we construct solutions of the linearized problem that grow in time in the Sobolev space $H^{4}$ H 4 , then we show that the RT equilibrium state is linearly unstable. With the help of the established unstable solutions of the linearized problem and error estimates between the linear and nonlinear solutions, we establish the nonlinear instability of the density, the horizontal and vertical velocities under $L^{1}$ L 1 -norm.

A PREY-PREDATOR MODEL WITH DIFFUSION AND A SUPPLEMENTARY RESOURCE FOR THE PREY IN A TWO‐PATCH ENVIRONMENT

2005 ◽  
Vol 9 (1) ◽  
pp. 9-24 ◽  
Author(s):  
J. Dhar
Keyword(s):  
Steady State ◽  
Asymptotic Stability ◽  
Prey Species ◽  
Steady State Solution ◽  
State Solution ◽  
Prey Population ◽  
Predator Species ◽  
Additional Resource ◽  
Linear And Nonlinear ◽  
Predator Model

In this paper, a prey‐predator dynamics, where the predator species partially depends upon the prey species, in a two patch habitat with diffusion and there is a non‐diffusing additional resource for the prey population, is modeled and analyzed. It is shown, that there exists a positive, monotonic, continuous steady state solution with continuous matching at the interface for both the species separately. Further, we obtain conditions for asymptotic stability for both linear and nonlinear cases. Šiame straipsnyje modeliuojama ir analizuojama plešr‐unu ir auku dinamika, laikant, kad plešr-unu populiacija dalinai priklauso nuo auku skačiaus. Areala sudaro dvi sritys, kuriose vyksta populiaciju individu difuzija, be to, aukoms yra išskirtas nedifunduojantis resursas. Irodyta, kad egzistuoja teigiamas, monotoniškas, tolydus stacionarusis sprendinys, tenkinantis tolydumo salyga abiems populiacijoms atskirai. Gautos asimptotinio stabilumo salygos tiesiniu ir netiesiniu atvejais.

scholarly journals Instability of the abstract Rayleigh–Taylor problem and applications

2020 ◽  
Vol 30 (12) ◽  
pp. 2299-2388 ◽  
Author(s):  
Fei Jiang ◽  
Song Jiang ◽  
Weicheng Zhan
Keyword(s):  
Initial Data ◽  
Stokes Problem ◽  
Strong Solutions ◽  
Viscous Flows ◽  
Existence Theory ◽  
Lagrangian Coordinates ◽  
Compatibility Conditions ◽  
Unstable Solutions ◽  
Taylor Problem ◽  
Proper Values

Based on a bootstrap instability method, we prove the existence of unstable strong solutions in the sense of [Formula: see text]-norm to an abstract Rayleigh–Taylor (RT) problem arising from stratified viscous fluids in Lagrangian coordinates. In the proof we develop a method to modify the initial data of the linearized abstract RT problem by exploiting the existence theory of a unique solution to the stratified (steady) Stokes problem and an iterative technique, such that the obtained modified initial data satisfy the necessary compatibility conditions on boundary of the original (nonlinear) abstract RT problem. Applying an inverse transform of Lagrangian coordinates to the obtained unstable solutions and taking then proper values of the parameters, we can further obtain unstable solutions of the RT problem in viscoelastic, magnetohydrodynamics (MHD) flows with zero resistivity and pure viscous flows (with/without interface intension) in Eulerian coordinates.

scholarly journals LINEAR AND NONLINEAR INSTABILITY OF FLOW IN CHANNEL OCCUPIED POROUS MEDIA

2017 ◽  
Vol 39 (3) ◽  
pp. 40-46
Author(s):  
A.A. Avramenko ◽  
N.P. Dmitrenko ◽  
Y.Y. Kovetskaya
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Kinematic Viscosity ◽  
Hydrodynamic Instability ◽  
Linear Instability ◽  
Nonlinear Instability ◽  
Linear Perturbations ◽  
Linear And Nonlinear

The paper investigates linear and nonlinear hydrodynamic instability of flow in channel ocuped porous medium. The effects of linear instability are considered using the method of linear perturbations. The nonlinear instability of the flow is considered using the renormalized expression for the coefficient of the kinematic viscosity.

scholarly journals Linear vs. nonlinear porosity estimation of NMR oil reservoir data

2010 ◽  
Vol 8 (1) ◽  
pp. 19-33
Author(s):  
Mohsen Abdou Abou Mandour ◽  
Mohamed Mohamed El-Affify ◽  
Mohamed Hassan Hassan ◽  
Amir Kamel Alramady
Keyword(s):  
Nuclear Magnetic Resonance ◽  
Regularization Parameter ◽  
Least Square ◽  
Oil Reservoir ◽  
Tight Sandstone ◽  
Linear Solution ◽  
T2 Distribution ◽  
Linear And Nonlinear ◽  
Linear Least Square ◽  
Nonlinear Solutions

Nuclear magnetic resonance is widely used to assess oil reservoir properties especially those that can not be evaluated using conventional techniques. In this regard, porosity determination and the related estimation of the oil present play a very important role in assessing the eco1nomic value of the oil wells. Nuclear Magnetic Resonance data is usually fit to the sum of decaying exponentials. The resulting distribution; i.e. T2 distribution; is directly related to porosity determination. In this work, three reservoir core samples (Tight Sandstone and two Carbonate samples) were analyzed. Linear Least Square method (LLS) and non-linear least square fitting using Levenberg-Marquardt method were used to calculate the T2 distribution and the resulting incremental porosity. Parametric analysis for the two methods was performed to evaluate the impact of number of exponentials, and effect of the regularization parameter (?) on the smoothing of the solution. Effect of the type of solution on porosity determination was carried out. It was found that 12 exponentials is the optimum number of exponentials for both the linear and nonlinear solutions. In the mean time, it was shown that the linear solution begins to be smooth at α = 0.5 which corresponds to the standard industrial value for the regularization parameter. The order of magnitude of time needed for the linear solution is in the range of few minutes while it is in the range of few hours for the nonlinear solution. Regardless of the fact that small differences exist between the linear and nonlinear solutions, these small values make an appreciable difference in porosity. The nonlinear solution predicts 12% less porosity for the tight sandstone sample and 4.5 % and 13 % more porosity in the two carbonate samples respectively.

Linear and nonlinear stability criteria for compressible MHD flows in a gravitational field

2013 ◽  
Vol 79 (5) ◽  
pp. 873-883 ◽  
Author(s):  
S. M. MOAWAD
Keyword(s):  
Gravitational Field ◽  
Variational Principles ◽  
Flow Speed ◽  
Equilibrium States ◽  
Order Partial Differential Equation ◽  
Equilibrium Equations ◽  
State Equations ◽  
Mhd Equilibrium ◽  
Linear And Nonlinear ◽  
Field Lines

AbstractThe equilibrium and stability properties of ideal magnetohydrodynamics (MHD) of compressible flow in a gravitational field with a translational symmetry are investigated. Variational principles for the steady-state equations are formulated. The MHD equilibrium equations are obtained as critical points of a conserved Lyapunov functional. This functional consists of the sum of the total energy, the mass, the circulation along field lines (cross helicity), the momentum, and the magnetic helicity. In the unperturbed case, the equilibrium states satisfy a nonlinear second-order partial differential equation (PDE) associated with hydrodynamic Bernoulli law. The PDE can be an elliptic or a parabolic equation depending on increasing the poloidal flow speed. Linear and nonlinear Lyapunov stability conditions under translational symmetric perturbations are established for the equilibrium states.

scholarly journals The linear and nonlinear instability of the Akhmediev breather

Nonlinearity ◽  
2021 ◽  
Vol 34 (12) ◽  
pp. 8331-8358
Author(s):  
P G Grinevich ◽  
P M Santini
Keyword(s):  
Nonlinear Instability ◽  
Linear And Nonlinear

scholarly journals Baroclinic and barotropic instabilities in planetary atmospheres - energetics, equilibration and adjustment

2019 ◽  
Author(s):  
Peter Read ◽  
Neil Lewis ◽  
Daniel Kennedy ◽  
Hélène Scolan ◽  
Fachreddin Tabataba-Vakili ◽  
...  
Keyword(s):  
Thermal Structure ◽  
Planetary Atmospheres ◽  
Stability Theorem ◽  
Unstable State ◽  
Nonlinear Instability ◽  
Background Flow ◽  
Flow Configuration ◽  
The Earth ◽  
Open Questions ◽  
Linear And Nonlinear

Abstract. Baroclinic and barotropic instabilities, are well known as the mechanisms responsible for the production of the dominant energy-containing eddies in the atmospheres of the Earth and several other planets, as well as the Earth's oceans. Here we consider insights provided by both linear and nonlinear instability theories into the conditions under which such instabilities may occur, with reference to forced and dissipative flows obtainable in the laboratory, in simplified numerical atmospheric circulation models and in the planets of our Solar System. The equilibration of such instabilities is also of great importance in understanding the structure and energetics of the observable circulation of atmospheres and oceans. Various ideas have been proposed concerning the ways in which baroclinic and barotropic instabilities grow to large amplitude and saturate, whilst also modifying their background flow and environment. This remains an area that continues to challenge theoreticians and observers, though some progress has been made. The notion that such instabilities may act under some conditions to adjust the background flow towards a critical state is explored here in the context of both laboratory systems and planetary atmospheres. Evidence for such adjustment processes is found relating to baroclinic instabilities under a range of conditions where the efficiency of eddy and zonal mean heat transport may mutually compensate to maintain a nearly invariant thermal structure in the zonal mean. In other systems, barotropic instabilities may efficiently mix potential vorticity to result in a flow configuration that is found to approach a marginally unstable state with respect to Arnol'd's second stability theorem. We discuss the implications of these findings and identify some outstanding open questions.

scholarly journals Instability of periodic traveling waves for the symmetric regularized long wave equation

2015 ◽  
Vol 219 ◽  
pp. 235-268
Author(s):  
Jaime Angulo Pava ◽  
Carlos Alberto Banquet Brango
Keyword(s):  
Sufficient Conditions ◽  
Traveling Wave Solutions ◽  
Linear Instability ◽  
Nonlinear Instability ◽  
Periodic Traveling Waves ◽  
Wave Solutions ◽  
Long Wave ◽  
Linearized Problem ◽  
Periodic Traveling Wave ◽  
The Stability

AbstractWe prove the linear and nonlinear instability of periodic traveling wave solutions for a generalized version of the symmetric regularized long wave (SRLW) equation. Using analytic and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so the linear instability of periodic profiles is obtained. An application of this approach is made to obtain the linear/nonlinear instability of cnoidal wave solutions for the modified SRLW (mSRLW) equation. We also prove the stability of dnoidal wave solutions associated to the equation just mentioned.

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