scholarly journals On classical solutions of Rayleigh–Taylor instability in inhomogeneous viscoelastic fluids

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zhidan Tan ◽  
Weiwei Wang
Keyword(s):  
Bounded Domain ◽  
Viscoelastic Fluids ◽  
Strong Solutions ◽  
Classical Solutions ◽  
Inverse Transformation ◽  
Lagrangian Coordinates ◽  
Elasticity Coefficient ◽  
Rayleigh Taylor Instability ◽  
Eulerian Coordinates ◽  
Rt Instability

Abstract We study the nonlinear Rayleigh–Taylor (RT) instability of an inhomogeneous incompressible viscoelastic fluid in a bounded domain. It is well known that there exist strong solutions of RT instability in $H^{2}$ H 2 -norm in inhomogeneous incompressible viscoelastic fluids, when the elasticity coefficient κ is less than some threshold $\kappa _{\mathrm{C}}$ κ C . In this paper, we prove the existence of classical solutions of RT instability in $L^{1}$ L 1 -norm in Lagrangian coordinates based on a bootstrap instability method with finer analysis, if $\kappa <\kappa _{\mathrm{C}}$ κ < κ C . Moreover, we also get classical solutions of RT instability in $L^{1}$ L 1 -norm in Eulerian coordinates by further applying an inverse transformation of Lagrangian coordinates.

scholarly journals A new upper bound for the largest growth rate of linear Rayleigh–Taylor instability

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Changsheng Dou ◽  
Jialiang Wang ◽  
Weiwei Wang
Keyword(s):  
Surface Tension ◽  
Growth Rate ◽  
Upper Bound ◽  
Viscous Fluids ◽  
Instability Condition ◽  
Interface Surface ◽  
Incompressible Viscous Fluids ◽  
Rayleigh Taylor Instability ◽  
Surface Tensor ◽  
Rt Instability

AbstractWe investigate the effect of (interface) surface tensor on the linear Rayleigh–Taylor (RT) instability in stratified incompressible viscous fluids. The existence of linear RT instability solutions with largest growth rate Λ is proved under the instability condition (i.e., the surface tension coefficient ϑ is less than a threshold $\vartheta _{\mathrm{c}}$ ϑ c ) by the modified variational method of PDEs. Moreover, we find a new upper bound for Λ. In particular, we directly observe from the upper bound that Λ decreasingly converges to zero as ϑ goes from zero to the threshold $\vartheta _{\mathrm{c}}$ ϑ c .

Two Classes of Nonlinear Singular Dirichlet Problems with Natural Growth: Existence and Asymptotic Behavior

2020 ◽  
Vol 20 (1) ◽  
pp. 77-93 ◽  
Author(s):  
Zhijun Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Classical Solutions ◽  
Natural Growth ◽  
Dirichlet Problems ◽  
Nonlinear Transformations

AbstractThis paper is concerned with the existence, uniqueness and asymptotic behavior of classical solutions to two classes of models {-\triangle u\pm\lambda\frac{|\nabla u|^{2}}{u^{\beta}}=b(x)u^{-\alpha}}, {u>0}, {x\in\Omega}, {u|_{\partial\Omega}=0}, where Ω is a bounded domain with smooth boundary in {\mathbb{R}^{N}}, {\lambda>0}, {\beta>0}, {\alpha>-1}, and {b\in C^{\nu}_{\mathrm{loc}}(\Omega)} for some {\nu\in(0,1)}, and b is positive in Ω but may be vanishing or singular on {\partial\Omega}. Our approach is largely based on nonlinear transformations and the construction of suitable sub- and super-solutions.

A Turbulent Heliosheath Driven by Rayleigh Taylor Instability

2021 ◽  
Author(s):  
Merav Opher ◽  
James Drake ◽  
Gary Zank ◽  
Gabor Toth ◽  
Erick Powell ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Solar Wind ◽  
Large Scale ◽  
Neutral Atom ◽  
Solar Magnetic Field ◽  
Characteristic Time Scale ◽  
Rayleigh Taylor Instability ◽  
Scale Turbulence ◽  
Magnetohydrodynamic Simulations ◽  
Rt Instability

Abstract The heliosphere is the bubble formed by the solar wind as it interacts with the interstellar medium (ISM). Studies show that the solar magnetic field funnels the heliosheath solar wind (the shocked solar wind at the edge of the heliosphere) into two jet-like structures1-2. Magnetohydrodynamic simulations show that these heliospheric jets become unstable as they move down the heliotail1,3 and drive large-scale turbulence. However, the mechanism that produces of this turbulence had not been identified. Here we show that the driver of the turbulence is the Rayleigh-Taylor (RT) instability caused by the interaction of neutral H atoms streaming from the ISM with the ionized matter in the heliosheath (HS). The drag between the neutral and ionized matter acts as an effective gravity which causes a RT instability to develop along the axis of the HS magnetic field. A density gradient exists perpendicular to this axis due to the confinement of the solar wind by the solar magnetic field. The characteristic time scale of the instability depends on the neutral H density in the ISM and for typical values the growth rate is ~ 3 years. The instability destroys the coherence of the heliospheric jets and magnetic reconnection ensues, allowing ISM material to penetrate the heliospheric tail. Signatures of this instability should be observable in Energetic Neutral Atom (ENA) maps from future missions such as IMAP4. The turbulence driven by the instability is macroscopic and potentially has important implications for particle acceleration.

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 The periodic quasigeostrophic equations: existence and uniqueness of strong solutions

1982 ◽  
Vol 91 (3-4) ◽  
pp. 185-203 ◽  
Author(s):  
A. F. Bennett ◽  
P. E. Kloeden
Keyword(s):  
Existence And Uniqueness ◽  
Hyperbolic Equations ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Coupled System ◽  
Classical Solutions ◽  
Schauder Estimates ◽  
Initial Vorticity ◽  
Relative Potential ◽  
Quasigeostrophic Equations

SynopsisThe periodic quasigeostrophic equations are a coupled system of a second order elliptic equation for a streamfunction and first order hyperbolic equations for the relative potential vorticity and surface potential temperatures, on a three-dimensional domain which is periodic in both horizontal spatial co-ordinates. Such equations are used in both numerical and theoretical studies in meteorology and oceanography. In this paper Schauder estimates and a Schauder fixed point theorem are used to prove the existence and uniqueness of strong, that is classical, solutions of the periodic quasigeostrophic equations for a finite interval of time, which is inversely proportional to the sum of the norms of the initial vorticity and surface temperatures.

Rayleigh-Taylor Instability of Newtonian and Oldroydian Viscoelastic Fluids in Porous Medium

1994 ◽  
Vol 49 (10) ◽  
pp. 927-930
Author(s):  
R. C. Sharma ◽  
V. K. Bhardwaj
Keyword(s):  
Porous Medium ◽  
Interfacial Tension ◽  
Viscoelastic Fluids ◽  
Taylor Instability ◽  
Horizontal Boundary ◽  
Wave Numbers ◽  
Unstable Configuration ◽  
Rayleigh Taylor Instability

AbstractThe Rayleigh-Taylor instability of viscous and viscoelastic (Oldroydian) fluids, separately, has been considered in porous medium. Two uniform fluids separated by a horizontal boundary and the case of exponentially varying density have been considered in both viscous and viscoelastic fluids. The effective interfacial tension succeeds in stabilizing perturbations of certain wave numbers (small wavelength perturbations) which were unstable in the absence of effective interfacial tension, for unstable configuration/stratification.

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