scholarly journals The viscous froth model: steady states and the high-velocity limit

2009 ◽  
Vol 465 (2108) ◽  
pp. 2391-2405 ◽  
Author(s):  
S. J. Cox ◽  
D. Weaire ◽  
G. Mishuris
Keyword(s):  
Steady State ◽  
High Velocity ◽  
Steady States ◽  
Two Dimensional ◽  
Isolated Points ◽  
Physical Effects ◽  
Straight Lines ◽  
Steady State Solutions ◽  
Finite Extent

The steady-state solutions of the viscous froth model for foam dynamics are analysed and shown to be of finite extent or to asymptote to straight lines. In the high-velocity limit, the solutions consist of straight lines with isolated points of infinite curvature. This analysis is helpful in the interpretation of observations of anomalous features of mobile two-dimensional foams in channels. Further physical effects need to be adduced in order to fully account for these.

scholarly journals Thermal Robustness of Entanglement in a Dissipative Two-Dimensional Spin System in an Inhomogeneous Magnetic Field

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 1066
Author(s):  
Gehad Sadiek ◽  
Samaher Almalki
Keyword(s):  
Magnetic Field ◽  
Steady State ◽  
Nearest Neighbor ◽  
Quantum Phase Transitions ◽  
Inhomogeneous Magnetic Field ◽  
Steady States ◽  
Inhomogeneous Field ◽  
Spin States ◽  
Two Dimensional ◽  
Spin Lattice

Recently new novel magnetic phases were shown to exist in the asymptotic steady states of spin systems coupled to dissipative environments at zero temperature. Tuning the different system parameters led to quantum phase transitions among those states. We study, here, a finite two-dimensional Heisenberg triangular spin lattice coupled to a dissipative Markovian Lindblad environment at finite temperature. We show how applying an inhomogeneous magnetic field to the system at different degrees of anisotropy may significantly affect the spin states, and the entanglement properties and distribution among the spins in the asymptotic steady state of the system. In particular, applying an inhomogeneous field with an inward (growing) gradient toward the central spin is found to considerably enhance the nearest neighbor entanglement and its robustness against the thermal dissipative decay effect in the completely anisotropic (Ising) system, whereas the beyond nearest neighbor ones vanish entirely. The spins of the system in this case reach different steady states depending on their positions in the lattice. However, the inhomogeneity of the field shows no effect on the entanglement in the completely isotropic (XXX) system, which vanishes asymptotically under any system configuration and the spins relax to a separable (disentangled) steady state with all the spins reaching a common spin state. Interestingly, applying the same field to a partially anisotropic (XYZ) system does not just enhance the nearest neighbor entanglements and their thermal robustness but all the long-range ones as well, while the spins relax asymptotically to very distinguished spin states, which is a sign of a critical behavior taking place at this combination of system anisotropy and field inhomogeneity.

Swirling flow states in finite-length diverging or contracting circular pipes

2017 ◽  
Vol 819 ◽  
pp. 678-712 ◽  
Author(s):  
Zvi Rusak ◽  
Yuxin Zhang ◽  
Harry Lee ◽  
Shixiao Wang
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Finite Length ◽  
Vortex Breakdown ◽  
Steady States ◽  
Inviscid Limit ◽  
Circular Pipes ◽  
Outlet Flow ◽  
Separation Zones ◽  
Steady State Solutions

The dynamics of inviscid-limit, incompressible and axisymmetric swirling flows in finite-length, diverging or contracting, long circular pipes is studied through global analysis techniques and numerical simulations. The inlet flow is described by the profiles of the circumferential and axial velocity together with a fixed azimuthal vorticity while the outlet flow is characterized by a state with zero radial velocity. A mathematical model that is based on the Squire–Long equation (SLE) is formulated to identify steady-state solutions of the problem with special conditions to describe states with separation zones. The problem is then reduced to the columnar (axially-independent) SLE, with centreline and wall conditions for the solution of the outlet flow streamfunction. The solution of the columnar SLE problem gives rise to the existence of four types of solutions. The SLE problem is then solved numerically using a special procedure to capture states with vortex-breakdown or wall-separation zones. Numerical simulations based on the unsteady vorticity circulation equations are also conducted and show correlation between time-asymptotic states and steady states according to the SLE and the columnar SLE problems. The simulations also shed light on the stability of the various steady states. The uniqueness of steady-state solutions in a certain range of swirl is proven analytically and demonstrated numerically. The computed results provide the bifurcation diagrams of steady states in terms of the incoming swirl ratio and size of pipe divergence or contraction. Critical swirls for the first appearance of the various types of states are identified. The results show that pipe divergence promotes the appearance of vortex-breakdown states at lower levels of the incoming swirl while pipe contraction delays the appearance of vortex breakdown to higher levels of swirl and promotes the formation of wall-separation states.

Steady-State Bifurcation for a Biological Depletion Model

2016 ◽  
Vol 26 (04) ◽  
pp. 1650066 ◽  
Author(s):  
Yan’e Wang ◽  
Jianhua Wu ◽  
Yunfeng Jia
Keyword(s):  
Steady State ◽  
Bifurcation Theory ◽  
Steady States ◽  
Two Dimensional ◽  
Implicit Function ◽  
Flux Boundary Condition ◽  
One Dimensional ◽  
Steady State Bifurcation ◽  
The Stability ◽  
Theoretical Results

A two-species biological depletion model in a bounded domain is investigated in which one species is a substrate and the other is an activator. Firstly, under the no-flux boundary condition, the asymptotic stability of constant steady-states is discussed. Secondly, by viewing the feed rate of the substrate as a parameter, the steady-state bifurcations from constant steady-states are analyzed both in one-dimensional kernel case and in two-dimensional kernel case. Finally, numerical simulations are presented to illustrate our theoretical results. The main tools adopted here include the stability theory, the bifurcation theory, the techniques of space decomposition and the implicit function theorem.

scholarly journals The Geometry of Power Systems Steady-State Equations– Part I: Power Surface

2020 ◽  
pp. 37-46
Author(s):  
B. Ayuev ◽  
V. Davydov ◽  
P. Erokhin ◽  
V. Neuymin ◽  
A. Pazderin
Keyword(s):  
Steady State ◽  
Power Systems ◽  
Power System ◽  
Power Flow ◽  
Analytical Study ◽  
State Theory ◽  
Steady States ◽  
State Equations ◽  
Feasibility Region ◽  
Steady State Solutions

Steady-state equations play a fundamental role in the theory of power systems and computation practice. These equations are directly or mediately used almost in all areas of the power system state theory, constituting its basis. This two-part study deals with a geometrical interpretation of steady-state solutions in a power space. Part I considers steady states of the power system as a surface in the power space. A power flow feasibility region is shown to be widely used in power system theories. This region is a projection of this surface along the axis of a slack bus active power onto a subspace of other buses power. The findings have revealed that the obtained power flow feasibility regions, as well as marginal states of the power system, depend on a slack bus location. Part II is devoted to an analytical study of the power surface of power system steady states.

scholarly journals Steady states of thin film droplets on chemically heterogeneous substrates

2020 ◽  
Vol 85 (6) ◽  
pp. 980-1020
Author(s):  
Weifan Liu ◽  
Thomas P Witelski
Keyword(s):  
Thin Films ◽  
Steady State ◽  
Finite Size ◽  
Steady States ◽  
One Dimension ◽  
Asymptotic Estimates ◽  
Perturbation Expansions ◽  
Flux Boundary Conditions ◽  
Axisymmetric Solutions ◽  
Steady State Solutions

Abstract We study steady-state thin films on chemically heterogeneous substrates of finite size, subject to no-flux boundary conditions. Based on the structure of the bifurcation diagram, we classify the 1D steady-state solutions that exist on such substrates into six different branches and develop asymptotic estimates for the steady states on each branch. Using perturbation expansions, we show that leading-order solutions provide good predictions of the steady-state thin films on stepwise-patterned substrates. We show how the analysis in one dimension can be extended to axisymmetric solutions. We also examine the influence of the wettability contrast of the substrate pattern on the linear stability of droplets and the time evolution for dewetting on small domains. Results are also applied to describe 2D droplets on hydrophilic square patches and striped regions used in microfluidic applications.

scholarly journals On the crossing of intermediate unstable steady state solutions for thermal ignition in a sphere

2001 ◽  
Vol 43 (1) ◽  
pp. 77-85 ◽  
Author(s):  
R. O. Weber ◽  
G. C. Wake ◽  
H. S. Sidhu ◽  
G. N. Mercer ◽  
B. F. Gray ◽  
...  
Keyword(s):  
Steady State ◽  
Unit Sphere ◽  
Steady States ◽  
Temperature Profiles ◽  
Thermal Ignition ◽  
Unstable Steady State ◽  
Steady State Temperature ◽  
Parameter Values ◽  
Steady State Solutions ◽  
Unstable Intermediate

AbstractSteady state solutions for spontaneous thermal ignition in a unit sphere are considered. The multiplicity of unstable, intermediate, steady state, temperature profiles is calculated and shown for selected parameter values. The crossing of the temperature profiles corresponding to the unstable, intermediate, steady states is exhibited in a particular case and is proven in general using elementary ideas from analysis. Estimates of the location of crossing points are given.

THE STABILITY OF SPATIALLY NONHOMOGENEOUS STEADY STATES IN A TUBULAR CHEMICAL REACTOR

1993 ◽  
Vol 03 (06) ◽  
pp. 1477-1486
Author(s):  
JAMES M. ROTENBERRY ◽  
ANTONMARIA A. MINZONI
Keyword(s):  
Steady State ◽  
Peclet Number ◽  
Chemical Reactor ◽  
Higher Order ◽  
Steady States ◽  
Complex Conjugate ◽  
Heat Of Reaction ◽  
Péclet Number ◽  
The Stability ◽  
Steady State Solutions

We study the axial heat and mass transfer in a highly diffusive tubular chemical reactor in which a simple reaction is occurring. The steady state solutions of the governing equations are studied using matched asymptotic expansions, the theory of dynamical systems, and by calculating the solutions numerically. In particular, the effect of varying the Peclet and Damköhler numbers (P and D) is investigated. A simple expression for the approximate location of the transition layer for large Peclet number is derived and its accuracy tested against the numerical solution. The stability of the steady states is examined by calculating the eigenvalues and eigenfunctions of the linearized equations. It is shown that a Hopf bifurcation of the CSTR model (i.e., the limit as the P approaches zero) can be continued up to order 1 in the Peclet number. Furthermore, it is shown numerically that for appropriate values of the Peclet number, the Damköhler number, and B (the heat of reaction) these Hopf bifurcations merge with the limit points of an "S–shaped" bifurcation curve in a higher order singularity controlled by the Bogdanov–Takens normal form. Consequently, there must exist a finite amplitude, nonuniform, stable periodic solution for parameter values near this singularity. The existence of higher order degeneracies is also explored. In particular, it is shown for D ≪ 1 that no value of P exists where two pairs of complex conjugate eigenvalues of the steady state solutions can cross the imaginary axis simultaneously.

Detonation in supersonic radial outflow

2014 ◽  
Vol 760 ◽  
pp. 313-341 ◽  
Author(s):  
Aslan R. Kasimov ◽  
Svyatoslav V. Korneev
Keyword(s):  
Steady State ◽  
Supersonic Flow ◽  
Radial Flow ◽  
Gaseous Detonation ◽  
Two Dimensional ◽  
Structure And Dynamics ◽  
Cellular Detonations ◽  
The Stability ◽  
Radial Outflow ◽  
Steady State Solutions

AbstractWe report on the structure and dynamics of gaseous detonation stabilized in a supersonic flow emanating radially from a central source. The steady-state solutions are computed and their range of existence is investigated. Two-dimensional simulations are carried out in order to explore the stability of the steady-state solutions. It is found that both collapsing and expanding two-dimensional cellular detonations exist. The latter can be stabilized by putting several rigid obstacles in the flow downstream of the steady-state sonic locus. The problem of initiation of standing detonation stabilized in the radial flow is also investigated numerically.

scholarly journals Stability and instability results for the 2D [alpha]-Euler equations

2017 ◽  
Author(s):  
◽  
Shibi Kapisthalam Vasudevan
Keyword(s):  
Steady State ◽  
Euler Equations ◽  
Nonlinear Term ◽  
Zero Mode ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Fourier Decomposition ◽  
Inviscid Incompressible Fluid ◽  
Stability And Instability ◽  
Steady State Solutions

We study stability and instability of time independent solutions of the two dimensional a-Euler equations and Euler equations; the a-Euler equations are obtained by replacing the nonlinear term (u [times] [del.])u in the classical Euler equations of inviscid incompressible fluid by the term (v [times] [del.])u, where v is the regularized velocity satisfying (1 -[alpha]2[delta])v [equals] u, and [alpha] [greater than] 0. In the first part of the thesis, for the a-model, we develop analogues of the classical Arnol'd type stability criteria based on the energy-Casimir method for several settings including multi connected domains, periodic channels, and others. In the second part of the thesis, we study stability of a particular steady state, the unidirectional solution of the a-Euler equation on the two dimensional torus, having only one non zero mode in its Fourier decomposition. Using continued fractions, we give a proof of instability of the steady state under fairly general conditions. In the third part of the thesis we study various properties of a family of elliptic operators introduced by Zhiwu Lin in his work on instability of steady state solutions of the two dimensional Euler equations. This involves Birman-Schwinger type operators associated with the linearization of the Euler equations about the steady state and certain perturbation determinants.

Sign in / Sign up

Export Citation Format

Share Document