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.

Nonlinear Peierls-Boltzmann Equation for Phonons: Structure and Stability of Solutions

1977 ◽  
Vol 32 (8) ◽  
pp. 805-812
Author(s):  
Fr. Kaiser
Keyword(s):  
Steady State ◽  
Steady State Solution ◽  
Phonon Transport ◽  
Model System ◽  
Steady States ◽  
Different Types ◽  
The Stability ◽  
Steady State Solutions ◽  
Different Order ◽  
Complicated Situation

Abstract Phonon transport in locally disturbed media is considered. The steady state solutions of the Peierls-Boltzmann type equations are studied. In particular, the flux-dependence of local excitations is investigated. It is proven that for a large class of scattering processes only two types of steady states are possible: a hysteresis type and a threshold one. 4 different types of factorization procedures are applied and it is shown that for these cases the steady states remain nearly unchanged. The stability conditions are reformulated in such a way that one can give a geometrical interpretation. The only stable solutions are nodes. The necessary modification of our model system to allow for limit cycles is indicated. Also, a more complicated situation, where the interaction Hamiltonian HI is a superposition of terms of different order, is investigated. The resulting steady state solution is again a hysteresis.

Approximate analysis of the containment of contaminated sites prior to remediation

2000 ◽  
Vol 42 (1-2) ◽  
pp. 319-324 ◽  
Author(s):  
H. Rubin ◽  
A. Rabideau
Keyword(s):  
Steady State ◽  
Peclet Number ◽  
Dimensionless Parameter ◽  
Contaminated Sites ◽  
Unsteady State ◽  
Péclet Number ◽  
Vertical Barrier ◽  
Time Period ◽  
Barrier Performance ◽  
Vertical Barriers

This study presents an approximate analytical model, which can be useful for the prediction and requirement of vertical barrier efficiencies. A previous study by the authors has indicated that a single dimensionless parameter determines the performance of a vertical barrier. This parameter is termed the barrier Peclet number. The evaluation of barrier performance concerns operation under steady state conditions, as well as estimates of unsteady state conditions and calculation of the time period requires arriving at steady state conditions. This study refers to high values of the barrier Peclet number. The modeling approach refers to the development of several types of boundary layers. Comparisons were made between simulation results of the present study and some analytical and numerical results. These comparisons indicate that the models developed in this study could be useful in the design and prediction of the performance of vertical barriers operating under conditions of high values of the barrier Peclet number.

Stability Considerations in Thermoelastic Contact

1980 ◽  
Vol 47 (4) ◽  
pp. 871-874 ◽  
Author(s):  
J. R. Barber ◽  
J. Dundurs ◽  
M. Comninou
Keyword(s):  
Steady State ◽  
Perturbation Method ◽  
Energy Function ◽  
First Principles ◽  
Dimensional Model ◽  
Contact Conditions ◽  
One Dimensional ◽  
Thermoelastic Contact ◽  
The Stability ◽  
Steady State Solutions

A simple one-dimensional model is described in which thermoelastic contact conditions give rise to nonuniqueness of solution. The stability of the various steady-state solutions discovered is investigated using a perturbation method. The results can be expressed in terms of the minimization of a certain energy function, but the authors have so far been unable to justify the use of such a function from first principles in view of the nonconservative nature of the system.

On general transformations and variational principles for the magnetohydrodynamics of ideal fluids. Part 4. Generalized isovorticity principle for three-dimensional flows

1999 ◽  
Vol 390 ◽  
pp. 127-150 ◽  
Author(s):  
V. A. VLADIMIROV ◽  
H. K. MOFFATT ◽  
K. I. ILIN
Keyword(s):  
Steady State ◽  
Variational Principles ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Steady States ◽  
Mhd Equations ◽  
Quadratic Functional ◽  
The First Variation ◽  
The Stability ◽  
And Variational Principles

The equations of magnetohydrodynamics (MHD) of an ideal fluid have two families of topological invariants: the magnetic helicity invariants and the cross-helicity invariants. It is first shown that these invariants define a natural foliation (described as isomagnetovortical, or imv for short) in the function space in which solutions {u(x, t), h(x, t)} of the MHD equations reside. A relaxation process is constructed whereby total energy (magnetic plus kinetic) decreases on an imv folium (all magnetic and cross-helicity invariants being thus conserved). The energy has a positive lower bound determined by the global cross-helicity, and it is thus shown that a steady state exists having the (arbitrarily) prescribed families of magnetic and cross-helicity invariants.The stability of such steady states is considered by an appropriate generalization of (Arnold) energy techniques. The first variation of energy on the imv folium is shown to vanish, and the second variation δ2E is constructed. It is shown that δ2E is a quadratic functional of the first-order variations δ1u, δ1h of u and h (from a steady state U(x), H(x)), and that δ2E is an invariant of the linearized MHD equations. Linear stability is then assured provided δ2E is either positive-definite or negative-definite for all imv perturbations. It is shown that the results may be equivalently obtained through consideration of the frozen-in ‘modified’ vorticity field introduced in Part 1 of this series.Finally, the general stability criterion is applied to a variety of classes of steady states {U(x), H(x)}, and new sufficient conditions for stability to three-dimensional imv perturbations are obtained.

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.

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.

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.

Dynamics and pattern formation in a modified Leslie–Gower model with Allee effect and Bazykin functional response

2017 ◽  
Vol 10 (05) ◽  
pp. 1750073 ◽  
Author(s):  
Peng Feng
Keyword(s):  
Steady State ◽  
Pattern Formation ◽  
Spatial Pattern ◽  
Numerical Simulations ◽  
Functional Response ◽  
Allee Effect ◽  
Turing Instability ◽  
Spatial Pattern Formation ◽  
The Stability ◽  
Steady State Solutions

In this paper, we study the dynamics of a diffusive modified Leslie–Gower model with the multiplicative Allee effect and Bazykin functional response. We give detailed study on the stability of equilibria. Non-existence of non-constant positive steady state solutions are shown to identify the rage of parameters of spatial pattern formation. We also give the conditions of Turing instability and perform a series of numerical simulations and find that the model exhibits complex patterns.

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