Nonlinear Stability, Thermoelastic Contact, and the Barber Condition

2000 ◽  
Vol 68 (1) ◽  
pp. 28-33 ◽  
Author(s):  
J. A. Pelesko
Keyword(s):  
Steady State ◽  
Steady State Solution ◽  
Rigid Wall ◽  
Stable Steady State ◽  
Dynamic Information ◽  
Long Time ◽  
Uniform Expansions ◽  
Short Time ◽  
Matching Techniques ◽  
Steady State Solutions

The behavior of a one-dimensional thermoelastic rod is modeled and analyzed. The rod is held fixed and at constant temperature at one end, while at the other end it is free to separate from or make contact with a rigid wall. At this free end we impose a pressure and gap-dependent thermal boundary condition. This condition, known as the Barber condition, couples the thermal and elastic problems. Such systems have previously been shown to undergo a bifurcation from a unique linearly stable steady-state solution to multiple steady-state solutions with alternating stability. Here, the system is studied using the asymptotic matching techniques of boundary layer theory to derive short-time, long-time, and uniform expansions. In this manner, the analysis is extended into the nonlinear regime and dynamic information about the history dependence and temporal evolution of the solution is obtained.

Nonlinear Stability Considerations in Thermoelastic Contact

1999 ◽  
Vol 66 (1) ◽  
pp. 109-116 ◽  
Author(s):  
J. A. Pelesko
Keyword(s):  
Steady State ◽  
Perturbation Technique ◽  
Multiple Scale ◽  
Steady State Solution ◽  
Rigid Wall ◽  
Stable Steady State ◽  
Dynamic Information ◽  
Multiple Steady State ◽  
Singular Perturbation Technique ◽  
Steady State Solutions

The behavior of a one-dimensional thermoelastic rod is modeled and analyzed. The rod is held fixed and at constant temperature at one end, while at the other end it is free to separate from or make contact with a rigid wall. At this free end a pressure and gap-dependent thermal boundary condition is imposed which couples the thermal and elastic problems. Such systems have previously been shown to undergo a bifurcation from a unique linearly stable steady-state solution to multiple steady-state solutions with alternating stability. Here, the system is studied using a two-timing or multiple-scale singular perturbation technique. In this manner, the analysis is extended into the nonlinear regime and dynamic information about the history dependence and temporal evolution of the solution is obtained.

scholarly journals STABILITY ANALYSIS OF A LOTKA–VOLTERRA TYPE PREDATOR–PREY SYSTEM INVOLVING ALLEE EFFECTS

2010 ◽  
Vol 52 (2) ◽  
pp. 139-145 ◽  
Author(s):  
HÜSEYİN MERDAN
Keyword(s):  
Steady State ◽  
Stability Analysis ◽  
Steady State Solution ◽  
Prey Population ◽  
Stable Steady State ◽  
Allee Effects ◽  
Predator Prey ◽  
Population Dynamics Model ◽  
Steady State Solutions ◽  
Model Subject

AbstractWe present a stability analysis of steady-state solutions of a continuous-time predator–prey population dynamics model subject to Allee effects on the prey population which occur at low population density. Numerical simulations show that the system subject to an Allee effect takes a much longer time to reach its stable steady-state solution. This result differs from that obtained for the discrete-time version of the same model.

An airfoil theory of bifurcating laminar separation from thin obstacles

1990 ◽  
Vol 216 ◽  
pp. 255-284 ◽  
Author(s):  
C. J. Lee ◽  
H. K. Cheng
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Steady State ◽  
Steady State Solution ◽  
Equation System ◽  
Branch Points ◽  
Laminar Separation ◽  
Kutta Condition ◽  
Numerical Studies ◽  
Steady State Solutions

Global interaction of the boundary layer separating from an obstacle with resulting open/closed wakes is studied for a thin airfoil in a steady flow. Replacing the Kutta condition of the classical theory is the breakaway criterion of the laminar triple-deck interaction (Sychev 1972; Smith 1977), which, together with the assumption of a uniform wake/eddy pressure, leads to a nonlinear equation system for the breakaway location and wake shape. The solutions depend on a Reynolds numberReand an airfoil thickness ratio or incidence τ and, in the domain$Re^{\frac{1}{16}}\tau = O(1)$considered, the separation locations are found to be far removed from the classical Brillouin–Villat point for the breakaway from a smooth shape. Bifurcations of the steady-state solution are found among examples of symmetrical and asymmetrical flows, allowing open and closed wakes, as well as symmetry breaking in an otherwise symmetrical flow. Accordingly, the influence of thickness and incidence, as well as Reynolds number is critical in the vicinity of branch points and cut-off points where steady-state solutions can/must change branches/types. The study suggests a correspondence of this bifurcation feature with the lift hysteresis and other aerodynamic anomalies observed from wind-tunnel and numerical studies in subcritical and high-subcriticalReflows.

scholarly journals Stability of the positive steady-state solutions of systems of nonlinear Volterra difference equations of population models with diffusion and infinite delay

2000 ◽  
Vol 23 (4) ◽  
pp. 261-270 ◽  
Author(s):  
B. Shi
Keyword(s):  
Steady State ◽  
Difference Equations ◽  
Infinite Delay ◽  
Steady State Solution ◽  
Population Models ◽  
Monotone Iterative ◽  
Positive Steady State ◽  
Infinite Delays ◽  
Steady State Solutions ◽  
Volterra Difference Equations

An open problem given by Kocic and Ladas in 1993 is generalized and considered. A sufficient condition is obtained for each solution to tend to the positive steady-state solution of the systems of nonlinear Volterra difference equations of population models with diffusion and infinite delays by using the method of lower and upper solutions and monotone iterative techniques.

scholarly journals ELIMINATION OF CHAOS IN MULTIMODE, INTRACAVITY-DOUBLED LASERS IN THE PRESENCE OF SPATIAL HOLE-BURNING

2001 ◽  
Vol 11 (10) ◽  
pp. 2637-2645 ◽  
Author(s):  
MONIKA E. PIETRZYK ◽  
MILTCHO B. DANAILOV
Keyword(s):  
Steady State ◽  
Steady State Solution ◽  
Hole Burning ◽  
Stable Steady State ◽  
Small Perturbations ◽  
Spatial Hole Burning ◽  
Saturation Coefficient ◽  
Amplitude Fluctuations ◽  
The Cross ◽  
State Configuration

In this paper possibilities of a stabilization of large amplitude fluctuations in an intracavity-doubled solid-state laser are studied. The modification of the cross-saturation coefficient by the effect of spatial hole-burning is taken into account. The stabilization of the laser radiation by an increase of the number of modes, as proposed in [James et al., 1990b; Magni et al., 1993], is analyzed. It is found that when the cross-saturation coefficient is modulated by the spatial hole-burning the stabilization is not always possible. We propose a new way of obtaining a stable steady-state configuration based on an increase of the strength of nonlinearity, which leads to a strong cancellation of modes, so that during the evolution all modes, but for a single one, are canceled. Such a steady-state solution is found to be stable with respect to small perturbations.

Gravity currents over fractured substrates in a porous medium

2007 ◽  
Vol 584 ◽  
pp. 415-431 ◽  
Author(s):  
DAVID PRITCHARD
Keyword(s):  
Porous Medium ◽  
Steady State ◽  
Gravity Current ◽  
Gravity Currents ◽  
Limited Extent ◽  
Form Part ◽  
Long Time ◽  
Self Similar ◽  
Steady State Solutions ◽  
A Current

We consider the behaviour of a gravity current in a porous medium when the horizontal surface along which it spreads is punctuated either by narrow fractures or by permeable regions of limited extent. We derive steady-state solutions for the current, and show that these form part of a long-time asymptotic description which may also include a self-similar ‘leakage current’ propagating beyond the fractured region with a length proportional to t1/2. We discuss the conditions under which a current can be completely trapped by a permeable region or a series of fractures.

Flows at Moderate Reynolds Numbers

2018 ◽  
Author(s):  
Sauro Succi
Keyword(s):  
Steady State ◽  
Reynolds Numbers ◽  
Complex Geometries ◽  
Computational Power ◽  
Flow Configuration ◽  
Major Interest ◽  
Long Time ◽  
Pulsatile Flows ◽  
Short Time ◽  
Very High

This chapter presents the application of LBE to flows at moderate Reynolds numbers, typically hundreds to thousands. This is an important area of theoretical and applied fluid mechanics, one that relates, for instance, to the onset of nonlinear instabilities and their effects on the transport properties of the unsteady flow configuration. The regime of Reynolds numbers at which these instabilities take place is usually not very high, of the order of thousands, hence basically within reach of present day computer capabilities. Nonetheless, following the full evolution of these transitional flows requires very long-time integrations with short time-steps, which command substantial computational power. Therefore, efficient numerical methods are in great demand. Also of major interest are steady-state or pulsatile flows at moderate Reynolds numbers in complex geometries, such as they occur, for instance, in hemodynamic applications. The application of LBE to such flows will also briefly be mentioned

scholarly journals Numerical prediction of steady state temperature based on transient measurements

2018 ◽  
Vol 240 ◽  
pp. 05024
Author(s):  
Ewa Pelińska-Olko ◽  
Marek Lewkowicz
Keyword(s):  
Experimental Data ◽  
Steady State ◽  
Heat Exchange ◽  
Time Range ◽  
Radiative Heat Exchange ◽  
Steady State Temperature ◽  
Long Time ◽  
Experimental Values ◽  
Short Time ◽  
Transient Measurements

We show how to use numerical analysis of short-time range experimental data for predicting the limit steady-state value of the investigated parameter. In this article the approach has been applied to a specific, although typical, thermal problem: determining the average steady-state temperature of a heater in the convective and radiative heat exchange with the environment. First, we describe a heat exchange experiment aimed at obtaining temperature experimental data in both short and long time range. Then we present a methodology for applying two methods, i.e., neural networks and least squares approximations, for obtaining predictions about the steady-state temperature values based on short time experimental data. The aim of the study is to compare the predictions to each other and to the long time experimental values, with the aim of determining the applicability range of the two methods.

Steady State Solutions to a Forced Nonlinear Oscillator: A Comparison of Results Using a Generalized Harmonic Balance Method and by Numerical Integrations

1993 ◽  
Author(s):  
Ben Noble ◽  
Julian J. Wu
Keyword(s):  
Steady State ◽  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Initial Point ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Original Equation ◽  
Long Time ◽  
Steady State Solutions ◽  
Generalized Harmonic Balance Method

Abstract Steady state solutions for nonlinear dynamic problems are interesting because (1) the long time behaviors of many problems are of practical concern, and, (2) these behaviors are often difficult to predict. This paper first presents a brief description of a generalized harmonic balance method (GHB) for steady state solutions to nonlinear problems via a nonlinear oscillator problem with a quadratic nonlinearity. Using this approach, steady state solutions are obtained for problems with several parameters: damping, nonlinearity and frequency (subharmonic, superharmonic and primary resonance). These results, plotted in time evolution curves and phase diagrams are compared with those obtained by numerically integrating the original differential equations. The effect of initial conditions on long time solutions is discussed. This investigation indicates that (1) the GHB steady state is an excellent approximate solution to that of the original equation if such a solution is numerically stable, and (2) the GHB steady state simply indicates a region of instability when the numerical solution to the original equation, using a point in that region as the initial point, is unstable.

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