THE STABILITY OF SOLUTE CRYSTALLIZATION EQUILIBRIUM

2005 ◽  
Vol 12 (04) ◽  
pp. 623-630
Author(s):  
SHIMIN ZHANG
Keyword(s):  
Stable Equilibrium ◽  
Unstable Equilibrium ◽  
Energy Peak ◽  
The Stability ◽  
Solute Crystallization

Solute crystallization equilibrium contains a stable equilibrium and an unstable equilibrium. The system must get across an energy peak of unstable equilibrium first, and then get into an energy valley of stable equilibrium during the solute crystallization. When solution is diluted, the energy peak becomes high, and the energy valley becomes shallow, the solute crystallization becomes difficult and the dissolution of the crystallized product becomes easy; when solution is diluted to a certain extent, the energy peak and the energy valley combine into one, and crystallization of the solute becomes impossible.

THE STABILITY OF VAPOR CONDENSATION EQUILIBRIUM

2005 ◽  
Vol 12 (03) ◽  
pp. 359-368
Author(s):  
SHIMIN ZHANG
Keyword(s):  
Partial Pressure ◽  
Stable Equilibrium ◽  
Vapor Condensation ◽  
Unstable Equilibrium ◽  
Energy Peak ◽  
Liquid Evaporation ◽  
The Stability

The system must get across an energy peak of unstable equilibrium during the condensation of pure vapor; as the supersaturated extent of vapor increases and the temperature decreases, the energy peak shortens and vapor condensation becomes easier. The system must get across an energy peak of unstable equilibrium first, and then get into an energy valley of stable equilibrium during the condensation of impure vapor; as the partial pressure of vapor decreases, the energy peak becomes taller, the energy valley more shallow, vapor condensation becomes more difficult and liquid evaporation becomes easier; when the partial pressure of vapor decreases to a certain extent, the energy peak and the energy valley combine into one, and vapor condensation becomes impossible.

THE STABILITY OF LIQUID EVAPORATION EQUILIBRIUM

2005 ◽  
Vol 12 (01) ◽  
pp. 115-121
Author(s):  
SHIMIN ZHANG
Keyword(s):  
Phase Equilibrium ◽  
Vapor Pressure ◽  
Stable Equilibrium ◽  
External Pressure ◽  
Unstable Equilibrium ◽  
Pure Liquid ◽  
Mechanical Equilibrium ◽  
Liquid Evaporation ◽  
The Stability ◽  
Constant External Pressure

For the evaporation of the pure liquid under the condition of constant temperature and constant external pressure, the phase equilibrium of the liquid vapor in the bubble and the liquid outside the bubble is always a kind of stable equilibrium whether there is air or not in the bubble. If there is no air in the bubble, the bubble and liquid cannot coexist in the mechanical equilibrium when the vapor pressure of the liquid in the bubble is less than or equal to the external pressure; the bubble and liquid can coexist in an unstable equilibrium of mechanics when the vapor pressure of the liquid is greater than the external pressure. If there is air in the bubble, the bubble and liquid can coexist in a stable equilibrium of mechanics when the vapor pressure of the liquid is less than or equal to the external pressure; the bubble and liquid can coexist in a stable and an unstable equilibrium of mechanics when the vapor pressure of the liquid is greater than the external pressure and less than a certain pressure pm; the bubble and liquid cannot coexist in the mechanical equilibrium when the vapor pressure of the liquid is equal to or greater than pm.

The Qualitative Analysis of Two Populations Commensalisms Model

2012 ◽  
Vol 524-527 ◽  
pp. 3705-3708
Author(s):  
Guang Cai Sun
Keyword(s):  
Qualitative Analysis ◽  
Equilibrium Point ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Mathematics Model ◽  
Stable Equilibrium Point ◽  
The Stability ◽  
Two Populations

This paper deals with the mathematics model of two populations Commensalisms symbiosis and the stability of all equilibrium points the system. It has given the conclusion that there is only one stable equilibrium point the system. This paper also elucidates the biology meaning of the model and its equilibrium points.

An Edge Dislocation in a Three-Phase Composite Cylinder Model

1991 ◽  
Vol 58 (1) ◽  
pp. 75-86 ◽  
Author(s):  
H. A. Luo ◽  
Y. Chen
Keyword(s):  
Edge Dislocation ◽  
Stable Equilibrium ◽  
Phase Model ◽  
Composite Cylinder ◽  
Two Phase ◽  
Trapping Mechanism ◽  
Three Phase ◽  
Cylinder Model ◽  
Stable Equilibrium Position ◽  
The Stability

An exact solution is given for the stress field due to an edge dislocation embedded in a three-phase composite cylinder. The force on the dislocation is then derived, from which a set of simple approximate formulae is also suggested. It is shown that, in comparison with the two-phase model adopted by Dundurs and Mura (1964), the three-phase model allows the dislocation to have a stable equilibrium position under much less stringent combinations of the material constants. As a result, the so-called trapping mechanism of dislocations is more likely to take place in the three-phase model. Also, the analysis and calculation show that in the three-phase model the orientation of Burgers vector has only limited influence on the stability of dislocation. This behavior is pronouncedly different from that predicted by the two-phase model.

scholarly journals A Four-Zone Model and Nonlinear Dynamic Analysis of Solution Multiplicity of Buoyancy Ventilation in Underground Building

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-24
Author(s):  
Yuxing Wang ◽  
Chunyu Wei
Keyword(s):  
Differential Equations ◽  
Equilibrium Point ◽  
Equilibrium Points ◽  
Underground Structure ◽  
Unstable Equilibrium ◽  
Phase Portraits ◽  
Zone Model ◽  
Unstable Equilibrium Point ◽  
Index 2 ◽  
The Stability

The solution multiplicity of natural ventilation in buildings is very important to personnel safety and ventilation design. In this paper, a four-zone model of buoyancy ventilation in typical underground building is proposed. The underground structure is divided to four zones, a differential equation is established in each zone, and therefore, there are four differential equations in the underground structure. By solving and analyzing the equilibrium points and characteristic roots of the differential equations, we analyze the stability of three scenarios and obtain the criterions to determine the stability and existence of solutions for two scenarios. According to these criterions, the multiple steady states of buoyancy ventilation in any four-zone underground buildings for different stack height ratios and the strength ratios of the heat sources can be obtained. These criteria can be used to design buoyancy ventilation or natural exhaust ventilation systems in underground buildings. Compared with the two-zone model in (Liu et al. 2020), the results of the proposed four-zone model are more consistent with CFD results in (Liu et al. 2018). In addition, the results of proposed four-zone model are more specific and more detailed in the unstable equilibrium point interval. We find that the unstable equilibrium point interval is divided into two different subintervals corresponding to the saddle point of index 2 and the saddle focal equilibrium point of index 2, respectively. Finally, the phase portraits and vector field diagrams for the two scenarios are given.

scholarly journals Computational Stability Analysis of Lotka-Volterra Systems

2016 ◽  
Vol 44 (2) ◽  
pp. 113-120
Author(s):  
Péter Polcz ◽  
Keyword(s):  
Stability Analysis ◽  
Lyapunov Functions ◽  
Stability Region ◽  
Stable Equilibrium ◽  
Linear Matrix ◽  
Stable Equilibrium Point ◽  
Computational Stability ◽  
Volterra Systems ◽  
Selection For ◽  
The Stability

Abstract This paper concerns the computational stability analysis of locally stable Lotka-Volterra (LV) systems by searching for appropriate Lyapunov functions in a general quadratic form composed of higher order monomial terms. The Lyapunov conditions are ensured through the solution of linear matrix inequalities. The stability region is estimated by determining the level set of the Lyapunov function within a suitable convex domain. The paper includes interesting computational results and discussion on the stability regions of higher (3,4) dimensional LV models as well as on the monomial selection for constructing the Lyapunov functions. Finally, the stability region is estimated of an uncertain 2D LV system with an uncertain interior locally stable equilibrium point.

On the Equilibrium of Cavitation Nuclei in Liquid-Gas Solutions

1981 ◽  
Vol 103 (3) ◽  
pp. 425-430 ◽  
Author(s):  
Y. S. Cha
Keyword(s):  
Stable Equilibrium ◽  
Thermodynamic Theory ◽  
Phase System ◽  
Saturated Liquid ◽  
Theoretical Justification ◽  
Two Phase ◽  
System Pressure ◽  
Two Component ◽  
Cavitation Nuclei ◽  
The Stability

The stability of a spherical bubble in a two-component two-phase system is examined by employing the thermodynamic theory of dilute solutions. It is shown that a bubble can remain in a state of stable equilibrium provided that the ratio of the total number of moles of the solute to the total number of moles of the solvent in the system is not extremely small and that the system pressure falls between an upper bound (dissolution limit) and a lower bound (cavitation limit). The results of the analysis provide a theoretical basis for the persistence of microbubbles in a saturated liquid-gas solution. Thus to a certain extent, the results also help to resolve the dilemma that exists in the field of cavitation due to (1) the necessity of postulating the existence of microbubbles; and (2) the lack of theoretical justification for the persistence of such bubbles in a liquid.

Fragility of Chaos in Multispecies Competition Influenced by Predation

2010 ◽  
Author(s):  
Lei Zhao ◽  
Huayong Zhang ◽  
Tousheng Huang ◽  
Xinqiang Zhu ◽  
Lu Han
Keyword(s):  
Phase Diagram ◽  
Numerical Simulations ◽  
Strange Attractor ◽  
Stable Equilibrium ◽  
Dynamical Model ◽  
Nonlinear Dynamical ◽  
Nonlinear Dynamical Model ◽  
The Stability ◽  
Multispecies Competition

In order to study the stability of chaotic behaviors, a nonlinear dynamical model of the competing multispecies with a predator is investigated. A series of numerical simulations is demonstrated via wave diagram and phase diagram. The results show that the chaos can change into either oscillation or ordinary equilibrium as the attacking rate of the predator increases. Moreover, chaos in the system becomes fragile and even vanishes when the attacking rate reaches 0.0186. This study also exhibits the transformation in phase diagram from a strange attractor to a stable equilibrium.

THE ROLE OF ADDITIONAL FOOD IN A PREDATOR–PREY MODEL WITH A PREY REFUGE

2016 ◽  
Vol 24 (02n03) ◽  
pp. 345-365 ◽  
Author(s):  
SUDIP SAMANTA ◽  
RIKHIYA DHAR ◽  
IBRAHIM M. ELMOJTABA ◽  
JOYDEV CHATTOPADHYAY
Keyword(s):  
Stable Equilibrium ◽  
Predator Population ◽  
Stable Limit ◽  
Prey Refuge ◽  
Additional Food ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Stability ◽  
The Impact

In this paper, we propose and analyze a predator–prey model with a prey refuge and additional food for predators. We study the impact of a prey refuge on the stability dynamics, when a constant proportion or a constant number of prey moves to the refuge area. The system dynamics are studied using both analytical and numerical techniques. We observe that the prey refuge can replace the predator–prey oscillations by a stable equilibrium if the refuge size crosses a threshold value. It is also observed that, if the refuge size is very high, then the extinction of the predator population is certain. Further, we observe that enhancement of additional food for predators prevents the extinction of the predator and also replaces the stable limit cycle with a stable equilibrium. Our results suggest that additional food for the predators enhances the stability and persistence of the system. Extensive numerical experiments are performed to illustrate our analytical findings.

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