Existence of regular solutions

2000 ◽  
pp. 9-47
Author(s):  
Vojislav Marić
Keyword(s):  
Regular Solutions

Thin structure of heterogeneous and multiphase fluid flows

2012 ◽  
Vol 9 (1) ◽  
pp. 175-180
Author(s):  
Yu.D. Chashechkin
Keyword(s):  
Large Scale ◽  
Fundamental System ◽  
Fluid Flows ◽  
Regular Solutions ◽  
Flow Models ◽  
Group Methods ◽  
Wide Range ◽  
Thin Structure ◽  
Multiphase Fluid ◽  
Complete Solutions

According to the results of visualization of streams, the existence of structures in a wide range of scales is noted: from galactic to micron. The use of a fundamental system of equations is substantiated based on the results of comparing symmetries of various flow models with the usage of theoretical group methods. Complete solutions of the system are found by the methods of the singular perturbations theory with a condition of compatibility, which determines the characteristic equation. A comparison of complete solutions with experimental data shows that regular solutions characterize large-scale components of the flow, a rich family of singular solutions describes formation of the thin media structure. Examples of calculations and observations of stratified, rotating and multiphase media are given. The requirements for the technique of an adequate experiment are discussed.

A discussion on equations of phase equilibrium between two regular binary phases

1981 ◽  
Vol 46 (6) ◽  
pp. 1433-1438
Author(s):  
Jan Vřešťál
Keyword(s):  
Phase Equilibrium ◽  
Linear Function ◽  
Concentration Dependence ◽  
Absolute Temperature ◽  
Enthalpy Change ◽  
Independent Parameter ◽  
Free Enthalpy ◽  
Regular Solutions ◽  
Regular Model ◽  
Concentration Dependences

The conditions of the existence of extreme on the concentration dependences of absolute temperature (x are mole fractions) T = Tα(xkα) and T = Tβ(xkβ) denoting equilibrium between two binary regular solutions are generally developed under two assumptions: 1) Free enthalpy change of pure components k = i, j at transition from phase α to β is a linear function of temperature. 2) Concentration dependence of excess free enthalpy (identical with enthalpy) of solutions α and β, respectively, is described in regular model by one concentration and temperature independent parameter for each individual phase.

Further study of a fourth-order elliptic equation with negative exponent

2011 ◽  
Vol 141 (3) ◽  
pp. 537-549 ◽  
Author(s):  
Zongming Guo ◽  
Zhongyuan Liu
Keyword(s):  
Fourth Order ◽  
Blow Up ◽  
Smooth Domain ◽  
Regular Solutions ◽  
Order Problem ◽  
Bounded Smooth Domain ◽  
Main Technique ◽  
Fourth Order Elliptic Equation ◽  
Fourth Order Problem ◽  
Bounded Smooth

We continue to study the nonlinear fourth-order problem TΔu – DΔ2u = λ/(L + u)2, –L < u < 0 in Ω, u = 0, Δu = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded smooth domain and λ > 0 is a parameter. When N = 2 and Ω is a convex domain, we know that there is λc > 0 such that for λ ∊ (0, λc) the problem possesses at least two regular solutions. We will see that the convexity assumption on Ω can be removed, i.e. the main results are still true for a general bounded smooth domain Ω. The main technique in the proofs of this paper is the blow-up argument, and the main difficulty is the analysis of touch-down behaviour.

COMPLEXES OF STANNIC IODIDE WITH AROMATIC HYDROCARBONS

1965 ◽  
Vol 43 (5) ◽  
pp. 1272-1278 ◽  
Author(s):  
J. F. Murphy ◽  
D. E. Baker
Keyword(s):  
Carbon Tetrachloride ◽  
Complex Formation ◽  
Aromatic Hydrocarbon ◽  
Aromatic Hydrocarbons ◽  
Extinction Coefficient ◽  
Regular Solutions ◽  
Benzene Toluene ◽  
Yield Values ◽  
Complex Stabilities ◽  
Iodide Complex

Spectrophotometric measurements on solutions of stannic iodide were found to provide evidence for complex formation with aromatic hydrocarbons. Calculations, based on spectra for mixed solutions of benzene and stannic iodide in carbon tetrachloride, yield values of 0.26 for the equilibrium constant (mole fraction), 28 400 1/mole cm for the molar extinction coefficient of the benzene – stannic iodide complex. Kinetic evidence indicates that the order of decreasing complex stabilities is from xylene to toluene to benzene. The formation of stannic iodide – aromatic hydrocarbon complexes provides an explanation for the discrepancy between measured solubilities of stannic iodide in benzene, toluene, and xylene, and the solubilities predicted by the Hildebrand theory of regular solutions.

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