Incompressible Limit of Solutions of Multidimensional Steady Compressible Euler Equations

A compactness framework is formulated for the incompressible limit of approximate solutions with weak uniform bounds with respect to the adiabatic exponent for the steady Euler equations for compressible fluids in any dimension. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass conservation and the vorticity. Another observation is that the incompressibility of the limit for the homentropic Euler flow is directly from the continuity equation, while the incompresibility of the limit for the full Euler flow is from a combination of all the Euler equations. As direct applications of the compactness framework, we establish two incompressible limit theorems for multidimensional steady Euler flows through infinitely long nozzles, which lead to two new existence theorems for the corresponding problems for multidimensional steady incompressible Euler equations.Comment: 17 pages; 2 figures. arXiv admin note: text overlap with arXiv:1311.398

Steady Euler Flows with Large Vorticity and Characteristic Discontinuities in Arbitrary Infinitely Long Nozzles

We establish the existence and uniqueness of smooth solutions with large vorticity and weak solutions with vortex sheets/entropy waves for the steady Euler equations for both compressible and incompressible fluids in arbitrary infinitely long nozzles. We first develop a new approach to establish the existence of smooth solutions without assumptions on the sign of the second derivatives of the horizontal velocity, or the Bernoulli and entropy functions, at the inlet for the smooth case. Then the existence for the smooth case can be applied to construct approximate solutions to establish the existence of weak solutions with vortex sheets/entropy waves by nonlinear arguments. This is the first result on the global existence of solutions of the multidimensional steady compressible full Euler equations with free boundaries, which are not necessarily small perturbations of piecewise constant background solutions. The subsonic-sonic limit of the solutions is also shown. Finally, through the incompressible limit, we establish the existence and uniqueness of incompressible Euler flows in arbitrary infinitely long nozzles for both the smooth solutions with large vorticity and the weak solutions with vortex sheets. The methods and techniques developed here will be useful for solving other problems involving similar difficulties.Comment: 43 pages; 2 figures; To be published in Advances in Mathematics (2019

Design of Air Conditioning Distributed Control System for an Office Building in Xi’an

AbstractIn this paper, the system 600 APOGEE (distributed control system) of Siemens building technologies is applied to an air conditioning controlling system in an office building. Firstly, control scheme is introduced. Then, considering the problem that air conditioning system usually loses contact with control in practice, some suggestion is presente

Food protein-stabilized nanoemulsions as potential delivery systems for poorly water-soluble drugs: preparation, in vitro characterization, and pharmacokinetics in rats

Nanoemulsions stabilized by traditional emulsifiers raise toxicological concerns for long-term treatment. The present work investigates the potential of food proteins as safer stabilizers for nanoemulsions to deliver hydrophobic drugs. Nanoemulsions stabilized by food proteins (soybean protein isolate, whey protein isolate, β-lactoglobulin) were prepared by high-pressure homogenization. The toxicity of the nanoemulsions was tested in Caco-2 cells using the 3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazoliumbromide viability assay. In vivo absorption in rats was also evaluated. Food protein-stabilized nanoemulsions, with small particle size and good size distribution, exhibited better stability and biocompatibility compared with nanoemulsions stabilized by traditional emulsifiers. Moreover, β-lactoglobulin had a better emulsifying capacity and biocompatibility than the other two food proteins. The pancreatic degradation of the proteins accelerated drug release. It is concluded that an oil/water nanoemulsion system with good biocompatibility can be prepared by using food proteins as emulsifiers, allowing better and more rapid absorption of lipophilic drugs