Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form

In this paper we consider minimizers of the functional$\min \{{\lambda }_1(\mathrm{\Omega })+\cdots +{\lambda }_k(\mathrm{\Omega })+\mathrm{\Lambda }|\mathrm{\Omega }|,:\mathrm{\Omega }\subset D\text{ open}\}$ where $D\subset {\mathbb{R}}^d$ is a bounded open set and where $0<{\lambda }_1(\mathrm{\Omega })\le \cdots \le {\lambda }_k(\mathrm{\Omega })$ are the first k eigenvalues on Ω of an operator in divergence form with Dirichlet boundary condition and with Hölder continuous coefficients. We prove that the optimal sets ${\mathrm{\Omega }}^{⁎}$ have finite perimeter and that their free boundary $\partial {\mathrm{\Omega }}^{⁎}\cap D$ is composed of a regular part, which is locally the graph of a $C^{1,\alpha }$-regular function, and a singular part, which is empty if $d<d^{⁎}$, discrete if $d=d^{⁎}$ and of Hausdorff dimension at most $d-d^{⁎}$ if $d>d^{⁎}$, for some $d^{⁎}\in \{5,6,7\}$.     

Morphologically controlled synthesis of 1-dimensional selenium dioxide and study of its application as catalyst for diesel fuel additive

In this research work, selenium dioxide (SeO₂) nanorods have been prepared by a solvothermal method in which a strong reducing agent (NaBH₄) was used to reduce precursor salt into SeO₂ nanorods. X-ray diffraction (XRD) technique was applied to observe the crystal structure which confirmed its tetragonal geometry. Moreover, morphology and particle size were studied by scanning electron microscopy (SEM). SEM fully described the 1-dimensional morphology of SeO₂ nanorods which then arranged themselves to create a 3-dimensional flower-like structure with an average particle size of 50 nm. Also, the catalytic activity of SeO₂ nanorods as diesel-additive was studied by defining different parameters such as fire and flash points, calorific value, cloud and pour points, specific gravity, and kinematic viscosity. Subsequently, SeO₂ nanorods proved to be an excellent diesel additive due to higher total heat content and lower value of kinematic viscosity which enhances the better performance of the diesel engine.     

Assembling hydrodynamic cloaks to conceal complex objects from drag

Transformation hydrodynamics and the corresponding metamaterials have been proposed as a means to exclude the drag force acting on an object. Here, we report a strategy to deploy the hydrodynamic cloaks in a more practical manner by assembling different-shaped cloaking parts. Our strategy is to first model a square-shaped cloak and a carpet cloak and then combine them to conceal a more complex-shaped space in the three-dimensional hydrodynamic flow. With the derivation of transformation hydrodynamics, the coordinate transformations for each hydrodynamic cloaking are demonstrated with the calculated viscosity tensors. The pressure and velocity fields of the square, triangular (carpet), and exemplary three-dimensional house-shaped cloaks are numerically simulated, thus showing a cloaking effect and reduced drag. This study suggests an efficient way of cloaking complex architectures from fluid-dynamic forces.     

Salting-in/Salting-out Mechanism of Carbon Dioxide in Aqueous Electrolyte Solutions

The solvation of carbon dioxide in sea water plays an important role in the carbon circle and the world climate. The salting-out/salting-in mechanism of CO₂ in electrolyte solutions still remains elusive at molecule level. The ability of ion salting-out/salting-in CO₂ in electrolyte solution follows Hofmeister Series and the change of water mobility induced by salts can be predicted by the viscosity B-coefficients. In this work, the chemical potential of carbon dioxide and the dynamic properties of water in aqueous NaCl, KF and NaClO₄ solutions are calculated and analyzed. According to the viscosity B-coefficients, NaClO₄ (0.012) should salt out the carbon dioxide relative to in pure water, but the opposite effect is observed for it. Our simulation results suggest that the salting-in effect of NaClO₄ is due to the strongly direct anion-CO₂ interaction. The inconsistency between Hofmeister Series and the viscosity B-coefficient suggests that it is not always right to indicate whether a salt belongs to salting-in or salting-out just from these properties of the salt solution in the absence of solute.     

Singular Solutions to Conformal Hessian Equations

The authors show that for any e ∈]0,1[,there exists an analytic outside zero solution to a uniformly elliptic conformal Hessian equation in a ball B C R5 which belongs to C1'ε(B) \ C1'ε+(B).     

Combination of RSM and NSGA-II algorithm for optimization and prediction of thermal conductivity and viscosity of bioglycol/water mixture containing SiO2 nanoparticles

The fluids containing nanoparticles have enhanced thermo-physical characteristics in comparison with conventional fluids without nanoparticles. Thermal conductivity and viscosity are thermo-physical properties that strongly determine heat transfer and momentum. In this study, the response surface method was firstly used to derive an equation for the thermal conductivity and another one for the viscosity of bioglycol/water mixture (20:80) containing silicon dioxide nanoparticles as a function of temperature as well as the volume fraction of silicon dioxide. Then, NSGA-II algorithm was used for the optimization and maximizing thermal conductivity and minimizing the nanofluid viscosity. Different fronts were implemented and 20th iteration number was selected as Pareto front. The highest thermal conductivity (0.576 W/m.K) and the lowest viscosity (0.61 mPa.s) were obtained at temperature on volume concentration of (80 °C and 2%) and (80 °C without nanoparticle) respectively. It was concluded that the optimum thermal conductivity and viscosity of nanofluid could be obtained at maximum temperature (80 °C) or a temperature close to this temperature. An increase in the volume fraction of silicon dioxide led to the enhancement of thermal conductivity but the solution viscosity was also increased. Therefore, the optimum point should be selected based on the system requirement.     

Ideal and non-ideal behaviour of {1-butyl-1-methylpyrrolydinium bis(trifluoromethylsulfonyl)imide + γ-butyrolactone} binary mixtures

Density, electrical conductivity and viscosity of binary liquid mixtures of 1-butyl-1-methylpyrrolydinium bis(trifluoromethylsulfonyl)imide, [bmpyrr][NTf₂], with γ-butyrolactone (GBL) were measured at temperatures from (293.15 to 323.15) K and at atmospheric pressure over the whole composition range. Excess molar volumes have been calculated from the experimental densities and fitted with the Redlich–Kister polynomial equation. These values are positive over the whole range of ionic liquid mole fraction and at all temperatures. In the range between 0.55 and 0.6 [bmpyrr][NTf₂] mole fraction, an ideal behaviour of the ionic liquid mixture with molecular solvent was observed for the first time. Other volumetric properties, such as isobaric thermal expansion coefficients, partial molar volumes and partial molar volumes at infinite dilution have been also calculated, in order to obtain information about interactions between GBL and selected ionic liquid. Positive values of these properties for both components also indicate weaker interactions between GBL and IL compared to the pure components. From the viscosity results, the Angell strength parameter was calculated and found to be 3.24 indicating that [bmpyrr][NTf₂] is a “fragile” liquid. From the volumetric and transport properties obtained, formation of the [bmpyrr]⁺ micellar structures was also discussed. All the results are compared to those obtained for imidazolium-based ionic liquid with GBL.     

Volumetric properties,viscosities and refractive indices of binary liquid mixtures of tetrafluoroborate-based ionic liquids with methanol at several temperatures

Densities, speeds of sound, viscosities and refractive indices of two binary systems 1-butyl-3-methylimidazolium tetrafluoroborate [bmim][BF₄] + methanol and 1-ethyl-3-methylimidazolium tetrafluoroborate [emim][BF₄] + methanol, as well as of all pure components, have been measured covering the whole range of compositions at T = (278.15 to 318.15) K and p = 101 kPa. From this data, excess molar volumes, excess isentropic compressibilities, viscosity deviations and refractive index deviations were calculated and fitted to extended versions of the Redlich–Kister equation. Estimated coefficients of these equations taking into account the dependence on composition and temperature simultaneously were also presented.     

表面活性剂对部分水解聚丙烯酰胺溶液抗盐性的影响

通过实验研究了阴离子表面活性剂(SDS)、非离子表面活性剂(OP-10)、两性表面活性剂(C12BE)浓度及KCl浓度对部分水解聚丙烯酰胺(阴离子型,HPAM)水溶液黏度的影响规律,进而分析各因素对聚合物溶液抗盐性的影响。实验结果表明:当表面活性剂浓度低于临界缔合浓度CAC时,聚合物溶液黏度变化不大;高于CAC后,随着表面活性剂浓度增大,聚合物溶液黏度急剧增加;当表面活性剂浓度达到聚合物饱和浓度PSP时,聚合物溶液黏度达到最大值;再加入阴离子和两性表面活性剂,将导致黏度降低,而加入非离子表面活性剂不再改变聚合物溶液的黏度,无机盐KCl对聚合物溶液有双重作用,低浓度KCl促进聚合物溶液黏度升高,高浓度KCl则导致聚合物溶液黏度急剧降低后趋于稳定,在相同KCl浓度下,三种表面活性剂的抗盐能力表现为:SDSOP-10C12BE。