Critical radius and dipole polarizability for a confined system

The analytical transfer matrix method （ATMM） is applied to calculating the critical radius τc and the dipole polarizability αd in two confined systems： the hydrogen atom and the Hulthēén potential. We find that there exists a linear relation between τe^1/2 and the quantum number nτ for a fixed angular quantum number l, moreover, the three bounds of αd（αd^K,αd^B,αd^U） satisfy an inequality：αd^K≤αd^B≤αd^U,A comparison betwen the ATMM,the exact numerical analysis, and the variational wavefunctions shows that our method works very well in the systems.

Critical bubble radius in solvent sublation

The complex compound of dithizone-Co（Ⅱ） was separated and concentrated from the aqueous phase to n-octanol by solvent sublation. From the analysis of the coalescence behavior of bubbles on water-organic interface, the conception of critical bubble radius was proposed, and the value of the critical bubble radius in the water-octanol system was obtained： 1.196 × 10^-3 m. The simulation of the mathematical model using CBR and experimental data is completed with perfect results, and the simulation of the mathematical model using CBR is very different with the classic one. The analytical results proved that the critical bubble radius should be adequately considered in mathematical model of solvent sublation.

Orifice plate cavitation mechanism and its influencing factors

The orifice plate energy dissipater is an economic and highly efficient dissipater. However, there is a risk of cavitaion around the orifice plate flow： In order to provide references for engineering practice, we examined the cavitation mechanism around the orifice plate and its influencing factors by utilizing mathematical analysis methods to analyze the flow conditions around the orifice plate in view of gas bubble dynamics. Through the research presented in this paper, the following can be observed： The critical radius and the critical pressure of the gas nucleus in orifice plate flow increase with its initial state parameter r0 ; the development speed of bubbles stabilizes at a certain value after experiencing a peak value and a small valley value; and the orifice plate cavitation is closely related to the distribution of the gas nucleus in flow. For computing the orifice plate cavitation number, we ought to take into account the effects of pressure fluctuation. The development time of the gas nucleus from the initial radius to the critical radius is about 107-10-5 s; therefore, the gas nucleus has sufficient time to develop into bubbles in the negative half-cycle of flow fluctuation. The orifice critical cavitation number is closely related to the orifice plate size, and especially closely related with the ratio of the orifice plate radius to the tunnel radius. The approximate formula for the critical cavitation number of the square orifice plate that only considers the main influencing factor was obtained by model experiments.

Open Manifolds with Nonnegative Ricci Curvature and Large Volume Growth

In this paper, we prove that if M is an open manifold with nonnegativeRicci curvature and large volume growth, positive critical radius, then sup Cp = ∞.As an application, we give a theorem which supports strongly Petersen's conjecture.

Theoretical analysis on capillary adhesion of microsized plates with a substrate

The stiction of a thin plate induced by the capillary force has attracted much attention in the broad range of applications. A novel method is presented to calculate the capillary adhesion problem of the plate through analytical method. The expressions of the surface energy, the strain energy and the total potential energy of the plate-substrate system have been analyzed and delineated. By means of continuum mechanics and the principle of minimum potential energy, the governing equation of the plate with an arbitrary shape and the corresponding transversality boundary condition due to the moving bound have been derived. Then the critical adhesion radius of the circular plate has been solved according to the supplementary transversality condition. Thus the deflections of the plates are analytically calculated with different critical adhesion radii. The results may be beneficial to the engineering application and the micro/nanomeasurement.