Simple Proofs of Upper and Lower Envelopes of Van Der Pauw’s Equation for Hall-Plates with an Insulated Hole and Four Peripheral Point-Contacts

For plane singly-connected domains with insulating boundary and four point-sized contacts, C0 …C3, van der Pauw derived a famous equation relating the two trans-resistances R01,23, R12,30 with the sheet resistance without any other parameters. If the domain has one hole van der Pauw’s equation becomes an inequality with upper and lower bounds, the envelopes. This was conjectured by Szymański et al. in 2013, and only recently it was proven by Miyoshi et al. with elaborate mathematical tools. The present article gives new proofs closer to physical intuition and partly with simpler mathematics. It relies heavily on conformal transformation and it expresses for the first time the trans-resistances and the lower envelope in terms of Jacobi functions, elliptic integrals, and the modular lambda elliptic function. New simple formulae for the asymptotic limit of a very large hole are also given.

The Principle of Square Hole Machining and It＇s Tooling Structure Design

In order to meet the rapid needs of processing square hole in mechanical equipment, the paper expounds the square hole processing method： planetary wheel method, and analyze the principle of tooling structure and process with computer graphics parameters design. The results that, as long as the appropriate parameters, using the above method not only can punch the square hole, can also be processed triangle, the five angle and hexagonal regular polygon holes. The square hole processing method can provide theoretical basis and engineering reliable reference for related engineering and technical personnel.

Experimental study of the critical sand starting velocity of gas-watersand flow in an inclined pipe

The purpose of this paper is to study the critical sand starting velocity and transformation law of flow pattern based on gas-water-sand three-phase flow in an inclined pipe.Firstly,the indoor simulation experiment system of gas-water-sand three-phase flow was used to test the conversion law of flow pattern based upon the different gas void fraction.Secondly,the influence of slug bubbles on sand migration was investigated according to distinctive hole deviation angles,gas void fraction and sand concentration.Finally,the critical sand starting velocity was tested based on dissimilar hole deviation angles,gas void fraction,sand concentration and sand particle size,and then the influence of the abovementioned key parameters on the sand starting velocity was debated based on the force analysis of the sand particles.The experimental results illustrated that when the gas void fraction was less than 5%,it was bubbly flow.When it increased from 5%to 30%,the bubbly flow and slug flow coexisted.When it was between 30%and 50%,the slug flow and agitated flow coexisted.When it reached 50%,it was agitated flow.Providing that the hole deviation angle was 90°,the phenomenon of overall migration and wavelike migration on the surface of sand bed was observed.On the contrary,the phenomenon of rolling and jumping migration was recognized.The critical sand starting velocity was positively correlated with the hole deviation angle and sand particle size,but negatively associated with the gas void fraction and sand concentration.This research can provide a certain reference for sand-starting production in the field of petroleum engineering.