Modified (2+1)-dimensional displacement shallow water wave system and its approximate similarity solutions

Recently, a new （2＋1）-dimensional shallow water wave system, the （2＋1）-dlmenslonal displacement shallow water wave system （2DDSWWS）, was constructed by applying the variational principle of the analytic mechanics in the Lagrange coordinates. The disadvantage is that fluid viscidity is not considered in the 2DDSWWS, which is the same as the famous Kadomtsev-Petviashvili equation and Korteweg-de Vries equation. Applying dimensional analysis, we modify the 2DDSWWS and add the term related to the fluid viscidity to the 2DDSWWS. The approximate similarity solutions of the modified 2DDSWWS （M2DDSWWS） is studied and four similarity solutions are obtained. For the perfect fluids, the coefficient of kinematic viscosity is zero, then the M2DDSWWS will degenerate to the 2DDSWWS.

A study on the flowability of gas displacing water in low-permeability coal reservoir based on NMR technology

Flowability of gas and water through low-permeability coal plays crucial roles in coalbed methane(CBM)recovery from coal reservoirs.To better understand this phenomenon,experiments examining the displacement of water by gas under different displacement pressures were systematically carried out based on nuclear magnetic resonance(NMR)technology using low-permeability coal samples of medium-high coal rank from Yunnan and Guizhou,China.The results reveal that both the residual water content(W_(r))and residual water saturation(S_(r))of coal gradually decrease as the displacement pressure(P)decreases.When P is 0-2 MPa,the decline rates of W_(r) and S_(r) are fastest,beyond which they slow down gradually.Coal samples with higher permeability exhibit higher water flowability and larger decreases in W_(r) and S_(r).Compared with medium-rank coal,high-rank coal shows weaker fluidity and a higher proportion of irreducible water.The relationship between P and the cumulative displaced water content(W_(c))can be described by a Langmuir-like equation,W_(c)=WLP/(PL+P),showing an increase in W_(c) in coal with an increase in P.In the low-pressure stage from 0 to 2 MPa,W_(c) increases most rapidly,while in the high-pressure stage(P>2 MPa),W_(c) tends to be stable.The minimum pore diameter(d′)at which water can be displaced under different displacement pressures was also calibrated.The d′value decreases as P increases in a power relationship;i.e.,d′the coal gradually decreases with the gradual increase in P.Furthermore,the d′values of most of the coal samples are close to 20 nm under a P of 10 MPa.

How did Archimedes discover the law of buoyancy by experiment？

After Archimedes and Vitruvius era, for more than 2000 years, it has been believed that the displaced water measurement of golden crown is impossible, and at his Eureka moment, Archimedes discovered the law of buoyancy （Proposition 7 of his principles） and proved the theft of a goldsmith by weighing the golden crown in water. A previous study showed that a small amount of displaced water was able to be measured with enough accuracy by the introduced method. Archimedes measured the weight of displaced water. He did not find the law of buoyancy but rather specific gravity of things at the moment. After which, Archimedes continued to measure the specific gravity of various solids and fluids. Through these measurements, he reached the discovery of the law of buoyancy directly by experiment. In this paper, the process to the discovery of Archimedes＇ principle （Proposition 5） is presented.