Seismic Wave Field Model Excited by a Finite Long Cylindrical Charge

The amplitude-frequency characteristic of a seismic wave excited by explosion sources directly affects the accuracy of seismic exploration. To reveal the effect law related to a cylindrical charge, the research proposes a seismic wavefield model excited by a long cylindrical charge. According to the characteristics of the blasting cavity generated by a finite length cylindrical charge, the seismic wavefield characteristics of a cylindrical charge excitation is obtained by superposing the seismic wavefield excited by a series of spherical charges. Numerical simulation results show that the calculation error of the blasting cavity characteristics of the theoretical model is within 10%. The comparison with field experimental results shows that the error of the model is within 9.4%. The velocity field of the excited seismic wave is almost the same as that of the spherical charge when the explosion distance to the cylindrical charge with finite length is 16-21 times longer than the charge length, but the frequency of the seismic wave is 30% higher than for a spherical charge. Moreover, the explosive velocity has a certain influence on the amplitude-frequency characteristic of the seismic wave excited by the cylindrical charge. The established theoretical model can accurately describe the amplitude-frequency characteristics of the seismic wavefield excited by a cylindrical charge with finite length.

A Composite Order Generalization of Modular Moonshine

We introduce a generalization of Brauer character to allow arbitrary finite length modules over discrete valuation rings. We show that the generalized super Brauer character of Tate cohomology is a linear combination of trace functions. Using this result, we find a counterexample to a conjecture of Borcherds about vanishing of Tate cohomology for Fricke elements of the Monster.

The meaning of the infinitely great

An infinitely small quantity is defined as a one-dimensional quantity of finite length but with sizes of space, while an infinitely great quantity is reached by the superposition or accumulation of infinitely many finite quantities, by the way of the change in direction. The change in direction indicates that there is a jump from a finite quantity to infinitely many finite quantities (infinitely great). The form of the manifestation of the infinitely great is one quantitative continuum that cannot be operated by any algorithms and all parts of space we see is this one quantitative continuum. .Any value are this single quantitative continuum that indicates the infinitely great and compresses any quantities outside of it to nothing. As a result, the infinite exact value of a circumferential length (π) has been obtained here.

Engineering Charts for Predicting Breakdown Pressure for Finite-Length Wellbore Intervals

In wellbore drilling, the drilling mud density needs to be carefully selected such that the mud pressure inside the wellbore will not exceed formation breakdown pressure to avoid wellbore fracturing and extensive mud losses. However, in the hydraulic fracturing treatment, the lesser the value of the formation breakdown pressure the more optimal is the operation. We found out in this study that the pumping schedule (e.g., pumping duration and pumping rate) are factors in optimizing the breakdown pressure. In addition, this work investigates the effects of the finite length between packers on the magnitude of the breakdown pressure in various geological formations. The time-dependent evolving stresses around the wellbore are solved in the framework of time-dependent poroelasticity theory. The breakdown pressure is predicted from the evolution of the circumferential effective stresses. The effects of injection rate, formation properties, borehole diameter and length, and pumping duration on the breakdown pressure are presented in the form of engineering charts, for representative in-situ stress.

Shock-induced interfacial instabilities of granular media

This paper investigates the shock-induced instability of the interfaces between gases and dense granular media with finite length via the coarse-grained compressible computational fluid dynamics–discrete parcel method. Despite generating a typical spike-bubble structure reminiscent of the Richtmyer–Meshkov instability (RMI), the shock-driven granular instability (SDGI) is governed by fundamentally different mechanisms. Unlike the RMI arising from baroclinic vorticity deposition on the interface, the SDGI is closely associated with the interfacial and bulk granular dynamics, which evolve with the transient coupling between particles and gases. Consequently, the SDGI follows a growth law distinctly different from that of the RMI, namely a semilinear slow regime followed by an exponentially expedited regime and a quadratic asymptotic regime. We further establish the instability criteria of the SDGI for granular media with infinite and finite lengths, which do not exist in the RMI. A scaling growth law of the SDGI for dense granular media with finite length is derived by normalizing the time with the rarefaction propagation time, which successfully collapses the data from cases with varying shock strength, particle column length and particle volume fraction and ought to hold for granular media with varying particle parameters. The effect of the initial perturbation magnitude can be properly considered in the scaling growth law by incorporating it into the length normalization.

Electrophoresis of tightly fitting spheres along a circular cylinder of finite length

Electrophoresis of a tightly fitting sphere of radius $a$ along the centreline of a liquid-filled circular cylinder of radius $R$ is studied for a gap width $h_0=R-a\ll a$ . We assume a Debye length $\kappa ^{-1}\ll h_0$ , so that surface conductivity is negligible for zeta potentials typically seen in experiments, and the Smoluchowski slip velocity is imposed as a boundary condition at the solid surfaces. The pressure difference between the front and rear of the sphere is determined. If the cylinder has finite length $L$ , this pressure difference causes an additional volumetric flow of liquid along the cylinder, increasing the electrophoretic velocity of the sphere, and an analytic prediction for this increase is found when $L\gg R$ . If $N$ identical, well-spaced spheres are present, the electrophoretic velocity of the spheres increases with $N$ , in agreement with the experiments of Misiunas & Keyser (Phys. Rev. Lett., vol. 122, 2019, 214501).