In the present work, both computational and experimental methods are employed to study the two-phase flow occurring in a model pump sump. The two-fluid model of the two-phase flow has been applied to the simulation of the three-dimensional cavitating flow. The governing equations of the two-phase cavitating flow are derived from the kinetic theory based on the Boltzmann equation. The isotropic RNG $k-\epsilon-k_{ca}$ turbulence model of two-phase flows in the form of cavity number instead of the form of cavity phase volume fraction is developed. The RNG $k-\epsilon-k_{ca}$ turbulence model, that is the RNG $k-\epsilon$ turbulence model for the liquid phase combined with the $k_{ca}$ model for the cavity phase, is employed to close the governing turbulent equations of the two-phase flow. The computation of the cavitating flow through a model pump sump has been carried out with this model in three-dimensional spaces. The calculated results have been compared with the data of the PIV experiment. Good qualitative agreement has been achieved which exhibits the reliability of the numerical simulation model.

The aim of this article is to investigate the solutions of generalized fractional partial differential equations involving Hilfer time fractional derivative and the space fractional generalized Laplace operators, occurring in quantum mechanics. The solutions of these equations are obtained by employing the joint Laplace and Fourier transforms, in terms of the Fox's $H$-function. Several special cases as solutions of one dimensional non-homogeneous fractional equations occurring in the quantum mechanics are presented. The results given earlier by Saxena et al. [Fract. Calc. Appl. Anal., 13(2) (2010), pp. 177-190] and Purohit and Kalla [J. Phys. A Math. Theor., 44 (4) (2011), 045202] follow as special cases of our findings.

We propose a class of non-semisimple matrix loop algebras consisting of 3×3 block matrices, and form zero curvature equations from the presented loop algebras to generate bi-integrable couplings. Applications are made for the AKNS soliton hierarchy and Hamiltonian structures of the resulting integrable couplings are constructed by using the associated variational identities.

Taking vibration converter of intelligent damper for drill strings as the study object, this paper analyzes the influential factors of motion state of the ball and conducts an explicit dynamics simulation by establishing a mechanics model of vibration converter. The study basis is Newton's laws of motion, D'Alembert's principle and Hertz contact theory. And we use world coordinate system, rotating coordinate system and Frenet coordinate system to deduce kinematics equations of vibration converter. The ultimate result demonstrates that the axial velocity and maximum contact stress change with the increment of ball diameter and helix angle. It also proves the validity of our derived kinematics and mechanical models and provides a good consultant value for the design and theoretical arithmetic of vibration converter for intelligent damper of drill strings.

In this paper, the Crank-Nicolson linear finite volume element method is applied to solve the distributed optimal control problems governed by a parabolic equation. The optimal convergent order $\mathcal{O}(h^2+k^2)$ is obtained for the numerical solution in a discrete $L^2$-norm. A numerical experiment is presented to test the theoretical result.

A cell conservative flux recovery technique is developed here for vertex-centered finite volume methods of second order elliptic equations. It is based on solving a local Neumann problem on each control volume using mixed finite element methods. The recovered flux is used to construct a constant free a posteriori error estimator which is proven to be reliable and efficient. Some numerical tests are presented to confirm the theoretical results. Our method works for general order finite volume methods and the recovery-based and residual-based a posteriori error estimators are the first result on a posteriori error estimators for high order finite volume methods.

Multiclass Lighthill-Whitham-Richards traffic models [Benzoni-Gavage and Colombo, Euro. J. Appl. Math., 14 (2003), pp. 587–612; Wong and Wong, Transp. Res. A, 36 (2002), pp. 827–841] give rise to first-order systems of conservation laws that are hyperbolic under usual conditions, so that their associated Cauchy problems are well-posed. Anticipation lengths and reaction times can be incorporated into these models by adding certain conservative second-order terms to these first-order conservation laws. These terms can be diffusive under certain circumstances, thus, in principle, ensuring the stability of the solutions. The purpose of this paper is to analyze the stability of these diffusively corrected models under varying reaction times and anticipation lengths. It is demonstrated that instabilities may develop for high reaction times and short anticipation lengths, and that these instabilities may have controlled frequencies and amplitudes due to their nonlinear nature.