Numerical simulations of a freely falling rigid sphere in bounded and unbounded water domains

Numerical studies were conducted on the hydrodynamics of a freely falling rigid sphere in bounded and unbounded water domains to investigate the drag coefficient, normalized velocity, pressure coefficient, and skin friction coefficient as a function of dimensionless time. The bounded domain was simulated by bringing the cylindrical water container's wall closer to the impacting rigid sphere and linking it to the blockage ratio (BR), defined as the ratio of the projection area of a freely falling sphere to that of the cross-section area of the cylindrical water container. Six cases of bounded domains (BR= 1%, 25%, 45%, 55%, 65%, and 75%) were studied. However, the unbounded domain was considered with a BR of 0.01%. In addition, the k–ω shear stress transport (SST) turbulence model was employed, and the computed results of the bounded domain were compared with those of other studies on unbounded domains. In the case of the bounded domain, which has a higher value of BR, a substantial reduction in normalized velocity and an increase in the drag coefficient were found. Moreover, the bounded domain yielded a significant increase in the pressure coefficient when the sphere was half-submerged; however, an insignificant effect was found on the skin friction coefficient. In the case of the unbounded domain, a significant reduction in the normalized velocity occurred with a decrease in the Reynold number (Re) whereas the drag coefficient increases with a decrease in Reynolds number.

Liouville-type results for positive solutions of pseudo-relativistic Schrödinger system

In this paper, we are concerned with the physically engaging pseudo-relativistic Schrödinger system: \[ \begin{cases} \left(-\Delta+m^{2}\right)^{s}u(x)=f(x,u,v,\nabla u) & \hbox{in } \Omega,\\ \left(-\Delta+m^{2}\right)^{t}v(x)=g(x,u,v,\nabla v) & \hbox{in } \Omega,\\ u>0,v>0 & \hbox{in } \Omega, \\ u=v\equiv 0 & \hbox{in } \mathbb{R}^{N}\setminus\Omega, \end{cases} \] where $s,t\in (0,1)$ and the mass $m>0.$ By using the direct method of moving plane, we prove the strict monotonicity, symmetry and uniqueness for positive solutions to the above system in a bounded domain, unbounded domain, $\mathbb {R}^{N}$ , $\mathbb {R}^{N}_{+}$ and a coercive epigraph domain $\Omega$ in $\mathbb {R}^{N}$ , respectively.

Stability of constant steady states of a chemotaxis model

The Cauchy problem for the parabolic–elliptic Keller–Segel system in the whole n-dimensional space is studied. For this model, every constant $$A \in {\mathbb {R}}$$ A ∈ R is a stationary solution. The main goal of this work is to show that $$A < 1$$ A < 1 is a stable steady state while $$A > 1$$ A > 1 is unstable. Uniformly local Lebesgue spaces are used in order to deal with solutions that do not decay at spatial variable on the unbounded domain.

A nonlocal problem for a generalized Trikomi equation with a spectral parameter in the unbounded domain

В данной статье изучена нелокальная задача для обобщенного уравнения Трикоми со спектральным параметром в неограниченной области эллиптическая часть которой является верхней полуплоскостью. Единственность решения поставленной задачи доказана методом интегралов энергии. Существование решения поставленной задачи доказана методом функций Грина и интегральных уравнений. In this article, we study a nonlocal problem for the generalized Tricomi equation with a spectral parameter in an unbounded domain, the elliptic part of which is the upper half-plane. The uniqueness of the solution to the problem posed is proved by the method of energy integrals. The existence of a solution to the problem is proved by the method of Green’s functions and integral equations.