Final fate of Kantowski–Sachs gravitational collapse

Although it is not a fundamental question, determining exact and general solutions for a given theory has advantages over a numerical integration in many specific cases. Of course, respecting the peculiarities of the problem. Revisiting the integration of the General Relativity Theory field equations for the Kantowski–Sachs spacetime describes a homogeneous but anisotropic universe whose spatial section has the topology of [Formula: see text], we integrate the equations for arbitrary curvature parameter and write the solutions considering the process of gravitational collapse. We took the opportunity and made some comments involving some features of the model such as energy density, shear, viscosity and the production of gravitational waves via Petrov classification.

Colloidal swimmers near curved and structured walls

We present systematic numerical simulations to understand the behavior of colloidal swimmers near walls of arbitrary curvature.

‘Unforced’ Navier–Stokes solutions derived from convection in a curved channel

Steady Boussinesq flow in a weakly curved channel driven by a horizontal temperature gradient is considered. Linear variation in the transverse direction is assumed so that the problem reduces to a system of ordinary differential equations. A series expansion in $G$, a parameter proportional to the Grashof number and the square root of the curvature, reveals a real singularity and anticipates hysteresis. Numerical solutions are found using path continuation and the bifurcation diagrams for different parameter values are obtained. Multivalued solutions are observed as $G$ and the Prandtl number vary. Often fields with the imposed structure that satisfy all the governing equations are insensitive to the boundary conditions and can be regarded as perturbations of the homogeneous (or ‘unforced’) problem. Four such unforced solutions are found. In two of these the velocity remains coupled with temperature which, formally, scales as $1/G$ as $G\rightarrow 0$. The other two are purely hydrodynamic. The existence of such solutions is due to the unbounded nature of the domain. It is shown that these occur not only for the Dean equations, but constitute previously unreported solutions of the full Navier–Stokes equations in an annulus of arbitrary curvature. Two additional unforced solutions are found for large curvature.