The stability of a two-dimensional inviscid flow in an annulus between two permeable cylinders is examined. The basic flow is irrotational, and both radial and azimuthal components of the velocity are non-zero. The direction of the radial flow can be from the inner cylinder to the outer one (the diverging flow) or from the outer cylinder to the inner one (the converging flow). It is shown that, independent of the direction of the radial flow, the basic flow is unstable to small two-dimensional perturbations provided that the ratio of the azimuthal component of the velocity to the radial one is sufficiently large. The instability is oscillatory and persists if the viscosity of the fluid is taken into consideration.

The stability of a two-dimensional viscous flow in an annulus with permeable walls with respect to small two-dimensional perturbations is studied. The basic steady flow is the most general rotationally invariant solution of the Navier-Stokes equations in which the velocity has both radial and azimuthal components, and the azimuthal velocity profile depends on the radial Reynolds number. It is shown that for a wide range of parameters of the problem, the basic flow is unstable to small two-dimensional perturbations. Neutral curves in the space of parameters of the problem are computed. Calculations show that the stability properties of this flow are determined by the azimuthal velocity at the inner cylinder when the direction of the radial flow is from the inner cylinder to the outer one and by the azimuthal velocity at the outer cylinder when the direction of the radial flow is reversed. This work is a continuation of our previous study of an inviscid instability in flows between rotating porous cylinders [K. Ilin and A. Morgulis, “Instability of an inviscid flow between porous cylinders with radial flow,” J. Fluid Mech. 730, 364–378 (2013)].

In this article, we study a Patlak–Keller–Siegel (PKS) model of a community of two species placed in the inhomogeneous environment. We employ PKS law for modeling tactic movement due to interspecific taxis and in response to the environmental fluctuations. These fluctuations can arise for natural reasons, e.g., the terrain relief, the sea currents and the food resource distribution, and there are artificial ones. The main result in the article elucidates the effect of the small-scale environmental fluctuations on the large-scale pattern formation in PKS systems. This issue remains uncharted, although numerous studies have addressed the pattern formation while assuming an homogeneous environment. Meanwhile, exploring the role of the fluctuating environment is substantial in many respects, for instance, for predicting the side effects of human activity or for designing the control of biological systems. As well, it is necessary for understanding the roles played in the dynamics of trophic communities by the natural environmental inhomogeneities—those mentioned above, for example. We examined the small-scale environmental inhomogeneities in the spirit of Kapitza’s theory of the upside-down pendulum, but we used the homogenization instead of classical averaging. This approach is novel for the dynamics of PKS systems (though used commonly for other areas). Employing it has unveiled a novel mechanism of exerting the effect from the fluctuating environment on the pattern formation by the drift of species arising upon the homogenization of the fluctuations.

Our aim is to study the effect of the outflow boundary conditions on the stability of incompressible flows in a domain with a permeable boundary. For this purpose, we examine the stability of the Couette flow with the radial throughflow between permeable cylinders. Most earlier studies of this flow employed the boundary conditions that prescribe all components of the flow velocity on both cylinders. Taking these boundary conditions as a reference point, we investigate the effect of imposing different outflow boundary conditions. These conditions prescribe the normal stress and either the tangential velocity or the tangential stress. It turns out that both sets of boundary conditions make the corresponding steady flows more unstable. In particular, it is shown that even the classical (purely azimuthal) Couette flow becomes unstable to two-dimensional perturbations if one of the cylinders is permeable and the normal stress (rather than normal velocity) is prescribed on that cylinder.

We consider a layer of an inviscid fluid with a free surface which is subject to vertical high-frequency vibrations. We derive three asymptotic systems of equations that describe slowly evolving (in comparison with the vibration frequency) free-surface waves. The first set of equations is obtained without assuming that the waves are long. These equations are as difficult to solve as the exact equations for irrotational water waves in a non-vibrating fluid. The other two models describe long waves. These models are obtained under two different assumptions about the amplitude of the vibration. Surprisingly, the governing equations have exactly the same form in both cases (up to the interpretation of some constants). These equations reduce to the standard dispersionless shallow-water equations if the vibration is absent, and the vibration manifests itself via an additional term which makes the equations dispersive and, for small-amplitude waves, is similar to the term that would appear if surface tension were taken into account. We show that our dispersive shallow-water equations have both solitary and periodic travelling wave solutions and discuss an analogy between these solutions and travelling capillary–gravity waves in a non-vibrating fluid.