We consider inviscid, incompressible shear flows with variable density and variable cross section. For this problem, we derived a new estimate for the growth rate of an unstable mode and a parabolic instability region which intersects semiellipse instability region under some condition.

This paper describes an adaptive preconditioner for numerical continuation of incompressible Navier–Stokes flows based on Stokes preconditioning [42] which has been used successfully in studies of pattern formation in convection. The preconditioner takes the form of the Helmholtz operator I–ΔtL which maps the identity (no preconditioner) for Δt≪1 to Laplacian preconditioning for Δt≫1. It is built on a first order Euler time-discretization scheme and is part of the family of matrix-free methods. The preconditioner is tested on two fluid configurations: three-dimensional doubly diffusive convection and a two-dimensional projection of a shear flow. In the former case, it is found that Stokes preconditioning is more efficient for , away from the values used in the literature. In the latter case, the simple use of the preconditioner is not sufficient and it is necessary to split the system of equations into two subsystems which are solved simultaneously using two different preconditioners, one of which is parameter dependent. Due to the nature of these applications and the flexibility of the approach described, this preconditioner is expected to help in a wide range of applications.

This paper establishes the existence of smooth solutions for the Doi–Edwards rheological model of viscoelastic polymer fluids in shear flows. The problem turns out to be formally equivalent to a K-BKZ equation but with constitutive functions spanning beyond the usual mathematical framework. We prove, for small enough initial data, that the solution remains in the domain of hyperbolicity of the equation for all t≥0.

Most classical work on the hydrodynamics of low-Reynolds-number swimming addresses deterministic locomotion in quiescent environments. Thermal fluctuations in fluids are known to lead to a Brownian loss of the swimming direction, resulting in a transition from short-time ballistic dynamics to effective long-time diffusion. As most cells or synthetic swimmers are immersed in external flows, we consider theoretically in this paper the stochastic dynamics of a model active particle (a self-propelled sphere) in a steady general linear flow. The stochasticity arises both from translational diffusion in physical space, and from a combination of rotary diffusion and so-called run-and-tumble dynamics in orientation space. The latter process characterizes the manner in which the orientation of many bacteria decorrelates during their swimming motion. In contrast to rotary diffusion, the decorrelation occurs by means of large and impulsive jumps in orientation (tumbles) governed by a Poisson process. We begin by deriving a general formulation for all components of the long-time mean square displacement tensor for a swimmer with a time-dependent swimming velocity and whose orientation decorrelates due to rotary diffusion alone. This general framework is applied to obtain the convectively enhanced mean-squared displacements of a steadily swimming particle in three canonical linear flows (extension, simple shear and solid-body rotation). We then show how to extend our results to the case where the swimmer orientation also decorrelates on account of run-and-tumble dynamics. Self-propulsion in general leads to the same long-time temporal scalings as for passive particles in linear flows but with increased coefficients. In the particular case of solid-body rotation, the effective long-time diffusion is the same as that in a quiescent fluid, and we clarify the lack of flow dependence by briefly examining the dynamics in elliptic linear flows. By comparing the new active terms with those obtained for passive particles we see that swimming can lead to an enhancement of the mean-square displacements by orders of magnitude, and could be relevant for biological organisms or synthetic swimming devices in fluctuating environmental or biological flows.