We demonstrate that upstream swimming of sperm emerges via an orientation disorder-order transition. The order parameter, the average orientation of the sperm head against the flow, follows a 0.5 power law with the deviation from the critical flow shear rate (γ − γc). This transition is successfully explained by a hydrodynamic bifurcation theory, which extends the sperm upstream swimming to a broad class of near surface micro-swimmers that possess front-back asymmetry and circular motion.

Experimental measurements of the force and torque on freely settling fibres are compared with predictions of the slender-body theory of Khayat & Cox (J. Fluid Mech., vol. 209, 1989, pp. 435–462). Although the flow is viscous dominated at the scale of the fibre diameter, fluid inertia is important on the scale of the fibre length, leading to inertial torques which tend to rotate symmetric fibres toward horizontal orientations. Experimentally, the torque on symmetric fibres is inferred from the measured rate of rotation of the fibres using a quasi-steady torque balance. It is shown theoretically that fibres with an asymmetric radius or mass density distribution undergo a supercritical pitch-fork bifurcation from vertical to oblique settling with increasing Archimedes number, increasing Reynolds number or decreasing asymmetry. This transition is observed in experiments with asymmetric mass density and we find good agreement with the predicted symmetry breaking transition. In these experiments, the steady orientation of the oblique settling fibres provides a means to measure the inertial torque in the absence of transient effects since it is balanced by the known gravitational torque.

We consider the linear stability and nonlinear evolution of two-dimensional shear flows that take the form of an unstratified plane Couette flow that is seeded with a localized ‘defect’ containing sharp density and vorticity variations. For such flows, matched asymptotic expansions furnish a reduced model that allows a straightforward and computationally efficient exploration of flows at sufficiently high Reynolds and Péclet numbers that sharp density and vorticity gradients persist throughout the onset, growth and saturation of instability. We are thereby able to study the linear and nonlinear dynamics of three canonical variants of stratified shear instability: Kelvin–Helmholtz instability, the Holmboe instability, and the lesser-considered Taylor instability, all of which are often interpreted in terms of the interactions of waves riding on sharp interfaces of density and vorticity. The dynamics near onset is catalogued; if the interfaces are sufficiently sharp, the onset of instability is subcritical, with a nonlinear state existing below the linear instability threshold. Beyond onset, both Holmboe and Taylor instabilities are susceptible to inherently two-dimensional secondary instabilities that lead to wave mergers and wavelength coarsening. Additional two-dimensional secondary instabilities are also found to appear for higher Prandtl numbers that take the form of parasitic Holmboe-like waves.

Abstract

In this paper, we study the role of shear-induced migration and particle-induced normal stresses in the formation and stability of a particle-laden, gravity-driven shallow flow. We first examine the modification of the base-state Nusselt flow due to the underlying microstructure, how shear-induced migration leads to viscosity stratification. We inspect the development of the base state via the boundary layer formation in the ‘shallow’ limit and find a reduction in entrance length with increasing bulk particle concentration and an increase in entrance length with increasing Péclet number ( $Pe_p = \dot {\gamma } a^2 / D_0$ , where $\dot{\gamma}$ is the average shear rate, a is the particle size and $D_0$ is the single particle diffusivity). A linear stability analysis is then performed on the fully developed state to identify two modes of instability typically found in gravity-driven falling films – the long-wave surface and the short-wave shear modes. We find that when the associated Péclet number is $Pe_p \ll 1$ , increasing bulk particle volume fraction delays the onset of instability for both the surface mode and shear mode. However, with $Pe_p = {O}(1)$ , we find an enhancement in both modes of instability. We also find that, beyond a critical Péclet number, for a fixed particle volume fraction, the surface mode is unstable even in the absence of fluid inertia. The enhanced destabilisation is attributed to the combined effects of base-state viscosity stratification and momentum forcing via particle concentration perturbations. We also show that the physics behind the enhancement of instability is independent of the choice of the constitutive model used to describe the dynamics of the particle phase, provided the chosen model has elements of shear-induced migration.