How fast must an oriented collection of extensile swimmers swim to escape the instability of viscous active suspensions? We show that the answer lies in the dimensionless combination $R=ρv_0^2/2σ_a$, where $ρ$ is the suspension mass density, $v_0$ the swim speed and $σ_a$ the active stress. Linear stability analysis shows that for small $R$ disturbances grow at a rate linear in their wavenumber $q$, and that the dominant instability mode involves twist. The resulting steady state in our numerical studies is isotropic hedgehog-defect turbulence. Past a first threshold $R$ of order unity we find a slower growth rate, of $O(q^2)$; the numerically observed steady state is {\it phase-turbulent}: noisy but {\it aligned} on average. We present numerical evidence in three and two dimensions that this inertia driven flocking transition is continuous, with a correlation length that grows on approaching the transition. For much larger $R$ we find an aligned state linearly stable to perturbations at all $q$. Our predictions should be testable in suspensions of mesoscale swimmers [D Klotsa, Soft Matter \textbf{15}, 8946 (2019)].
Granular suspensions present a transition from a Newtonian rheology in the Stokes limit to a Bagnoldian rheology when inertia is increased. A custom rheometer which can be run in a pressure or a volume-imposed mode is used to examine this transition in the dense regime close to jamming. By varying systematically the interstitial fluid, shear rate, and packing fraction in volume-imposed measurements, we show that the transition takes place at a Stokes number of 10 independent of the packing fraction. Using pressure-imposed rheometry, we investigate whether the inertial and viscous regimes can b…
Non-Brownian suspensions present a transition from Newtonian behavior in the zero-shear limit to a shear thickening behaviour at a large shear rate, none of which is clearly understood so far. Here, we carry out numerical simulations of such an athermal dense suspension under shear, at an imposed confining pressure. This set-up is conceptually identical to the recent experiments of Boyer and co-workers [Phys. Rev. Lett. 107,188301 (2011)]. Varying the interstitial fluid viscosities, we recover the Newtonian and Bagnoldian regimes and show that they correspond to a dissipation dominated by visc…
The coalescence of liquid drops has conventionally been thought to have just two regimes when the drops are brought together slowly in vacuum or air: a viscous regime corresponding to the Stokes-flow limit and a later inertially-dominated regime. Recent work [Proc. Natl. Acad. Sci. 109, 6857 (2012)] found that the Stokes-flow limit cannot be reached in the early moments of coalescence, because the inertia of the drops cannot be neglected then. Instead, the drops are described by an "inertially limited viscous" regime, where surface tension, inertia, and viscous forces all balance. The dynamics…
For the Saffman-Taylor instability, the inertia of the fluid may become important for large Reynolds numbers Re. We investigate the effects of inertia on the width of the viscous fingers experimentally. We find that, due to inertia, the finger width can increase with increasing speed, contrary to what happens at small Re. We find that inertial effects need to be considered above a critical Weber number We. In this case it can be shown that the finger width is governed by a balance between viscous forces and inertia. This allows us to define a modified control parameter 1/B', which takes the co…
A scalar constitutive law was recently obtained for dense granular flows from a two-grain argument, both in the inertial regime (grain inertia) and in the viscous regime. As the resulting law is not exactly the same in both regimes, we here provide an expression for the crossover between both regimes.
Velocity relaxation of an elastic sphere immersed in a viscous incompressible fluid is studied on the basis of the equations of linear elasticity and the linearized Navier-Stokes equations. It is found that both translational motion after a sudden impulse and rotational motion after a sudden twist show jittery behavior in the long-time regime, with many reversals of velocity if the sphere is sufficiently flexible. In the extreme long-time regime the translational and rotational velocity relaxation functions each decay with a universal algebraic long-time tail. The added mass and the added mome…
Hysteresis is a major feature of the solid-liquid transition in granular materials. This property, by allowing metastable states, can potentially yield catastrophic phenomena such as earthquakes or aerial landslides. The origin of hysteresis in granular flows is still debated. However, most mechanisms put forward so far rely on the presence of inertia at the particle level. In this paper, we study the avalanche dynamics of non-Brownian suspensions in slowly rotating drums and reveal large hysteresis of the avalanche angle even in the absence of inertia. By using micro-silica particles whose in…
Recent experiments with a rotating probe immersed in weakly fluidized granular materials show a complex behavior on a wide range of timescales. Viscous-like relaxation at high frequency is accom- panied by an almost harmonic dynamical trapping at intermediate times, with possibly anomalous long time behavior in the form of super-diffusion. Inspired by the Itinerant Oscillator model for dif- fusion in molecular liquids, and other models with coupled thermostats acting at different timescales, here we discuss a new model able to account for fast viscous relaxation, dynamical trapping and super-d…
Three regimes of granular avalanches in fluids are put in light depending on the Stokes number St which prescribes the relative importance of grain inertia and fluid viscous effects, and on the grain/fluid density ratio r. In gas (r >> 1 and St > 1, e.g., the dry case), the amplitude and time duration of avalanches do not depend on any fluid effect. In liquids (r ~ 1), for decreasing St, the amplitude decreases and the time duration increases, exploring an inertial regime and a viscous regime. These regimes are described by the analysis of the elementary motion of one grain.
In this communication, we present a new lengthscale-based rheology for dense sheared particle suspensions as they transition from inertial- to viscous-dominated. We derive a lengthscale ratio using straightforward physics-based considerations for a particle subjected to pressure and drag forces. In doing so, we demonstrate that an appropriately chosen length-scale ratio intrinsically provides a consistent relationship between normal stress and system proximity to its ''jammed'' or solid-like state, even as a system transitions between inertial and viscous states, captured by a variable Stokes …
Micro- and nano-swimmers moving in a fluid solvent confined by structures that produce entropic barriers are often described by overdamped active Brownian particle dynamics, where viscous effects are large and inertia plays no role. However, inertial effects should be considered for confined swimmers moving in media where viscous effects are no longer dominant. Here, we study how inertia affects the rectification and diffusion of self-propelled particles in a two-dimensional asymmetric channel. We show that most of the particles accumulate at the channel walls as the masses of the particles in…
We probe the three dimensional flow structure and rheology of gravitational (non-density matched) suspensions for a range of driving rates in a split-bottom geometry. We establish that for sufficiently slow flows, the suspension flows as if it were a dry granular medium, and confirm recent theoretical modeling on the rheology of split-bottom flows. For faster driving, the flow behavior is shown to be consistent with the rheological behavior predicted by the recently developed "inertial number approaches for suspension flows.
Motivated from the classical expressions of the mean squared displacement and the velocity autocorrelation function of Brownian particles suspended either in a Newtonian viscous fluid or trapped in a harmonic potential, we show that for all time-scales the mean squared displacement of Brownian microspheres with mass $m$ and radius $R$ suspended in any linear, isotropic viscoelastic material is identical to the creep compliance of a linear mechanical network that is a parallel connection of the linear viscoelastic material with an inerter with distributed inertance, $m_R=\frac{m}{\text{6}πR}$. …
Finite difference method in prolate spheroidal coordinates for freely suspended spheroidal particles in linear flows of viscous and viscoelastic fluids
