Mathieu Jenny

Instabilities and transition of a sphere falling or ascending freely in a Newtonian fluid

The scenario of transition to chaos for a sphere falling or ascending under the action of gravity in a Newtonian fluid is investigated by numerical simulation. The mathematical formulation is parameterized using two non-dimensional parameters: the solid/fluid density ratio and the generalized Galileo number expressing the ratio between the gravity–buoyancy and viscosity effects. The study is carried out fully in this two-parameter space. The results show that for all density ratios the vertical fall or ascension becomes unstable via a regular axisymmetry breaking bifurcation. This bifurcation sets in slightly earlier for light spheres than for dense ones. A steady oblique fall or ascension follows before losing stability and giving way to an oscillating oblique movement. The secondary Hopf bifurcation is shown not to correspond to that of a fixed sphere wake for density ratios lower than 2.5, for which the oscillations have a significantly lower frequency. Trajectories of falling spheres become chaotic directly from the oblique oscillating regime. Ascending spheres present a specific behaviour before reaching a chaotic regime. The periodically oscillating oblique regime undergoes a subharmonic transition yielding a low-frequency oscillating ascension which is vertical in the mean (zigzagging regime). In all these stages of transition, the trajectories are planar with a plane selected randomly during the axisymmetry breaking. The chaotic regime appears to result from an interplay of a regular and of an additional Hopf bifurcation and the onset of the chaotic regime is accompanied by the loss of the remaining planar symmetry. The asymptotic chaotic states present an intermittent character, the relaminarization phases letting the subcritical plane and periodic trajectories reappear.

Nonvertical ascension or fall of a free sphere in a Newtonian fluid

It is shown that the system represented by a free sphere ascending or falling in a Newtonian fluid under the action of gravity buoyancy undergoes a regular, symmetry breaking bifurcation making the trajectory deviate from the vertical direction. The instability threshold expressed in terms of the asymptotic Reynolds number lies below that of a fixed sphere wake. The instability is shown to saturate and reach a fixed point corresponding to a straight oblique ascension (fall).

Primary instability of a Taylor-Couette flow with a radial stratification and radial buoyancy

We consider the primary instability onset of a Taylor-Couette flow with a radial stratification and radial volumic force. Our system corresponds to a simplified geophysical model that takes into account the effect of the buoyancy opposed to the centrifugal force. Three parameters of the physical problem, which correspond to the buoyancy, the diffusivity, and the rotational velocity, are considered to describe the onset of the primary instability. Using an efficient numerical method, we compute the threshold value of the Taylor number, which depends on the Froude and Schmidt numbers, and describe the most unstable mode provided by the linear analysis.

Shear-banding and Taylor-Couette instability in thixotropic yield stress fluids

We consider the flow of thixotropic yield stress fluids between two concentric cylinders. To account for the fluid thixotropy, we use Houskas model [Houska, Ph.D. thesis, Czech Technical University, Prague, 1981] with a single structural parameter driven by a kinetic equation. Because of the yield stress and the geometric inhomogeneity of the stress, only a part of the material in the gap may flow. Depending on the breakdown rate of the structural parameter, the constitutive relation can lead to a nonmonotonic flow curve. This nonmonotonic behavior is known to induce a discontinuity in the slope of the velocity profile within the flowing material, called shear banding. Thus, for fragile structures, a shear-banded flow characterized by a very sharp transition between the flowing and the static regions may be observed. For stronger structures, the discontinuity disappears and a smooth transition between the flowing and the static regions is observed. The consequences of the thixotropy on the linear stability of the azimuthal flow are studied in a large range of parameters. Although the thixotropy allows shear banding in the base flow, it does not modify fundamentally the linear stability of the Couette flow compared to a simple yield stress fluid. The apparent shear-thinning behavior depends on the thixotropic parameters of the fluid and the results about the onset of the Taylor vortices in shear-thinning fluids are retrieved. Nevertheless, the shear banding modifies the stratification of the viscosity in the flowing zone such that the critical conditions are mainly driven by the width of the flowing region.

Numerical Solution to Horizontal Zero-inertia, Viscous Dam-Break Problem

Debris flows such as avalanches and lahars differ from the classical dam-break problem of hydraulics due to the relative importance of viscous versus inertial forces in the momentum balance. An equation of motion describing debris flow in the limit of zero inertia is developed and solved using a converged finite difference numerical, in two limits: short time and long time. These solutions are then combined into a single, universal model.

Influence of Instabilities on the Trajectory of a Light Sphere

Recent numerical simulations of the wake of a fixed sphere have confirmed that hydrodynamic forces are likely to have a significant impact on the trajectory of a freely falling (or ascending) sphere. An ideally spherical body ceases to follow a straight vertical trajectory at the Reynolds number (based on its velocity U and diameter d) corresponding to the onset of the primary instability responsible for the breaking of axisymmetry in a fixed sphere wake, i.e. at Re = 212. This instability has been shown to generate a steady non axisymmetric flow with a symmetry plane containing the asymptotic flow velocity, the orientation of which is arbitrary, i.e. selected by any small perturbation at the instability onset. In this communication, we present further work focussed on the experimental investigation of the effect of instabilities on the trajectory of a free sphere. The axisymmetry breaking results in a lift and torque, the vector of lift lying in the symmetry plane and the torque being normal to this plane. This leads to the conclusion that a free-falling (ascending) sphere will be deviated from its vertical trajectory as soon as its Reynolds number reaches the threshold of 212. Moreover, the trajectory will be deflected in an arbitrarily selected vertical plane. An experimental setup has been implemented to investigate this effect. It consists of a 2.5 m high water tank with a .5 times .5 m cross section placed in an air-conditioned chamber allowing to control finely the asymptotic Reynolds number of small spheres (on the order of a mm in diameter) by varying the water temperature. Spheres of densities close to that of water, both lighter and heavier, are considered. The trajectories are investigated fully in three dimensions by processing of images of two cameras following the sphere movement. The preliminary results, presented here for polypropylene spheres lighter than water, confirm the numerically and theoretically predicted effect. After a short acceleration phase roughly in vertical direction the primary instability deflects the trajectories each time in a different vertical plane. The investigation of the fixed sphere wake showed the onset of a secondary Hopf-type instability at Re ≈ 275. The same type of instability develops clearly for free spheres. Unlike for the fixed sphere, the secondary instability is observed to dominate and to yield a wavy trajectory with a vertical mean direction.Copyright

Rheology of fiber suspensions using MRI

The suspensions of non-Brownian fibers are of interest for many applications. Although many studies concerning suspensions are available in the literature, most of them concern suspensions of spherical particles. In this paper, global and local rheology of fiber suspensions are explored near the jamming transition. A critical volume fraction is extracted from the experimental data. The value of this critical volume fraction is in agreement with the expected value of the concentration of rigid rods above which the isotropic phase becomes unstable. Moreover, non-reversible effects of the shearing are observed in flow curves because of the non-Brownian behavior of the studied fibers.

Bulk and local rheology in a dense and vibrated granular suspension

In this paper, we investigate experimentally the dynamics of particles in dense granular suspensions when both shear and external vibrations are applied. We study in detail how vibrations affect particle reorganization at the local scale and modify the apparent rheology. The nonlocal nature of the rheology when no vibrations are applied is evidenced, in agreement with previous numerical studies from the literature. It is also shown that vibrations induce structural reorganizations, which tend to homogenize the system and cancel the nonlocal properties.

Eigenmodes of a rotating spherical Stokes flow with a radial stratification and radial buoyancy

The eigenmodes of a rotating spherical Stokes flow with a radial stratification are computed. The radial buoyancy is taken into account to fit to a geophysical model. We study the stability of the eigenmodes in the parameters space. Indeed, the flow depends on the Froude and the Schmidt numbers which describe respectively the buoyancy and the mass diffusivity.