Y. Couder

EXPERIMENTAL AND NUMERICAL STUDY OF VORTEX COUPLES IN TWO-DIMENSIONAL FLOWS

Two-dimensional turbulence is investigated experimentally in thin liquid films. This study shows the spontaneous formation of couples of opposite-sign vortices in von Karman wakes. The structure of these couples, their behaviour and their role in turbulent flows is then studied using both a numerical simulation and laboratory results.

On the hydrodynamics of soap films

Abstract Several experiments aiming at the exploration of the hydrodynamical properties of soap films are presented. Their interpretation takes into account the very specific equation of state of these films. It is shown that on short time scales each element of the film moves as a whole so that the film can be considered as a two-dimensional fluid with a local density proportional to its thickness. When set horizontally, quasi two-dimensional turbulent flows can be obtained. The film behaves as an incompressible fluid whenever the motions occur at velocities small compared to the velocity of its elastic waves. An estimate of the role of air friction is given. The static quasi equilibrium of a film when set vertical is discussed. Phenomena equivalent to the rise of buoyant bubbles can be obtained. It is shown that lee waves can also be generated confirming that a vertical soap film has the dynamical properties of a two-dimensional density stratified fluid.

Turning a plant tissue into a living cell froth through isotropic growth

The forms resulting from growth processes are highly sensitive to the nature of the driving impetus, and to the local properties of the medium, in particular, its isotropy or anisotropy. In turn, these local properties can be organized by growth. Here, we consider a growing plant tissue, the shoot apical meristem of Arabidopsis thaliana. In plants, the resistance of the cell wall to the growing internal turgor pressure is the main factor shaping the cells and the tissues. It is well established that the physical properties of the walls depend on the oriented deposition of the cellulose microfibrils in the extracellular matrix or cell wall; this order is correlated to the highly oriented cortical array of microtubules attached to the inner side of the plasma membrane. We used oryzalin to depolymerize microtubules and analyzed its influence on the growing meristem. This had no short-term effect, but it had a profound impact on the cell anisotropy and the resulting tissue growth. The geometry of the cells became similar to that of bubbles in a soap froth. At a multicellular scale, this switch to a local isotropy induced growth into spherical structures. A theoretical model is presented in which a cellular structure grows through the plastic yielding of its walls under turgor pressure. The simulations reproduce the geometrical properties of a normal tissue if cell division is included. If not, a “cell froth” very similar to that observed experimentally is obtained. Our results suggest strong physical constraints on the mechanisms of growth regulation.

Experimental and numerical investigation of a forced circular shear layer

In a previous article we introduced a dissipative circular geometry in which stationary states of the shear flow instability were obtained. We show here that the dynamical behaviour of this flow depends strongly on the aspect ratio of the cell. In large cells, where the number of vortices is large, transitions from a mode with m vortices to a mode with ( m −1) vortices occur through localized processes. In contrast to that situation, in small cells, transition takes place after a series of bifurcations which correspond to the successive breaking of all the symmetries of the flow. We show that, provided an adequate forcing term is introduced, a two-dimensional numerical simulation of this flow is sufficient to recover all the dynamical processes which characterize the experimental flow.

A shear-flow instability in a circular geometry

A circular shear zone is created in a thin layer of fluid. The Kelvin-Helmholtz instability induces regular, steady patterns of m vortices. The experimental conditions are such that neither the centrifugal nor the Coriolis forces play a role in the motion. The state of the flow is defined by a Reynolds number, the value of which is controlled by the imposed velocities. The pattern of vortices can be characterized by its wavevector k or by m , the order of its symmetry. As k is quantized, its evolution, due to an increase or a decrease of the controlled stress, leads to transitions between patterns of different m. The transitory states between different symmetries are investigated. The experiments are performed with a soap film which provides a new type of visualization of an air flow.

Dendritic growth in the Saffman-Taylor experiment

Saffman-Taylor experiments are conducted in a circular axisymmetric Hele-Shaw cell. We show that when a small pre-existent isolated bubble comes into contact with the tip of a finger, this finger starts growing faster, its radius of curvature at the extremity is reduced, and it takes a parabolic shape. When its velocity is large, this finger is affected by dendritic instabilities. Although they occur here in an isotropic system, these dendrites are directly comparable to those observed in crystallization fronts.

The Saffman-Taylor instability: from the linear to the circular geometry

The Saffman–Taylor fingers are studied in cells that have the form of sectors of a disk. The less viscous fluid can be injected at the apex (divergent flow) or at the periphery (convergent flow). As in the linear geometry, at large velocities, a unique finger tends to occupy a well determined fraction λ of the cell angular width. This fraction is a function of the angle of the cell, being larger than 0.5 in the divergent case and smaller in the convergent case. In both cases these fractions tend linearly toward λ=0.5 when the angle of the cell tends to zero. In support of recent theories, these results show how the selection is changed when the geometry induces an increase or a decrease of the curvature of the profiles. The formation of fingers in the circular geometry is revisited. In a divergent flow, the circular front appears to break into independent parts so that each finger grows as if it were contained in a sector shaped cell. The rate of occupancy of the cell by one of the fluids as a function of...

On the tip-splitting instability of viscous fingers

The instabilities of Saman{Taylor viscous ngers are revisited experimentally in the standard linear channel as well as in wedges of angle 0. The local destabilization of a nger occurs by a splitting of its tip and results in the formation of two branches separated by a fjord. It is shown that, in a rst approximation, the central line of a fjord follows a curve normal to the successive proles of stable ngers. These normal curves are computed analytically for the Saman{Taylor nger in a linear cell and numerically for the wedges. The length of a fjord is critically sensitive to the position of the initial destabilization of the nger. The nearer to the tip it occurs, the longer the fjord will be. Assuming a uniform spatial distribution of the disturbances in the central part of the nger front it is possible to predict the size distribution of the lateral branches. In the linear channel the probability of branches larger than the channel width is negligible. For wedges of increasing angle the probability of large secondary branches increases. Finally, for wedges with 0 larger than approximately 90 innite fjords separating two long-lived structures are observed. Our experimental results also suggest a generalization of the denition of virtual cells. With this new denition it is possible to show that the increasing complexity of the patterns corresponds to a hierarchy of virtual cells of various sizes.

Transition to Chaos by Spatio-Temporal Intermittency in Directional Viscous Fingering

We present a statistical description of the transition to chaos in a directional viscous fingering experiment. In this extended one-dimensional system the order-disorder transition follows a scenario of spatio-temporal intermittency and appears as a second-order transition. The critical exponents of the transition are determined and compared with other systems exhibiting a similar behaviour. The values of these exponents are discussed.

An experiment on two aspects of the interaction between strain and vorticity

Presented here are two results concerning the interaction between vorticity and strain. Both are obtained experimentally by investigating the hyperbolic flow created in Taylors four-roll mill. This pure straining flow becomes intrinsically unstable through a supercritical bifurcation to form an array of counter-rotating vortices aligned in the stretching direction. The dimensionless parameter characterizing the flow is the internal Reynolds number Re = γΔ 2 /v based on the velocity gradient γ and on the gap between the rollers A, and the threshold value is Re c = 17