Philippe Maïssa

Observation of negative-frequency waves in a water tank: a classical analogue to the Hawking effect?

The conversion of positive-frequency waves into negative-frequency waves at the event horizon is the mechanism at the heart of the Hawking radiation of black holes. In black-hole analogues, horizons are formed for waves propagating in a medium against the current when and where the flow exceeds the wave velocity. We report on the first direct observation of negative-frequency waves converted from positive-frequency waves in a moving medium. The measured degree of mode conversion is significantly higher than that expected from the theory.

Horizon effects with surface waves on moving water

Surface waves on a stationary flow of water are considered in a linear model that includes the surface tension of the fluid. The resulting gravity-capillary waves experience a rich array of horizon effects when propagating against the flow. In some cases, three horizons (points where the group velocity of the wave reverses) exist for waves with a single laboratory frequency. Some of these effects are familiar in fluid mechanics under the name of wave blocking, but other aspects, in particular waves with negative co-moving frequency and the Hawking effect, were overlooked until surface waves were investigated as examples of analogue gravity (Schutzhold R and Unruh W G 2002 Phys. Rev. D 66 044019). A comprehensive presentation of the various horizon effects for gravity-capillary waves is given, with emphasis on the deep water/ short wavelength case kh1, where many analytical results can be derived. A similarity of the state space of the waves to that of a thermodynamic system is pointed out.

Pattern formation in bubbles emerging periodically from a liquid free surface

Abstract.Patterns formed by centimeter scale bubbles on the free surface of a viscous liquid are investigated in a cylindrical container. These bubbles emerge periodically at the surface and interact with each other in the central zone. Their radial emission, due to interaction and radial surface flow, leads to the formation of a variety of patterns. Different star-like and spiral patterns appear spontaneously by increasing the bubble emergence frequency. It is found that these patterns are due to a constant angular shift in the bubble emission direction. Measurements of this angular shift show a supercritical bifurcation accompanied by a transition from a pattern of two opposed straight arms to spiral patterns. By applying the tools and concepts from the study of leaf arrangement in botany (phyllotaxis), the recognized patterns and the mechanism of the pattern formation are discussed. Close similarities to the leaf arrangement are found in the behavior of the angular shift and the patterns. These findings suggest that the observed patterns are formed by a packing mechanism of successively appearing elements (bubbles), which is similar to that of the leaves at the earliest stage of phyllotaxis.

Wave blocking phenomenon of surface waves on a shear flow with a constant vorticity

Propagation of gravity-capillary surface waves on a background shear flow with a constant vorticity is studied and compared with the case when the background flow is uniform in depth. Under the assumption that the background flow gradually varies in the horizontal direction, the primary attention is paid to the wave blocking phenomenon; the effect of vorticity on this phenomenon is studied in detail. The conditions for wave blocking are obtained and categorized for different values of the governing dimensionless parameters: Froude number, dimensionless vorticity, and surface tension.

Axisymmetric annular curtain stability

A temporal stability analysis was carried out to investigate the stability of an axially moving viscous annular liquid jet subject to axisymmetric disturbances in surrounding co-flowing viscous gas media. We investigated in this study the effects of inertia, surface tension, the gas-to-liquid density ratio, the inner-to-outer radius ratio and the gas-to-liquid viscosity ratio on the stability of the jet. With an increase in inertia, the growth rate of the unstable disturbances is found to increase. The dominant (or most unstable) wavenumber decreases with increasing Reynolds number for larger values of the gas-to-liquid viscosity ratio. However, an opposite tendency for the most unstable wavenumber is predicted for small viscosity ratio in the same inertia range. The surrounding gas density, in the presence of viscosity, always reduces the growth rate, hence stabilizing the flow. There exists a critical value of the density ratio above which the flow becomes stable for very small viscosity ratio, whereas for large viscosity ratio, no stable flow appears in the same range of the density ratio. The curvature has a significant destabilizing effect on the thin annular jet, whereas for a relatively thick jet, the maximum growth rate decreases as the inner radius increases, irrespective of the surrounding gas viscosity. The degree of instability increases with Weber number for a relatively large viscosity ratio. In contrast, for small viscosity ratio, the growth rate exhibits a dramatic dependence on the surface tension. There is a small Weber number range, which depends on the viscosity ratio, where the flow is stable. The viscosity ratio always stabilizes the flow. However, the dominant wavenumber increases with increasing viscosity ratio. The range of unstable wavenumbers is affected only by the curvature effect.

Non-axisymmetric annular curtain stability

A stability analysis of non-axisymmetric annular curtain is carried out for an axially moving viscous jet subject in surrounding viscous gas media. The effect of inertia, surface tension, gas-to-liquid density ratio, inner-to-outer radius ratio, and gas-to-liquid viscosity ratio on the stability of the jet is studied. In general, the axisymmetric disturbance is found to be the dominant mode. However, for small wavenumber, the non-axisymmetric mode is the most unstable mode and the one likely observed in reality. Inertia and the viscosity ratio for non-axisymmetric disturbances show a similar stability influence as observed for axisymmetric disturbances. The maximum growth rate in non-axisymmetric flow, interestingly, appears at very small wavenumber for all inertia levels. The dominant wavenumber increases (decreases) with inertia for non-axisymmetric (axisymmetric) flow. Gas-to-liquid density ratio, curvature effect, and surface tension, however, exhibit an opposite influence on growth rate compared to axisymmetric disturbances. Surface tension tends to stabilize the flow with reductions of the unstable wavenumber range and the maximum growth rate as well as the dominant wavenumber. The dominant wavenumber remains independent of viscosity ratio indicating the viscosity ratio increases the breakup length of the sheet with very little influence on the size of the drops. The range of unstable wavenumbers is affected only by curvature in axisymmetric flow, whereas all the stability parameters control the range of unstable wavenumbers in non-axisymmetric flow. Inertia and gas density increase the unstable wavenumber range, whereas the radius ratio, surface tension, and the viscosity ratio decrease the unstable wavenumber range. Neutral curves are plotted to separate the stable and unstable domains. Critical radius ratio decreases linearly and nonlinearly with the wavenumber for axisymmetric and non-axisymmetric disturbances, respectively. At smaller Weber numbers, a wider unstable domain is predicted for non-axisymmetric modes. For both axisymmetric and non-axisymmetric modes, the disturbance frequency is found to be the same and equal to the negative of axial wavenumber. Finally, comparison between theory and existing experiment leads to good qualitative agreement. A more accurate comparison is not possible given the difference in flow conditions.

Horizon effects for surface waves in wave channels and circular jumps

Surface waves in classical fluids experience a rich array of black/white hole horizon effects. The dispersion relation depends on the characteristics of the fluid (in our case, water and silicon oil) as well as on the fluid depth and the wavelength regime. In some cases, it can be tuned to obtain a relativistic regime plus high-frequency dispersive effects. We discuss two types of ongoing analogue white-hole experiments: deep water waves propagating against a counter-current in a wave channel and shallow waves on a circular hydraulic jump.