The effect of rotation on the Rayleigh-Bénard stability threshold

The standard method used to solve the Rayleigh-Bénard linear stability problem for a rotating fluid leads to a complex expression which can only be evaluated numerically. Here the problem is solved by a different method similar to that used in a recent paper on the non-rotating case [A. Prosperetti, “A simple analytic approximation to the Rayleigh-Bénard stability threshold,” Phys. Fluids 23, 124101 (2011)10.1063/1.3662466]. In principle the method leads to an exact result which is not simpler than the standard one. Its value lies in the fact that it is possible to obtain from it an approximate explicit analytic expression for the dependence of the Rayleigh number on the wave number of the perturbation and the rate of rotation at marginal stability conditions. Where the error can be compared with exact results in the literature, it is found not to exceed a few percent over a very broad Taylor number range. The relative simplicity of the approach permits us, among others, to account for the effects of a finite thermal conductivity of the plates, which have not been studied befor

A brief summary of L. van Wijngaarden's work up till his retirement

This paper attempts to provide an overview of Professor Leen van Wijngaarden's scientific work by briefly summarizing a number of his papers. The review is organized by topic and covers his work on pressure waves in bubbly liquids, bubble dynamics, two-phase flow, standing waves in resonant systems, and flow cavitation noise. A list of publications up till his retirement in March 1997 is provided in the Appendix

A numerical method for the dynamics of non-spherical cavitation bubbles

A boundary integral numerical method for the dynamics of nonspherical cavitation bubbles in inviscid incompressible liquids is described. Only surface values of the velocity potential and its first derivatives are involved. The problem of solving the Laplace equation in the entire domain occupied by the liquid is thus avoided. The collapse of a bubble in the vicinity of a solid wall and the collapse of three bubbles with collinear centers are considered

On the characteristics of the equations of motion for a bubbly flow and the related problem of critical flow

For the study of transients in gas-liquid flows, the equations of the so-called separated flow model are inadequate, because they possess, in the general case where gas and liquid move at different velocities, complex characteristics. This paper is concerned with the equations of motion for bubbly flow. The equations are discussed with emphasis on the aspects of relative motion and the characteristics are calculated. It is found that all characteristics are real. The results are used to establish a relation between gas velocity, liquid velocity, void fraction and sound velocity at critical flow. This relation agrees very well with experimental data for these quantities as measured by Muir and Eichhorn in the throat of a converging-diverging nozzle

Oscillations of a gas pocket on a liquid-covered solid surface

The dynamic response of a gas bubble entrapped in a cavity on the surface of a submerged solid subject to an acoustic field is investigated in the linear approximation. We derive semi-analytical expressions for the resonance frequency, damping and interface shape of the bubble. For the liquid phase, we consider two limit cases: potential flow and unsteady Stokes flow. The oscillation frequency and interface shape are found to depend on two dimensionless parameters: the ratio of the gas stiffness to the surface tension stiffness, and the Ohnesorge number, representing the relative importance of viscous forces. We perform a parametric study and show, among others, that an increase in the gas pressure or a decrease in the surface tension leads to an increase in the resonance frequency until an asymptotic value is reached

Inert gas accumulation in sonoluminescing bubbles

In this paper we elaborate on the idea [Lohse et al., Phys. Rev. Lett. 78, 1359-1362 (1997)] that (single) sonoluminescing air bubbles rectify argon. The reason for the rectification is that nitrogen and oxygen dissociate and their reaction products dissolve in water. We give further experimental and theoretical evidence and extend the theory to other gas mixtures. We show that in the absence of chemical reactions (e.g., for inert gas mixtures) gas accumulation in strongly acoustically driven bubbles can also occur.Comment: J. Chem. Phys., in press (to appear in November 1997), 30 pages, 15 eps-figure

Effective velocity boundary condition at a mixed slip surface

This paper studies the nature of the effective velocity boundary conditions for liquid flow over a plane boundary on which small free-slip islands are randomly distributed. It is found that, to lowest order in the area fraction $\beta$ covered by free-slip regions with characteristic size $a$, a macroscopic Navier-type slip condition emerges with a slip length of the order of $a\beta$. The study is motivated by recent experiments which suggest that gas nano-bubbles may form on solid walls and may be responsible for the appearance of a partial slip boundary conditions for liquid flow. The results are also relevant for ultra-hydrophobic surfaces exploiting the so-called ``lotus effect''.Comment: 14 pages, 1 figur

Phase Diagrams for Sonoluminescing Bubbles

Sound driven gas bubbles in water can emit light pulses. This phenomenon is called sonoluminescence (SL). Two different phases of single bubble SL have been proposed: diffusively stable and diffusively unstable SL. We present phase diagrams in the gas concentration vs forcing pressure state space and also in the ambient radius vs gas concentration and vs forcing pressure state spaces. These phase diagrams are based on the thresholds for energy focusing in the bubble and two kinds of instabilities, namely (i) shape instabilities and (ii) diffusive instabilities. Stable SL only occurs in a tiny parameter window of large forcing pressure amplitude $P_a \sim 1.2 - 1.5$atm and low gas concentration of less than $0.4\%$ of the saturation. The upper concentration threshold becomes smaller with increasing forcing. Our results quantitatively agree with experimental results of Putterman's UCLA group on argon, but not on air. However, air bubbles and other gas mixtures can also successfully be treated in this approach if in addition (iii) chemical instabilities are considered. -- All statements are based on the Rayleigh-Plesset ODE approximation of the bubble dynamics, extended in an adiabatic approximation to include mass diffusion effects. This approximation is the only way to explore considerable portions of parameter space, as solving the full PDEs is numerically too expensive. Therefore, we checked the adiabatic approximation by comparison with the full numerical solution of the advection diffusion PDE and find good agreement.Comment: Phys. Fluids, in press; latex; 46 pages, 16 eps-figures, small figures tarred and gzipped and uuencoded; large ones replaced by dummies; full version can by obtained from: http://staff-www.uni-marburg.de/~lohse

Orthogonal, solenoidal, three-dimensional vector fields for no-slip boundary conditions

Viscous fluid dynamical calculations require no-slip boundary conditions. Numerical calculations of turbulence, as well as theoretical turbulence closure techniques, often depend upon a spectral decomposition of the flow fields. However, such calculations have been limited to two-dimensional situations. Here we present a method that yields orthogonal decompositions of incompressible, three-dimensional flow fields and apply it to periodic cylindrical and spherical no-slip boundaries.Comment: 16 pages, 2 three-part figure

A note on the effective slip properties for microchannel flows with ultra-hydrophobic surfaces

A type of super-hydrophobic surface consists of a solid plane boundary with an array of grooves which, due to the effect of surface tension, prevent a complete wetting of the wall. The effect is greatest when the grooves are aligned with the flow. The pressure difference between the liquid and the gas in the grooves causes a curvature of the liquid surface resisted by surface tension. The effects of this surface deformation are studied in this paper. The corrections to the effective slip length produced by the curvature are analyzed theoretically and a comparison with available data and related mathematical models is presented.Comment: 19 pages, 5 figure