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Steady two-dimensional Rayleigh–Bénard convection between stress-free isothermal boundaries is studied via numerical computations. We explore properties of steady convective rolls with aspect ratios ${\\rm \\pi} /5\\leqslant \\varGamma \\leqslant 4{\\rm \\pi}$, where $\\varGamma$ is the width-to-height ratio for a pair of counter-rotating rolls, over eight orders of magnitude in the Rayleigh number, $10^3\\leqslant Ra\\leqslant 10^{11}$, and four orders of magnitude in the Prandtl number, $10^{-2}\\leqslant Pr\\leqslant 10^2$. At large $Ra$ where steady rolls are dynamically unstable, the computed rolls display $Ra \\rightarrow \\infty$ asymptotic scaling. In this regime, the Nusselt number $Nu$ that measures heat transport scales as $Ra^{1/3}$ uniformly in $Pr$. The prefactor of this scaling depends on $\\varGamma$ and is largest at $\\varGamma \\approx 1.9$. The Reynolds number $Re$ for large-$Ra$ rolls scales as $Pr^{-1} Ra^{2/3}$ with a prefactor that is largest at $\\varGamma \\approx 4.5$. All of these large-$Ra$ features agree quantitatively with the semi-analytical asymptotic solutions constructed by Chini & Cox (Phys. Fluids, vol. 21, 2009, 083603). Convergence of $Nu$ and $Re$ to their asymptotic scalings occurs more slowly when $Pr$ is larger and when $\\varGamma$ is smaller.

‘Storm oil’ – nearly water-insoluble oil poured into the ocean and acting as a surfactant – has been used since ancient times to smooth the waves on the ocean. It was first scientifically described by Benjamin Franklin (Phil. Trans. R. Soc. Lond., vol. 64, 1774, pp. 445–460). In a recent paper, by combining highly controlled experiments in a wave tank and direct numerical simulations, Erinin et al. (J. Fluid Mech., vol. 972, 2023, R5) have now beautifully revealed the strong effect of soluble surfactants on the dynamics of plunging breakers. Remarkably, it is not the change in surface tension which mainly matters, but the surface tension gradient which emerges through compression and dilation of the plunging breaker surface.

The general principles of evidence synthesis are that the relevant literature has to be sought and collected, omitting nothing of relevance, recognising the difficulties posed by the different types of research from which the evidence can be adduced and the highly variable quality of the available studies. The pieces of information or data included in the publications found and designated to be included in the review have to be extracted and evaluated, and the results have to be synthesised, interpreted, and presented. The terms used to describe the various types of evidence synthesis are bewildering and heterogeneous; they need re-evaluation, perhaps in a systematic review.

The common answer to the question “Why is the sky blue?” is usually Rayleigh scattering. In 1953 Edward Hulburt demonstrated that Rayleigh scattering accounts for 1/3 and ozone absorption for 2/3 of the blue colour of the zenith sky at sunset. In this study, an approach to quantify the contribution of ozone to the blue colour of the sky for different viewing geometries is implemented using the radiative transfer model SCIATRAN and the CIE (International Commission on Illumination) XYZ 1931 colour system. The influence of ozone on the blue colour of the sky is calculated for solar zenith angles of 10–90∘ and a wide range of viewing geometries. For small solar zenith angles, the influence of ozone on the blue colour of the sky is minor, as expected. However, the effect of ozone increases with increasing solar zenith angle. The calculations for the Sun at the horizon confirm Hulburt's estimation with remarkably good agreement. More stratospheric aerosols reduce the ozone contribution at and near the zenith for the Sun at the horizon. The exact contribution of ozone depends strongly on the assumed total ozone column. The calculations also show that the contribution of ozone increases with increasing viewing zenith angle and total ozone column. Variations in surface albedo as well as full treatment of polarised radiative transfer were found to have only minor effects on the contribution of ozone to the blue colour of the sky. Furthermore, with an observer at 10 km altitude an increase in the ozone influence can be seen.

We report a numerical analysis of the unforced break-up of free cylindrical threads of viscous Newtonian liquid whose interface is coated with insoluble surfactants, focusing on the formation of satellite droplets. The initial conditions are harmonic disturbances of the cylindrical shape with a small amplitude$\\unicode[STIX]{x1D716}$, and whose wavelength is the most unstable one deduced from linear stability theory. We demonstrate that, in the limit$\\unicode[STIX]{x1D716}\\rightarrow 0$, the problem depends on two dimensionless parameters, namely the Laplace number,$La=\\unicode[STIX]{x1D70C}\\unicode[STIX]{x1D70E}_{0}\\bar{R}/\\unicode[STIX]{x1D707}^{2}$, and the elasticity parameter,$\\unicode[STIX]{x1D6FD}=E/\\unicode[STIX]{x1D70E}_{0}$, where$\\unicode[STIX]{x1D70C}$,$\\unicode[STIX]{x1D707}$and$\\unicode[STIX]{x1D70E}_{0}$are the liquid density, viscosity and initial surface tension, respectively,$E$is the Gibbs elasticity and$\\bar{R}$is the unperturbed thread radius. A parametric study is presented to quantify the influence of$La$and$\\unicode[STIX]{x1D6FD}$on two key quantities: the satellite droplet volume and the mass of surfactant trapped at the satellite’s surface just prior to pinch-off,$V_{sat}$and$\\unicode[STIX]{x1D6F4}_{sat}$, respectively. We identify a weak-elasticity regime,$\\unicode[STIX]{x1D6FD}\\lesssim 0.05$, in which the satellite volume and the associated mass of surfactant obey the scaling law$V_{sat}=\\unicode[STIX]{x1D6F4}_{sat}=0.0042La^{1.64}$for$La\\lesssim 2$. For$La\\gtrsim 10$,$V_{sat}$and$\\unicode[STIX]{x1D6F4}_{sat}$reach a plateau of about$3\\,\\%$and$2.9\\,\\%$, respectively,$V_{sat}$being in close agreement with previous experiments of low-viscosity threads with clean interfaces. For$La<7.5$, we reveal the existence of a discontinuous transition in$V_{sat}$and$\\unicode[STIX]{x1D6F4}_{sat}$at a critical elasticity,$\\unicode[STIX]{x1D6FD}_{c}(La)$, with$\\unicode[STIX]{x1D6FD}_{c}\\rightarrow 0.98$for$La\\lesssim 0.2$, such that$V_{sat}$and$\\unicode[STIX]{x1D6F4}_{sat}$abruptly increase at$\\unicode[STIX]{x1D6FD}=\\unicode[STIX]{x1D6FD}_{c}$for increasing$\\unicode[STIX]{x1D6FD}$. The jumps experienced by both quantities reach a plateau when$La\\lesssim 0.2$, while they decrease monotonically as$La$increases up to$La=7.5$, where both become zero.

Thermoacoustic instabilities are one of the most challenging problems faced by gas turbine and rocket motor manufacturers. The key instability mechanism is described by the Rayleigh criterion. The Rayleigh criterion does not directly show how to alter a system to make it more stable. This is the objective of sensitivity analysis. Because thermoacoustic systems have many design parameters, adjoint sensitivity analysis has been proposed to obtain all the sensitivities with one extra calculation. Although adjoint sensitivity analysis can be carried out in both the time and the frequency domain, the frequency domain is more natural for a linear analysis. Perhaps surprisingly, the Rayleigh criterion has not yet been rigorously derived and comprehensively interpreted in the frequency domain. The contribution of this theoretical paper is threefold. First, the Rayleigh criterion is interpreted in the frequency domain with integral formulae for the complex eigenvalue. Second, the first variation of the Rayleigh criterion is calculated both in the time and frequency domain, both with and without Lagrange multipliers (adjoint variables). The Lagrange multipliers are physically related to the system’s observables. Third, an adjoint Rayleigh criterion is proposed. The paper also points out that the conclusions of Juniper ( Phys. Rev. Fluids , vol. 3, 2018, 110509) apply to the first variation of the Rayleigh criterion, not to the Rayleigh criterion itself. The mathematical relations of this paper can be used to compute sensitivities directly from measurable quantities to enable optimal design.

It is well known that when a horizontal layer of fluid is heated from below, a thermal boundary layer of less dense and hot fluid rises up. When this boundary layer becomes unstable, convective motion in the fluid above sets in. Forecasting when instabilities take place is essential. When a salt dissolved in a fluid saturating a porous medium heated from below is considered, simultaneous mass diffusion and thermal diffusion occur. Unlike the diffusion of heat, the diffusion of salt can take place only through the fluid phase, so an additional physical effect has to be considered: the Soret effect, that is the mass flux created by a temperature gradient. In the present paper the onset of convection in a rotating layer of bi-disperse porous medium saturated by a binary fluid mixture, taking into account the Soret effect , is analysed. Linear stability analysis is performed in order to determine the instability thresholds for the onset of convection via a steady state (stationary convection) and via an oscillatory state (oscillatory convection). Nonlinear stability analysis is performed to obtain the global stability threshold with respect to the L 2 -norm.

We calculate the steady acoustic flow – the steady drift of fluid mass or acoustic streaming appearing along the path of an acoustic stimulus – in porous media. In particular, we suggest a mechanism to explain acoustic contributions to mass transport in porous media at geological, unit operation and lab-on-a-chip length scales. We study several cases of steady acoustic flow for a planar acoustic wave whose wavelength is large compared with the pore size. We commence our analysis at the ideal limit of same acoustic properties in the solid and fluid. The effective flow may then be treated intuitively according to the Darcy equation for flow through porous media in addition to a correction for the average azimuth of the pores compared with the acoustic path. We further consider the framework of a rigid porous frame, where the presence of a flow forcing mechanism resulting from the viscous dissipation of the acoustic wave at the solid surface of the pores hinders the intuitive application of the Darcy equation. However, we show that the steady acoustic flow in this case may be written as a quasi-Darcy-type equation. The analysis is conducted by a detailed calculation of the transport of mass through cylindrical pores of similar size but arbitrary azimuth compared with the acoustic path. We consider large, medium and small pore diameter limits relative to the viscous penetration length of the acoustic wave near the pore surface.

\"Cumulative meta-analysis\" describes a statistical procedure to calculate, retrospectively, summary estimates from the results of similar trials every time the results of a further trial in the series had become available. In the early 1990 s, comparisons of cumulative meta-analyses of treatments for myocardial infarction with advice promulgated through medical textbooks showed that research had continued long after robust estimates of treatment effects had accumulated, and that medical textbooks had overlooked strong, existing evidence from trials. Cumulative meta-analyses have subsequently been used to assess what could have been known had new studies been informed by systematic reviews of relevant existing evidence and how waste might have been reduced. We used a systematic approach to identify and summarise the findings of cumulative meta-analyses of studies of the effects of clinical interventions, published from 1992 to 2012. Searches were done of PubMed, MEDLINE, EMBASE, the Cochrane Methodology Register and Science Citation Index. A total of 50 eligible reports were identified, including more than 1,500 cumulative meta-analyses. A variety of themes are illustrated with specific examples. The studies showed that initially positive results became null or negative in meta-analyses as more trials were done; that early null or negative results were over-turned; that stable results (beneficial, harmful and neutral) would have been seen had a meta-analysis been done before the new trial; and that additional trials had been much too small to resolve the remaining uncertainties. This large, unique collection of cumulative meta-analyses highlights how a review of the existing evidence might have helped researchers, practitioners, patients and funders make more informed decisions and choices about new trials over decades of research. This would have led to earlier uptake of effective interventions in practice, less exposure of trial participants to less effective treatments, and reduced waste resulting from unjustified research.

More than 2,300 years ago, Aristotle pondered the shapes of beach stones and proposed that the abrasion of a pebble is more rapid at locations farther from its center, because those regions can receive greater abrasive impulses. He concluded that beach stones evolve toward spheres. But the pebbles used in designs from Stone Age ceremonies (see images to the right) to modern art are not spherical at all. Rather, they have the familiar oval shape of the stones shown in the image on page 170 of a beach on Banks Peninsula in New Zealand. The first laboratory studies of the shapes of beach stones evolving under frictional abrasion began in the 20th century with the experimental and theoretical work of Robert Strutt, the fourth Lord Rayleigh (son and biographer of the third Lord Rayleigh, who was awarded a 1904 Nobel prize for his discovery of argon).