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A linear and weakly nonlinear analysis of convection in a layer of shear-thinning fluids between two horizontal plates heated from below is performed. The objective is to examine the effects of the nonlinear variation of the viscosity with the shear rate on the nature of the bifurcation, the planform selection problem between rolls, squares and hexagons, and the consequences on the heat transfer coefficient. Navier’s slip boundary conditions are used at the top and bottom walls. The shear-thinning behaviour of the fluid is described by the Carreau model. By considering an infinitesimal perturbation, the critical conditions, corresponding to the onset of convection, are determined. At this stage, non-Newtonian effects do not come into play. The critical Rayleigh number decreases and the critical wavenumber increases when the slip increases. For a finite-amplitude perturbation, nonlinear effects enter in the dynamic. Analysis of the saturation coefficients at cubic order in the amplitude equations shows that the nature of the bifurcation depends on the rheological properties, i.e. the fluid characteristic time and shear-thinning index. For weakly shear-thinning fluids, the bifurcation is supercritical and the heat transfer coefficient increases, as compared with the Newtonian case. When the shear-thinning character is large enough, the bifurcation is subcritical, pointing out the destabilizing effect of the nonlinearities arising from the rheological law. Departing from the onset, the weakly nonlinear analysis is carried out up to fifth order in the amplitude expansion. The flow structure, the modification of the viscosity field and the Nusselt number are characterized. The competition between rolls, squares and hexagons is investigated. Unlike Albaalbaki & Khayat (J. Fluid. Mech., vol. 668, 2011, pp. 500–550), it is shown that in the supercritical regime, only rolls are stable near onset.

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Motivated by recent experimental results obtained in a low-Prandtl-number fluid (Jaletzky 1999), we study theoretically the rotating cylindrical annulus model with rigid boundary conditions. A boundary layer theory is presented which allows a systematic study of the linear properties of the system in the asymptotic regime of very fast rotation rates. It shows that the Stewartson layers have a (de)stabilizing influence at (high) low Prandtl numbers. In the weakly nonlinear regime and for low Prandtl numbers, a strong retrograde mean flow develops at quadratic order. The Poiseuille part of this mean flow is determined by an equation obtained by averaging of the Navier–Stokes equation. It thus gives rise to a new global-coupling term in the envelope equation describing modulated waves, which can be used for other systems. The influence of this global-coupling term on the sideband instabilities of the waves is studied. In the strongly nonlinear regime, the waves restabilize against these instabilities at small rotation rates, but they are destabilized by a short-wavelength mode at larger rotation rates. We also find an inversion in the dependence of the amplitude on the Rayleigh number at low Prandtl numbers and intermediate rotation rates.

A general reformulation of the Reynolds stresses created by two-dimensional waves breaking a translational or a rotational invariance is described. This reformulation emphasizes the importance of a geometrical factor: the slope of the separatrices of the wave flow. Its physical relevance is illustrated by two model systems: waves destabilizing open shear flows; and thermal Rossby waves in spherical shell convection with rotation. In the case of shear-flow waves, a new expression of the Reynolds-Orr amplification mechanism is obtained, and a good understanding of the form of the mean pressure and velocity fields created by weakly nonlinear waves is gained. In the case of thermal Rossby waves, results of a three-dimensional code using no-slip boundary conditions are presented in the nonlinear regime, and compared with those of a two-dimensional quasi-geostrophic model. A semi-quantitative agreement is obtained on the flow amplitudes, but discrepancies are observed concerning the nonlinear frequency shifts. With the quasi-geostrophic model we also revisit a geometrical formula proposed by Zhang to interpret the form of the zonal flow created by the waves, and explore the very low Ekman-number regime. A change in the nature of the wave bifurcation, from supercritical to subcritical, is found. IntroductionIn hydrodynamic stability theory and turbulence modelling, it is natural and customary to separate the velocity field into a mean flow V and a fluctuating part v. In the Navier-Stokes equation for V , the nonlinear term expressing the feedback of the fluctuating flow onto the mean flow is usually written as the divergence of the Reynolds stress tensorwhere the angle brackets indicate a suitable averaging. A good understanding or modelling of τ is therefore required to explain the form of the mean flow, and other mean properties of the flow, such as the flow rate and head losses, in the case of an open system for instance. The tensor τ is also quite important for energy since in purely hydrodynamical systems its contraction with the mean strain rate tensor is the only possible source of growth of the fluctuating kinetic energy, as shown in a landmark paper by Reynolds (1895) Busse (1983) in the case of a fluctuating field corresponding to a columnar quasi-two-dimensional wave. He noted in his § 3 (see also his figure 3), a link between the variations of the 'phase function' of the wave and the relevant cross-diagonal Reynolds stress, which was revisited by Zhang (1992). The reformulation proposed by Zhang for the Reynolds stress, however, is limited to a special form of the streamfunction. In the somewhat more general case of a two-dimensional, x, y, fluctuating field, Pedlosky (1987) established ( § 7.3, p. 502) a link between the product v x v y that controls the most important Reynolds stress, i.e. the cross-diagonal stress τ xy , and the slope of the streamlines of v. Pedlosky offered no simple formula for the average v x v y , however.The primary aim of this paper is to complement these pioneering works by propo...

Motivated by the experimental results of Liu and Ecke (1997, 1999), different models are developed to analyze the weakly nonlinear dynamics of the traveling-wave sidewall modes appearing in rotating Rayleigh-Bénard convection. These models assume fully rigid boundary conditions for the velocity field. At the linear level, this influences most strongly the critical frequencies: they appear to be proportional to the logarithm of the Coriolis number, which is twice the inverse of the Ekman number. An annular flow domain is considered. This multiply connected geometry is shown to lead generally to the existence of a global mean-flow mode proportional to the average, over the azimuthal coordinate, of the square of the modulus of the envelope of the waves. Because this mode feeds back on the active wave modes at cubic order, the resulting Ginzburg-Landau envelope equation contains a nonlocal term. This new term, however, vanishes in the large-gap limit relevant to the experiments of Liu and Ecke. As compared with previous theoretical work, the present models lead to reduced discrepancies with the results of these experiments concerning the coefficients of the envelope equation. It is also shown that the new nonlocal effects may be realized experimentally in a small-gap annular geometry if a small-Prandtl-number fluid is used, despite the fact that no regime of Benjamin-Feir instability is predicted to occur.

The onset of convection in a rotating cylindrical annulus with sloping conical boundaries is studied in the case where this slope increases with the radius. The critical modes assume the form of drifting spiralling columns attached to the inner cylindrical wall at moderate and large Prandtl numbers, but they become attached to the outer wall at low Prandtl numbers. These latter ‘equatorially attached’ modes are multicellular at intermediate rotation rates. Through a perturbation analysis which is validated by a numerical code, we show that all equatorially attached modes are quasi-inertial modes and analyse the physical mechanisms leading to multicells. This is done for both stress-free and no-slip boundary conditions. At finite amplitudes the convection generates a Reynolds stress which leads to the development of a mean zonal flow, and a geometrical analysis of the mechanisms leading to this zonal flow is presented. The influence of Ekman friction on the zonal flow is also studied.