Hexagons and triangles in the Rayleigh–Bénard problem: quintic-order equations on a hexagonal lattice

Fujimura, Kikuo; Yamada, Syouko

doi:10.1098/rspa.2007.0340

Cited by 7 publications

(3 citation statements)

References 29 publications

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“…Its real part s r is the growth-rate of the disturbance and the imaginary part allows to define the axial phase velocity. Substituting (40) into (38) and 39:…”

Section: Linear Stability Analysismentioning

confidence: 99%

Taylor-vortex flow in shear-thinning fluids

et al. 2019

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The present paper deals with the Taylor-Couette flow of shear-thinning fluids. It focuses on the first principles understanding the influence of the viscosity stratification and the nonlinear variation of the effective viscosity µ with the shear rateγ on the flow structure in the Taylor vortex flow regime. A wide gap configuration (η = 0.4) is mainly considered. A weakly nonlinear analysis, using the amplitude expansion method at high order is adopted as a first approach to study nonlinear effects. For the numerical computation, the shear-thinning behavior is described by the Carreau model. The rheological parameters are varied in a wide range. The results indicate that the flow field undergoes a significant change as shear-thinning effects increase. First, vortices are squeezed against the inner wall and the center of the patterns are shifted axially towards the radial outflow boundaries (z = 0, z/λ z = 1). This axial shift leads to increasing concentration of vorticity at these positions. The outflow becomes more stronger than the inflow and the inflow zone, where the vorticity is low, increases accordingly. Nevertheless, the strength of the vortices relative to the velocity of the inner cylinder is weaker. Second, the pseudo-Nusselt number, ratio of the torque to that obtained in the laminar flow, decreases. Third, higher harmonics become more relevant and grow faster with Reynolds number. Finally, the modification of the viscosity field is described.

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“…Its real part s r is the growth-rate of the disturbance and the imaginary part allows to define the axial phase velocity. Substituting (40) into (38) and 39:…”

Section: Linear Stability Analysismentioning

confidence: 99%

Taylor-vortex flow in shear-thinning fluids

et al. 2019

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“…Amplitude equations at quintic order as given by Hoyle (2006), Fujimura & Yamada (2008), are recalled, and their stationary solutions and stability in the supercritical regime, in the range of validity of our analysis, is studied.…”

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confidence: 99%

Weakly nonlinear analysis of Rayleigh–Bénard convection in shear-thinning fluids: nature of the bifurcation and pattern selection

et al. 2015

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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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“…It occurs because the symmetry about the mid plane of the system is broken as happens in case of non-Boussinesq convection. Amplitude equation of quintic order (Fujimura and Yamada 2008) shows coexistence of hexagons with rolls above the critical value irrespective of the symmetric nature of the boundaries. Matthews (1998) proved that regular hexagon cannot exist in finite domain if the aspect ratio of a rectangular container is a rational number.…”

Section: Introductionmentioning

confidence: 96%

Energy-conserving model of hexagonal pattern in Rayleigh-Bénard convection

Mondal¹,

Das²

2022

Fluid Dyn. Res.

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We have constructed an energy-conserving sixteen mode dynamical system to model hexagonal pattern in Rayleigh-Bénard convection of Boussinesq fluids with symmetric stress-free thermally conducting boundaries. The model shows stable roll pattern at the onset of convection. Hexagon is found to appear in the system via sausage and (or) stationary rhombus patterns. Both up and down hexagons arise periodically or chaotically with roll, sausage and rhombus patterns. Hexagonal patterns exist for all values of the Prandtl number, 1 ≤ Pr ≤ 5 explored here. However the pattern is more prominent for small Pr and k < kc , where k denotes the wave number. The plot of Nusselt number matches with previous theoretical result. In dissipationless limit, the total energy and the unavailable energy are constants though the kinetic energy, the potential energy and the available energy vary with time. The derived model does not diverge for large values of Rayleigh number Ra.

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Hexagons and triangles in the Rayleigh–Bénard problem: quintic-order equations on a hexagonal lattice

Cited by 7 publications

References 29 publications

Taylor-vortex flow in shear-thinning fluids

Taylor-vortex flow in shear-thinning fluids

Weakly nonlinear analysis of Rayleigh–Bénard convection in shear-thinning fluids: nature of the bifurcation and pattern selection

Energy-conserving model of hexagonal pattern in Rayleigh-Bénard convection

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