Stability boundaries of roll and square 
convection in binary fluid mixtures with positive 
separation ratio

Huke, B.; Lücke, M.; Büchel, P.; Jung, Ch.

doi:10.1017/s0022112099007648

Cited by 26 publications

(31 citation statements)

References 25 publications

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“…[17,18,31,47,48,59], and, on the other hand, realistic and physical exploration of the spatial complexity of the Soret and other problems in a large container for negative [19,20,23,24,25,27,29,28,32,33,34,35,36,37,42,44,43,49,50,51,52,53,54,55,56,57,58] and positive [22,30,29,38,39,40,41,44,45,46,52,54,59,60,61] values of the separation ratio. The first line of research has led to an understanding of the mechanisms producing global bifurcations, period-doubling, and chaos in two-dimensional convection in a confined geometry.…”

Section: Introductionmentioning

confidence: 99%

Thermosolutal and binary fluid convection as a 2×2 matrix problem

Tuckerman

2001

Physica D: Nonlinear Phenomena

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We describe an interpretation of convection in binary fluid mixtures as a superposition of thermal and solutal problems, with coupling due to advection and proportional to the separation parameter S. Many of the properties of binary fluid convection are then consequences of generic properties of 2 × 2 matrices. The eigenvalues of 2 × 2 matrices varying continuously with a parameter r undergo either avoided crossing or complex coalescence, depending on the sign of the coupling (product of off-diagonal terms). We first consider the matrix governing the stability of the conductive state. When the thermal and solutal gradients act in concert (S > 0, avoided crossing), the growth rates of perturbations remain real and of either thermal or solutal type. In contrast, when the thermal and solutal gradients are of opposite signs (S < 0, complex coalescence), the growth rates become complex and are of mixed type. Surprisingly, the kinetic energy of nonlinear steady states is also governed by an eigenvalue problem very similar to that governing the growth rates. More precisely, there is a quantitative analogy between the growth rates of the linear stability problem for infinite Prandtl number and the amplitudes of steady states of the minimal five-variable Veronis model for arbitrary Prandtl number. For positive S, avoided crossing leads to a distinction between low-amplitude solutal and high-amplitude thermal regimes. For negative S, the transition between real and complex eigenvalues leads to the creation of branches of finite amplitude, i.e. to saddle-node bifurcations. The codimension-two point at which the saddle-node bifurcations disappear, leading to a transition from subcritical to supercritical pitchfork bifurcations, is exactly analogous to the Bogdanov codimension-two point at which the Hopf bifurcations disappear in the linear problem.

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Section: Introductionmentioning

confidence: 99%

Thermosolutal and binary fluid convection as a 2×2 matrix problem

Tuckerman

2001

Physica D: Nonlinear Phenomena

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“…However, roll structures do in general still exist in mixtures as stable structures for these parameters, not at onset, but at larger r. Figure 2 shows the CR instability boundaries of rolls in a parameter interval where an exchange of stability between rolls and squares at onset is predicted in [15]. One sees that the curvature of the CR boundary at the critical point diverges -a feature that follows also from cubic amplitude equations, see [16] -when this exchange occurs. For the parameters P r = 10, ψ = 0.01 of Fig.…”

Section: Stability Properties Of Rollsmentioning

confidence: 92%

“…In y-direction, however, all perturbation wavenumbers, say, b ≥ 0 have to be investigated. For a discussion of the effect of mean flow perturbations, e. g., δΦ 00n we refer to [16].…”

Section: Technical Remarksmentioning

confidence: 99%

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Convective Patterns in Binary Fluid Mixtures with Positive Separation Ratios

Huke

Lücke

2002

Thermal Nonequilibrium Phenomena in Fluid Mixtures

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Abstract. We summarize our findings about laterally periodic convection structures in binary mixtures in the Rayleigh-Bénard system for positive Soret effect. Stationary roll, square, and crossroll solutions and their stability are determined with a multimode Galerkin expansion. The oscillatory competition of squares and rolls in the form of crossroll oscillations is reviewed. They undergo a subharmonic bifurcation cascade where the oscillation period grows in integer steps as a consequence of an entrainment process.

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“…& Lücke (2002); Le Gal et al (1985); Dominguez-Lerma et al (1995); Barten et al (1989); Schöpf & Zimmermann (1993); Knobloch & Moore (1988); Cross & Kim (1988); Kolodner et al (1986); Fütterer & Lücke (2002); Touiri et al (1996); Ahlers & Rehberg (1986); Walden et al (1985)). In particular, the roll convection and the corresponding stability mechanisms are well known (see Huke et al (2000)). For the system with porous medium, however, the standard of knowledge is much less developed, although there has been some work concerning the stability of the ground state and monocellular flow by Charrier-Mojtabi et al (2007), Elhajjar et al (2008) and Sovran et al (2001).…”

Section: Introductionmentioning

confidence: 99%

Roll convection of binary fluid mixtures in porous media

et al. 2010

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We investigate theoretically the nonlinear state of ideal straight rolls in the RayleighBénard system of a fluid layer heated from below with a porous medium using a Galerkin method. Applying the Oberbeck-Boussinesq approximation, binary mixtures with positive separation ratio are studied and compared to one-component fluids. Our results for the structural properties of roll convection resemble qualitatively the situation in the Rayleigh-Bénard system without porous medium except for the fact that the streamlines of binary mixtures are deformed in the so-called Soret regime. The deformation of the streamlines is explained by means of the Darcy equation which is used to describe the transport of momentum. In addition to the properties of the rolls, their stability against arbitrary infinitesimal perturbations is investigated. We compute stability balloons for the pure fluid case as well as for a wide parameter range of Lewis numbers and separation ratios which are typical for binary gas and fluid mixtures. The stability regions of rolls are found to be restricted by a crossroll, a zigzag and a new type of oscillatory instability mechanism, which can be related to the crossroll mechanism.

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Stability boundaries of roll and square convection in binary fluid mixtures with positive separation ratio

Cited by 26 publications

References 25 publications

Thermosolutal and binary fluid convection as a 2×2 matrix problem

Thermosolutal and binary fluid convection as a 2×2 matrix problem

Convective Patterns in Binary Fluid Mixtures with Positive Separation Ratios

Roll convection of binary fluid mixtures in porous media

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