Instability Benard

Kauffman also considers that autocatalytic reactions are a necessary precondition for biogenesis processes, as they can wind themselves up via self-amplifying feedback processes until a critical boundary has been reached. The next step would then be the transition from autocatalysis to self-organisation, similar to the transition from unstructured water to convection cells (the Benard instability) (Davies, 2000). 

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We can illustrate the salient features of convective dispersal by choosing a simple velocity distribution in a rectangular convection cell (0associated with the onset of Benard instability in the conditions of Boussinesq approximations (e.g., Turcotte and Schubert, 1982). Let us make the calculation for the so-called free-slip conditions, which permit free movement along the boundaries, both vertical and horizontal, such as a convection cell which would be limited by no rigid boundary. From Turcotte and Schubert (1982), we take the velocity field to be... 

Remark. Instability and bistability are defined as properties of the macroscopic equation. The effect of the fluctuations is merely to make the system decide to go to one or the other macroscopically stable point. Similarly the Taylor instability and the Benard cells are consequences of the macroscopic hydrodynamic equations. ) Fluctuations merely make the choice between different, equally possible macrostates, and, in these examples, determine the location of the vortices or of the cells in space. (In practice they are often overruled by extraneous influences, such as the presence of a boundary.) Statements that fluctuations shift or destroy the bistability are obscure, because on the mesoscopic level there is no sharp separation between stable and unstable systems. Some authors call a mesostate (i.e., a probability distribution P) bistable when P has two maxima, however flat. This does not correspond to any observable fact, however, unless the maxima are well-separated peaks, which can each be related to separate macrostates, as in (1.1). 

Remark. A great deal of attention has been paid in recent years to non-equilibrium stationary processes that are unstable and also extended in space. They can give rise to different phases that exist side by side, so that translation symmetry is broken. The name dissipative structures has been coined for them, and the prime examples are the Benard cells and the Zhabotinski reactions, but they also occur in biology and meteorology. However, these are features of the macroscopic equations. They are only relevant for fluctuation theory inasmuch as the fluctuation becomes very large at the point where the instability sets in. The critical fluctuations in XIII.5 are an example. There are many more varieties, in particular in the case of more variables. 

HYDRODYNAMIC AND HYDROMAGNETIC STABILITY. S. Chandrasekhar. Lucid examination of the Rayleigh-Benard problem clear coverage of the theory of instabilities causing convection. 704pp. 5b x 8b. 64071-X Pa. 12.95... 

In Fig. 3.5, visualization sequences are shown for the Case 3. In this case of non-rotating translating cylinder, no violent instability was seen to occur for two reasons. Firstly the imposed disturbance field, as given by Eqn. (3.3.1) has no captive vortex i.e.F = 0) as the cylinder does not rotate while translating. Secondly, if there are shed vortices present, they will be very weak and Benard- Karman vortex street is seen to affect the flow weakly far downstream of the translating cylinder - only at earlier times. 

Provansal, M., Mathis, C. and Boyer, L. (1987). Benard-von Karman instability transient and forced regimes. J. Fluid Mech., 182, 1-22. 

Dissipative structures arise only in strongly nonequilibrium systems, with the states described by nonhnear equations for internal macro parameters. The emergence of the Benard cells in fluids can be described using non hnear differential equations of hydrodynamics coupled with Lyapunov s analysis of the instability of the respective solutions. It is shown that the solution of hydrodynamic equations related to a resting fluid and normal heat transfer becomes unstable at AT > AT, and a new stable convection mode is established in the fluid. 

The first problem considered is the classic problem of Rayleigh-Benard convection -namely the instability that is due to buoyancy forces in a quiescent fluid layer that is heated... 

It is worthwhile illustrating the proof of the principle of exchange of stabilities for the Rayleigh-Benard problem. Not only will this allow us to discuss the derivation of instability criteria for the case of no-slip boundaries, but the approach to proving this principle can also be applied to other problems. 

In this section, we consider the classic problem of a fluid layer of depth d, with an upper surface that is an interface with air that is maintained at an ambient temperature 7o. The fluid layer is heated from below, and we shall assume that the lower fluid boundary is isothermal with temperature T (> To). This problem sounds exactly like the Rayleigh-Benard problem with a free upper surface. However, we consider the fluid layer to be very thin (i.e., d small) so that the Rayleigh number, which depends on d3, is less than the critical value for this configuration. Nevertheless, as previously suggested, the fluid layer may still undergo a convective motion that is due to Marangoni instability. 

The governing equations for the linear stability theory are the same as for the Rayleigh-Benard problem, namely (12-215), except that it is customary to drop the buoyancy terms because these are of secondary importance for very thin fluid layers where Marangoni instabilities are present but Ra <neutral state. Assuming that... 

Problem 12-11. Marangoni Instability (The Principle of Exchange of Stabilities). Following the procedure that was outlined in Section F for the Rayleigh-Benard problem, prove that the principle of exchange of stabilities is valid for the Marangoni instability problem (Section H). 

Problem 12-13. Raleigh-Benard Convection - Nonisothermal Boundaries. The assumption of isothermal boundaries in the buoyancy-driven convection instability problem is generally an oversimplification. A more realistic picture is that the upper and lower surfaces are in contact with reservoirs that are maintained at a constant temperature so that the thermal boundary conditions are better approximated as... 

Problem 12-16. The Effect of Shear Flow on Rayleigh-Benard Instability. We wish to consider the effect of shear on the Rayleigh-Benard buoyancy-driven instability. The problem that we will analyze is identical to that outlined in Section H (for a pair of rigid boundaries) except that there is a simple shear flow in a direction that we can designate as x, driven by motion of the upper boundary, i.e.,... 

Problem 12-17. Buoyancy-Driven Instability of a Fluid Layer in a Porous Medium Based on Darcy s Law. We consider the classical Rayleigh-Benard problem of a fluid layer that is heated from below, except in this case, the fluid is within a porous medium so that the equations of motion are replaced with the Darcy equations, which were discussed in Subsection Cl of this chapter. Hence the averaged velocity within the porous medium is given by Darcy s law... 

J. H. lienhard, An Improved Approach to Conductive Boundary Conditions for the Rayleigh-Benard Instability, J. Heat Transfer (109) 378-387,1987. 

The mechanism of Benard cell formation, also termed the Marangoni instability, was first elucidated and demonstrated theoretically by Pearson (1958) who, unaware of Block s experimental work, showed that if there was an adverse temperature gradient of sufficient magnitude across a thin liquid film with a free surface that such a layer could become unstable and lead to cellular convection. Following Pearson, the instability mechanism is illustrated in Fig. 10.6.2. There a small disturbance is assumed to cause the film of initially... 

Figure 10.6.2 Instability mechanism for Benard cell formation induced by a surface tension gradient.

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