Instabilities Rayleigh—Benard

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.,... 

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

While the cold container bottom yields a stably stratified configuration right from the start, the unstable density stratification at the cool top of the container may result in a Rayleigh-Benard instability [93]. In the third and fourth inset of Figure 9.2, the instability is observed as cold liquid draining down from the top into the bulk. 

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... 

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 12-17. 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... 

Newell-Whitehead-Segel equation, 23 Non-potential effects, 41 Orientational instability, 283 Pattern formation, 1,11 Phase field, 168 Polymerization wave, 235, 239 Polymerization waves, 236, 238 Propagating front, 260-261 Quantum dots, 123-124 Rayleigh-Benard convection, 61 SHS, 247-248 Smectics, 57 Spiral wave, 47 Stochastic oscillations, 92 Stripes, 2, 10 Surface diffusion, 126... 

Convection instabilities in simple isotropic fluids, like Rayleigh-Benard convection (for a recent review see Bodensctiatz et are completely under-... 

It has been extensively demonstrated that thick liquid films rupture due to the formation (nucleation) of a pinhole in the film which spreads outward radially until the film is destroyed. The nucleation of the pinhole could result from either the presence of fluid instabilities in the film (for example, Marangoni, Rayleigh-Taylor or Benard instabilities) [33] or else by the presence of immiscible microphases which could serve as nucleation sites. 

If either the monomer or the polymer, or both, are liquid natural convection, caused by the heat liberated by the exothermic reaction, can occur. Consider first the case when the monomer is liquid and the polymer is solid (cf. Section 1). We will discuss separately upward and downward propagating fronts. If the front propagates upward, then the chemical reaction heats the monomer from below which reminds of the classical Rayleigh-Benard problem. If the Rayleigh number is sufficiently large, then the planar front loses its stability and stationary natural convection above the front occurs. For descending planar fronts there is no such convective instability. An approximate analytical approach allows one to find stability conditions for the propagating reaction front and to determine the modes which appear when the front loses stability [22]. 

It is usual in the literature (see, for instance, Drazin and Reid 1981), to call Rayleigh-Benard (R-B) shallow convection the instability problem produced mainly by buoyancy (with eventually a Marangoni and Biot effects in a non-deformable free surface). 

Landau, L. D. (1944) On the problem of turbulence. C. R. Dokl. Acad. Sci. URSS 44, 311 Libchaber, A., Maurer, J. (1980) Une experience de Rayleigh-Benard de geometrie reduite Multiplication, accrochage et demultiplication de frequences. J. Phys. (Paris) Colloq 41, C3, 51 Lin, J., Kahn, P. B. (1982) Phase and amplitude instability in delay-diffusion population models. J. Math. Biol. 13, 383... 

Pattern formation in hydrodynamic instabilities has been studied intensely over the last decades [1, 2], Although Rayleigh-Benard convection (RBC) in simple fluids has been the prime example [3], the rich variety of scenarios found in nematic hquid crystals (LCs) has attracted increased attention. 

It is now well known that many systems undergo an instability leading to stationary periodic structures when they are driven sufficiently far from thermal equilibrium lll. Classical examples are provided by the Rayleigh-Benard (R.B) or the Marangoni instabilities in isotropic fluids. 

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