Rolls, convection

An entirely different type of transport is formed by thermal convection and conduction. Flow induced by thermal convection can be examined by the phaseencoding techniques described above [8, 44, 45] or by time-of-flight methods [28, 45]. The latter provide less quantitative but more illustrative representations of thermal convection rolls. The origin of any heat transport, namely temperature gradients and spatial temperature distributions, can also be mapped with the aid of NMR techniques. Of course, there is no direct encoding method such as those for flow parameters. However, there are a number of other parameters, for example, relaxation times, which strongly depend on the temperature so that these parameters can be calibrated correspondingly. Examples are described in Refs. [8, 46, 47], for instance. 

Figure 2.9.9(a) shows a schematic representation of a thermal convection cell in Rayleigh-Benard configuration [8]. With a downward temperature gradient one expects convection rolls that are more or less distorted by the tortuosity of the fluid filled pore space. In the absence of any flow obstacles one expects symmetrical convection rolls, such as illustrated by the numerical simulation in Figure 2.9.9(b). 

The spatial temperature distribution established under steady-state conditions is the result both of thermal conduction in the fluid and in the matrix material and of convective flow. Figure 2. 9.10, top row, shows temperature maps representing this combined effect in a random-site percolation cluster. The convection rolls distorted by the flow obstacles in the model object are represented by the velocity maps in Figure 2.9.10. All experimental data (left column) were recorded with the NMR methods described above, and compare well with the simulated data obtained with the aid of the FLUENT 5.5.1 [40] software package (right-hand column). Details both of the experimental set-up and the numerical simulations can be found in Ref. [8], The spatial resolution is limited by the same restrictions associated with spin... 

Fig. 2.9.9 (a) Schematic cross section of a compartments at the top and bottom, respec-convection cell in Rayleigh-Benard configura- tively. (b) Velocity contour plot of typical tion. In the version examined in Refs. [8, 44], a convection rolls expected in the absence of any fluid filled porous model object of section flow obstacles (numerical simulation). 

This model bears the imprint of its own limitations as a good model should. It is assumed that there is no movement, yet algae are growing in the upper layers of the water and presumably making the medium more dense. This suggest that sedimentation or convective roll-over should be considered. We shall not go into such generalizations here, but merely leave this example with the comment that the same equations turn up in the theory of a continuous reactor in which a belt of material is to be impregnated with a solution, whose solute is immobilized on sites in the belt. 

Clever. R.M., Finite Amplitude Longitudinal Convection Rolls in an Inclined Layer", J. Heat Trans.. Vol. 95. p. 407, 1973. 

Here 0 are parameters. Ed Lorenz (1963) derived this three-dimensional system from a drastically simpl ified model of convection rolls in the atmosphere. The same equations also arise in models of lasers and dynamos, and as we ll see in Section 9.1, they exactly describe the motion of a certain waterwheel (you might like to build one yourself). 

Like the waterwheel, the Lorenz system (1) has two types of fixed points. The origin (x, y, z ) = (0,0,0) is a fixed point for all values of the parameters. It is like the motionless state of the waterwheel. For r > 1, there is also a symmetric pair of fixed points x = y = b(r -1), z = r -1. Lorenz called them C and C. They represent left- or right-turning convection rolls (analogous to the steady rotations of the waterwheel). As r —> 1, C and coalesce with the origin in a pitchfork bifurcation. 

At the short end of the scale (of the order of 10 cm) the motion may be due to convective instability in a shallow surface layer as explained by Gem-merich and Hasse (1992). They made field observations of the surface motion by spraying powdered corkwood on the surface, watching how it collected on a time scale of 10 seconds or less, into various structures depending on the condition. Lines were by far the most common structure observed. They may, for example, be caused by convective rolls along the wind drift direction. 

R. M. Clever, Finite Amplitude Longitudinal Convection Rolls in Inclined Layers, /. Heat Transfer (95) 407-408,1973. 

In the presence of a mean flow, the generalized Swift-Hohenberg equation includes the advection of the convective rolls. It can be written, e.g., in the form... 

Electrically driven convection in nematic liquid crystals [6,7,16] represents an alternative system with particular features listed in the Introduction. At onset, EC represents typically a regular array of convection rolls associated with a spatially periodic modulation of the director and the space charge distribution. Depending on the experimental conditions, the nature of the roll patterns changes, which is particularly reflected in the wide range of possible wavelengths A found. In many cases A scales with the thickness d of the nematic layer, and therefore, it is convenient to introduce a dimensionless wavenumber as q = that will be used throughout the paper. Most of the patterns can be understood in terms of the Carr-Helfrich (CH) mechanism [17, 18] to be discussed below, from which the standard model (SM) has been derived... 

For a/e > 0.0057 EC occurs superimposed onto the Freedericksz state (secondary instability) at a higher voltage Uc > Upi- The standard model can also be applied here by carrying out numerical linear stability analysis of the Freedericksz distorted state, and one is faced with similar modifications and difficulties as mentioned in case C before. The ea-dependent Uc and Qc, presented by the dotted hnes in Fig. lO(a-b), have been calculated numerically. It should be noted that the convection rolls are now oriented parallel to the initial director alignment, contrary to the normal rolls in case A or C. 

Fig. 3.11.4. A fluctuation that is spatially periodic in x results in a secondary velocity that is also spatially periodic. This can have a destabilizing effect and generate convective rolls.

Hy favours rolls with axes normal to y, but in the field-free case the rolls degenerate into a square pattern that may be regarded as a linear superposition of crossed convection rolls. When AT is increased well beyond A7 a complex hexagonal structure is found with a nematic-isotropic interface if the temperature of the upper plate is large enough. 

Fig. 3.13.6. The formation of oblique convective rolls in electrohydrodynamic instability. Below the point M, which may be called a triple point , the transition takes place directly to oblique rolls (/= 10 Hz). Beyond M, normal rolls are obtained first, followed by undulatory rolls which then change continuously into oblique rolls (/ = 60 Hz). (Ribotta et...

Fig.ll Deformation of a one-armed spiral wave due to the effect of convective rolls. 

Then they have clearly demonstrated the importance of evaporation in the generation of these structures. Indeed on closure of the reactor the shapes fade away but reform on opening. Furthermore, using the schlieren technique they obtained photographs of stria-tions (prepatterns) [6 ] existing in the fluid prior to irradiation. The existence of these vermiculated convective rolls was decisively proven by depositing ink on the bottom of the dish. 

As a consequence in this series of experiments the photochemical reaction serves only to reveal preexisting convective motions which are induced by evaporative cooling [ 7] On the other hand since similar prepatterns have been characterized in the pure solvents reported by Avnir and al., we think that the same phenomenon is responsible for the onset of structures in these open systems too. Similarly, any surface reaction (for instance with oxygen) yielding a coloured product will provide a way to visualize bulk motions as the coloured product is then advected on the convective rolls. Such a mechanism is probably also responsible for the onset of most of the stationary mosaic patterns observed in shallow layers of chemical systems [8 ]. 

Figure 10. Convective director structures in a nematic side group polymer, (a) Schematic diagram of the experimental set-up for continuous rotation about an axis perpendicular to the magnetic field, (b) Optical texture after two hours of rotation, (c) NMR spectra obtained during one revolution after two hours, (d) Schematic diagram of convection rolls evolving due to nonlinear coupling between director rotation and viscous flow, (e) Director distribution P( P) extracted from the NMR spectra in (c). For details see [123].

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