Cellular convection

L. Biihler, S. H. Davis. Flow induced changes of the morphological stability in directional solidification localized morphologies. J Crystal Growth 756 629, 1998 Y.-J. Chen, S. H. Davis, Directional solidification of a binary alloy into a cellular convective flow (unpublished). Applied Math Technical Report No. 9708, Northwestern University, Evanston, IL 60208. 

At low Rayleigh numbers, Wragg (W6) found a smaller Ra dependence, resembling more the dependence in laminar free convection. In this range of Ra numbers, a cellular flow pattern is believed to exist, analogous to that of thermal and surface tension-driven cellular convection (Benard cells F3). In the range where the convection is turbulent, the Ra1/3 dependence has been confirmed over seven powers of Ra by Ravoo (R9), who used a centrifuge to vary the body force at constant bulk composition. 

Mass transfer rates are increased in the presence of eruptions because the interfacial fluid is transported away from the interface by the jets. For mass transfer from drops with the controlling resistance in the continuous phase, the maximum increase in the transfer rate is of the order of three to four times (S8), not greatly different from the estimate of Eq. (10-4) for cellular convection. This may indicate that equilibrium is attained in thin layers adjacent to the interface during the spreading and contraction. When the dispersed-phase resistance controls, on the other hand, interfacial turbulence may increase the mass transfer rate by more than an order of magnitude above the expected value. This is almost certainly due to vigorous mixing caused by eruptions within the drop. 

Various other instances of hydrodynamic and electrohydrodynamic instabilities in nematic and, to a lesser extent, smectic liquid crystals have been investigated. No attempt is made here to review this work. For the present discussion, it is sufficient to note that (a) most of the work has dealt with oriented layers having anisotropic properties, and (b) some interesting instabilities arise in oriented layers which do not occur for isotropic materials. An example of the latter is cellular convection in a fluid layer confined between horizontal plates maintained at different temperatures. With an isotropic fluid, convection can arise only if the lower plate is hotter than the upper plate. Then, fluid near the lower plate is less dense and tends to rise while fluid near the upper plate is denser and tends to sink. With an oriented layer, however, convection can arise even when the upper plate is hotter if the anisotropy of thermal conduction properties is of a particular type (8). 

Hyde et al. (24) found that the peak apparent viscosity of lamellar liquid crystal fell by about two orders of magnitude as alcohol chain length was reduced from 15 to five in water-alcohol-Teepol systems. However, the complete picture of how various types of amphiphilic compounds and their mixtures influence viscosity is not available. In particular, it is not known under what conditions fairly low viscosities of liquid crystals can be achieved although Hallstrom and Friberg (22) report viscosities of about 0.2 poise for some compositions in the water— monocaprylin-tricaprylin system. As indicated previously, low viscosities increase the possibilities for occurrence of hydrodynamic instabilities involving cellular convection. 

The consequent drop in tension will give rise to surface flow away from the disturbance which, thus, reinforces the initial flux disturbance. We postulate the presence of cellular convection on both sides of the interface promoting heat transfer in the liquid film and mixing in the vapor which tends to disperse the more volatile component. 

Orell, A. and Westwater, J.W., Spontaneous interfacial cellular convection accompanying mass transfer etylene glycol-acetic acid-ethyl acetate, AIChE J., 8, 350-356, 1962. 

C. V. Stemling and L. E. Scriven, Interfacial turbulence hydrodynamic instability and the Marangoni effect, AIChE J. 5, 514 (1959) L. E. Scriven and C. V Stemling, On cellular convection driven by surface tension gradients effects of mean surface tension and surface viscosity, J. Fluid Mech. 19, 321 (1964). 

Cellular Convection Induced by Surface Tension Gradients... 

In the preceding section, we have examined a variety of steady thermocapillary and diffusocapillary flows. Not all such flows are stable and in fact surface tension variations at an interface can be sufficient to cause an instability. We consider here the cellular patterns that arise with liquid layers where one boundary is a free surface along which there is a variation in surface tension. It is well known that an unstable buoyancy driven cellular convective motion can result when a density gradient is parallel to but opposite in direction to a body force, such as gravity. An example of this type of instability was discussed in Section 5.5 in connection with density gradient centrifugation. 

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... 

Fig. 3. Steady cellular convection as observed by Bdnard (B8) (cf. A4) (courtesy of Ministry of Air, Paris) (a) top view (b) side view (c) perspective view.

Fig. 4. The effect of fluid motion present at the inception of cellular convection (B4) (a) the effect of initial translational motion (b) the effect of initial rotational motion.

As will be shown below, Benard s work also provided for many years ahead the experimental foundation for the study of cellular convection. His dramatic results inspired the work of many others, chief among them a group of French experimentalists, and a number of English theoretical physicists led by Lord Rayleigh. 

Cellular convection was not treated until 1952 when Pillow (P4) formulated the nonlinear problem for two-dimensional flow between two horizontal plates. He obtained steady velocity profiles for the two-dimensional rolls by neglecting the effects of viscosity and heat conduction except in the region very near the boundaries. Using an approximate theory, he predicted a 5/4-power dependence of heat transfer rate on the temperature difference, which is the proportionality found experimentally. 

Kuo (K2) has recently obtained a solution to the nonlinear equations of cellular convection by expanding the dependent variables in series of orthogonal functions and by expanding the coefficients of these functions in a power series of an amplitude parameter. His solution also predicts a heat transfer rate proportional to the 5/4 power of the temperature difference. 

Nield, D.A. Surface tension and buoyancy effects in cellular convection. J. Fluid Mech. 19, 341-352 (1964)... 

Orell A. and Westwater J.W. (1962). Spontaneous Interfacial Cellular Convection Accompanying Mass Transfer Ethylene Glycol - Acetic Acid - Ethyl Acetate. Journal of the American Institute of Chemical Engineers 8(3) 350-356. 

FIGURE 6.1 Cellular convection produced by surface tension gradients in a thin liquid layer heated from below. Reproduced from Avsec (1939) with permission from the French Ministry of Defense. 

FIGURE 6.2 Origin of cellular convection driven by interfacial tension gradients in a thin layer of liquid heated from below or cooled from above. Warm fluid reaching the interface at P reduces the interfacial tension locally. 

---