Long-Wavelength Instability in Surface-Tension-Driven Bénard Convection

Abstract

Laboratory studies reveal a deformational instability that leads to a drained region (dry spot) in an initially flat liquid layer (with a free upper surface) heated uniformly from below. This long-wavelength instability supplants hexagonal convection cells as the primary instability in viscous liquid layers that are sufficiently thin or are in microgravity. The instability occurs at a temperature gradient 34% smaller than predicted by linear stability theory. Numerical simulations show a drained region qualitatively similar to that seen in the experiment.PACS numbers: 47.20.Dr, 47.20.Ky, 47.54.+r, 68.15.+e Bénard's observation in 1900 [1] of hexagonal convection patterns launched the modern study of convection, pattern formation, and instabilities; yet understanding of the surface-tension-driven regime in which Bénard performed his experiments is still far from complete. Block [2] and Pearson [3] first showed how temperature-induced surface tension gradients (thermocapillarity) caused the instability observed in Bénard's experiments. Pearson's linear stability analysis with a nondeformable liquid-gas interface yielded an instability at wavenumber q = 1.99 (scaled by the mean liquid depth d) and Marangoni number M c = 80, where M ≡ σ T △ T d/ρνκ (see Fig. 1) expresses the competition between the destabilization by thermocapillarity and the stabilization by diffusion.A deformable free surface allows a second type of primary instability in which a perturbation creates a nonuniform liquid depth and temperature profile [4,5]. Thermocapillarity causes cool, elevated regions to pull liquid from warm, depressed regions (see Fig. 1). The instability appears with a long wavelength since surface tension stabilizes short wavelengths. Linear stability analyses that allow for deformation reveal this instability (see Fig. 2) at zero wavenumber (q = 0) and M c = 2 3 G [5,6], where the Galileo number, G ≡ gd 3 /νκ (g is the acceleration of gravity), gives the relative strengths of the stabilizing mechanisms of diffusion and gravity and thus determines which instability will form. For sufficiently thin, viscous liquid layers or small g (e.g., microgravity), 2 3 G < 80, so the long-wavelength mode should become the primary instability (see Fig. 2). This longwavelength instability has not been experimentally investigated.In this Letter we describe experimental observations of the onset of the long-wavelength instability and compare these observations to linear stability theory. The instability leads to a large-scale drained region with diameter ∼ 100d. The qualitative features of the instability are compared to nonlinear theory based on a longwavelength evolution equation. We also explore the competition between the long-wavelength and hexagonal instabilities and study the physical mechanism that selects which pattern will appear. For a range of liquid depths, both patterns coexist.

Thermocapillary effect on the dynamics of viscous beads on vertical fiber

“…2 Re(T t + uT r + wT z ) = 1 Pr T rr + T r r + 2 T zz , (19) where the Reynolds number is defined as Re = ρV L/μ, and the Prandtl number is defined as Pr = ν/κ. Assuming 2 1 and Re = O(1) or smaller, we can neglect the contributions of the inertial terms.…”

“…This long-wavelength instability mode (the S mode) was first predicted and investigated theoretically by Scriven and Sternling [17] and Smith [18]. VanHook et al [19,20] investigated both experimentally and theoretically on the long-wave instability of a thin liquid layer heated from below or cooled from above. In the experiments, the long-wave instability mode takes the form of a localized depression ("dry spot") or a localized elevation ("high spot"), depending on the thickness and thermal conductivity of the gas layer above the liquid.…”

Morphology changes in the evolution of liquid two-layer films

“…The main focus lies thereby on front instabilities of moving contact lines 1,2,3 or on instabilities of the free liquid-gas interface of a flat film 4,5,6 .…”

The Effect of Crystal-Melt Surface Energy on the Stability of Ultra-Thin Melt Films

“…Finally, we comment that the emergence of a long wave instability due to the crystal-melt surface energy has mathematical features similar to the emergence of a long-wavelength in- stability and dry-patch formation in surface-energy-driven Benard convection [34], although the physical processes are not related.…”