Unveiling unique scaling behavior in miscible, volatile Marangoni spreading.

Abstract

We present a novel observation of the expansion of the outer tip radius of a fast-spreading ethanol-water film spreading over a deep substrate of water. The experimentally measured radius of the outer tip of the film (<i>r</i>_o) and its velocity (<i>U</i>_o) display a complex scaling with time and drop properties. The variation showed by <i>r</i>_o differed from the commonly observed scalings of <i>t</i>^3/4 and <i>t</i>^1/4. We propose novel scaling laws for <i>r</i>_o and <i>U</i>_o by expressing <i>r</i>_o as the sum of the radius of the stable part of the film <i>r</i>_f and the length of the unstable part <i>l</i>_p at the periphery of the stable part of the film, that had azimuthally uniformly spaced plumes. The radius of the stable part of the film scales as <i>r</i>_f ∼ <i>t</i>^1/4 since, while the film expands, the Marangoni stresses are balanced by viscous stresses within the film thickness. At the same time, <i>l</i>_p ∼ <i>t</i>^3/4 since the plumes grow at the periphery of the stable part of the film, with the driving surface tension stresses balanced by the viscous stresses in a shear layer below the plumes. Combining these two scaling laws yielded a novel, two-term scaling law for <i>r</i>_o, which is close to a single power-law scaling <i>r</i>_o ∼ <i>t</i>^1/2. We obtain an expression for the dimensionless mean outer tip radius as , where <i>t</i>* = <i>t</i>/<i>t</i>_<i>ξ</i>, <i>t</i>_<i>ξ</i> = (<i>r</i>_d⁴<i>ρ</i>_w<i>μ</i>_w/Δ<i>σ</i>²)^1/3 being the time scale. Similarly, we show that the dimensionless velocity scales as with the variables <i>λ</i>₁ and <i>λ</i>₂ being functions of <i>t</i>* and drop properties. These proposed scaling laws are shown to match our measurements, thereby validating the phenomenology of such miscible, volatile spreading.