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 (ro) and its velocity (Uo) display a complex scaling with time and drop properties. The variation showed by ro differed from the commonly observed scalings of t3/4 and t1/4. We propose novel scaling laws for ro and Uo by expressing ro as the sum of the radius of the stable part of the film rf and the length of the unstable part lp 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 rf ∼ t1/4 since, while the film expands, the Marangoni stresses are balanced by viscous stresses within the film thickness. At the same time, lp ∼ t3/4 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 ro, which is close to a single power-law scaling ro ∼ t1/2. We obtain an expression for the dimensionless mean outer tip radius as , where t* = t/tξ, tξ = (rd4ρwμw/Δσ2)1/3 being the time scale. Similarly, we show that the dimensionless velocity scales as with the variables λ1 and λ2 being functions of t* and drop properties. These proposed scaling laws are shown to match our measurements, thereby validating the phenomenology of such miscible, volatile spreading.