Spreading of Viscous Fluid Drops on a Solid Substrate Assisted by Thermal Fluctuations

We study the spreading of viscous drops on a solid substrate, taking into account the effects of thermal fluctuations in the fluid momentum. A nonlinear stochastic lubrication equation is derived and studied using numerical simulations and scaling analysis. We show that asymptotically spreading drops admit self-similar shapes, whose average radii can increase at rates much faster than these predicted by Tanner’s law. We discuss the physical realizability of our results for thin molecular and complex fluid films, and predict that such phenomenon can in principal be observed in various flow geometries.

Figure 1

Schematic representation of a one-dimensional spreading drop, confined in a channel of width $W$. The precursor layer of height $b$ is also shown.

Figure 2

Average shape of the droplet $⟨\tilde{h}(\tilde{x},\tilde{t})⟩$ over 50 realizations as a function of time for zero temperature (left) and $\sigma ={10}^{-2}$ (right). The right picture clearly shows the enhanced spreading of the droplet in the presence of thermal fluctuations. The numerical parameters used in this simulation are $\Delta \tilde{t}=0.01$, $\Delta \tilde{x}=0.05$, and $\tilde{b}=0.01$.

Figure 3

Results of volume-conserving dynamics in one-dimensional geometry. (a) Lateral scale of the droplet (10) as a function of time. Solid lines represent averages over 50 realizations of (9) with $\sigma ={10}^{-2}$, $5\times {10}^{-3}$, ${10}^{-3}$ (from top to bottom), while the dashed line is the noiseless ($\sigma =0$) dynamics. Dotted lines correspond to the power laws ${\tilde{t}}^{1/4}$ and ${\tilde{t}}^{1/7}$. (b) Averaged profile of the droplet for times $\tilde{t}={10}^{3n/4}$, $n=0,1,2,3,4$. (c) Rescaled droplet profiles.