Droplet Spreading on a Substrate

The present work focuses on spreading dynamics of thin viscous droplets on a planar and smooth substrate in a small Reynolds number regime. The droplets are affected by gravity, surface tension and viscous forces. For triple-phase zone, the moving contact-line models are provided to remove the singularity at the edge of the droplet. Our aim of this study is to predict the experimental results and extend the analysis to spreading on a porous substrate. Besides, we quantify the role of rheological parameters, for example, exponent for power-law liquids, on the spreading dynamics. The mathematical models for a Newtonian droplet and power-law droplet spreading over different substrates are derived using the lubrication theory. The flow inside the saturated porous media is described by the Darcy\u27s law for Newtonian liquids or modified Darcy\u27s law for power-law fluids, assuming a discontinuous wetting front separating the saturated from the unsaturated domain. In the cases for spreading over porous media, we use the Beavers and Joseph boundary condition for tangential velocity on the surface of the substrate. In the end of the theoretical derivation, we have a fourth order nonlinear diffusion partial differential equation for the profile of droplet and a second order nonlinear diffusion partial differential equation for the wetting front for the case of porous substrate. The governing equations are solved numerically using a finite difference formulation, augmented by the use of Newton-Raphson iteration scheme to treat the nonlinearity. We choose a backward-Euler method for the time advancement algorithm. Finally, numerical results are presented to demonstrate the dependence of spreading exponent on rheological properties of fluids and the results are validated by the Tanner\u27s law when the liquid is Newtonian. The dependence well matches the analytical relation (Starov et al. [2003]) despite different choices of contact-line conditions. For the case of shear-thinning droplet spreading on a solid substrate, the numerical results are compared with the experimental data for xanthane droplet (Rafai, Bonn, & Boudaoud [2004]). Our analysis for spreading over an impermeable substrate can capture the behavior of the Tanner\u27s law when the permeability number is zero. For cases of spreading over a porous substrate, the evolution of the contact radius and central height of the droplet follows the Tanner\u27s law during the initial spreading period. The dependence of the evolution of radius and central height on the permeability number is reported in this study. Finally, we compare the evolution of contact radius of a PDMS droplet against the experimental results reported in Denesuk et al. [1993] and the numerical results from Alleborn & Raszillier [2004]