Shear flow past two-dimensional droplets pinned or moving on an adhering channel wall at moderate Reynolds numbers: a numerical study

Abstract

Numerical simulations are presented of shear flow past two-dimensional droplets adhering to a wall, at moderate Reynolds numbers. The results were obtained using a level-set method to track the interface, with measures to eliminate any errors in the conservation of mass of droplets. First, the case of droplets whose contact lines are pinned is considered. Data are presented for the critical value of the dimensionless shear rate (Weber number, $\hbox{\it We}$), beyond which no steady state is found, as a function of Reynolds number, $\hbox{\it Re}$. $\hbox{\it We}$ and $\hbox{\it Re}$ are based on the initial height of the droplet and shear rate; the range of Reynolds numbers simulated is $\hbox{\it Re} \leq 25$. It is shown that, as $\hbox{\it Re}$ is increased, the critical value $\hbox{\it We}_c$ changes from $\hbox{\it We}_c\propto \hbox{\it Re}$ to $\hbox{\it We}_c\approx$ const., and that the deformation of droplets at $\hbox{\it We}$ just above $\hbox{\it We}_c$ changes fundamentally from a gradual to a sudden dislodgement. In the second part of the paper, drops are considered whose contact lines are allowed to move. The contact-line singularity is removed by using a Navier-slip boundary condition. It is shown that macroscale contact angles can be defined that are primarily functions of the capillary number based on the contact-line speed, instead of the value of $\hbox{\it We}$ of the shear flow. It is shown that a Cox–Voinov-type expression can be used to describe the motion of the downstream contact line. A qualitatively different relation is tested for the motion of the upstream contact line. In a third part of this paper, results are presented for droplets moving on a wall with position-dependent sliplength or contact-angle hysteresis window, in an effort to stabilize or destabilize the drop.