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Depinning of liquid droplets on substrates by flow of a surrounding immiscible fluid is central to applications such as cross-flow microemulsification, oil recovery and waste cleanup. Surface roughness, either natural or engineered, can cause droplet pinning, so it is of both fundamental and practical interest to determine the flow strength of the surrounding fluid required for droplet depinning on rough substrates. Here, we develop a lubrication-theory-based model for droplet depinning on a substrate with topographical defects by flow of a surrounding immiscible fluid. The droplet and surrounding fluid are in a rectangular channel, a pressure gradient is imposed to drive flow and the defects are modelled as Gaussian-shaped bumps. Using a precursor-film/disjoining-pressure approach to capture contact-line motion, a nonlinear evolution equation is derived describing the droplet thickness as a function of distance along the channel and time. Numerical solutions of the evolution equation are used to investigate how the critical pressure gradient for droplet depinning depends on the viscosity ratio, surface wettability and droplet volume. Simple analytical models are able to account for many of the features observed in the numerical simulations. The influence of defect height is also investigated, and it is found that, when the maximum defect slope is larger than the receding contact angle of the droplet, smaller residual droplets are left behind at the defect after the original droplet depins and slides away. The model presented here yields considerably more information than commonly used models based on simple force balances, and provides a framework that can readily be extended to study more complicated situations involving chemical heterogeneity and three-dimensional effects. 

The influence of defect geometry on the critical inclination angle required for droplet depinning on inclined substrates is studied. 

The formation and evolution of a heterogeneous flow and flow reversal are examined in highly elastic, gel-like wormlike micelles (WLMs) formed from an amphiphilic triblock poloxamer P234 in 2M NaCl. A combination of linear viscoelastic, steady shear, and creep rheology demonstrate that these WLMs have a yield stress and exhibit viscoelastic aging, similar to some soft glassy materials. Nonlinear shear rheology and rheoparticle tracking velocimetry reveal that these poloxamer WLMs undergo a period of strong elastic recoil and flow reversal after the onset of shear startup. As flow reversal subsides, a fluidized high shear rate region and a nearly immobile low shear rate region of fluid form, accompanied by wall slip and elastic instabilities. The features of this flow heterogeneity are reminiscent of those for aging yield stress fluids, where the heterogeneous flow forms during the initial stress overshoot and is sensitive to the inherent stress gradient of the flow geometry. Additionally, macroscopic bands that form transiently above a critical shear rate become “trapped” due to viscoelastic aging in the nearly immobile region. This early onset of the heterogeneous flow during the rapidly decreasing portion of the stress overshoot differs from that typically observed in shear banding WLMs and is proposed to be necessary for observing significant flow reversal. Exploring the early-time, transient behavior of this WLM gel with rheology similar to both WLM solutions and soft glassy materials provides new insights into spatially heterogeneous flows in both of these complex fluids.