Deformation and motion of a thin sessile drop in shear flow

Drop deformation and motion driven by shear flows are widespread phenomena in nature and underpin a variety of technological applications. We investigate the deformation, motion and contact-line dynamics of two-dimensional drops in steady shear flows within the framework of lubrication theory. For pinned drops with 180 Superscript ring 180 ∘ 180^∘ hysteresis, we identify a critical shear rate, beyond which the upstream side becomes unstable and a thinning film develops. At late stage, the thinning film connects to the bulk drop through a self-similar capillary transition structure. For moving drops with 0 Superscript ring 0 ∘ 0^∘ hysteresis, the dynamical behaviours can be classified into three regions: in the low-shear region, the steady-state drop velocity increases linearly with the shear number upper S S S ; in the intermediate-shear region, the drop velocity exhibits a non-monotonic dependence on upper S S S ; in the high-shear region, the steady-state drop profile can be clearly divided into the entrained film, dimple and capillary ridge. Moreover, during the drop evolution under high shear, a Landau–Levich–Derjaguin-like (LLD-like) film emerges and exhibits a scaling law that differs from the classical LLD law. These results provide a unified theoretical and numerical framework for understanding shear-driven drop dynamics and may offer insights into analogous three-dimensional systems.