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 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msup> <mml:mn>180</mml:mn> <mml:mo>∘</mml:mo> </mml:msup> </mml:math> $180^\circ$ 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 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msup> <mml:mn>0</mml:mn> <mml:mo>∘</mml:mo> </mml:msup> </mml:math> $0^\circ$ 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 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>S</mml:mi> </mml:math> $S$ ; in the intermediate-shear region, the drop velocity exhibits a non-monotonic dependence on upper S <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>S</mml:mi> </mml:math> $S$ ; 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.