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Fluid Flow Analysis of a Two-Dimensional Sessile Drop in Linear Shear Flow
Hao-Kun Chu and S. S. Sadhal

The fluid mechanics of a viscous liquid drop placed in a gaseous linear shear flow is analyzed in the Stokes-flow regime by using the bipolar coordinate system. This study is motivated by investigations on protein crystal growth for which it is known that shear flow in the protein solution affects the nucleation rate. It is of interest to be able to generate and control shear flow in the sessile-drop geometry which is commonly used for crystal growth and aggregation. While drops for such experiments are usually of the spherical-cap shape, the enormous analytical complexity has led us to deal with a two-dimensional case first, i.e., the cylindrical equivalent of the cap. With the bipolar coordinate system, the circular interface can be exactly identified by constant value of one of the coordinates, and lends itself to satisfying the relevant continuity conditions at the gas-liquid and the solid-liquid interfaces of the system. The analysis yields an exact solution in the creeping-flow regime in the form of a Fourier integral. The numerical evaluation of the integral provides the flow streamlines along with the shear stress distribution within the drop. It is expected that these results will be useful in controlling and qualifying the shear rate in the drop.

Keywords: Sessile drops, fluid shear, Stokes flow, protein crystallization

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