Modeling shear-influenced deformation of drops in a converging channel using lubrication theory

Fluid flows at low Reynolds number through planar converging channels impose extensional strain on suspended particles along the centerline of the flow. However, when the particles are near the wall, the no-slip condition for the velocity imposes an additional shear component to the total strain. If the suspended particles are deformable -- like drops -- their shape and size can be tuned by simply controlling the flow rate. In this study, we apply concepts from lubrication theory and kinematics of mixing to describe the deformation of drops undergoing shear-influenced deformation. The model considers the initial location of the drop and time spent by it in the flow and generates a space-time phase space that predicts all possible particle trajectories. We show how to calculate the drop deformation from the phase space and experimentally validate the model by tracking the deformation of Silicone oil drops in a flow of Castor oil through a converging channel. Our model accurately predicts the net drop deformation for different combinations of initial drop location and time spent in the flow by the drop. The model can be conveniently used to calculate drop deformation in flows where the single-phase flow solution is available.