Modeling the dynamics of a drop moving in a plane channel with a viscous barrier

This article examines the influence of a highly viscous region on the deformation of a single droplet moving in a thermoviscous fluid flow with an anomalous viscosity-temperature dependence. Numerical modeling is performed using the phase field method, consisting of the Cahn-Hilliard-Navier-Stokes system of equations, supplemented by an equation for the temperature field. The results of numerical modeling for laminar flow with a steady-state temperature field in a flat channel are presented. The influence of a viscous barrier located in the middle of the channel on the droplet velocity and its deformation is analyzed for three capillary numbers. Graphs of the droplet velocity dependence on the capillary number are plotted. It is shown that for all cases of capillary numbers, the viscous barrier formed in the channel imparts additional acceleration to the droplet. For all calculations, the total normalized mass of the system was calculated over time, and the maximum error is shown to not exceed 5%. The droplet shape at the channel exit is shown for different cases of a viscous barrier. It is shown that if the capillary number is high, inclusions of the surrounding liquid can form in the droplet. However, if the droplet moves through a channel with a viscous barrier, the high-viscosity region prevents the formation of inclusions of other liquids due to more intense droplet deformation. It is established that the relative velocity of a droplet in a flow of an anomalously viscous liquid is higher than in a flow of liquid with a constant minimum viscosity. Graphs of the droplet’s relative velocity versus the liquid anomaly parameter are presented for three capillary numbers. It is shown that the higher the liquid anomaly parameter, the higher the relative velocity of the droplet. It is established that the viscous barrier formed in the channel accelerates the droplet and increases its relative velocity.