Transit Time Theory for a Droplet Passing through a Slit in Pressure-Driven Low Reynolds Number Flows

Soft objects squeezing through small apertures are crucial for many in vivo and in vitro processes. Red blood cell transit time through splenic inter-endothelial slits (IESs) plays a crucial role in blood filtration and disease progression, while droplet velocity through constrictions in microfluidic devices is important for effective manipulation and separation processes. As these transit phenomena are not well understood, we sought to establish analytical and numerical solutions of viscous droplet transit through a rectangular slit. This study extends from our former theory of a circular pore because a rectangular slit is more realistic in many physiological and engineering applications. Here, we derived the ordinary differential equations (ODEs) of a droplet passing through a slit by combining planar Poiseuille flow, the Young-Laplace equations, and modifying them to consider the lubrication layer between the droplet and the slit wall. Compared to the pore case, we used the Roscoe solution instead of the Sampson one to account for the flow entering and exiting a rectangular slit. When the surface tension and lubrication layer were negligible, we derived the closed-form solutions of transit time. When the surface tension and lubrication layer were finite, the ODEs were solved numerically to study the impact of various parameters on the transit time. With our solutions, we identified the impact of prescribed pressure drop, slit dimensions, and droplet parameters such as surface tension, viscosity, and volume on transit time. In addition, we also considered the effect of pressure drop and surface tension near critical values. For this study, critical surface tension for a given pressure drop describes the threshold droplet surface tension that prevents transit, and critical pressure for a given surface tension describes the threshold pressure drop that prevents transit. Our solutions demonstrate that there is a linear relationship between pressure and the reciprocal of the transit time (referred to as inverse transit time), as well as a linear relationship between viscosity and transit time. Additionally, when the droplet size increases with respect to the slit dimensions, there is a corresponding increase in transit time. Most notably, we emphasize the initial antagonistic effect of surface tension which resists droplet passage but at the same time decreases the lubrication layer, thus facilitating passage. Our results provide quantitative calculations for understanding cells passing through slit-like constrictions and designing droplet microfluidic experiments.