Zeitschrift für Physikalische Chemie und Biophysik
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Abstrakt

A Theory of Transit Time for Droplets Squeezing through a Circular and Slit Constrictions

Ethan Sun*, Lubna Shah, Zhangli Peng

We derived analytical equations and solutions of transit time of a viscous droplet passing through small pores and slits at the microscale under constant prescribed pressure. These mathematical analyses were motivated by the vital processes of biological cells passing through small pores in blood vessels and sinusoids, and droplets passing through artificially designed pores in microfluidics. First, we derived the Ordinary Differential Equations (ODEs) of a droplet passing through a circular pore by combining the Sampson solution, Poiseuille flow, and Young-Laplace equations. If the surface tension is negligible, we derived the closed-form solutions of transit time. If the surface tension is finite, by solving these ODEs numerically, we studied the effects of pressure, pore dimensions, surface tension, viscosity, and drop size on the transit time. Furthermore, we extended our studies from a circular pore to a slit, which is more realistic in many physiological and engineering applications. Using these analytical models, we found that the transit time is linearly proportional to viscosity, approximately linearly proportional to length and drop volume, and approximately inversely linear for pressure only when the surface tension is negligible. It is also a highly nonlinear function of pore radius and slit width. We also compared the transit time of a circular pore and a slit with the same cross-sectional area. Our results show that the transit time is always longer in the slit than the circular pore for most cases in practical applications. Our results will provide quantitative calculations for designing droplet microfluidics and understanding cells passing through constrictions.