Analytical theory for a droplet squeezing through a circular pore in creeping flows under constant pressures

Abstract

We derived equations and closed-form solutions of transit time for a viscous droplet squeezing through a small circular pore with a finite length at microscale under constant pressures. Our analyses were motivated by the vital processes of biological cells squeezing through small pores in blood vessels and sinusoids and droplets squeezing through pores in microfluidics. First, we derived ordinary differential equations (ODEs) of a droplet squeezing through a circular pore by combining Sampson flow, Poiseuille flow, and Young–Laplace equations and took into account the lubrication layer between the droplet and the pore wall. Second, for droplets wetting the wall with small surface tension, we derived the closed-form solutions of transit time. For droplets with finite surface tension, we solved the original ODEs numerically to predict the transit time. After validations against experiments and finite element simulations, we studied the effects of pressure, viscosity, pore/droplet dimensions, and surface tension on the transit time. We found that the transit time is inversely linearly proportional to pressure when the surface tension is low compared to the critical surface tension for preventing the droplet to pass and becomes nonlinear when it approaches the critical tension. Remarkably, we showed that when a fixed percentage of surface tension to critical tension is applied, the transit time is always inversely linearly proportional to pressure, and the dependence of transit time on surface tension is nonmonotonic. Our results provided a quick way of quantitative calculations of transit time for designing droplet microfluidics and understanding cells passing through constrictions.

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

“…Instead, the external fluid forms a lubrication layer between the droplet and each wall as shown in Figure 2, where h is the thickness of each layer. For a 2D droplet in a Hele-Shaw flow in a channel, it was shown that h is related to the channel height T and to the capillary number Ca as: Like the Sampson term shown in our previous work [5], the Roscoe term can be split into two terms of two half problems (the last two terms in Equation ( 14)). Equations ( 7) and ( 14) can be solved for 𝑉 and 𝑉 .…”

“…Here, we determine the durations of the droplet transit stages and of its total transit through the slit, based on the pressure drops calculated above. To do so, we adapted the procedures used in the circular pore case [5]. For stages I and V, the time needed for the droplet to form or retract a curved half-ellipsoidal shaped droplet head is very small, and so they are considered negligible for this study.…”

“…There does not exist any other published paper that derives analytical equations for droplet transit time through rectangular slits to compare our results with. However, we conducted a comparison with the model developed by Tang et al which analyzes droplet motion through a pore with a circular cross-section [5]. We conducted this comparison by utilizing a rectangular slit constriction as well as a circular pore constriction with the same cross-sectional area of 2.5 pm 2 , while keeping all other parameters such as surface tension and viscosity constant (Figure 7).…”

“…Although our computationally predicted transit time matches well with our experimental measurements, the simulations are expensive and less insightful than analytical theory. Additionally, in another recent work, we developed an analytical theory for a droplet passing through a circular pore [5], but the idealized circular pore is significantly different from the IES geometry we studied experimentally and computationally [2]. In this study, we extended our analytical theory of a circular pore to a slit for the transit time of a droplet.…”

“…When studying Stoke's flow, Dassios and Vafeas developed a 3D concentric sphere model for particles in creeping flows and derived analytical expressions for the velocity, the total pressure, the angular velocity, and the stress tensor fields [21]. In addition, we recently developed an analytical study of droplet transit through a circular pore [5].…”