Pressure of a viscous droplet squeezing through a short circular constriction: An analytical model

The model of a droplet squeezed through a narrow-constricted channel has many applications in pathology, chip/filter/membrane design, drug delivery, etc. Understanding the transient physics of the squeezing process is important in the design and optimization of many micro flow systems. However, available models often ignore the influence of droplet viscosity, and they usually feature low numerical efficiency by solving Navier-Stokes equations. In the present research, we developed a low-dimension analytical model to predict the pressure of squeezing a viscous droplet through a circular constricted channel with acceptable fidelity and low computational cost. Our approach is as follows. We first adapt the Hagen-Poiseuille law to predict the viscosity effect of droplet squeezing. Next, we obtain an analytical expression for the extra pressure caused by only the curvature change obtained. Finally, the general expression of squeezing pressure taking consideration of viscosity and surface tension is expressed. The analytical model we developed is in great agreement with the numerical solutions of the Navier-Stokes equation at a low Reynolds number and low capillary number. These findings have fundamental significance for future applications in engineering and industry.