Numerical studies of geometry effects of a two-dimensional microfluidic four-roll mill on droplet elongation and rotation

A microfluidic analog of the four-roll mill has been designed to generate all flow types including extensional flow, shear flow and rotational flow [Lee et al. Appl. Phys. Lett. 90, 074103 (2007)]. Boundary element methods are powerful tools to study interfacial dynamics of droplets in microchannels due to its high facility in describing the complex geometries and the various boundary conditions. Through a two-dimensional (2D) spectral boundary element method, effects of the geometry and the volume flow rate on the elongation and rotation of droplets in this microfluidic four-roll mill (MFRM) are investigated in this paper. Behaviors of deforming or rotating droplets trapped in a MFRM are determined by the size of orifices connecting the channels (inlets and outlets) and the central cavity, and the volume flow rate of the continuous phases (CP) pumped into and out of the device, especially for the rotation of droplets. Through an approximate analysis of order of magnitude, a simple conclusion that the average angular velocity of the rotation is proportional to the volume flow rate of CP at inlets and the radius of the central cavity is presented, which is testified by the numerical calculation. Besides, the appropriate radius of the central cavity which directly determines the size of the orifices is specified through the analysis of the flow fields. These results are helpful in designing a MFRM with a much larger depth than the width of the micro-channel for the purpose of microfluidic rheometry.