A volume-of-fluid formulation for the study of co-flowing fluids governed by the Hele-Shaw equations

Abstract

We present a computational framework to address the flow of two immiscible viscous liquids which co-flow into a shallow rectangular container at one side, and flow out into a holding container at the opposite side. Assumptions based on the shallow depth of the domain are used to reduce the governing equations to one of Hele-Shaw type. The distinctive feature of the numerical method is the accurate modeling of the capillary effects. A continuum approach coupled with a volume-of-fluid formulation for computing the interface motion and for modeling the interfacial tension in Hele-Shaw flows are formulated and implemented. The interface is reconstructed with a height-function algorithm. The combination of these algorithms is a novel development for the investigation of Hele-Shaw flows. The order of accuracy and convergence properties of the method are discussed with benchmark simulations. A microfluidic flow of a ribbon of fluid which co-flows with a second liquid is simulated. We show that for small capillary numbers of O(0.01), there is an abrupt change in interface curvature and focusing occurs close to the exit.

Stability and hysteresis of Faraday waves in Hele-Shaw cells

“…where the first term indicates the principal radii of curvature of the wave profile ζ and the second term represents the out-of-plane curvature of the meniscus which can also contribute to the surface tension force (see figure 1). For regular Hele-Shaw flow, when multi-phase flow occurs, the out-of-plane interface shape is always assumed to be semicircular and the contact angle θ is naturally 180 • (Saffman & Taylor 1958;McLean & Saffman 1981;Park & Homsy 1984;Afkhami & Renardy 2013). Nevertheless, this assumption is not appropriate for Faraday waves in Hele-Shaw cells.…”

Capillary focusing close to a topographic step: shape and instability of confined liquid filaments

“…We use a volumeof-fluid (VOF) based method to directly solve the governing equations. Our numerical model, described in detail by Afkhami and Renardy (2013), has the distinctive feature of being capable of, accurately and robustly, modeling the surface tension force. In addition, the accurate interface reconstruction, the second-order curvature computation, and the use of adaptive mesh refinement allow for resolved description of the interface, enabling us to investigate the complex features of interface profiles extracted from experiments, which was not possible in prior studies.…”

Experimental and simulation studies of the shape and motion of an air bubble contained in a highly viscous liquid flowing through an orifice constriction

“…Use of the harmonic mean in place of the arithmetic mean was first highlighted when considering heat transfer through two materials of differing thermal conductivity (Patankar, 1980). Furthermore, it has been demonstrated that the harmonic mean viscosity is an appropriate parameter to use in certain classes of multiphase flow problem (Wörner, 2012), such as the flow of two immiscible liquids in a 2D Hele-Shaw cell (Afkhami and Renardy, 2013).…”