Numerical and experimental studies on the viscous folding in diverging microchannels

Abstract

We performed numerical and experimental studies on the viscous folding in diverging microchannel flows which were recently reported by Cubaud and Mason (Phys Rev Lett 96:114501, 2006a). We categorized the flow patterns as “stable”, “folding,” and “chaotic” depending on channel shape, flow ratio, and viscosity ratio between two fluids. We focused on the effect of kinematic history on viscous folding, in particular, by changing the shape of diverging channels: 90°, 45°, and hyperbolic channel. In experiments, the proposed power–law relation ($f\sim \dot{\gamma }^{1},$ where f is the folding frequency, and $\dot{\gamma }$ is the characteristic shear rate) by Cubaud and Mason (Phys Rev Lett 96:114501, 2006a) was found to be valid even for hyperbolic channel. The hyperbolic channel generated moderate flows with smaller folding frequency, amplitude, and a delay of onset of the folding compared with other two cases, which is considered to be affected by compressive stress when compared to the simulation results. In each channel, the folding frequency increases and the amplitude decreases as the thread width decreases since higher compressive stress is applied along the thin thread. The secondary folding was also reproduced in the simulation, which was attributed to locally heterogeneous development of compressive stresses along the thread. This study proves that the viscous folding can be controlled by the design of flow kinematics and of the compressive stresses at the diverging region.

Appendix

In order to consider different physical properties of two-phase fluids, Heaviside function should be defined in a computational domain. In order to predict the Heaviside function, two approaches were introduced: one using a distance from the interface (Udaykumar et al. 1997), and the other using a solution from Poisson equation (Tryggvason et al. 2001). When the interface is positioned in the computational domain without a contact, both methods provide nearly same solution. However, when the interface confronts the boundary of the computational domain, the Poisson equation gives incorrect solution near the contact point since the boundary condition for the Poisson equation is unphysical (Fig. 8a), whereas the distance-based method shows an exact Heaviside function as shown in Fig. 8b. Therefore, in this study, the Heaviside function was determined by the distance-based method rather than by directly solving a Poisson equation as in our previous studies (Chung et al. 2008, 2009a, b) since the interface confronts the boundary of the flow domain at flow focusing region and outflow region.

Fig. 8

Comparison of Heaviside function obtained by different methods when the interface confronts the boundary of the flow domain (64 × 64 mesh)