Mathias Nagel

A new prediction of wavelength selection in radial viscous fingering involving normal and tangential stresses

We reconsider the radial Saffman-Taylor instability, when a fluid injected from a point source displaces another fluid of higher viscosity in a Hele-Shaw cell, where the fluids are confined between two neighboring flat plates. The advancing fluid front is unstable and forms fingers along the circumference. The so-called Brinkman equation is used to describe the flow field, which also takes into account viscous stresses in the plane of the confining plates and, unlike the Darcy equation, not only viscous stresses due to the confining plates. We show why in-plane stresses cannot always be neglected and how they appear naturally in the potential flow problem. The dispersion relation obtained with the Brinkman equation agrees better with the experimental results than the classical linear stability analysis of radial fingering in Hele-Shaw cells that uses Darcys law as a model for the fluid motion.

Boundary elements method for microfluidic two-phase flows in shallow channels

Abstract In the following work we apply the boundary element method to two-phase flows in shallow microchannels, where one phase is dispersed and does not wet the channel walls. These kinds of flows are often encountered in microfluidic Lab-On-A-Chip devices and characterized by low Reynolds and low capillary numbers. Assuming that these channels are homogeneous in height and have a large aspect ratio, we use depth-averaged equations to describe these two-phase flows using the Brinkman equation, which constitutes a refinement of Darcy’s law. These partial differential equations are discretized and solved numerically using the boundary element method, where a stabilization scheme is applied to the surface tension terms, allowing for a less restrictive time step at low capillary numbers. The convergence of the numerical algorithm is checked against a static analytical solution and on a dynamic test case. Finally the algorithm is applied to the non-linear development of the Saffman–Taylor instability and compared to experimental studies of droplet deformation in expanding flows.

A numerical study of droplet trapping in microfluidic devices

Microfluidic channels are powerful means of control of minute volumes such as droplets. These droplets are usually conveyed at will in an externally imposed flow which follows the geometry of the micro-channel. It has recently been pointed out by Dangla et al. [“Trapping microfluidic drops in wells of surface energy,” Phys. Rev. Lett. 107(12), 124501 (2011)] that the motion of transported droplets may also be stopped in the flow, when they are anchored to grooves which are etched in the channels top wall. This feature of the channel geometry explores a direction that is usually uniform in microfluidics. Herein, this anchoring effect exploiting the three spatial directions is studied combining a depth averaged fluid description and a geometrical model that accounts for the shape of the droplet in the anchor. First, the presented method is shown to enable the capture and release droplets in numerical simulations. Second, this tool is used in a numerical investigation of the physical mechanisms at play in th...

Oscillations of confined fibres transported in microchannels

We investigate the trajectories of rigid fibres as they are transported in a pressure-driven flow, at low Reynolds number, in shallow Hele-Shaw cells. The transverse confinement and the resulting viscous friction on these elongated objects, as well as the lateral confinement (i.e. the presence of lateral walls), lead to complex fibre trajectories that we characterize with a combination of microfluidic experiments and simulations using modified Brinkman equations. We show that the transported fibre behaves as an oscillator for which we obtain and analyse a complete state diagram.