Nonlinear unstable viscous fingers in Hele–Shaw flows. II. Numerical simulation

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Abstract
The nonlinear stages of two-dimensional immiscible displacement processes in Hele–Shaw flows are investigated by means of large scale numerical simulations based on a purely Lagrangian vortex method. The vortex sheet at the interface between the two fluid phases is discretized into circular arcs with a continuous distribution of circulation, which renders our numerical technique highly accurate. A complicated unsteady growth mechanism is observed for the emerging viscous fingers, involving a combination of spreading, shielding, and tip splitting. As the surface tension is further reduced, smaller length scales arise and the fingertip exhibits a new splitting pattern in which three new lobes emerge instead of two. Monitoring the velocity as well as the radius of curvature at the fingertip demonstrates that the instability of the finger evolves in an oscillatory fashion. The two-lobe and the three-lobe splitting can thus be explained as different manifestations of the same instability mode. Comparison with experiment shows good qualitative but only fair quantitative agreement. By imposing a constraint on the curvature at the fingertip, experimental results, which show fingers of width considerably smaller than half the cell width and exhibit ‘‘dendritic’’ instability modes, are reproduced.

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