Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations

@article{Morrow2019NumericalIO,
  title={Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations},
  author={Liam C. Morrow and Timothy J. Moroney and Scott W. McCue},
  journal={Journal of Fluid Mechanics},
  year={2019},
  volume={877},
  pages={1063 - 1097}
}
Viscous fingering experiments in Hele-Shaw cells lead to striking pattern formations which have been the subject of intense focus among the physics and applied mathematics community for many years. In recent times, much attention has been devoted to devising strategies for controlling such patterns and reducing the growth of the interfacial fingers. We continue this research by reporting on numerical simulations, based on the level set method, of a generalised Hele-Shaw model for which the… 

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Three-dimensional direct numerical simulations of the full Navier–Stokes equations are conducted, using a phase field method to enforce surface tension at the interface, to verify the theory and explore the effect of depth gradient on the interface (in)stability.

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TLDR
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In this work, linear stability theory and variational calculus are used to determine closed-form expressions for the proper time-dependent injection rates Q(t) required to either minimize the interface disturbances or to control the number of emerging fingers.

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