Some numerical methods for the Hele-Shaw equations

Journal Article · Tue Mar 01 00:00:00 EST 1994 · Journal of Computational Physics; (United States)

DOI:https://doi.org/10.1006/jcph.1994.1046· OSTI ID:7242387

Whitaker, N ^[1]

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1. Univ. of Massachusetts, Amherst (United States)

Tryggvason and Aref used a boundary integral method and the vortex-in-cell method to evolve the interface between two fluids in a Hele-Shaw cell. The method gives excellent results for intermediate values of the nondimensional surface tension parameter. The results are different from the predicted results of McLean and Saffman for small surface tension. For large surface tension, there are some numerical problems. In this paper, we implement the method of Tryggvason and Aref but use the point vortex method instead of the vortex-in-cell method. A parametric spline is used to represent the interface. The finger widths obtained agree well with those predicted by McLean and Saffman. We conclude the the method of Tryggvason and Aref can provide excellent results but that the vortex-in-cell method may not be the method of choice for extreme values of the surface tension parameter. In a second method, we represent the interface with a Fourier representation. In addition, an alternative way of discretizing the boundary integral is used. Our results are compared to the linearized theory and the results of McLean and Saffman and are shown to be highly accurate. 21 refs., 4 figs., 2 tabs.

• OSTI ID: 7242387
• Journal Information: Journal of Computational Physics; (United States), Vol. 111:1; ISSN 0021-9991
• Country of Publication: United States
• Language: English

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