The effect of surface tension on the shape of fingers in a Hele Shaw cell

Abstract

The experimental results of Saffman & Taylor (1958) and Pith (1980) on fingering in a Hele Shaw cell are modelled by two-dimensional potential flow with surface-tension effects included at the interface. Using free streamline techniques, the shape of the free surface is expressed rn the solution of a nonlinear integro-differential equation. The equation is solved numerically and the solutions are compared with experimental results. The shapes of the profiles are very well predicted, but the dependence of finger width on surface tension is not quantitatively accurate, although the qualitative behaviour is correct. A conflict between the numerics and a formal singular perturbation analysis is noted but not resolved. The stability of the steady finger to small disturbances is also examined. Linearized stability analysis indicates that the twodimensional fmgers are not stabilized by the surface-tension effect, which disagrees with the experimental observations. A possible rewon for the discrepancy between theory and experiment is suggested.

On the motion of unsteady translating bubbles in an unbounded Hele-Shaw cell

“…[25][26][27][28] Many formulations of the numerical methods are based on boundary integral equations (BIEs) due to the simplicity of the underlying equations, where, for example, a spectrally accurate spatial discretisation can be employed. 28 In many cases, the boundary integral equations are formulated using Green's second identity and letting the point of observation approach the interface between the two fluids.…”

Exact results with the J-integral applied to free-boundary flows

“…This restriction is in fact typical of free-boundary problems where a technique suitable for one situation completely fails in another case. As an example, conformal mapping techniques McLean & Saffman (1980), Saffman & Taylor (1958) and Taylor & Saffman (1959) have been shown to be efficient only for the isobaric interface. The J-integral provides exact results for infinite voids and bounds for finite-size voids.…”

“…This result is in fact a direct consequence of the uniqueness of the solution without surface tension. As a consequence, the deviation from 2 3 is linear in γ while, for viscous fingering it deviates from 1 2 by a γ 2/3 expansion (McLean & Saffman 1980). From our study of a limited list of examples, the technique fails to deliver a definitive value of λ in the cases for which surface tension is a singular perturbation.…”

“…It is worthwhile recalling that all the standard methods of complex analysis like the Hodograph methods with conformal mapping (Saffman & Taylor 1958;Taylor & Saffman 1959;McLean & Saffman 1980) and Scharwz's method failed to find this result although they were very successful for the Saffman-Taylor problem even in unusual contexts (see for example viscous fingering in the wedge geometry, Ben Amar 1991). It turns out that the boundary condition in (4.2) seems to be responsible for the failure of such analytic function treatments since this condition breaks the 'conformal symmetry': it balances the modulus of a complex function:…”

Numerical solution of the hele-shaw equations

“…Our model can be used to investigate steady state finger shapes proposed by McLean & Saffman (1981). We might also investigate convergence of our algorithm.…”