Hele Shaw flows with a free boundary produced by the injection of fluid into a narrow channel

Abstract

A blob of Newtonian fluid is sandwiched in the narrow gap between two plane parallel surfaces so that, a t some initial instant, its plan-view occupies a simply connected domain D0. Further fluid, with the same material properties, is injected into the gap at some fixed point within D0, so that the blob begins to grow in size. The domain D occupied by the fluid at some subsequent time is to be determined.It is shown that the growth is controlled by the existence of an infinite number of invariants of the motion, which are of a purely geometric character. For sufficiently simple initial domains D0 these allow the problem to be reduced to the solution of a finite system of algebraic equations. For more complex initial domains an approximation scheme leads to a similar system of equations to be solved.

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

“…For example, in the case of Laplacian fields, if we write φ (1+2) = φ (1) + φ (2) , where both φ (1) and φ (2) are Laplacian, then by subtraction we can identify 12) which is conserved for all paths C surrounding any domain within which either φ (1) or φ (2) has singularities, and is zero in absence of singularities. Later, we show an example of this in deriving the Richardson (1972) moments in Hele-Shaw flows with source points. In our applications of (2.7) to fingering problems, most of the terms may vanish because of φ n .…”

“…Richardson moments An infinite set of conserved integrals has been discovered by Richardson (1972) for Hele-Shaw flows induced by injection of fluid at a point, and one might wonder if there is a relation to the class of integrals discussed here. Richardson's concern is with simply connected blobs of fluid occupying a time-dependent domain with zero pressure at their boundary.…”

“…These are integrals over a two-dimensional domain whereas the conserved integrals that we discuss here are along a one-dimensional path within a two-dimensional domain. For further discussion of this and related conservation integrals, see Richardson (1972Richardson ( , 1992, Tanveer & Vasconcelos (1995), Cummings, King & Howison (1997), Crowdy & Tanveer (1998) and Crowdy (1999).…”

“…We may apply (2.12) by letting φ (1) = φ (1) (x, y, t) be the solution to Richardson's injection problem, for which the corresponding complex potential Φ(z) has the form (Richardson 1972)…”

“…Choosing Φ (2) (z) = z n , or iz n , with n > 0, this provides a new derivation of Richardson's (1972) moment conservation.…”