Statistical approach for radial fingering in a Hele Shaw cell

(Received 24 July 1992, accepted 3 August 1992)

Abstract
We have used a statistical method to describe an unsteady radial fingering pattern. We determine a complete set of equations for the growth of the microstructure envelope which couples envelope radius, individual growth rate of fingers and the distribution function in the cell size space. In a preliminary approach, we neglect the important screening effect which may be responsible for the commonly observed fractal structures. In that case, the structure is compact, i.e. the relative width of fingers of order unity. The self-similar regime for which the time dependence of the envelope radius is in t^1/2 is then still a solution. The corresponding distribution function in the -space varies with t^1/4. Contrary to regular structures for which any wavelength and thus any growth velocity of the envelope can be a solution, the self-similar solution found here determines unambiguously the growth velocity of the radial pattern. This velocity is a function of the amplitude of the noise present in the Hele Shaw cell. Numerical integration of the equations is performed and the time evolution of the distribution function starting from various initial conditions analysed.