Flory approximant for the fractal dimension of the viscous-finger pattern: Transient and asymptotic behaviors

We present a ‘‘Flory-type’’ mean-field theory to calculate the fractal dimension D of the radial viscous finger pattern formed by the less viscous fluid in a Hele-Shaw cell. We predict (a) the relation between the interfacial tension σ between the two fluids and the onset of the asymptotic behavior and (b) the variation of the effective fractal dimension with time in the transient regime for a fixed σ.