Experiments such as those depicted in Fig. 1 have three-dimensional analogs which have been studied [13, 14, 15] almost as extensively as their more easily interpreted two-dimensional variants. Recent advances in admittance spectroscopy  and image analysis  make it possible to analyze the shapes of evolving three-dimensional patterns. High speed magnetic resonance imaging also has been applied to the study of fluid flow in porous media. Three dimensional radial pattern formation in which branches grow outward from the end of a conduit constitute another class of systems to which our growth model should pertain.
In the full three-dimensional radial geometry model, the unperturbed spherical interface advances with velocity
An infinitesimal perturbation to the growth front with a spherical harmonic
grows at the relative rate
where . Figure 2 shows a plot of the growth rate as a function of the perturbation harmonic for a range of dissipation values. The growth rate is negative for values of l less than a critical mode number which depends on the degree of dissipation in the growth channels. Such long wavelength perturbations shrink as the aggregate grows. The greater , therefore, the more nearly spherical the evolving pattern appears. In Fig. 3 this marginally stable harmonic number is plotted as a function of the amount of dissipation ( ) and anisotropy ( ) in the system. As seen from the plot, dissipation and current confinement act in concert to stabilize the dense branching morphology in the three dimensional radial geometry.
In the limit , we can solve for the critical aggregate radius beyond which the envelope will be stable against perturbations of harmonic number l:
In the anisotropic limit ( ), the expression simplifies to
For the DBM is stable against perturbations of harmonic number greater than l. Even for , an initially disordered core pattern can cross over into a regime where densely branched growth will be stable, provided the enveloping equipotential still is reasonably smooth by the time the interface has advanced to a mean position . The size at which the envelope of the growing pattern becomes stable scales with the size of the inner boundary, , and depends strongly on the amount of dissipation in the system. The position at which such a crossover to stable dense radial growth might occur for and appears in Fig. 4 as a rapid increase in with . Under certain conditions, decreases again for very near R and a second crossover from stable DBM to unstable growth is possible.