Dissipation occurs naturally in the growth channels of many physical pattern forming systems. In electrochemical deposition, for example, the deposited metal has a measurable resistance; the advancing fluid in viscous fingering systems similarly has a finite viscosity. A complete theory for pattern formation in such systems would account for field gradients within the detailed branching geometry of the evolving pattern. In the absence of such a theory, we construct a self-consistent model by assuming that a dense branching pattern has already formed and investigating its stability against small deformations. We treat the region behind the smooth advancing envelope as an effective medium whose transport properties mimic those of actual patterns at least in an average sense.
Our two-sided model consists of a scalar field both in the patterned aggregate region (region 1) and in the region outside (region 2). Within the aggregate region, the field satisfies an anisotropic Laplace's equation
where and are respectively the conductivities along and perpendicular to the branches; and denote the corresponding components of the Laplacian. This macroscopic conductivity anisotropy originates from the branched structure of the DBM and reflects the currents' preference to flow along the branches rather than between them. It should not be confused with the microscopic crystalline anisotropy which is responsible for stabilizing the dendritic morphology against tip splitting.
Outside the pattern, we assume the field satisfies the diffusion equation
This reduces to the usual Laplace's equation in the quasistatic limit. The conductivity in this region is isotropic. We assume that the interface advances at a rate proportional to the local current density, , where b is a system-dependent material parameter and is the position of the interface. We further assume that the currents arise from gradients in the field according to Fick's law:
The system is driven out of equilibrium by a constant potential difference applied across the boundaries at and :
Assuming continuity in both the field and current across the interface at ,
allows us to solve for the interface's evolution.
In the context of electrochemical deposition, Eqs. (1)-(7) might be interpreted as describing a pair of arbitrarily shaped electrodes held at a fixed voltage difference in contact with an electrolyte of conductivity . The field then would correspond roughly to the electrochemical potential at position . For viscous fingering, represents the local pressure field, and the system is driven by a constant pressure difference between the boundaries. While this simple growth model glosses over most system-dependent details which might dominate a system's behavior under some operating conditions, its behavior is rich enough to shed light on generic mechanisms of pattern formation under diffusive control.
We find it useful to define two dimensionless control parameters: the conductivity anisotropy
and the conductivity contrast
Large anisotropy is indicated by small values of , which physically correspond to stronger confinement of currents to the branches. Smaller values of similarly correspond to stronger conductivity contrast between the invading and displaced phases. For viscous fingering, the condition corresponds to injecting a viscous fluid into an inviscid fluid. The interface is intrinsically stable under these conditions and no branches form. By contrast, corresponds to the DLA-like case in which the aggregate surface is an equipotential and the interface is intrinsically unstable at all wavelengths. We focus instead on the more interesting intermediate range .
Analysis of this model can either proceed numerically on a computer, or analytically via linear stability analysis. Previously , 2D radial computer simulations of this model were performed, showing DLA and DBM-like growth. Here, we concentrate on linear stability analysis to probe the predictions of our model in various geometries. Following the procedure described by Mullins and Sekerka  we first assume that a radial or planar DBM has formed of a certain size and that its envelope separates regions 1 and 2 in the above formulation. The envelope is then distorted with a perturbation of infinitesimal amplitude . Since a general infinitesimal perturbation can be built up by linear superposition of any complete set of functions, we examine only sinusoidal modulations in the 2D radial and planar geometries, and spherical harmonics in the 3D radial geometry. The response of the system to linear order in is then calculated in the form of a dimensionless growth rate of the perturbation
where labels the position of the unperturbed envelope, and its velocity.