  Attractive colloidal dispersions, suspensions of fine particles which
aggregate and frequently form a space spanning elastic gel are ubiquitous
materials in society with a wide range of applications. The colloidal networks
in these materials can exist in a mode of free settling when the network weight
exceeds its compressive yield stress. An equivalent state occurs when the
network is held fixed in place and used as a filter through which the
suspending fluid is pumped. In either scenario, hydrodynamic instabilities
leading to loss of network integrity occur. Experimental observations have
shown that the loss of integrity is associated with the formation of eroded
channels, so-called streamers, through which the fluid flows rapidly. However,
the dynamics of growth and subsequent mechanism of collapse remain poorly
understood. Here, a phenomenological model is presented that describes
dynamically the radial growth of a streamer due to erosion of the network by
rapid fluid back flow. The model exhibits a finite-time blowup -- the onset of
catastrophic failure in the gel -- due to activated breaking of the
inter-colloid bonds. Brownian dynamics simulations of hydrodynamically
interacting and settling colloids in dilute gels are employed to examine the
initiation and propagation of this instability, which is in good agreement with
the theory. The model dynamics are also shown to accurately replicate
measurements of streamer growth in two different experimental systems. The
predictive capabilities and future improvements of the model are discussed and
a stability-state diagram is presented providing insight into engineering
strategies for avoiding settling instabilities in networks meant to have long
shelf lives.
