  Nonlinear phenomena including multiple equilibria and spontaneous
oscillations are common in fluid networks containing either multiple phases or
constituent flows. In many systems, such behavior might be attributed to the
complicated geometry of the network, the complex rheology of the constituent
fluids, or, in the case of microvascular blood flow, biological control. In
this paper we investigate two examples of a simple three-node fluid network
containing two miscible Newtonian fluids of differing viscosities, the first
modeling microvascular blood flow and the second modeling stratified laminar
flow. We use a combination of analytic and numerical techniques to identify and
track saddle-node and Hopf bifurcations through the large parameter space. In
both models, we document sustained spontaneous oscillations and, for an
experimentally relevant example of parameter analysis, investigate the
sensitivity of these oscillations to changes in the viscosity contrast between
the constituent fluids and the inlet flow rates. For the case of stratified
laminar flow, we detail a physically realizable set of network parameters that
exhibit rich dynamics. The tools and results developed here are general and
could be applied to other physical systems.
