  When a fluid comprised of multiple phases or constituents flows through a
network, non-linear phenomena such as multiple stable equilibrium states and
spontaneous oscillations can occur. Such behavior has been observed or
predicted in a number of networks including the flow of blood through the
microcirculation, the flow of picoliter droplets through microfluidic devices,
the flow of magma through lava tubes, and two-phase flow in refrigeration
systems. While the existence of non-linear phenomena in a network with many
inter-connections containing fluids with complex rheology may seem
unsurprising, this paper demonstrates that even simple networks containing
Newtonian fluids in laminar flow can demonstrate multiple equilibria.
  The paper describes a theoretical and experimental investigation of the
laminar flow of two miscible Newtonian fluids of different density and
viscosity through a simple network. The fluids stratify due to gravity and
remain as nearly distinct phases with some mixing occurring only by diffusion.
This fluid system has the advantage that it is easily controlled and modeled,
yet contains the key ingredients for network non-linearities. Experiments and
3D simulations are first used to explore how phases distribute at a single
T-junction. Once the phase separation at a single junction is known, a network
model is developed which predicts multiple equilibria in the simplest of
networks. The existence of multiple stable equilibria is confirmed
experimentally and a criteria for their existence is developed. The network
results are generic and could be applied to or found in different physical
systems.
