Laminar flow of two miscible fluids in a simple network

Abstract

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

Observations of spontaneous oscillations in simple two-fluid networks

“…Rather than determining the phase separation and effective viscosity functions separately through experiment, we determine their combination in a single experiment. The flat region in the pressure-flow curves (Q A /Q in > 0.77) correspond to vessels B and C essentially containing all water [26].…”

“…The T-junction has outlet branches which are 90 degrees relative to each other and the orientation of this diverging node as shown in Fig. 1 is important for the results we obtain [26]. The content of vessel C is monitored by imaging every 5 seconds.…”

“…In the model, the boundary condition to the vessels downstream of the inlet diverging node are provided by a constitutive law which states the phase separation function. For our stratified flow system, this phase separation behavior was measured in prior work [26]. The phase separation function provides the volume fraction in the downstream tubes as a function of the flow into and out of the node, Q in and Q A .…”

“…This fluid system provides controllable laboratory experiments and the simple network geometries are amenable to analysis. Through theory and experiment, we have shown that the existence of phase separation at a single junction and non-linear mixture viscosity in this system can lead to multiple stable equilibrium states within the network [25,26]. We recently conducted a theoretical study of dynamics in networks with this fluid system [27].…”

“…The fluid system is two miscible Newtonian fluids of differing viscosities and densities such that there is stratified flow within each tube [26]. The network is only a single loop with one inlet and one outlet.…”