Spontaneous Oscillations in Simple Fluid Networks

Abstract

Abstract. 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.

Observations of spontaneous oscillations in simple two-fluid networks

“…In dimensionless terms this convection model depends upon the ratio of the nominal resistances, r C /r B and r A /r B , the ratio of the volume of the vessels, V A /V C and V B /V C , the viscosity contrast µ 2 /µ 1 , and Φ in . In our previous theoretical paper, we found that the region of parameter space where the convective model shows instability is dominated by the case when the diameter of vessel C is large relative to all others [27]. The large diameter introduces a simplifying limit where r C /r B → 0, V A /V C → 0, and V B /V C → 0.…”

“…The sizing of the lengths and diameters in the network were guided by the model predictions in our previous work [27]. The diameters of the cylindrical vessels was set to d A = 0.8 mm = 1/32 in, d B = 0.51 mm = 1/50 in, and d C = 3.2 mm = 1/8 in.…”

“…Through theory and experiment, we have shown that the existence of phase sep-aration 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]. We used a combination of analytic and numerical techniques to identify and track saddlenode and Hopf bifurcations through the large parameter space.…”

Oscillations and Multiple Equilibria in Microvascular Blood Flow

“…This approach has proved fruitful in the past. Prior theoretical work using similar network models, but different constitutive laws for stratified flow of Newtonian fluids with different viscosity (Karst et al 2014) led to experiments which confirmed the existence of oscillations (Storey et al 2015). Here, however, the predictions of observable oscillatory behavior with blood flow occur over a relatively narrow range of parameters.…”

“…We find a starting point for the continuation by plotting the solution curves of Eqs. 39, 40 along an equilibrium curve and visually identifying intersections, see Karst et al (2014) for a more complete description of the technique.…”