Non-Newtonian Flow in Bifurcated Piping and Arterial Networks

Abstract

Flows in multi-branch piping systems are modeled for linear Newtonian and nonlinear Power Law, Bingham Plastic and Herschel-Bulkley fluids. The nonlinear models are often used to describe multiphase fluids consisting of liquid and solid phases, including cement slurries and drilling muds in petroleum engineering. Rheological and heterogeneous multiphase effects are important to blood flows in single and bifurcated arteries, capillaries and veins – biological applications are also emphasized to highlight their increasing importance in medical research. Unfortunately, conventional engineering approaches, many oversimplified, employ idealized assumptions for total energy conservation. Others are unrealistic and based on empirical “head loss” coefficients related to fittings, roughness, bend and expansion effects. In our approach, mass conservation is assumed, leading to momentum and energy reductions and a credible predictive algorithm is devised. Bifurcation results are given for several fluid rheologies, either in closed analytical form or using numerical algorithms based on closed form solutions. When the entry flow rate and all outlet pressures are given, along with needed geometric details, solutions give pressure drop versus flow rate relations useful in pumping and power calculations. Also calculated are pressure levels that ensure safe, burst-free operations and wall viscous shear stress values that support cleaning and remediation operations. Analytical formulas are derived for circular flow cross-sections while numerical extensions are summarized for general clogged flows (applications in piping conduits of arbitrary cross-sectional geometry). Example calculations are given demonstrating the usefulness and versatility of the new methods.