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Technical Brief

A Modification of Murray's Law for Shear-Thinning Rheology

[+] Author and Article Information
Patrick M. McGah

Department of Mechanical Engineering,
University of Washington,
Stevens Way, Box 352600,
Seattle, WA 98195
e-mail: pmcgah@u.washington.edu

Massimo Capobianchi

Professor Department of Mechanical Engineering, Gonzaga University,
502 East Boone Avenue,
Spokane, WA 99258
e-mail: capobianchi@gonzaga.edu

1Corresponding author.

Manuscript received September 5, 2014; final manuscript received December 29, 2014; published online March 10, 2015. Assoc. Editor: Hai-Chao Han.

J Biomech Eng 137(5), 054503 (May 01, 2015) (6 pages) Paper No: BIO-14-1440; doi: 10.1115/1.4029504 History: Received September 05, 2014; Revised December 29, 2014; Online March 10, 2015

This study reformulates Murray's well-known principle of minimum work as applied to the cardiovascular system to include the effects of the shear-thinning rheology of blood. The viscous behavior is described using the extended modified power law (EMPL), which is a time-independent, but shear-thinning rheological constitutive equation. The resulting minimization problem is solved numerically for typical parameter ranges. The non-Newtonian analysis still predicts the classical cubic diameter dependence of the volume flow rate and the cubic branching law. The current analysis also predicts a constant wall shear stress throughout the vascular tree, albeit with a numerical value about 15–25% higher than the Newtonian analysis. Thus, experimentally observed deviations from the cubic branching law or the predicted constant wall shear stress in the vasculature cannot likely be attributed to blood's shear-thinning behavior. Further differences between the predictions of the non-Newtonian and the Newtonian analyses are highlighted, and the limitations of the Newtonian analysis are discussed. Finally, the range and limits of applicability of the current results as applied to the human arterial tree are also discussed.

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Figures

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Fig. 1

Viscosity versus shear rate for the least-squares fit of EMPL model (solid line) to the experimental data (symbols) of Chien et al. [16]

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Fig. 2

Cost function versus radius for a flow rate of 6 mL/min using b = 778 erg/cm3/s for both Newtonian and non-Newtonian cases

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Fig. 4

Wall shear stress versus optimal radius using b = 778 erg/cm3/s for both Newtonian and non-Newtonian cases

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Fig. 5

Velocity profiles at the optimal radius using b = 778 erg/cm3/s for both Newtonian and non-Newtonian cases

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Fig. 3

Volume flow rate versus optimal radius for a non-Newtonian fluid in the case where b = 778 erg/cm3/s. The slope of the curve is equal to 3 to within a margin 10−12, indicating that Murray's cubic relationship is valid even when non-Newtonian behavior is considered.

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