Hemorheology - Constitutive Equations

Constitutive Equations

The relationships between shear stress and shear rate for blood must be determined experimentally and expressed by constitutive equations. Given the complex macro-rheological behavior of blood, it is not surprising that a single equation fails to completely describe the effects of various rheological variables (e.g., hematocrit, shear rate). Thus, several approaches to defining these equations exist, with some the result of curve-fitting experimental data and others based on a particular rheological model.

• Newtonian fluid model where has a constant viscosity at all shear rates. This approach is valid for high shear rates where the vessel diameter is much bigger than the blood cells.

• Bingham fluid model takes into account the aggregation of red blood cells at low shear rates. Therefore, it acts as an elastic solid under threshold level of shear stress, known as yield stress.

• Einstein model where η₀ is the suspending fluid Newtonian viscosity, "k" is a constant dependent on particle shape, and H is the volume fraction of the suspension occupied by particles. This equation is applicable for suspensions having a low volume fraction of particles. Einstein showed k=2.5 for spherical particles.

• Casson model where ""a" and "b" are constants; at very low shear rates, b is the yield shear stress. However, for blood, the experimental data can not be fit over all shear rates with only one set of constants "a" and "b", whereas fairly good fit is possible by applying the equation over several shear rate ranges and thereby obtaining several sets of constants.

• Quemada model where k₀, k_∞ and γ_c are constants. This equation accurately fits blood data over a very wide range of shear rates.