Blood viscoelasticity and thixotropy from stress formation and relaxation measurements: A unified model

Abstract

A Maxwell-type equation, which involves both time-dependent viscosity $\mathit{\eta }\mathrm{(}\mathit{t}\mathrm{)}$ and elastic modulus $\mathit{G}\mathrm{(}\mathit{t}\mathrm{)}$ is proposed as governing the shear stress difference ${\mathit{\sigma }}^{\prime }=\mathit{\sigma }-{\mathit{\sigma }}_{\mathit{y}}$ , where σ and ${\mathit{\sigma }}_{\mathit{y}}$ are total and yield shear stresses respectively.

A (reaction kinetics) equation for a structure variable λ is written which describes the evolution of blood structure (network ⇄ rouleau ⇄ RBC) in stress formation and stress relaxation measurements and which also depends on the (constant) applied shear rate ${\dot{\mathit{\gamma }}}_1$ . The time dependent viscosity is assumed to involve the solution $\mathit{\lambda }\mathrm{(}\mathit{t}\mathrm{,}{\dot{\mathit{\gamma }}}_1$ ) of the rate equation in the same manner than its equilibrium value ${\mathit{\lambda }}_{\mathit{e}\mathit{q}}\mathrm{(}{\dot{\mathit{\gamma }}}_1\mathrm{)}$ enters in a viscosity equation $\mathit{\eta }\mathrm{(}{\mathit{\lambda }}_{\mathit{e}\mathit{q}}\mathrm{)}$ yet proposed by one of the authors. A simple relation $\mathit{G}\mathrm{(}\mathit{\lambda }\mathrm{)}$ completes the structural description.

General solutions $\mathit{\sigma }\mathrm{(}\mathit{t}\mathrm{)}$ of the Maxwell-type equation are discussed in the case of stress relaxation (after the cessation of steady flow) and stress formation (under constant shear rate). Especially, the latter exhibits the well-known “overshoot” behavior. Moreover, the long time behavior of the former allows the quantification of the yield shear stress. Lastly one example of application to blood measurements is discussed.