5.16 Viscous Fluids

The most common family of constitutive relations employed for viscous fluids, including Newtonian fluids, is given by

\[ \begin{equation} \boldsymbol{\tau}\left(J,\mathbf{D}\right)=\left(\kappa-\frac{2}{3}\mu\right)\left(\tr\mathbf{D}\right)\,\mathbf{I}+2\mu\,\mathbf{D}\,,\label{eq:viscous-stress} \end{equation} \]

where \(\mu\) and \(\kappa\) are, respectively, the dynamic shear and bulk viscosity coefficients (both positive), which may generally depend on \(J\) and, for non-Newtonian fluids, on invariants of \(\mathbf{D}\). In practice, most constitutive models for non-Newtonian viscous fluids only use a dependence on \(\dot{\gamma}=\sqrt{2\mathbf{D}:\mathbf{D}}\), since it is the only non-zero invariant in viscometric flows ¹. In this case, substituting eq.\eqref{eq:viscous-stress} into eq.(3.5-9) produces

\[ \begin{equation} \begin{aligned}\boldsymbol{\mathcal{C}}^{\tau} & =\left(\kappa-\frac{2}{3}\mu\right)\mathbf{I}\otimes\mathbf{I}+\frac{2}{\dot{\gamma}}\left(\frac{\partial\kappa}{\partial\dot{\gamma}}-\frac{2}{3}\frac{\partial\mu}{\partial\dot{\gamma}}\right)\left(\tr\mathbf{D}\right)\mathbf{I}\otimes\mathbf{D}\\ & +2\left(\frac{2}{\dot{\gamma}}\frac{\partial\mu}{\partial\dot{\gamma}}\mathbf{D}\otimes\mathbf{D}+\mu\,\mathbf{I}\odot\mathbf{I}\right)\,. \end{aligned} \label{eq:viscous-tangent-D} \end{equation} \]

The term containing \(\mathbf{I}\otimes\mathbf{D}\) is the only one that does not exhibit major symmetry. In Newtonian fluids, \(\mu\) and \(\kappa\) are independent of \(\mathbf{D}\); in incompressible fluids they are independent of \(J\) (since \(J=1\) remains constant and \(\tr\mathbf{D}=0\)). Thus, for both of these cases the term containing \(\mathbf{I}\otimes\mathbf{D}\) drops out and \(\boldsymbol{\mathcal{C}}^{\tau}\) exhibits major symmetry.

Similarly, using eq.\eqref{eq:viscous-stress}, the tangent \(\boldsymbol{\tau}_{J}^{\prime}\) in eq.(3.5-11) reduces to

\[ \begin{equation} \boldsymbol{\tau}_{J}^{\prime}=\left(\frac{\partial\kappa}{\partial J}-\frac{2}{3}\frac{\partial\mu}{\partial J}\right)\left(\tr\mathbf{D}\right)\,\mathbf{I}+2\frac{\partial\mu}{\partial J}\,\mathbf{D}\,.\label{eq:viscous-tangent-J} \end{equation} \]

Explicit forms for the dependence of \(\mu\) or \(\kappa\) on \(J\) are not illustrated here, since viscosity generally shows negligible dependence on pressure (thus \(J\)) over typical ranges of pressures in fluids, hence \(\boldsymbol{\tau}_{J}^{\prime}\approx\mathbf{0}\) in most analyses.

Many fluid mechanics textbooks employ Stoke's condition (\(\kappa=0\)) for the purpose of equating the elastic pressure \(p\) with the mean normal stress \(-\frac{1}{3}\tr\boldsymbol{\sigma}\) ²; in FEBio, \(\kappa\) is kept as a user-defined material property, which may be set to zero if desired. A common example of a non-Newtonian fluid is the Carreau model, where \(\boldsymbol{\tau}=2\mu\left(\dot{\gamma}\right)\mathbf{D}\), which is a special case of eq.\eqref{eq:viscous-stress}, with \(\kappa=2\mu/3\) and

\[ \begin{equation} \mu=\mu_{\infty}+\left(\mu_{0}-\mu_{\infty}\right)\left(1+\left(\lambda\dot{\gamma}\right)^{2}\right)^{\left(n-1\right)/2}\,,\label{eq:Carreau-model-viscosity} \end{equation} \]

where \(\lambda\) is a time constant, \(n\) is a parameter governing the power-law response, \(\mu_{0}\) is the viscosity when \(\dot{\gamma}=0\) and \(\mu_{\infty}\) is the viscosity as \(\dot{\gamma}\to\infty\). Other common models of non-Newtonian viscous fluids are summarized in ³, though it should be noted that some of these models produce infinite values when evaluating \(\dot{\gamma}^{-1}\partial\mu/\partial\dot{\gamma}\) as \(\dot{\gamma}\to0\), which is problematic in the evaluation of \(\boldsymbol{\mathcal{C}}^{\tau}\) in eq.\eqref{eq:viscous-tangent-D}.

For nearly incompressible fluids, a simple constitutive relation may be adopted for the pressure,

\[ \begin{equation} p\left(J\right)=K\left(1-J\right)\,,\label{eq:ideal-fluid} \end{equation} \]

where \(K\) is the bulk modulus of the fluid in the limit when \(J=1\). It follows that \(p^{\prime}\left(J\right)=-K\) and \(p^{\prime\prime}\left(J\right)=0\) in eq.(3.5-10). This constitutive relation is adopted for nearly-incompressible CFD analyses in FEBio, though alternative formulations may be easily implemented.

---

1. Reddy, J. N.. "An introduction to continuum mechanics : with applications." Cambridge University Press, pp. xiv, 354 p. (2008). ↩

2. Panton, Ronald L. "Incompressible flow." John Wiley \& Sons (2006). ↩

3. Cho, Y I; Kensey, K R. "Effects of the non-{Newtonian} viscosity of blood on flows in a diseased arterial vessel. {Part} 1: {Steady} flows." Biorheology, vol. 28, pp. 241-62 (1991). ↩