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5.15 Viscous Fluids
The most common family of constitutive relations employed for viscous fluids, including Newtonian fluids, is given by where and are, respectively, the dynamic shear and bulk viscosity coefficients (both positive), which may generally depend on and, for non-Newtonian fluids, on invariants of . In practice, most constitutive models for non-Newtonian viscous fluids only use a dependence on , since it is the only non-zero invariant in viscometric flows . In this case, substituting eq.(5.15-1) into eq.(3.5.2-6) produces The term containing is the only one that does not exhibit major symmetry. In Newtonian fluids, and are independent of ; in incompressible fluids they are independent of (since remains constant and ). Thus, for both of these cases the term containing drops out and exhibits major symmetry.
Similarly, using eq.(5.15-1), the tangent in eq.(3.5.2-8) reduces to Explicit forms for the dependence of or on are not illustrated here, since viscosity generally shows negligible dependence on pressure (thus ) over typical ranges of pressures in fluids, hence in most analyses.
Many fluid mechanics textbooks employ Stoke's condition ( ) for the purpose of equating the elastic pressure with the mean normal stress ; in FEBio, 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 , which is a special case of eq.(5.15-1), with and where is a time constant, is a parameter governing the power-law response, is the viscosity when and is the viscosity as . Other common models of non-Newtonian viscous fluids are summarized in , though it should be noted that some of these models produce infinite values when evaluating as , which is problematic in the evaluation of in eq.(5.15-2).
For nearly incompressible fluids, a simple constitutive relation may be adopted for the pressure, where is the bulk modulus of the fluid in the limit when . It follows that and in eq.(3.5.2-7). This constitutive relation is adopted for nearly-incompressible CFD analyses in FEBio, though alternative formulations may be easily implemented.