Introduction to Complex Fluids

In this chapter we introduce the fundamental concepts in Newtonian and complex fluid mechanics, beginning with the basic underlying assumptions in continuum mechanical modeling. The equations of mass and momentum conservation are derived, and the Cauchy stress tensor makes its first of many appearances. The Navier–Stokes equations are derived, along with their inertialess limit, the Stokes equations. Models used to describe complex fluid phenomena such as shear-dependent viscosity and viscoelasticity are then discussed, beginning with generalized Newtonian fluids. The Carreau–Yasuda and power-law fluid models receive special attention, and a mechanical instability is shown to exist for highly shear-thinning fluids. Differential constitutive models of viscoelastic flows are then described, beginning with the Maxwell fluid and Kelvin–Voigt solid models. After providing the foundations for objective (frame-invariant) derivatives, the linear models are extended to mathematically sound nonlinear models including the upper-convected Maxwell and Oldroyd-B models and others. A derivation of the upper-convected Maxwell model from the kinetic theory perspective is also provided. Finally, normal stress differences are discussed, and the reader is warned about common pitfalls in the mathematical modeling of complex fluids.