2.12 Newtonian fluid

In Sec. 2.6 we introduced forces in a fluid. At the macroscopic scale of a continuum, we characterise a fluid’s response to applied forces through constitutive models.

The Newtonian (or linear viscous) fluid is the most common constitutive model that represents the behaviour of many liquids and gases. It states that a fluid at rest (or uniform velocity) does not sustain shear stress eqn; it can be expressed by

|----------------------------| ☐☐☐ = 2 devD and ☐☐☐ = ☐☐☐ pI| ------------------------------ \relax \special {t4ht=

PICT\relax \special {t4ht=

The model is a continuum representation9 of Newton’s law of viscosity10 which states that shear stress eqn is proportional to velocity gradient by the dynamic viscosity eqn.

It is a specific case of the more general Stokesian fluid, defined as eqn, where eqn is deformation rate, Eq. (2.33 ). The Newtonian model assumes: (1) the fluid is isotropic, i.e. the value of eqn is independent of the direction in which it is measured; (2) zero bulk viscosity associated with a change in volume o the fluid.

Using eqn in Eq. (2.41 ) ensures that the shear stress is induced by deformation only. Taking the deviatoric part, eqn, ensures viscous stresses are not generated by volume changes, which are represented by eqn.

This is due to the “eqn” operator subtracting eqn from each diagonal component of eqn, giving a total of eqn(eqn) from all 3 diagonal components. From Eq. (2.35 ), eqn

Pressure gradient

When we substitute Eq. (2.41) in eqn in Eq. (2.19 ), the pressure part is eqn. The term eqn is equivalent to gradient eqn.

PICT\relax \special {t4ht=

Like divergence of stress in Eq. (2.22 ), the gradient of pressure represents the pressure flux across the surface per unit volume as eqn, according to

 1 Z rp = liVm!0 --V- (dS p): S \relax \special {t4ht=
Since eqn holds for any variable eqn, a gradient term is conservative, like divergence, and can be converted to a surface integral under an equivalent Gauss’s Theorem
Z Z (dS ) = r dV : S V \relax \special {t4ht=

9attributed to Adhémar Jean Claude Barré de Saint-Venant, 1843, and George Gabriel Stokes, 1845; derived previously using molecular models by Claude-Louis Navier, Mémoire sur les lois du mouvement des fluides, 1822.
10Isaac Newton, Philosophiae Naturalis Principia Mathematica, 1687.

Notes on CFD: General Principles - 2.12 Newtonian fluid