10. Constitutive Equations of Fluid Mechanics

10.1. Newtonian Fluids

A Newtonian fluid is the model of a fluid as a continuous medium whose stress tensor can be written as the sum of the hydrostatic pressure stress tensor \(-p \mathbb{I}\) - the only contribution holding in statics - and a viscous stress tensor \(\mathbb{S}\)

\[\mathbb{T} = -p \mathbb{I} + \mathbb{S} \ ,\]

and the viscous stress tensor is isotropic and linear in the first-order spatial derivatives of the velocity field,

(10.1)\[\mathbb{S} = 2 \mu \mathbb{D} + \lambda (\nabla \cdot \vec{u}) \mathbb{I} \ ,\]

being \(\mu, \lambda\) the viscosity coefficients, and \(\mathbb{D}\) the strain velocity tensor (1.4). Thus, the definition

Definition 10.1 (Newtonian fluid)

A Newtonian fluid is a continuous medium whose stress tensor reads

(10.2)\[\mathbb{T} = - p \mathbb{I} + 2 \mu \mathbb{D} + \lambda (\nabla \cdot \vec{u}) \mathbb{I} \ .\]

Note

The expression (10.1) of the viscosity stress tensor is the most general expression of a 2-nd order symmetric isotropic tensor proportional to 1-st order derivatives of a vector field.