Newtonian fluid

A Newtonian fluid has viscous stress proportional to the local rate-of-strain tensor, with coefficients independent of that rate. For an isotropic fluid these coefficients are viscosities. Its Newtonian fluid stress tensor specifies the corresponding constitutive relation; the material and its stress tensor are distinct concepts.

For isotropic incompressible flow of a Newtonian fluid, the stress tensor is the displayed expression, with pressure $p$, dynamic viscosity $\mu$, velocity $\mathbf{u}$, and identity matrix $I$. In Couette flow, its off-diagonal component is the shear stress.

Shear stress is the tangential component of traction on a surface. For leading unidirectional flow $u(y)$ in a Newtonian fluid, $\tau =\mu du/dy$.

The shear velocity is the velocity scale $u_{\ast }=\sqrt{|\tau |/\rho }$ associated with a shear stress $\tau$ in a fluid of mass density $\rho$. It is a stress scale, not automatically the actual local fluid velocity at a grain.

The rate-of-strain tensor is the symmetric part of the velocity gradient,

The shear rate is the scalar rate at which adjacent material layers acquire shear strain. In simple shear $u_x=u_x(y)$, its signed value is $\dot{\gamma }=d{u}_x/dy$.

A simple shear flow has velocity $\mathbf{u}=(\dot{\gamma }y,0,0)$ in Cartesian coordinates. Its velocity gradient has one nonzero off-diagonal entry.

The spin tensor is the antisymmetric part of the velocity gradient, $\mathbf{\Omega }=(\nabla \mathbf{u}-(\nabla \mathbf{u}{)}^T)/2$. It describes local rigid-body rotation.

For a planar velocity field in polar coordinates,