Newtonian fluid

A Newtonian fluid is a fluid in which shear stress is linearly proportional to the velocity gradient in the direction perpendicular to the plane of shear. The constant of proportionality is known as the viscosity. For a Newtonian fluid, the viscosity by definition depends only on temperature and pressure, and also the chemical composition of the fluid if the fluid is not a pure substance. If the fluid is incompressible and viscosity is constant across the fluid, the equation governing the shear stress, in the Cartesian coordinate system, is

$\tau _{ij}=\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}-{\frac {2}{3}}\delta _{ij}\nabla \cdot \mathbf {v} \right)$

where

$\tau _{ij}$ is the shear stress on the $i^{th}$ face of a fluid element in the $j^{th}$ direction

$v_{i}$ is the velocity in the $i^{th}$ direction

$x_{j}$ is the $j^{th}$ direction coordinate

$\delta _{ij}$ is the Kronecker delta

If a fluid does not obey this relation, it is termed a non-Newtonian fluid.