Newtonian fluid

A Newtonian fluid (named for Isaac Newton) is a fluid that flows like water—its 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.

A simple equation to describe Newtonian fluid behaviour is

$\tau =\mu {\frac {dv}{dx}}$

where

$\tau$ is the shear stress exerted by the fluid ("drag")

$\mu$ is the fluid viscosity - a constant of proportionality

${\frac {dv}{dx}}$ is the velocity gradient perpendicular to the direction of shear

In common terms, this means the fluid continues to flow, regardless of the forces acting on it. For example, water is Newtonian, because it continues to exemplify fluid properties no matter how fast it is stirred or mixed. Contrast this with a non-Newtonian fluid, in which stirring can leave a "hole" behind (that gradually fills up over time - this behaviour is seen in materials such as pudding, oobleck, or, to a less rigorous extent, sand), or cause the fluid to become thinner, the drop in viscosity causing it to flow more (this is seen in non-drip paints).

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), not on the forces acting upon it.

If the fluid is incompressible and viscosity is constant across the fluid, the equation governing the shear stress, in the Cartesian coordinate system, is

$\tau _{ij}=\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}\right)$

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

If a fluid does not obey this relation, it is termed a non-Newtonian fluid, of which there are several types.