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# Newtonian fluid

Continuum mechanics
Conservation of mass
Conservation of momentum
Navier-Stokes equations
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A Newtonian fluid (named for Isaac Newton) is a fluid that flows like water—its stress versus rate of strain curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.

A simple equation to describe Newtonian fluid behaviour is

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

where

τ is the shear stress exerted by the fluid ("drag") [Pa]
μ is the fluid viscosity - a constant of proportionality [Pa·s]
$\frac{du}{dx}$ is the velocity gradient perpendicular to the direction of shear [s−1]

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, which brush on easily but become more viscous when on walls).

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

$\tau_{ij}=\mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i} \right)$

with comoving stress tensor $\mathbb{P}$ (also written as $\mathbf{\sigma}$)

$\mathbb{P}_{ij}= - p \delta_{ij} + \mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i} \right)$

where, by the convention of tensor notation,

τij is the shear stress on the ith face of a fluid element in the jth direction
ui is the velocity in the ith direction
xj is the jth direction coordinate

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