Newtonian fluid

Continuum mechanics
Key topics
Conservation of mass
Conservation of momentum
Navier-Stokes equations
Classical mechanics
Stress · Strain · Tensor
Solid mechanics
Solids · Elasticity
Fluid mechanics
Fluids · Fluid statics
Fluid dynamics · Viscosity · Newtonian fluids
Non-Newtonian fluids
Surface tension
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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.

Definition

A simple equation to describe Newtonian fluid behaviour is

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

where

${\displaystyle \tau }$ is the shear stress exerted by the fluid ("drag") [Pa]
${\displaystyle \mu }$ is the fluid viscosity - a constant of proportionality [Pa·s]
${\displaystyle {\frac {du}{dx}}}$ 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, starch in water (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

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

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

${\displaystyle \mathbb {P} _{ij}=-p\delta _{ij}+\mu \left({\frac {\partial u_{i}}{\partial x_{j}}}+{\frac {\partial u_{j}}{\partial x_{i}}}\right)}$

where, by the convention of tensor notation,

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

If a fluid does not obey this relation, it is termed a non-Newtonian fluid, of which there are several types, including polymer solutions, molten polymers, many solid suspensions and most highly viscous fluids.