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A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time.^[1][2][3][4] Stresses are proportional to the rate of change of the fluid's velocity vector.

A fluid is Newtonian only if the tensors that describe the viscous stress and the strain rate are related by a constant viscosity tensor that does not depend on the stress state and velocity of the flow. If the fluid is also isotropic (mechanical properties are the same along any direction), the viscosity tensor reduces to two real coefficients, describing the fluid's resistance to continuous shear deformation and continuous compression or expansion, respectively.

Newtonian fluids are the easiest mathematical models of fluids that account for viscosity. While no real fluid fits the definition perfectly, many common liquids and gases, such as water and air, can be assumed to be Newtonian for practical calculations under ordinary conditions. However, non-Newtonian fluids are relatively common and include oobleck (which becomes stiffer when vigorously sheared) and non-drip paint (which becomes thinner when sheared). Other examples include many polymer solutions (which exhibit the Weissenberg effect), molten polymers, many solid suspensions, blood, and most highly viscous fluids.

Newtonian fluids are named after Isaac Newton, who first used the differential equation to postulate the relation between the shear strain rate and shear stress for such fluids.

Definition

An element of a flowing liquid or gas will endure forces from the surrounding fluid, including viscous stress forces that cause it to gradually deform over time. These forces can be mathematically first order approximated by a viscous stress tensor, usually denoted by ${\displaystyle \tau }$.

The deformation of a fluid element, relative to some previous state, can be first order approximated by a strain tensor that changes with time. The time derivative of that tensor is the strain rate tensor, that expresses how the element's deformation is changing with time; and is also the gradient of the velocity vector field ${\displaystyle v}$ at that point, often denoted ${\displaystyle \nabla v}$.

The tensors ${\displaystyle \tau }$ and ${\displaystyle \nabla v}$ can be expressed by 3×3 matrices, relative to any chosen coordinate system. The fluid is said to be Newtonian if these matrices are related by the equation ${\displaystyle {\boldsymbol {\tau }}={\boldsymbol {\mu }}(\nabla v)}$ where ${\displaystyle \mu }$ is a fixed 3×3×3×3 fourth order tensor that does not depend on the velocity or stress state of the fluid.

Incompressible isotropic case

For an incompressible and isotropic Newtonian fluid in laminar flow only in the direction x (i.e. where viscosity is isotropic in the fluid), the shear stress is related to the strain rate by the simple constitutive equation

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

• ${\displaystyle \tau }$ is the shear stress ("skin drag") in the fluid,
• ${\displaystyle \mu }$ is a scalar constant of proportionality, the dynamic viscosity of the fluid
• ${\displaystyle {\frac {du}{dy}}}$ is the derivative in the direction y, normal to x, of the flow velocity component u that is oriented along the direction x.

In case of a general 2D incompressibile flow in the plane x, y, the Newton constitutive equation become:

$${\displaystyle \tau _{xy}=\mu \left({\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial x}}\right)}$$

• ${\displaystyle \tau _{xy}}$ is the shear stress ("skin drag") in the fluid,
• ${\displaystyle {\frac {\partial u}{\partial y}}}$ is the partial derivative in the direction y of the flow velocity component u that is oriented along the direction x.
• ${\displaystyle {\frac {\partial v}{\partial x}}}$ is the partial derivative in the direction x of the flow velocity component v that is oriented along the direction y.

We can now generalize to the case of an incompressible flow with a general direction in the 3D space, the above constitutive equation becomes

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

• ${\displaystyle x_{j}}$ is the ${\displaystyle j}$th spatial coordinate
• ${\displaystyle v_{i}}$ is the fluid's velocity in the direction of axis ${\displaystyle i}$
• ${\displaystyle \tau _{ij}}$ is the ${\displaystyle j}$-th component of the stress acting on the faces of the fluid element perpendicular to axis ${\displaystyle i}$. It is the ij-th component of the shear stress tensor

or written in more compact tensor notation

$${\displaystyle {\boldsymbol {\tau }}=\mu \left(\nabla \mathbf {u} +\nabla \mathbf {u} ^{T}\right)}$$

${\displaystyle \nabla \mathbf {u} }$

An alternative way of stating this constitutive equation is:

where

$${\displaystyle {\boldsymbol {\varepsilon }}={\tfrac {1}{2}}\left(\mathbf {\nabla u} +\mathbf {\nabla u} ^{\mathrm {T} }\right)}$$

This constitutive equation is also called the Newtonian law of viscosity.

The total stress tensor ${\displaystyle {\boldsymbol {\sigma }}}$ can always be decomposed as the sum of the isotropic stress tensor and the deviatoric stress tensor (${\displaystyle {\boldsymbol {\sigma }}'}$):

${\displaystyle {\boldsymbol {\sigma }}={\frac {1}{3}}Tr({\boldsymbol {\sigma }})\mathbf {I} +{\boldsymbol {\sigma }}'}$

In the incompressible case, the isotropic stress is simply proportional to the thermodynamic pressure ${\displaystyle p}$: