Newtonian fluid

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Definition

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

The deformation of that fluid element, relative to some previous state, can be approximated to first order 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 $v$ at that point, often denoted $\nabla v$.

The tensors $\tau$ and $\nabla v$ 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 $\mathbf {\tau } =\mathbf {\mu } (\nabla v)$ where $\mu$ 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, the viscous stress is related to the strain rate by the simpler equation

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

where

$\tau$ is the shear stress ("drag") in the fluid,

$\mu$ is a scalar constant of proportionality, the shear viscosity of the fluid

${\frac {du}{dy}}$ is the derivative of the velocity component that is parallel to the direction of shear, relative to displacement in the perpendicular direction.

If the fluid is incompressible and viscosity is constant across the fluid, this equation can be written in terms of an arbitrary coordinate system as

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

where

$x_{j}$ is the $j$th spatial coordinate

$v_{i}$ is the fluid's velocity in the direction of axis $i$

$\tau _{ij}$ is the $j$th component of the stress acting on the faces of the fluid element perpendicular to axis $i$.

One also defines a total stress tensor $\mathbf {\sigma }$, that combines the shear stress with conventional (thermodynamic) pressure $p$. The stress-shear equation then becomes

$\mathbf {\sigma } _{ij}=-p\delta _{ij}+\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}\right)$

or written in more compact tensor notation

$\mathbf {\sigma } =-p\mathbf {I} +\mu \left(\nabla \mathbf {v} +\nabla \mathbf {v} ^{T}\right)$

where $\mathbf {I}$ is the identity tensor.

For anisotropic fluids

More generally, in a non-isotropic Newtonian fluid, the coefficient $\mu$ that relates internal friction stresses to the spatial derivatives of the velocity field is replaced by a nine-element viscous stress tensor $\mu _{ij}$.

There is general formula for friction force in a liquid: The vector differential of friction force is equal the viscosity tensor increased on vector product differential of the area vector of adjoining a liquid layers and rotor of velocity:

${d}\mathbf {F} {=}\mu _{ij}\,\mathbf {dS} \times \mathrm {rot} \,\mathbf {u}$

where $\mu _{ij}$ – viscosity tensor. The diagonal components of viscosity tensor is molecular viscosity of a liquid, and not diagonal components – turbulence eddy viscosity.

Newtonian law of viscosity

The following equation illustrates the relation between shear rate and shear stress:

$\tau =\mu {du \over dy}$,

where:

• τ is the shear stress;
• μ is the viscosity, and
• ${\textstyle {\frac {du}{dy}}}$is the shear rate.

If viscosity is constant, the fluid is Newtonian.

Power law model

In blue a Newtonian fluid compared to the dilatant and the pseudoplastic, angle depends on the viscosity.

The power law model is used to display the behavior of Newtonian and non-Newtonian fluids and measures shear stress as a function of strain rate.

The relationship between shear stress, strain rate and the velocity gradient for the power law model are:

$\tau =-m\left\vert {\dot {\gamma }}\right\vert ^{n-1}{\frac {dv_{x}}{dy}}$,

where

• $\left\vert {\dot {\gamma }}\right\vert ^{n-1}$is the absolute value of the strain rate to the (n-1) power;

• ${\textstyle {\frac {dv_{x}}{dy}}}$ is the velocity gradient;

• n is the power law index.

If

• n < 1 then the fluid is a pseudoplastic.
• n = 1 then the fluid is a Newtonian fluid.
• n > 1 then the fluid is a dilatant.

Fluid model

The relationship between the shear stress and shear rate in a casson fluid model is defined as follows:

${\sqrt {\tau }}={\sqrt {\tau _{0}}}+S{\sqrt {dV \over dy}}$

where τ₀ is the yield stress and

$S={\sqrt {\frac {\mu }{(1-H)^{\alpha }}}}$,

where α depends on protein composition and H is the Hematocrit number.

Examples

Water, air, alcohol, glycerol, and thin motor oil are all examples of Newtonian fluids over the range of shear stresses and shear rates encountered in everyday life. Single-phase fluids made up of small molecules are generally (although not exclusively) Newtonian.

See also

• Fluid mechanics
• Non-Newtonian fluid