Newtonian fluid

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 τ .

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 ∇ v .

The tensors τ and ∇ 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 τ = μ ( ∇ v ) where μ 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

τ = μ d u d y

where

τ is the shear stress ("drag") in the fluid, μ is a scalar constant of proportionality, the shear viscosity of the fluid d u d y 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

τ i j = μ ( ∂ v i ∂ x j + ∂ v j ∂ x i )

where

x j is the j th spatial coordinate v i is the fluid's velocity in the direction of axis i τ i j 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 σ ) that combines the shear stress with conventional (thermodynamic) pressure p . The stress-shear equation then becomes

σ i j = − p δ i j + μ ( ∂ v i ∂ x j + ∂ v j ∂ x i )

For anisotropic fluids

More generally, in a non-isotropic Newtonian fluid, the coefficient μ that relates internal friction stresses to the spatial derivatives of the velocity field is replaced by a nine-element viscosity tensor μ i j .

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 F = μ i j d S × r o t u

where μ i j – viscosity tensor. The diagonal components of viscosity tensor is molecular viscosity of a liquid, and not diagonal components – turbulence eddy viscosity.

Examples

Water, air, milk, alcohol, glycerol, and thin motor oil are examples of (practically) Newtonian fluids that have different density properties, which allows a different shear stress and rate flow graphs within a given interval.