A generalized Newtonian fluid is an idealized
fluid
In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...
for which the
shear stress
Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...
is a function of
shear rate
In physics, mechanics and other areas of science, shear rate is the rate at which a progressive shear strain is applied to some material, causing shearing to the material. Shear rate is a measure of how the velocity changes with distance.
Simple ...
at the particular time, but not dependent upon the history of deformation. Although this type of fluid is
non-Newtonian (i.e. non-linear) in nature, its
constitutive equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two or more physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance o ...
is a generalised form of the
Newtonian fluid
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. Stresses are proportional to the rate of cha ...
. Generalised Newtonian fluids satisfy the following
rheological equation:
:
where
is the
shear stress
Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...
, and
is the
shear rate
In physics, mechanics and other areas of science, shear rate is the rate at which a progressive shear strain is applied to some material, causing shearing to the material. Shear rate is a measure of how the velocity changes with distance.
Simple ...
. The quantity
represents an ''
apparent viscosity'' or ''
effective viscosity'' as a function of the shear rate.
The most commonly used types of generalized Newtonian fluids are:
*
Power-law fluid
In continuum mechanics, a power-law fluid, or the Ostwald–de Waele relationship, is a type of generalized Newtonian fluid. This mathematical relationship is useful because of its simplicity, but only approximately describes the behaviour of a re ...
*
Cross fluid
*
Carreau fluid
*
Bingham fluid
It has been shown that
lubrication theory
In fluid dynamics, lubrication theory describes the flow of fluids (liquids or gases) in a geometry in which one dimension is significantly smaller than the others. An example is the flow above air hockey tables, where the thickness of the air ...
may be applied to all generalized Newtonian fluids in both two and three dimensions.
See also
*
Navier–Stokes equations
The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ...
References
{{Non-Newtonian fluids
Non-Newtonian fluids