The Oldroyd-B model is a constitutive model used to describe the flow of
viscoelastic
In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly ...
fluids.
This model can be regarded as an extension of the
upper-convected Maxwell model The upper-convected Maxwell (UCM) model is a generalisation of the Maxwell material for the case of large deformations using the upper-convected time derivative. The model was proposed by James G. Oldroyd. The concept is named after James Clerk Max ...
and is equivalent to a fluid filled with elastic bead and spring dumbbells.
The model is named after its creator
James G. Oldroyd
James Gardner Oldroyd (25 April 1921 – 22 November 1982''The Times'' November 25, 1982 page 26 "Deaths") was a British mathematician and noted rheologist. He formulated the Oldroyd-B model to describe the viscoelastic behaviour of non-Newtonia ...
.
The model can be written as:
where:
*
is the deviatoric part of the
stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
;
*
is the relaxation time;
*
is the retardation time =
;
*
is the
upper-convected time derivative In continuum mechanics, including fluid dynamics, an upper-convected time derivative or Oldroyd derivative, named after James G. Oldroyd, is the rate of change of some tensor property of a small parcel of fluid that is written in the coordinate ...
of stress tensor:
*
is the fluid velocity;
*
is the total
viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
composed of solvent and polymer components,
;
*
is the deformation rate tensor or rate of strain tensor,