Deborah number

The Deborah number (De) is a dimensionless number, often used in rheology to characterize the fluidity of materials under specific flow conditions. It is based on the premise that given enough time even a solid-like material will flow. The flow characteristics are not inherent properties of the material alone, but a relative property which depends on two fundamentally different characteristic times.

Formally, the Deborah number is defined as the ratio of the relaxation time characterizing the time it takes for a material to adjust to applied stresses or deformations, and the characteristic time scale of an experiment (or a computer simulation) probing the response of the material:

where t_c refers to the stress relaxation time , and t_p refers to the time scale of observation.

This incorporates both the elasticity and viscosity of the material. At lower Deborah numbers, the material behaves in a more fluidlike manner, with an associated Newtonian viscous flow. At higher Deborah numbers, the material behavior enters the non-Newtonian regime, increasingly dominated by elasticity and demonstrating solidlike behavior.

For example, for a Hookean elastic solid, the relaxation time ${\displaystyle t_{\mathrm {c} }}$ will be infinite and it will vanish for a Newtonian viscous fluid. For liquid water, ${\displaystyle t_{\mathrm {c} }}$ is typically 10⁻¹² s, for lubricating oils passing through gear teeth at high pressure it is of the order of 10⁻⁶ s and for polymers undergoing plastics processing, the relaxation time will be of the order of a few seconds. Therefore, depending on the situation, these liquids may exhibit elastic properties, departing from purely viscous behavior.

...
Wikipedia