# Viscoelastic scales

## Deborah number

The Deborah number $D_e$ is a dimensionless number, which describes how fluid a material appears. It is equal to the ratio of the relaxation time $\tau$ divided by the observation time t:

$D_e = \tau/t$.

The relaxation time of a material is an intrinsic property which describes how long it takes a material to return to its original state after undergoing stress. Materials with smaller Deborah numbers will appear fluid for an observed timescale. This number is primarily used to compare different Rheological systems or experiments on systems.

The history of this constant's name is rather interesting. A biblical prophet, Deborah, said "The mountains flowed before the Lord" in her famous song after the victory over the Philistines. Years later, some physicists thought that there was a great deal of truth to this statement. On a human timescale, mountains rarely appear to flow, but on geological timescales, mountains do flow. They decided to name the dimensionless constant that describes a material's fluid under a given timescale after Deborah. (The name of the Deborah number and the birth of Rheology is described in further detail in Physics Today January 1964)

Once a material has a Deborah number of about one, it becomes a non-Newtonian fluid. This is often the case for polymer flows, and a great deal of work has gone into understanding nontrivial flow of polymers.

This termonology has been even borrowed by biology. In the Journal of Plankton research, there is a group which uses a Deborah number to describe the (time scale of a process)/(timescale of deformaiton) of plankton ecology. Note that this is a slightly different definition of a Deborah number, but I assume that they are referring it as such because since it describes the timescale of a process relative to observation time. I think that they describe this better than I can so here is a excerpt from their abstact:

Because microzones smear Out along the shear, to prevent nutrient-seekers and predators using them as scent trails, organisms may convolute their microzones by swimming, particularly across the shear. In a predator-prey model, it has been shown that when De, (shear rate) (time taken to swim radius of detection sphere), >2.6, not all the perceived prey is accessible. More economical hunting strategies and those allowing access to more of the perceived prey, require better sensory and navigational abilities. When De >2.6, the predator will perceive a greater flux of accessible prey when it swims across the shear than when it swims in the other two dimensions. De may help to understand many more biological processes in deforming media. (Selection and control of Deborah numbers in plankton ecology, Ian R. Jenkinson and Tim Wyatt, V14, N12 1992).

## Péclet number

The Péclet number is a dimensionless number used in calculations involving convective heat transfer in fluid dynamics and mass dispersion. In fluid dynamics, it is the ratio of the thermal energy convected to the fluid to the thermal energy conducted within the fluid. If Pe is small, conduction is important and in such a case, the major source of conduction could be down the walls of a tube. The Péclet number is the product of the Reynolds number and the Prandtl number. It depends on the heat capacity, density, velocity, characteristic length and heat transfer coefficient.

$\mathrm{Pe}_L = \frac{L V}{\alpha} = \mathrm{Re}_L \cdot \mathrm{Pr}.$

where

• L - characteristic length
• V - Velocity
• α - Thermal diffusivity $= \frac{k}{\rho c_p}$
• k - Thermal conductivity
• ρ - Density
• $c_p$ - Heat capacity

In the case of mass dispersion, it is the product of Reynolds number with the Schmidt number.

$\mathrm{Pe}_L = \frac{L V}{D} = \mathrm{Re}_L \cdot \mathrm{Sc}$

where

• L - characteristic length
• V - Velocity
• D - mass diffusivity

## Reynolds number

Reynolds number is a dimensionless number $Re$ used in fluid mechanics as the measure of the ratio of inertial forces to viscous forces. It quantifies the relative importance of these two effects for given flow conditions. When Reynolds number is low, viscous effects dominate the inertial forces and the flow is laminar. On the other hand, when inertial forces dominate viscous effects the flow is turbulent. But, since rheology is dealing with fluids of viscosity that is not constant, Reynolds number cannot be easily computed in most cases.

Reynolds number is generally the following:

$Re={{\rho {\bold \mathrm V} D} \over {\mu}} = {{{\bold \mathrm V} D} \over {\nu}} = {{{\bold \mathrm Q} D} \over {\nu}A}$

where we have the following quantities

• ${\mathrm V}$ is the mean fluid velocity (m/s)
• ${D}$ is the diameter (m)
• ${\mu}$ is the dynamic viscosity of the fluid (N·s/m²)
• ${\nu}$ is the kinematic viscosity ($\nu = \mu /$ρ) (m²/s)
• ${\rho}$ is the density of the fluid (kg/m³)
• ${Q}$ is the volumetric flow rate (m³/s)
• ${A}$ is the pipe cross-sectional area (m²)

For:

- a flow in circular pipe $D$ is diameter of the pipe

- a flow in rectangular duct $D=\frac{4 A}{P}$ where $A$ is cross-sectional area of the duct, and $P$ is wetted perimeter

- a flow between two plane parallel surfaces $D$ is distance between the plates

Typical values of Reynolds number are the following:

• Spermatozoa ~ 1×10−2
• Blood flow in brain ~ 1×102
• Blood flow in aorta ~ 1×103
• Typical pitch in Major League Baseball ~ 2×105
• A large ship ~ 5×109