# Viscoelastic scales

## Introduction

The motion of complex fluids is diffusional not ballistic. The time, $\tau \,\!$, to travel a distance, $r_{0}\,\!$, is:

For a simple, ballistic fluid: $\tau \sim \frac{r_{0}}{\left\langle v \right\rangle }\,\!$
For a complex, diffusional fluid: $\tau \sim \frac{r_{0}^{2}}{D}\,\!$

The average velocity, $\left\langle v \right\rangle \,\!$, comes from the average kinetic energy, $\frac{1}{2}mv^{2}=\frac{3}{2}kT \,\!$; the diffusion coefficient from an Einstein equation, $D=\frac{kT}{6\pi \eta a}\,\!$. Typically, $\frac{1}{D}\simeq a\times 10^{18}\frac{s}{m^{3}}\,\!$

Assuming a relaxation distance of $a\sim r_{0} \,\!$ $\tau \simeq 10^{18}\frac{s}{m}r_{0}^{3} \,\!$
Assuming a range of sizes, $r_{0}\simeq 10^{-8}\to 10^{-5}m \,\!$ The time scales are, $\tau \sim 10^{-6}\to 10^{3}s \,\!$

## Deborah number

One of Maxwell’s ideas: viscous flow is the decay of elastically stored energy; molecules in a low energy state are displaced to a higher energy state and time is required for the energy to “relax”; a time called the relaxation time, $\tau$. The observation time is t.

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.

$D_{e}>>1\,\!$ $D_{e}\sim O\left( 1 \right)\,\!$ $D_{e}<<1\,\!$
Solid-like Viscoelastic Liquid-like

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).

## Do Cathedral Glasses Flow?

We mentioned this in class, and I thought it was an interesting topic. This was a misconception for a long time in the science community. Even Norde makes this mistake in our reading this week: "However, given sufficient time even solids are viscous: glass windows in medieval cathedrals are often ticker at the bottom than at the top..." (p.344)

This was shows to be false with an article from the American Journal of Physics only in 1998. "The conclusion is that window glasses may flow at ambient temperature only over incredibly long times, which exceed the limits of human history." (Edgar Dutra Zanotto Am. J. Phys. 66(5), May 1998.)

The observed thickening is actually due to a processing method used during those times: "The difference in thickness sometimes observed in antique windows probably results from glass manufacturing methods... Until the 19th century, the only way to make window glass was to blow molten glass into a large globe then flatten it into a disk. Whirling the disk introduced ripples and thickened the edges. For structural stability, it would make sense to install those thick portions in the bottom of the pane..." [[1]]

## Péclet number

The Péclet number is a dimensionless number used to compare the time for diffusion to the time for viscous flow. Taking the average displacement to be the particle radius, then an average characteristic time is:

$t_{a}=\frac{a^{2}}{D}=\frac{6\pi \eta _{0}a^{3}}{kT}$

A characteristic time for shear flow is the inverse of the shear rate: ${1}/{{\dot{\gamma }}}\; \,\!$. The ratio of the two characteristic time is the Péclet number:

$P_{e}=\frac{\text{time for diffusion}}{\text{time for viscous flow}}=\frac{t_{a}}{{1}/{{\dot{\gamma }}}\;}=\frac{{a^{2}}/{D}\;}{{1}/{{\dot{\gamma }}}\;}=\frac{6\pi \eta _{0}a^{3}\dot{\gamma }}{kT} \,\!$

It is sometimes better to replace the viscosity with the shear stress:

$P_{e}=\frac{6\pi a^{3}\sigma }{kT} \,\!$

The Péclet number can be thought of as the ratio of applied stress to thermal stress.

The Péclet number is also 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

## Stress and time scales

 The Péclet number can be thought of as the ratio the time for diffusion to the time for viscous flow or of the ratio of applied stress to thermal stress: \begin{align}  & P_{e}=\frac{\text{time for diffusion}}{\text{time for viscous flow}}=\frac{{a^{2}}/{D}\;}{{1}/{{\dot{\gamma }}}\;}=\frac{6\pi \eta _{0}a^{3}\dot{\gamma }}{kT} \\ & \text{ }=\frac{6\pi a^{3}\sigma }{kT}=\frac{\sigma }{{kT}/{6\pi a^{3}}\;}=\frac{\text{mechanical stress}}{\text{thermal stress}} \\  \end{align}\,\! The larger the particle, the less stress is needed to make the Péclet number large; that is, when the particle size is large, the more likely we are to see viscoelasticity. The Deborah number is the ratio of the relaxation time to the observation time: $D_{e}=\frac{\text{structural relaxation time}}{\text{experimental measurement time}}\,\!$ For viscoelastic measurements to give interesting information; the Deborah number should be large and the Péclet number small. The result of stress needs to be “recovered” on the scale of diffusion time.

## Taylor number

As the shear rate increases, secondary flows can occur – pronounced in rotational flow.

Nevertheless these are stable flows.

Couette geometry. Taylor vortices.
Goodwin, Fig. 1.7, p. 11
Goodwin, Fig. 1.8, p. 12.

If $\Omega _{c}\,\!$ is the angular velocity, then the Taylor number is:

$T_{a}=\Omega _{c}\left( \frac{R_{o}+R_{i}}{2} \right)^{{1}/{2}\;}\frac{\rho \left( R_{o}-R_{i} \right)^{{3}/{2}\;}}{2\eta }\,\!$

## 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

At high shear rates in rheological measurements, secondary flows can become chaotic and turbulence follows.

 For Couette flow $R_{e}=\frac{\Omega _{c}\left( R_{o}+R_{i} \right)\left( R_{o}-R_{i} \right)\rho }{2\eta }\,\!$ In terms of shear rate: $R_{e}\approx \frac{\dot{\gamma }\left( R_{o}-R_{i} \right)^{2}\rho }{\eta }\,\!$ For cone and plate geometry, where the cone angle, $\alpha \,\!$, is in degrees. $R_{e}\approx \frac{\dot{\gamma }\rho }{\eta }\left( \frac{\pi R\alpha }{180} \right)^{2}\,\!$

### Fluid flow at low Reynolds number

At low Reynolds number the inertial terms in the Navier-Stokes equation are relatively unimportant; the flow pattern is determined by a balance of the viscous forces and the pressure gradients in the fluid. As a consequence, expressions for the velocity, drag forces, and the like do not involve the fluid density directly. One of the nice features of flow at low Reynolds numbers is that the flow is laminar-the viscosity quickly damps out any turbulent effects. Two simple low Reynolds number flows are Poiseuille flow and Stokes' flow, both of which have practical applications.

A very interesting and famous publication on Life at Low Reynolds Number by Dr. EM Purcell can be found at: http://jilawww.colorado.edu/perkinsgroup/Purcell_life_at_low_reynolds_number.pdf