**Nanomedicine,
Volume I: Basic Capabilities**

**©
1999 Robert A. Freitas Jr. All Rights
Reserved.**

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

**9.4.1.1 Biofluid Viscosity**

Viscosity is a measure of the resistance of a fluid to shearing
when the fluid is in motion. Consider a plate of surface area A moving parallel
to a fixed plane surface with constant velocity v, being pushed laterally by
a constant force F. A fluid of viscosity h fills
the volume between the two surfaces, which are separated by a distance d. The
fluid layer nearest the moving plate also moves at velocity v and the layer
nearest the stationary plane remains stationary. In between the two surfaces,
the velocity increases linearly with distance, establishing a constant gradient.
This gradient is usually called the shear rate (essentially a size-normalized
velocity), or 'g = v/d (m/sec-m, or sec^{-1}).
The absolute viscosity may then be defined by:

where h has MKS units of N-sec/m^{2},
Pascal-sec, or, more simply, kg/m-sec. In a Newtonian fluid, the shear stress
F/A (N/m^{2}) increases linearly with shear rate 'g,
so that viscosity h is constant over a wide range
of shear rates. Many common fluids such as air, water, saline and blood serum
closely approximate the ideal Newtonian fluid. The viscosity of solutions of
molecules is related to the diffusion coefficient; see Eqn.
3.5.

The viscosities of some common materials are given in Table
9.4. The viscosity of an ideal gas is independent of density and independent
of pressure between 0.01-10 atm; at higher pressures, intermolecular interactions
lead to higher viscosities.^{390}
Theory predicts that gas viscosity ~ T^{1/2} (e.g., rises with the square-root
of temperature T), but a somewhat larger temperature exponent is obtained experimentally
for real gases. In contrast, the viscosity of liquids increases with rising
pressure (typically by a factor of 2-3 after moving from 1 atm up to 1000 atm
for organic liquids) and decreases with rising temperature. In normal liquids
the temperature dependence is approximated by Andrade's formula which gives:

where for example the activation energy for viscosity E_{v}
~ 25 zJ and the constant k_{v} ~ 2.1 x 10^{-6} kg/m-sec for
pure water at 1 atm. Nonelectrolytes dissolved in water generally cause viscosity
to rise, while solvation of electrolytes may increase or decrease viscosity
(the effect is typically ~10% or less for a 1M solution). Large asymmetric solute
molecules increase viscosity more than an equal mass of small spherical molecules.
The threshold between solid and liquid is generally taken as h
~ 10^{14} kg/m-sec.^{364}

Most biofluids are viscoelastic and non-Newtonian, with apparent
viscosity h_{a} varying with shear rate and
displaying other nonlinear characteristics such as hysteresis, relaxation, and
creep.^{362} Saliva behaves more like
an elastic body than like water. Because of its high molecular weight, DNA solution
is viscoelastic even at low concentrations. Sex glands produce viscoelastic
fluids, including semen and uterine cervical mucus^{1392}
(h_{a} varies ~20% during the menstrual cycle).^{3575-3577}
Human synovial fluid becomes increasingly incompressible at higher pressures,
with h_{a} ~ 10 kg/m-sec at 'g
~ 0.1 sec^{-1} (e.g., knee flexion during very slow walking) declining
to h_{a} ~ 0.1 kg/m-sec at 'g
~ 10 sec^{-1} (very fast running), and to h_{a}
~ 0.001 kg/m-sec at 'g** **~ 10,000 sec^{-1
}(experimental).^{362} Viscoelasticity
is also an important property of respiratory tract mucus, which typically shows
an h_{a} ~ 1 kg/m-sec at low shear rates
near 'g ~ 0.1-1 sec^{-1}, falling to h_{a}
~ 0.01 kg/m-sec at high shear rates near 'g ~ 100-1000
sec^{-1}.^{362} Even ice is
a viscoelastic material, with h_{a} ~ 10^{10}10^{13}
kg/m-sec (estimated as maximum shear stress divided by shear rate from data
in Sinha et al^{1609}) for 'g
~ 10^{-3}-10^{-7} sec^{-1} at 262 K; ice viscosity varies
with temperature (Table
9.4) as described by Andrade's formula, with E_{v} ~ 110 zJ as determined
experimentally for pure ice,^{1609}
and taking k_{v} ~ 6.3 x 10^{-4} kg/m-sec at high shear rate.

Cytoplasm is a complex viscoelastic material having a continuous
liquid phase (the cytosol) plus various suspended particles, granules, and membranous
structures. More precisely, the cytomatrix is a mixed-phase body composed of
a fibrillar network penetrated by a solution.^{1408}
As a result, viscosity is different in the various phases. From flow behavior,
the viscosity of *E. coli* protoplasm was estimated as ~1000 kg/m-sec;
from measured diffusion rates of sucrose, dextran, and b-galactosidase,
the apparent viscosity was 3-4 x 10^{-3} kg/m-sec.^{1407}
Cytoplasm is inhomogeneous and anisotropic at many levels of organization.

Last updated on 21 February 2003