Geoscience Reference
In-Depth Information
Poiseuille who did pioneer work on viscous flow): these
are 10 1 Pa s. Viscosity is a scalar quantity, possessing
magnitude but not direction. The most succinct formal
definition goes something like “the force needed to
maintain unit velocity difference between unit areas of a
substance that are unit distance apart.”
The ratio of molecular viscosity to density, confusingly
termed kinematic viscosity , is given the symbol,
Line
z
Foam
cube
Pulley
x ( u )
g = du / dz = t / g
t
u
(nu) and
has dimensions m 2 s 1 , often quoted in Stokes (St), one
stoke being 10 4 m 2 s 1 . Authors sometimes forget to specify
which viscosity they are using, so always check carefully.
z
g
t
1 kg
3.9.2
Controls on viscosity
Fig. 3.40 Leonardo's implicit analog model for the action of viscosity
in resisting an applied force. In this case the force is exerted on the
top unit area of a foam cube. In continuous fluid deformation, as
distinct from the finite displacement of solids, the displacement in x
is the velocity, u (as shown).
As for density it is important to realize that Newtonian
viscosity is a material property of pure homogeneous
substances: the warning italic letters signifying caveats,
exceptions, and potential sources of confusion;
Specific conditions of T and P must be quoted when a value
for viscosity is quoted. Some variations of molecular
(dynamic) viscosity with temperature are given in Fig. 3.42.
Natural materials are often impure, with added contam-
inants; particles may also be of variable chemical composi-
tion. For example, the viscosity of molten magma is highly
dependent upon Si content (Section 5.1), and the viscos-
ity of an aqueous suspension of silt or clay differs radically
from that of pure water (Fig. 3.42).
For a given applied stress,
shear strain is proportional
to viscosity; it varies linearly
and continuously with time
and is irreversible
Shear stress and rate of
strain are linearly related by
the viscosity coefficient; zero
stress gives zero strain and
any finite stress gives strain
Fluid 1
Fluid 1
Fluid 2
Fluid 2
3.9.3 Maxwell's view of viscosity as a
transport coefficient
0
0
Shear strain, e
Rate of shear strain, d e /d t
In fluid being sheared past a stationary interface, those
molecules furthest from the interface have a greater
forward (drift) momentum transferred to their random
thermal motions as they are dragged along. Under steady
conditions (i.e. shear is continuously applied) the combi-
nation of forward drift due to shear and random thermal
molecular agitation (very much faster) must set up a con-
tinuous forward velocity gradient; molecules constantly
diffuse drift momentum as they collide with slower mov-
ing molecules closer to the interface where momentum is
dissipated as heat. We see clearly from this approach why
Maxwell viewed molecular viscosity as a momentum
diffusion transport coefficient, analogous to the transport
of both conductive heat and mass (Section 4.18).
Thermal effects thus have a great control on the value of
viscosity. Although it is a little more difficult to imagine
the viscous transport of momentum in a solid, we can
nevertheless measure the angle of shear achieved by a
Defectus lubricatus is a material property of any fluid,
with a constant value for the pure fluid appropriate
only under specified conditions of T and P
Fig. 3.41 Newtonian fluids.
or
(eta). This is equal to the ratio between the applied
shearing stress,
(tau), that causes deformation and the
resulting displacement gradient or rate of vertical strain ,
d u /d z . We call a fluid Newtonian when this ratio is finite
and linear for all values (Fig. 3.41). We shall briefly exam-
ine the behavior of non-Newtonian substances in
Section 3.15. From knowledge of the units involved in
,
and d u /d z , check that the dimensions of viscosity are
ML 1 T 1 , and the units, N s m 2 or Pa s. Viscosity is
sometimes quoted in units of poises (named in honor of
 
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