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6
10
0
Roscoe; theory
well-sorted
Bagnold;
granular
shear
5
Poorly
sorted
10
-1
4
m
r
=
(1 - 1.35
c
)
-2.5
3
2
10
-2
Einstein; theory
(v. dilute)
1
0
0.1
0.2
0.3
0.4
Concentration of spheres by volume fraction (
c
)
10
-3
Brine
(20% NaCl)
Fig. 3.43
The variation of relative dynamic viscosity (with
respect to pure water at zero solids concentration,
0
) with
solid sphere concentration according to two theoretical
models; Einstein is for vanishingly small
c
, Roscoe for finite
c
.
The Bagnold curve is for experimental data on the
behavior of spheres under shear when solid-solid reactions
are induced by the shear and intragranular collisions are
produced.
Water
10
-4
Air
Methane
10
-5
of viscosity in terms of the diffusion of momentum by
viscous forces is again essential. Thus any swinging pendu-
lum put into motion and then left, once corrected for fric-
tion around the bearings, slows down (is damped)
progressively; the time required for damping being
inversely proportional to viscosity. As Einstein later
explained in a relation between viscosity and diffusion, the
damping is due to molecular collisions between fluid and
the pendulum mass moving through it. This makes it eas-
ier to conceptualize the reason why solid suspensions have
increased viscosity over pure fluid alone (Fig. 3.43).
Maxwell and Einstein were able to show from similar
molecular collisional arguments why experimentally
determined viscosities of liquids were inversely propor-
tional to temperature while the viscosity of gas is broadly
independent of pressure.
10
0
10
40 60 100 200
400
Temperature (
ยบ
C)
Fig. 3.42
The dynamic viscosity of some common pure substances as
a function of temperature.
shear couple acting on an elastic solid in just the same way
(Section 1.26; see Fig. 3.84).
It is simplest to grasp why
solid
liquid
gas
from
the point of view of molecular
kinetic theory
(Section 4.18)
applied to the states of matter. Thus decreasing concentra-
tions of molecules cause deformation or flow to be easier
as the molecules are more widely spaced. Maxwell's view
3.10
Viscous force
past an interface, most simply a stationary solid obstruc-
tion to the flow or another fluid of similar or contrasting
material and kinematic properties. Such physical systems
are clearly common in Nature.
In Section 2.4 on motion we neglected frictional effects
arising from viscosity. Here we consider the simplest type
of viscous fluid flow and ask how net forces might come
about. The flows are steady Newtonian systems moving
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