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The most important distinction between a solid
such as steel and a viscous fluid such as water or
air is that shear stress in a solid material is
proportional to shear strain, and the material
ceases to deform when equilibrium is reached,
whereas the shear stress in a viscous fluid is
proportional to the time rate of strain. The
proportionality factor for the solid is the shear
modulus. The proportionality factor for the
viscous fluid is the dynamic, or absolute,
viscosity. For example, the shear stress of a fluid
near a wall is given by τ=μ(dv/dy)
where τ(tau) is the shear stress, μ(mu) is
the dynamic viscosity, and dv/dy is the
time rate of strain, which is also the
velocity gradient normal to the wall. Thus
the definition of the viscosity, μ, is the
ratio of the shear stress to the velocity
gradient, μ= τ/(dv/dy).
 Consider the flow shon in fig.2.1 this velocity
distribution is typical of that for laminar
(nonturbulent) flow next to a solid boundary.
Several observations relation to this figure will
help you appreciate the interaction between
viscosity and velocty disribution.

     Figure 2.1
     Velocity distribution next to a boundary.
First, the velocity gradient ant the boundary is finite. The
curve of velocity variation cannot be tangent to the
boundary because this would imply an infinite velocity
gradient and, in turn, and infinite shear stress, whech is
impossible. Second, a velocity gradient that becomes less
steep (dv/dy becomes smaller ) whith distance from the
boundary has a maximum shear stress at the boundary, and
the shear stress decreases with distance from the boundary.
Also note that the velocity of the fluid is zero at the
stationary boundary. This is characteistic of all flows dealt
with in this basic txt. That is, at the boundary surface the
fluid has the velocity of the boundary –no slip occurs.
Many of the equations of fluid mechanics
include the combination μ/ρ in
them .Because it occurs so frequently,
this combination has been given the
special name kinematic viscosity (so
called because the force dimension
cancels out in the combination μ/ρ ). The
symbol used to identify kinematic viscosity
is ν(nu).
Whenever shear stress is applied to fluids, motion
occurs. This is the basic difference between
fluids and solids. Solids can resist shear stress in
a static condition, but fluidsdeform continuously
under the action of a shear stress. Another
important characteristic of fluids is that the
viscous resistance is independent of the normal
force (pressure) acting within the fluid. In
constrast, for two solids sliing relative to each
other, the shearing resistance is totally dependent
on the normal force between the two.
The manner in which
viscous forces are
produced can be seen in
the conveyor-belt
analogy. Consider a type
of transit system in which
people are carried from
one part of a city to
another on conveyor
belts(Fig.2.2a) . People
ride the fast-moving belt
from left to right-an
equilibrium condition
exists. Next visualixe the
action when the people        Figure 2.2
step off the fast belt onto   Conveyor-belt transportation
a slower-moving belt (Fig.    system.
2.2b) .
Before they step off the fast-moving belt, each
possesses a certain amount of momentum in the
x direction. But as soon sa they acquire the
wpeed of the slower belt, their momentum in the
x direction is reduced by a significant amount. It
is knownfrom basic mechanics that a change in
monentum of a body results from an external
force acting on that body. In our example, it is
the slower belt that exers a force in the negative z
direction as each person steps on the belt.
Conversely, as each person steps on the slower belt, a forc
eis exerted on the belt in the positive x direction. Now, if
the people step from the faster belt to the slower 퓨딧 at a
rather steady rate, then a rather continyuous force is
exerted on the slower belt . In effect, by the action of the
people moving in the negative y direction, they produe a
force on the slow belt in the positive zx direction. One may
think of this as shear force in the x direction. In a sililar
manner, it can be visualixed that if people stepped from the
slow-moving belt to the faster one, a “ shear force” in the
negative x direction vwould be imposed on the faster belt.
If the people were continuously going both ways
(back and forth) from one the direction of motion )
on the slow belt and a like retarding force on the
fast belt. Furthermore, as the relative speed
between the belts changes (analogous to a
change in velocity gradient ) , the shear force is
increased or decreased in direct proportion to the
increase or decrease in relative velocity. Thus if
both belts were mad to have the same speed, the
shear force would be zero.
In fluid flow we can think of streams of fluid traveling in a
given general direction, such as in a pipe, with the fluid
nearer the pipe center traveling faster9analogous to the
faster belt) while the fluid nearer the wall is traveling more
slowly. The interaction between streams, in the case of gas
flow, occurs when the molecules of gas travel back and
forth between adjacent streams, thus creating a shear
stress in the fluid. Because the rate of activity (back-and-
forth motion)of the gas molecules increases with an
increase with the temperature of the gas, such is indeed
the case, as can be seen in gig.2.3
For liquids, the shear stress is involved
with the cohesive forces between
molecules. Thes eforce s decrease with
temprature, which resuts in a decrease in
viscosity with an increase in temperature
(see Fig.2.3) The variation of viscosity
(dynamic and kinematic ) for other fluids
is given in figs,A.2 and A.3 in the
Figure 2.3
viscosity for
air and crude
Units of Viscosity

From Eq. (2.6) tit can be seen that thee
  units of μ are N.s/m2
    μ= τ/(dv/dy)=N/m2/((m/s)/m)=N.s/m2

ν are m2/s
    ν= μ/ ρ=N.s/m2/(N. s2 /m4)=m2/s

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