A fluid in motion is a complex phenomenon, as it possesses potential, kinetic, and gravitational energy besides causing friction viscous forces to come into play.

**VISCOSITY**

In
previous section, the behavior of fluids at rest is discussed. Successive
discussions will bring out the influence of fluid motion on different
properties. A fluid in motion is a complex phenomenon, as it possesses
potential, kinetic, and gravitational energy besides causing friction viscous forces
to come into play. Therefore, it is necessary to consider the case of an ideal
liquid to simplify the task. An ideal liquid is incompressible (i.e., bulk
modulus is infinity) and in which no shearing forces can be maintained (i.e.,
the coefficient of viscosity is zero).

Most
of the fluids offer resistance towards motion. A frictional force acts at the
contact surface when a fluid moves relative to a solid or when two fluids move
relative to each other. This resistance to fluid motion is similar to the friction
produced when a solid moves on a surface. The internal friction existing
between the layers of a moving fluid is viscosity. So, viscosity is defined as
‘the property of a fluid to oppose the relative motion between its layers’.

Consider
a liquid flowing through a horizontal surface with two neighboring layers. The
upper layer tends to accelerate the lower layer and in turn, the lower layer
tends to retard the upper layer. As a result, a backward tangential force is
set-up. This tends to destroy the relative motion. This accounts for the
viscous behavior of fluids.

Consider
a liquid flowing steadily over a horizontal fixed layer (Figure 7.15). The
velocities of the layers increase uniformly as we move away from the fixed
layer. Consider any two parallel layers A and B. Let *v* and *v + dv *be the
velocities of the neighboring* *layers
at distances *x* and *x + dx* respectively from the fixed
layer.

The
force of viscosity *F* acting
tangentially between two layers is given by Newton’s First law. This force is
proportional to (i) area *A* of the
liquid and (ii) the velocity gradient dv/dx

Where
the constant of proportionality η is called the coefficient of viscosity of the
liquid and the negative sign implies that the force is frictional and it
opposes the relative motion. The dimensional formula for coefficient of
viscosity is [*ML*^{−} ^{1} *T*^{−}^{1}]_{}

A
metal plate of area 2.5×10^{-4}*m*^{2}
is placed on a 0.25×10^{-3}*m*
thick layer of castor oil. If a force of 2.5 N is needed to move the plate with
a velocity 3×10^{-2}*m s*^{-1}*,* calculate the coefficient of viscosity
of castor oil.

Given:
A=2.5×10^{-4} *m*^{2}, *dx* = 0.25×10^{-3}*m*, *F*=2.5*N *and*
dv *= 3×10^{-2}* m s*^{-1}

*Solution*

The
flow of fluids occurs in different ways. It can be a steady or streamlined
flow, unsteady or turbulent flow, compressible or incompressible flow or even
viscous or non- viscous flow. For example, consider a calm flow of water
through a river. Careful observation reveals that the velocity of water at
different locations of the river is quite different. It is almost faster at the
center and slowest near the banks. However, the velocity of the particle at any
point is constant. For better understanding, assume that the velocity of the
particle is about 4 meter per second at the center of the river. Hence it will
be of the same value for all other particles crossing through this point. In a
similar way, if the velocity of the particle flowing near the bank of the river
is 0.5 meter per second, then the succeeding particles flowing through it will
have the same value.

When
a liquid flows such that each particle of the liquid passing through a point
moves along the same path with the same velocity as its predecessor then the
flow of liquid is said to be a *streamlined*
*flow*. It is also referred to as
steady or* *laminar flow. The actual
path taken by the particle of the moving fluid is called a streamline, which is
a curve, the tangent to which at any point gives the direction of the flow of
the fluid at that point as shown in Figure 7.16. It is named so because the
flow looks like the flow of a stream or river under ideal conditions.

If
we assume a bundle of streamlines having the same velocity over any cross
section perpendicular to the direction of flow then such bundle is called a ‘*tube of *

Thus, it is important to note
that any* *particle in a tube of flow
always remains in the tube throughout its motion and cannot mix with liquid in
another tube. Always the axis of the tube of flow gives the streamline. The
streamlines always represent the trajectories of the fluid particles. The flow
of fluid is streamlined up to a certain velocity called critical velocity. This
means a steady flow can be achieved at low flow speeds below the critical
speed.

When
the speed of the moving fluid exceeds the critical speed, *v*_{c} the motion becomes turbulent. In this case, the
velocity changes both in magnitude and direction from particle to particle and
hence the individual particles do not move in a streamlined path. Hence, the
path taken by the particles in turbulent flow becomes erratic and
whirlpool-like circles called eddy current or eddies (Figure 7.17 (a) and (b)).
The flow of water just behind a boat or a ship and the air flow behind a moving
bus are a few examples of turbulent flow.

The
distinction between the two types of motion can be easily demonstrated by
injecting a jet of ink axially in a wide tube through which water flows. When
the velocity of the fluid is small the ink will move in a straight line path.
Conversely, when the velocity is increased beyond a certain value, the ink will
spread out showing the disorderliness and hence the motion becomes turbulent.
The zig-zag motion results in the formation of eddy currents and as a
consequence, much energy is dissipated.

We
have learnt that the flow of a fluid becomes steady or laminar when the
velocity of flow is less than the critical velocity v_{c} otherwise,
the flow becomes turbulent. Osborne Reynolds (1842-1912) formulated an equation
to find out the nature of the flow of fluid, whether it is streamlined or
turbulent.

It
is a dimensionless number called ‘*Reynold’s*
*number*’. It is denoted by the symbol
R_{c}* *or K.* *In the equation, ρ denotes the density
of the fluid, *v* the velocity of the
flowing fluid, D is the diameter of the pipe in which the fluid flow, and η is
the coefficient of viscosity of the fluid. The value of R_{c} remains
the same in any system of units.

Hence,
Reynold’s number R_{c} is a critical variable, which decides whether
the flow of a fluid through a cylindrical pipe is streamlined or turbulent. In
fact, the critical value of R_{c} at which the turbulent sets is found
to be the same for geometrically similar flows. For example, when two liquids
(say oil and water) of different densities and viscosities flow in pipes of
same shapes and sizes, the turbulence sets in at almost the same value of R_{c}.
The above fact leads to the *Law of similarity*
which states that when there are two geometrically similar flows, both are
essentially equal to each other, as long as they embrace the same Reynold’s
number. The *Law of similarity* plays a
very important role in technological applications.

The
shape of ships, submarines, racing cars, and airplanes are designed in such a
way that their speed can be maximized.

To
understand terminal velocity, consider a small metallic sphere falling freely
from rest through a large column of a viscous fluid.

The
forces acting on the sphere are (i) gravitational force of the sphere acting
vertically downwards, (ii) upthrust U due to buoyancy and (iii) viscous drag
acting upwards (viscous force always acts in a direction opposite to the motion
of the sphere).

Initially,
the sphere is accelerated in the downward direction so that the upward force is
less than the downward force. As the velocity of the sphere increases, the
velocity of the viscous force also increases. A stage is reached when the net
downward force balances the upward force and hence the resultant force on the
sphere becomes zero. It now moves down with a constant velocity.

The
maximum constant velocity acquired by a body while falling freely through a
viscous medium is called the terminal velocity V_{T}. In the Figure
7.18, a graph is drawn with velocity along y- axis and time along x- axis. It
is evident from the graph

Consider
a sphere of radius *r* which falls
freely through a highly viscous liquid of coefficient of viscosity η. Let the density of the material of the sphere be ρ and the density of the fluid be σ.

Here,
it should be noted that the terminal speed of the sphere is directly
proportional to the square of its radius. If σ is greater than ρ, then the term (ρ *-* σ*)* becomes negative leading to a negative terminal
velocity. That is why air bubbles rise up through water or any fluid. This is
also the reason for the clouds in the sky to move in the upward direction.

When
a body falls through a viscous medium, it drags the layer of the fluid
immediately in contact with it. This produces a relative motion between the
different layers of the liquid. Stoke performed many experiments on the motion
of small spherical bodies in different fluids and concluded that the viscous
force F acting on a spherical body of radius r depends directly on

i)
radius (*r*) of the sphere

ii)
velocity (*v*) of the sphere and

ii)
coefficient of viscosity η of the liquid

Therefore
F ∝ η^{x}*r* * ^{y}v^{z}*
⇒

Using
dimensions, the above equation can be written as

*[MLT *^{–}* *^{2}]* *=* k [ML*^{−}^{1}*T
*^{–}*
*^{1}]*
^{x} *×[

On
solving, we get x=1, y=1, and z=1 Therefore, F=kη *rv*

Experimentally, Stoke found that the value of k = 6π

**F = 6π****η****
rv
(7.23)**

This
relation is known as Stoke’s law

Since
the raindrops are smaller in size and their terminal velocities are small,
remain suspended in air in the form of clouds. As they grow up in size, their
terminal velocities increase and they start falling in the form of rain.

This
law explains the following:

a)
Floatation of clouds

b)
Larger raindrops hurt us more than the smaller ones

c)
A man coming down with the help of a parachute acquires constant terminal
velocity.

Poiseuille
analyzed the steady flow of liquid through a capillary tube. He derived an
expression for the volume of the liquid flowing per second through the
capillary tube.

As
per the theory, the following conditions must be retained while deriving the
equation.

·
The
flow of liquid through the tube is streamlined.

·
The
tube is horizontal so that gravity does not influence the flow

·
The
layer in contact with the wall of the tube is at rest

· The pressure is uniform over any cross section of the tube

We
can derive Poiseuille’s equation using dimensional analysis. Consider a liquid
flowing steadily through a horizontal capillary tube. Let *v*=(V/t) be the volume of the
liquid flowing out per second through a capillary tube. It depends on (1)
coefficient of viscosity (η) of the liquid, (2) radius of the tube (r), and (3)
the pressure gradient (P/*l*) .

Then,

where,
*k* is a dimensionless constant.
Therefore,

So,
equating the powers of M, L, and T on both sides, we get

*a *+* c *= 0, −*a *+*
b *−2*c *=3, and −*a *−2*c
*= −1

We
have three unknowns *a*, *b*, and *c.* We have three equations, on solving, we get

*a *= −1,* b *= 4, and* c *= 1* *

Therefore,
equation (7.24) becomes,

Experimentally,
the value of k is shown to be π/8 , we have

The
above equation is known as *Poiseuille’s*
*equation *for the flow of liquid
through a* *narrow tube or a capillary
tube. This relation holds good for the fluids whose velocities are lesser than
the critical velocity (v_{c}).

The
importance of viscosity can be understood from the following examples.

1)
The oil used as a lubricant for heavy machinery parts should have a high
viscous coefficient. To select a suitable lubricant, we should know its
viscosity and how it varies with temperature [Note: As temperature increases,
the viscosity of the liquid decreases]. Also, it helps to choose oils with low
viscosity used in car engines (light machinery).

2)
The highly viscous liquid is used to damp the motion of some instruments and is
used as brake oil in hydraulic brakes.

3)
Blood circulation through arteries and veins depends upon the viscosity of
fluids.

4)
Millikan conducted the oil drop experiment to determine the charge of an
electron. He used the knowledge of viscosity to determine the charge.

Tags : Properties of Matter , 11th Physics : UNIT 7 : Properties of Matter

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