2. Viscosity exists only in fluids and is a term used to describe the friction
which the fluid exhibits against its own motion. Therefore, viscosity is an
internal friction of a fluid. It means how much force is required to slide one
layer of the fluid over another one.
5.1 Definition
Figure 1. (a) The particles in an ideal (non-viscous) fluid all move through
the pipe with the same velocity. (b) In a viscous fluid, the velocity of the
fluid particles is zero at the surface of the pipe and increases to a
3. When an ideal (non-viscous) fluid flows through a pipe, the fluid layers
slide past one another with no resistance.
If the pipe has a uniform cross section, each layer has the same velocity,
as shown in Figure 1a. In contrast, the layers of a viscous fluid have
different velocities, as Figure 1b indicates.
The fluid has the greatest velocity at the center of the pipe, whereas the
layer next to the wall doesn’t move because of adhesive forces between
molecules and the wall surface.
To better understand the concept of viscosity, consider a layer of liquid
between two solid surfaces, as in Figure 2.
The lower surface is fixed in position, and the top surface moves to the
right with a velocity v under the action of an external force F.
Because of this motion, a portion of the liquid is distorted from its original
shape, ABCD, at one instant to the shape AEFD a moment later. The
force required to move the upper plate and distort the liquid is proportional
to both the area A in contact with the fluid and the speed v of the fluid.
Further, the force is inversely proportional to the distance d between the
4. Figure 2. A layer of liquid between two solid surfaces in which the lower
surface is fixed and the upper surface moves to the right with a velocity v.
(1)
where η is a constant depending upon the nature of the liquid and its
temperature, and is called the coefficient of viscosity. If the two layers are
very close together, then denoting the distance between them by (dx) and the
relative velocity by (dv), we get:F = - η A
dx
dv
----------------------- (2)
5. where the minus (-) sign indicates the decreasing character of the velocity
gradient (dv / dx).
If A = 1 cm2 and (dv/dx) = 1 , then:
F = η (numerically)
Therefore, η is thus defined as the tangential force per unit area required to
maintain a unit velocity gradient, i.e. a unit relative velocity between two
layers of a unit distance apart. In SI system, the units of η are , while the
corresponding cgs units are . This called poise, where: 1
poise = 1 = 0.1
5.2 Flow of fluids through narrow tubes:
Consider a narrow tube of length ‘L’ and radius ‘a’. If the flow of the fluid is
laminar, the velocity of the fluid is maximum at the axis of the tube and zero
at the wall. Let us consider the flow of cylindrical element of radius ‘r’ and
length ‘L’, as shown in Fig. (3).
6. The force on the left end of the element is P1π r2, and that on
the right end is P2π r2. the net force is thus:-
F = (P1 - P2) π r2 -----------------------
(3)
Since the element does not accelerate, this force must be
balanced with the viscous retarding force, F, at the surface of
this element. The area over which the viscous force acted is
the surface area of the cylindrical element of radius ‘r’.
A = 2 π r L
and F = - η A
Thus, the viscous force is:
F = - η (2 π r L) ----------------- (4)
Where is the velocity gradient. Equating this force given by
eqn. (4) to that given in eqn.(3), we find that:-
Δv =
7. This shows that the velocity changes more and more rapidly
with ‘r’ when we go from the centre (r=0) to the tube wall
(r=a). To find the velocity distribution through the cross
section of the tube, then:-
v =
v = -------------- (5)
This equation shows that:-
(1) At the centre (r = 0), the velocity has a maximum value, as :-
v = (P1-P2)a2/4ηL
(2) At the wall (r = a), the velocity has a minimum value of zero.
(3) The average velocity = (vmax. +vmin.)/2
(4) vmax. α a2 and vmax. α (P1-P2)/L
8. V = ------------- (6)
This relation is called Poisuill’s law, which shows that the volume rate of flow,
V, is inversely proportional to η, as might be expected. It is also proportional
to the pressure gradient along the tube, and it varies as the fourth power of
radius (a4).
Figure 5. Velocity profile of a fluid flowing
through a uniform pipe of circular cross section.
The rate of flow is given by Poiseuille’s law.
9. Equation (6) can be used to determine ‘η’ of a liquid. as
η = --------- (7)
From Poiseuille’s law, we see that in order to maintain a constant flow
rate, the pressure difference across the tube has to increase if the
viscosity of the fluid increases.
This fact is important in understanding the flow of blood through the
circulatory system. The viscosity of blood increases as the number of red
blood cells rises.
Blood with a high concentration of red blood cells requires greater
pumping pressure from the heart to keep it circulating than does blood of
lower red blood cell concentration.
Note that the flow rate varies as the radius of the tube raised to the fourth
power. Consequently, if a constriction occurs in a vein or artery, the heart
will have to work considerably harder in order to produce a higher
pressure drop and hence maintain the required flow rate.
10. Example 1:
A patient receives a blood transfusion through a needle of radius 0.20 mm
and length 2.0 cm. The density of blood is 1050 kg/m3. The bottle
supplying the blood is 0.500 m above the patient’s arm. What is the rate of
flow through the needle?
Solution:
The pressure difference is:
Substitute into Poiseuille’s law gives
