energy to the following liquid. Thus actual driving pressure is only P Q d
R
~
2
2 2
p
and theliquid emerges out at the outlet end with appreciable velocity and does not just trickle out.
(iii) The flow is accelerated near the entrance and becomes steady only after a certain distance from the inlet end. This error can be corrected by taking the length of tube as (L + 1.64R) instead of L. Hence tubes of longer length will give better results.
(iv) The formula does not hold for gases.
6.12 STOKES LAW FOR VISCOUS DRAG ON MOVING BODIES
When a small body is allowed to move through a viscous fluid, the layers of the fluid in contact with the body move with the velocity of the moving body, while those at large distance from it are at rest. Thus the motion of a body through the fluid creates relative motion between different layers of the fluid near it. Viscous forces are developed which oppose the relative motion between different layers of the fluid and hence the medium opposes the motion of the body. This opposing force increases with the velocity of the body. Stoke has shown that for a small sphere of radius r moving slowly with velocity v through a homogeneous fluid of infinite extension, the viscous retarding force is given by
F = 6phrv
Where h is the coefficient of viscosity of the fluid.
It is to be noted that an equilibrium is established when this viscous drag on the body is balanced by driving force. In that case net force on the body becomes zero and it starts moving with constant velocity called its terminal velocity.
Calculation of terminal velocity: Let us consider a small ball, whose radius is r and density is
r, falling freely in a liquid (or gas), whose density is s and coefficient of viscosity h. When it attains a terminal velocity v, it is subjected to two forces.
(i) effective force acting downward = V (r – s) g = 4 3
3
p r s
r (-
)g(ii) viscous force acting upward = 6ph rv
\ Since the ball is moving with a constant velocity v i.e., there is no acceleration in it, the net force acting on it must be zero. That is
6ph rv = 4 3 3
p r s
r (-
)g v = 2 9 2 r g (r s
)h-
Thus terminal velocity of the ball is directly proportional to the square of its radius.
6.13 EFFECT OF VARIOUS FACTORS ON VISCOSITY OF FLUIDS
(i) Temperature: The coefficient of viscosity of liquids decreases rapidly with rise in tem- perature. The effect is so marked that it would be practically meaningless to state the value of viscosity of a liquid without mentioning the temperature.
On the other hand, in case of gases, the viscosity increases with the rise in tempera- ture. This can be explained on the basis of the kinetic theory of gases.
(ii) Pressure: The coefficient of viscosity of liquids, in general, increases with increase of pressure. However, in case of water there is a decrease in viscosity for the first two hundred atmospheric pressure. The increase of viscosity with pressure is much more in case of very viscous liquids than in case of fairly mobile liquids. The coefficient of viscosity of all gases increases with increase of pressure. At moderate pressures the coefficient of viscosity is independent of pressure. At low pressure it is proportional to pressure. At very high pressure, the coefficient of viscosity of gases increases with increase with pressure.
(iii) Impurity: The viscosity of a liquid is also sensitive to impurities. For solutions in some cases the coefficient of viscosity is less than that of the pure solvent while in other cases it is greater. There is no set rule for this change. In case of mixtures the coefficient of viscosity is generally less than the arithmetic mean of the coefficients of viscosity of the components of the mixture.
6.14 OBJECT
Determination of the viscosity of water by method of capillary flow. [Poiseuilles method] Apparatus used: Capillary tube fitted on a board with a manometer and side tubes, constant level reservoir, measuring cylinder, a stopwatch traveling microscope.
Formula used: The coefficient of viscosity h of a liquid is given by the formula.
h =
p
q
PR4 8l
=p r
(h g) Q R 8 4 kg/(m - sec) or poiseWhere R = radius of the capillary tube
Q = volume of water collected per second
= length of the capillary tube
r = density of liquid (r = 1 × 103 kg / m3 for water) h = difference of levels in manometer
Description of the apparatus: The apparatus used
is shown in the Fig. 6.11. Water from the constant level reservoir flows to the union X; thence through a capillary tube of known length of a graduated jar. From the unions X and Y two pieces of rubber tub- ing make connections to the manometer. The dif- ference of the levels E and F gives the value of the pressure difference between the ends of the capil- lary tube K.
Manipulations:
1. Arrange the flow of water in such a way that the emergent water is a slow trickle or a succession of drops. This is done to ensure streamlined motion.
Fig. 6.10
Fig. 6.11
2. When every thing is steady, collect water for two minutes in the graduated jar. Note the quantity collected. From this find Q, the amount of water passing per second.
3. Find the difference in the level of the water in the manometer and from this calculate ‘P’. 4. Vary the flow of the water slightly by raising or lowering the reservoir and when every
thing is steady, repeat (2) and (3). Thus make 5 sets of Q and P. Take the mean of P
Q.
5. Measure the length ‘l’ of the tube K and also internal diameter of the sample provided.
6. Draw a graph between h and Q and find the value of h
Q