Let us pour equal amounts of water and castor oil in two identical funnels. It is observed that water flows out of the funnel very quickly whereas the flow of castor oil is very slow. This is because of the frictional force acting within the liquid. This force offered by the adjacent liquid layers is known as viscous force and the phenomenon is called viscosity. Viscosity is the property of the fluid by virtue of which it opposes relative motion between its different layers. Both liquids and gases exhibit viscosity but liquids are much more viscous than gases.

Co-efficient of viscosity
Consider a liquid to flow steadily through a pipe as shown in the Fig. below. The layers of the liquid which are in contact with the walls of the pipe have zero velocity. As we move towards the axis, the velocity of the liquid layer increases and the centre layer has the maximum velocity v. Consider any two layers P and Q separated by a distance dx. Let dv be the difference in velocity between the two layers.

Streamline flow
The flow of a liquid is said to be steady, streamline or laminar if every particle of the liquid follows exactly the path of its preceding particle and has the same velocity of its preceding particle at every point. Let abc be the path of flow of a liquid and v1, v2 and v3 be the velocities of the liquid at the points a, b and c respectively. During a streamline flow, all the particles arriving at ‘a’ will have the same velocity v1 which is directed along the tangent at the point ‘a’. A particle arriving at b will always have the same velocity v2. This velocity v2 may or may not be equal to v1. Similarly all the particles arriving at the point c will always have the same velocity v3. In other words, in the streamline flow of a liquid, the velocity of every particle crossing a particular point is the same.

The streamline flow is possible only as long as the velocity of the fluid does not exceed a certain value. This limiting value of velocity is called critical velocity.

Turbulent flow
When the velocity of a liquid exceeds the critical velocity, the path and velocities of the liquid become disorderly. At this stage, the flow loses all its orderliness and is called turbulent flow. Some examples of turbulent flow are :
● After rising a short distance, the smooth column of smoke from an incense stick breaks up into irregular and random patterns.
● The flash – flood after a heavy rain. Critical velocity of a liquid can be defined as that velocity of liquid upto which the flow is streamlined and above which its flow becomes turbulent.

Reynold’s number
Reynolds number is a pure number which determines the type of flow of a liquid through a pipe. It is denoted by NR.
It is given by the formula
NR = vc ρ D/η
where vc is the critical velocity, ρ is the density, η is the co-efficient of viscosity of the liquid and D is the diameter of the pipe. If NR lies between 0 and 2000, the flow of a liquid is said to be streamline. If the value of NR is above 3000, the flow is turbulent. If NR lies between 2000 and 3000, the flow is neither streamline nor turbulent, it may switch over from one type to another. Narrow tubes and highly viscous liquids tend to promote stream line motion while wider tubes and liquids of low viscosity lead to tubulence.

Stoke’s law (for highly viscous liquids)
When a body falls through a highly viscous liquid, it drags the layer of the liquid immediately in contact with it. This results in a relative motion between the different layers of the liquid. As a result of this, the falling body experiences a viscous force F. Stoke performed many experiments on the motion of small spherical bodies in different fluids and concluded that the viscous force F acting on the spherical body depends on
● Coefficient of viscosity η of the liquid
● Radius a of the sphere and
● Velocity v of the spherical body. Dimensionally it can be proved that
F = k ηav
Experimentally Stoke found that
k = 6π
∴ F = 6π ηav
This is Stoke’s law.

Expression for terminal velocity
Consider a metallic sphere of radius ‘a’ and density ρ to fall under gravity in a liquid of density σ . The viscous force F acting on the metallic sphere increases as its velocity increases. A stage is reached when the weight W of the sphere becomes equal to the sum of the upward viscous force F and the upward thrust U due to buoyancy (Fig. below). Now, there is no net force acting on the sphere and it moves down with a constant velocity v called terminal velocity.

Experimental determination of viscosity of highly viscous liquids
The coefficient of highly viscous liquid like castor oil can be determined by Stoke’s method. The experimental liquid is taken in a tall, wide jar. Two marking B and C are marked as shown in Fig. below. A steel ball is gently dropped in the jar. The marking B is made well below the free surface of the liquid so that by the time ball reaches B, it would have acquired terminal velocity v. When the ball crosses B, a stopwatch is switched on and the time taken t to reach C is noted. If the distance BC is s, then terminal velocity v = s/t.

Application of Stoke’s law
Falling of rain drops: When the water drops are small in size, their terminal velocities are small. Therefore they remain suspended in air in the form of clouds. But as the drops combine and grow in size, their terminal velocities increases because v α a2. Hence they start falling as rain.
Poiseuille’s equation
Poiseuille investigated the steady flow of a liquid through a capillary tube. He derived an expression for the volume of the liquid flowing per second through the tube. Consider a liquid of co-efficient of viscosity η flowing, steadily through a horizontal capillary tube of length l and radius r. If P is the pressure difference across the ends of the tube, then the volume V of the liquid flowing per second through the tube depends on η, r and the pressure gradient (P/l).

Determination of coefficient of viscosity of water by Poiseuille’s flow method
A capillary tube of very fine bore is connected by means of a rubber tube to a burette kept vertically. The capillary tube is kept horizontal as shown in Fig. 5.19. The burette is filled with water and the pinch – stopper is removed. The time taken for water level to fall from A to B is noted. If V is the volume between the two levels A and B, then volume of liquid flowing per second is V t. If l and r are the length and radius of the capillary tube respectively, then

If ρ is the density of the liquid then the initial pressure difference between the ends of the tube is P1 = h1ρg and the final pressure difference P2 = h2ρg. Therefore the average pressure difference during the flow of water is P where

Viscosity – Practical applications
The importance of viscosity can be understood from the following examples.
(i) The knowledge of coefficient of viscosity of organic liquids is used to determine their molecular weights.
(ii) The knowledge of coefficient of viscosity and its variation with temperature helps us to choose a suitable lubricant for specific machines. In light machinery thin oils (example, lubricant oil used in clocks) with low viscosity is used. In heavy machinery, highly viscous oils (example, grease) are used.