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Viscosity


We know that when water contained in a beaker is set into rotation with a glass rod or a finger, it starts rotating in the form of co-axial cylindrical layers as shown in Figure. When we stop stirring, the speeds of different layers are seen to decrease gradually and finally the water comes to rest, showing that an internal friction comes into play, which tries to destroy the relative motion between the layers.
 
The property of liquids (or gases) by virtue of which an internal resistance or friction comes into play when a liquid is in motion is called viscosity. The internal friction tries to destroy the motion of the liquid.
Or
The property of liquids (or gases) by virtue of which a backward dragging force a viscous drag acts tangentially on the layers of the liquid on motion and it tries to stop the motion is called viscosity.

Cause of viscosity


If in the above experiment, we set the water into motion and then place our finger quite near the glass surface, the water layers in contact with the glass surface appear to be almost stationary. But as we move our finger towards center, layers are found to be moving with greater and greater speeds and the innermost layer with the highest speed. Thus, there is a relative motion between different layers of the water. Same is true of other fluids (liquids or gases) set into motion. Such a flow is called a laminar flow. The result of this relative motion between the layers is that there appears a viscous drag or backward dragging force, which tries to destroy the motion. To maintain the motion, an external agency has to overcome this drag.

Co-efficient of Viscosity


Consider a liquid flowing over a horizontal solid surface in the form of parallel layers. The layer in contact with the solid surface is found to be at rest and as we move up, the velocity of the layers goes on increasing and the layer at the top possesses maximum velocity. Let us consider two parallel layers P and Q at distances x, x+dx from the solid surface moving with the velocities v and v + dv respectively.
Then, denotes the rate of change of velocity with distance in the direction of increasing distance and it is called the velocity gradient. The relative motion between the two layers can take place only, if some external force acts between them. Due to viscosity, a force F acts in opposite direction to destroy the relative motion. According to Newton, the viscous force F depends upon the following factors:
  1. It is directly proportional to the area of the layers in contact i.e.
         F α A
  2. It is directly proportional to the velocity gradient between the layers i.e.
         F α
Combining these two factors, we have
        F α A
        F = - η A
 
Where η is a constant depending upon the nature of the liquid and it is called the coefficient of viscosity. The negative sign shows that the viscous force is directed in a direction opposite to the direction of the motion of the liquid.

If A = 1 and = 1, then from equation we have η is a constant depending upon the nature of the liquid and is called the coefficient of viscosity. The negative sign shows that the viscous force is directed in a direction opposite to the direction of the motion of the liquid.

If A = 1 and = 1, then from equation we have  η = F

Hence, the coefficient of viscosity of a liquid may be defined as the tangential viscous force, which maintains a unit velocity gradient between its two parallel layers, each of unit area.

In cgs system, the unit of η is dyn s cm-2 or g cm-1 s-1 and is called poise. The coefficient of viscosity of a liquid is said to be one poise, if a tangential force of 1 dyne maintains a velocity gradient of 1 cm s-1 cm-1 between two parallel layers each 1 cm2 in area.

In SI system, the unit of η is N s m-2 or kg m-1 s-1 and is called decapoise. The coefficient of viscosity of a liquid is said to be 1 decapoise, if a tangential force of 1 N maintains a velocity gradient of 1 ms-1 m-1 between two parallel layers, each 1 m2 in area.

1 decapoise = 1 N sm-2 = (105 dyne) x s x (100 cm)-2 = 10 poise.

The dimensional formula of coefficient of viscosity is [ML-1T-1]

Stokes' Law

When a spherical body moves through a viscous medium (e.g. rain drop falling through air), the body produces a relative motion between layers of the fluid. The layers in contact with the body moves with the velocity of the body itself and the layer next to it, has lesser velocity, the next still lesser and so on, while the layer at a considerable distance from the body remains at rest. As a result of this relative motion, a backward dragging force is set up, which opposes the motion of the body. The backward dragging force increases with increase in velocity of the moving body and finally if the body is small, the backward dragging force soon balances the driving force and it starts moving with a constant velocity known as the terminal velocity of the body.
 
Stokes was the first to prove that the backward dragging force acting on a small spherical body is directly proportional to (i) terminal velocity of the body (ii) radius of the body and (iii) the coefficient of viscosity of the viscous medium i.e.
    F α η r V
Or F = 6π η r V

Here, 6 π is constant of proportionality.
 
As the body falls, the following three forces acts on the body:
  1. Weight of the body acting vertically downwards, which is given by
    mg = π r3 pg
  2. Upward thrust equal to the weight of the fluid displaced acting vertically upwards, which is given by
    U = π r3σ g

    Therefore, effective or apparent weight of the body acting vertically downwards,

    mg' = mg - U
    mg' = π r3 (p - σ ) g
  3. the viscous force acting vertically upwards (in a direction opposite to the motion of the body). The viscous force goes on increasing as the velocity of the falling body increases. A stage comes, when the weight of the body becomes just equal to the sum of the upthrust and the viscous force. At that stage, the body falls with constant maximum velocity, called the terminal velocity. If v is terminal velocity attained, then according to Stokes’ law, the viscous force on the body is given by
     
         F = 6 π η r v
     
     
    In equilibrium, the effective weight of the body is equal to the viscous force on attaining the terminal velocity i.e.
     π r3 (ρ - σ ) = 6 π η r v ------(i)
                  

    This equation is the expression for terminal velocity. In case, the density of the medium through which the spherical body falls is very small (for example, body falling through air), then density of the medium (σ ) may be neglected as compared to the density of the material (p) of the spherical body. In such a case, the equation reduces to
                  

Discussion of the result


From the equation (i) it follows that the terminal velocity attained by a spherical body:
  1. is directly proportional to the square of its radius. It implies that the bigger bodies attain larger terminal velocities. For this reason, the rain drops do not fall with same velocity. The bigger drops come down with greater velocity in comparison to the smaller drops. It may be pointed out that the terminal velocity attained is in no way connected to the height, from which the drops are falling.
  2. is directly proportional to the difference between the densities of body and the fluid i.e. (ρ - σ)
    1. In case, ρ > σ , the R.H.S of the equation (i) will be positive and hence the spherical body will attain terminal velocity in downward direction.
    2. In case ρ < σ , the R.H.S of equation (i) will be negative and hence the spherical body will possess negative terminal velocity. It will mean that instead of coming down, the spherical body will start rising up. For this reason, air bubble in a liquid and clouds in sky are seen to move in upward direction rising up. For this reason, air bubble in a liquid and clouds in sky are seen to move in upward direction.
  3. is inversely proportional to the coefficient of viscosity.
    It means more viscous the fluid, smaller is the terminal velocity, it will attain.




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