Co-efficient of Viscosity:
To obtain a quantitative expression for viscosity, let us consider the velocities of the layers P and Q situated at distance x and x + dx respectively from the stationary horizontal surface S (Figure). Let the velocities of layers P and Q be v and v + dv respectively and the velocity gradient between the two layers is dv/dx . The viscous force between the two layers is proportional to:
(a) the surface area A of the layer on which it acts, and
(b) the velocity gradient dv/dx .
Therefore, we can write the viscous force F as:
F ∝ A (dv/ dx)
or, F = - η A (dv /dx)
where η is the constant of proportionality known as the co-efficient of viscosity. It depends on the nature of the liquid. The negative sign in Eq) signifies that the viscous force acts in a direction opposite to the flow of liquid. Further, if A = 1 and dv/dx = 1 , we have F =η . That is, the co-efficient of viscosity (η) of a liquid can be defined as the tangential viscous force per unit area acting between layers of a liquid in which unit velocity gradient is maintained in a direction normal to the layers. To determine the dimensions of η, we could write from Eq.:
η= (F / A)/(dv / dx)
= (Force / Area)/(Velocity / Distance)
= (MLT- 2 / L2)/(LT-1/ L)
= ML-1 T-1
The absolute cgs unit of the co-efficient of viscosity is "Poise" and SI unit is N m- 2 s or Pa s or kg m-1 s-1 . Also, 1 Poise = 10-1 Pa s.
Above description of viscosity is valid only when the flow of fluid is laminar, that is, the velocity of liquid has a small value. The flow of fluid becomes turbulent if the value of velocity is too low or when it is too high. In other words, the flow of liquid remains laminar only for a range of velocity called critical velocities. You will learn it now.