Viscosity and Newton’s Law of Viscous Force

Viscosity and Newton’s Law of Viscous Force

In case of steady flow of a fluid when a layer of fluid slips or tends to slip on adjacent layers in contact, the two layers exert tangential force on each other which tries to destroy the relative motion between them. The property of a fluid due to which it opposes the relative motion between its different layers is called viscosity and the force between the layers opposing the relative motion is called viscous force.

Consider the two layers CD and MN of the liquid at distance x and x + dx from the fixed surface AB, having the velocities v and v + dv respectively. Then, \(\frac{dv}{dx}\) denotes the rate of change of velocity with distance and is known as velocity gradient.

According to Newton’s hypothesis, the tangential force F acting on a plane parallel layer is proportional to the are of the plane A and the velocity gradient \(\frac{dv}{dx}\) in a direction normal to the layer, i.e.,

\(F\,\,\propto \,\,A\) and \(F\,\,\propto \,\,\frac{dv}{dx}\) \(\therefore \,\,\,\,F\propto \,\,A\frac{dv}{dx}\) \(\Rightarrow F=-\eta A\frac{dv}{dx}\)

Where, \(\eta \) is a constant called the coefficient of viscosity, negative sign is employed because viscous force acts in a direction opposite to the flow of liquid.

If \(A=1,\,\frac{dv}{dx}=1,\,then\,\,\eta =F\)

Hence, the coefficient of viscosity is defined as the viscous force acting per unit area between two layers moving with unit velocity gradient.