Using Newton’s law of viscosity for streamline flow, derive an expression for coefficient of viscosity.

Answer: \(\eta = \dfrac{F}{A\,(dv/dx)}\)

In streamline flow of a liquid over a fixed surface, the liquid moves in parallel layers; the layer in contact with the surface is at rest and the velocity increases with distance \(x\) from it. Adjacent layers exert a tangential (viscous) force on each other opposing their relative motion.

Newton’s law of viscosity: the viscous force \(F\) between two layers is

• directly proportional to the area of the layers in contact, \(F \propto A\), and
• directly proportional to the velocity gradient, \(F \propto \dfrac{dv}{dx}\).

where \(\eta\) is the coefficient of viscosity. Hence

If \(A = 1\) m\(^2\) and \(dv/dx = 1\) s\(^{-1}\), then \(\eta = F\): the coefficient of viscosity is the viscous force per unit area per unit velocity gradient. SI unit N s/m\(^2\) (pascal second), dimensions \([M^1L^{-1}T^{-1}]\). (Strictly, \(F = -\eta A\,dv/dx\), the minus sign showing that the force opposes relative motion.)