Viscosity and Newton's Law of Viscosity

Viscosity is the property of a fluid by which it offers internal resistance -- essentially, a kind of internal friction -- to the relative motion between its own adjacent layers. Just as solid surfaces sliding over one another experience ordinary friction, adjacent layers of a real fluid sliding past one another (as in streamline flow) experience a tangential dragging force, each layer trying to slow down the layer moving faster than it, and speed up the layer moving slower than it, next to it.

Newton's law of viscosity puts this idea into a precise, quantitative form. Consider a fluid in streamline flow between two parallel layers, one moving faster than the other. The tangential (viscous) force $F$ needed to maintain a constant relative velocity between the two layers is found, experimentally, to be:

• directly proportional to the area $A$ of contact between the layers, and
• directly proportional to the velocity gradient, $dv/dx$, i.e. the rate at which the fluid's velocity changes with perpendicular distance $x$ measured across the layers (a large velocity gradient means the fluid's speed changes rapidly from one layer to the next close by; a small velocity gradient means it changes only gradually).

Combining these two proportionalities with a constant of proportionality $\eta$ (the Greek letter eta), Newton's law of viscosity is written as:

The constant $\eta$ is called the coefficient of viscosity of the fluid (often simply called its "viscosity"). Rearranging, $\eta = \dfrac{F/A}{dv/dx}$, so the coefficient of viscosity is the tangential (viscous) force per unit area required to maintain a unit velocity gradient between two layers of the fluid. Since $F/A$ has SI unit $\text{N/m}^2 = \text{Pa}$ and $dv/dx$ has SI unit $\text{(m/s)/m} = \text{s}^{-1}$, the coefficient of viscosity $\eta$ has SI unit $\text{Pa}\cdot\text{s}$ (pascal-second), sometimes also called the poiseuille (Pl) in honour of Jean Léonard Marie Poiseuille, who studied viscous flow through narrow tubes.

A fluid that obeys Newton's law of viscosity exactly, with a constant value of $\eta$ regardless of the velocity gradient applied to it, is called a Newtonian fluid -- water, air, and most simple liquids and gases are, to a very good approximation, Newtonian. …