What is Newtonian viscosity? — Physics

Definition 4.1 University Physics — Year 2 · Chapter 4 — Viscous Flows

In a plane shear flow $\vect v = v_x(y)\,\vect e_x$ , the fluid above a surface $y =$ const exerts on the fluid below it (through a surface element $\dd S$ ) the tangential force

\dd\vect F = \eta\,\frac{\dd v_x}{\dd y}\,\dd S\;\vect e_x ,

dragging the slower layer forward and being dragged back in return. The shear stress is $\tau = \eta\,\dd v_x/\dd y$ ; the coefficient $\eta$ (unit $\mathrm{Pa}\,\mathrm{s}$ ) is the dynamic viscosity of the fluid, and $\nu = \eta/\rho$ ( $\mathrm{m}^{2}/\mathrm{s}$ ) its kinematic viscosity. A fluid for which $\eta$ does not depend on the shear rate is Newtonian: water, air, oils, alcohol; blood, paint, ketchup and polymer melts are not. Orders of magnitude at $20{}^{\circ}\mathrm{C}$ : air $\eta = 1.8 \times 10^{-5}\,\mathrm{Pa}\,\mathrm{s}$ ( $\nu = 1.5 \times 10^{-5}\,\mathrm{m}^{2}/\mathrm{s}$ ), water $1.0 \times 10^{-3}\,\mathrm{Pa}\,\mathrm{s}$ ( $\nu = 1.0 \times 10^{-6}\,\mathrm{m}^{2}/\mathrm{s}$ ), olive oil $0.08\,\mathrm{Pa}\,\mathrm{s}$ , glycerol $1.5\,\mathrm{Pa}\,\mathrm{s}$ , honey $\sim10\,\mathrm{Pa}\,\mathrm{s}$ . Liquids get thinner when heated, gases thicker.

Left: plane Couette flow between a fixed plate and a plate moving at

U

— a linear profile, the same shear stress on every layer. Right: the viscous forces on a slab of fluid; they cancel unless the velocity profile is curved.

Examples

Example 4.4 (Shear stress on a table)

A $0.1\,\mathrm{mm}$ film of oil ( $\eta = 0.1\,\mathrm{Pa}\,\mathrm{s}$ ) between a plate and a table, the plate sliding at $1\,\mathrm{m}/\mathrm{s}$ : $\tau = \eta U/h = 1\,\mathrm{kPa}$ , a force of $100\,\mathrm{N}$ on a tenth of a square metre — the principle of the lubricated bearing, where the viscous drag replaces dry friction a hundred times larger. Honey on a spoon, $10\,\mathrm{Pa}\,\mathrm{s}$ , $1\,\mathrm{mm}$ thick, draining at $1\,\mathrm{mm}/\mathrm{s}$ : $\tau = 10\,\mathrm{Pa}$ , exactly the weight of a millimetre of honey per unit area — which is why it drains at that speed.

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