Viscosity

Viscosity describes how much a fluid resists flowing when its layers move relative to each other. This resistance comes from internal stresses within the fluid, which, in an ideal case, are directly proportional to how fast the fluid is being deformed. In liquids, this resistance is due to the cohesive forces between molecules; in gases, it results from molecules colliding with one another. With the exception of superfluids, every fluid has some viscosity, meaning viscous effects are always present in real fluid flows.

For everyday liquids, viscosity matches the informal idea of "thickness"—syrup is more viscous than water. Scientifically, viscosity is defined as a force multiplied by time divided by area, so its SI unit is the newton-second per square meter, also called the pascal-second.

When a viscous fluid is pushed through a tube, it moves faster near the center than near the walls. To keep the flow going, some stress—like a pressure difference between the tube's ends—is required. This is because force must overcome the friction between fluid layers sliding past each other. For a steady flow rate in a tube, the needed force is proportional to the fluid's viscosity.

In general, a fluid's viscosity depends on its state—such as temperature, pressure, and how fast it is being deformed. However, for some fluids, this dependence is negligible in certain conditions. For example, the viscosity of a Newtonian fluid does not change significantly with the rate of deformation.

Zero viscosity—meaning no resistance to shear stress—occurs only at very low temperatures in superfluids. Otherwise, the second law of thermodynamics requires all fluids to have positive viscosity. A fluid with zero viscosity is called ideal or inviscid. For non-Newtonian fluids, viscosity can be time-independent (pseudoplastic, plastic, or dilatant flows) or time-dependent (thixotropic or rheopectic flows).

**Etymology** The word "viscosity" comes from the Latin *viscum* ("mistletoe"), which also referred to a sticky glue made from mistletoe berries.

**Dynamic viscosity** In materials science and engineering, understanding the forces or stresses involved in deforming a material is often important. For a simple spring, Hooke's law says force is proportional to displacement from equilibrium—these are elastic stresses. In other materials, stresses arise from the rate of deformation over time; these are viscous stresses. For instance, in water, the stresses from shearing depend not on how far the fluid has been sheared, but on how quickly the shearing happens.

Viscosity is the material property that relates viscous stresses to the rate of change of deformation (the strain rate). This is easiest to visualize in a simple shearing flow, like planar Couette flow. In Couette flow, a fluid is trapped between two large parallel plates: one fixed, the other moving at a constant speed. If the top plate moves slowly enough to avoid turbulence, the fluid particles move parallel to it in steady state, with speed varying from zero at the bottom plate to the top plate's speed at the top. Each fluid layer moves faster than the one below it, and friction between layers creates a force resisting their relative motion. The fluid applies a force opposite to the top plate's motion and an equal opposite force on the bottom plate, so an external force is needed to keep the top plate moving at constant speed.

In many fluids, the flow velocity changes linearly from zero at the bottom to the top plate's speed. The force \( F \) on the top plate is proportional to the speed \( u \), the area \( A \) of each plate, and inversely proportional to their separation \( y \):

\[ F = \mu A \frac{u}{y}. \]

The proportionality factor \( \mu \) (Greek letter mu) is the dynamic viscosity, often simply called viscosity. Its dimensions are (mass/length)/time, giving SI units of kilogram per meter per second.