Viscosity
Pour honey and water side by side on a tilted surface. Water races ahead while honey crawls. Both are fluids, yet they behave very differently — the reason is viscosity. Viscosity is the internal resistance of a fluid to flow, arising from friction between neighbouring layers of fluid moving at different speeds. The more viscous a fluid, the more it resists being deformed by shear forces. At the microscopic level, viscosity comes from momentum transfer between fluid molecules. In liquids, molecules are close together and intermolecular attractions dominate — so viscosity decreases as temperature rises (those attractions are easier to overcome when molecules have more thermal energy). In gases, molecules rarely interact directly; viscosity arises from molecules bouncing between faster and slower layers, which actually increases with temperature. This opposite behaviour is a clean experimental test of the two mechanisms. Viscosity is not just a laboratory curiosity. Engineers use it to design engine lubricants, doctors measure blood viscosity to detect cardiovascular disease, and geophysicists model the viscosity of Earth's mantle (about 10²¹ Pa·s!) to understand plate tectonics. Understanding viscosity gives you the key to predicting how any real fluid — from air to magma — moves under a force.
- The Sliding Layers Model
- Shear Stress and the Definition of Viscosity
- Dynamic vs Kinematic Viscosity
- Poiseuille's Law: Viscous Flow in a Pipe
- Stokes' Law: Drag on a Sphere
The Sliding Layers Model
Imagine the fluid as a stack of very thin horizontal sheets — like a deck of playing cards. The bottom layer is held stationary (stuck to a fixed surface), and each layer above it moves slightly faster. The top layer moves fastest because it is dragged by the moving surface above it. This gradient of velocities across the fluid depth is called a velocity gradient or shear rate, dv/dy.
Shear Stress and the Definition of Viscosity
Each faster layer tries to drag the slower layer beneath it, and the slower layer resists. This internal friction force per unit area is the shear stress, τ (tau). Isaac Newton proposed that for many fluids, τ is directly proportional to the velocity gradient: τ = η (dv/dy). The constant of proportionality η (eta) is the dynamic viscosity. Fluids that obey this linear relationship are called Newtonian fluids (water, air, most oils). Fluids like ketchup or blood plasma can deviate — they are non-Newtonian.
Dynamic vs Kinematic Viscosity
Dynamic viscosity η measures the absolute resistance to shear and has SI units of Pa·s (pascal-seconds), also written N·m⁻²·s or kg·m⁻¹·s⁻¹. An older unit, the poise (P), is still common: 1 Pa·s = 10 P. Kinematic viscosity ν = η/ρ (where ρ is density) measures how viscosity affects flow relative to inertia. Its SI unit is m²/s; the older unit is the stoke (St). Both quantities matter: engineers prefer kinematic viscosity when inertial effects (like turbulence) are important.
Poiseuille's Law: Viscous Flow in a Pipe
When a viscous fluid flows through a cylindrical pipe of radius R and length L under a pressure difference ΔP, the volume flow rate Q is given by Poiseuille's Law: Q = πR⁴ΔP / (8ηL). This is enormously sensitive to radius — double the pipe radius and flow increases 16-fold (factor of 2⁴). That is why clogged arteries, which reduce effective radius, dramatically reduce blood flow even with the same blood pressure difference.
Stokes' Law: Drag on a Sphere
A solid sphere of radius r moving at speed v through a fluid of viscosity η experiences a viscous drag force F = 6πηrv (Stokes' Law). When this drag force plus buoyancy balances gravity, the sphere reaches terminal velocity. Stokes' Law is used in the falling-ball viscometer, a standard device for measuring fluid viscosity. It also explains why fine dust particles fall almost imperceptibly slowly through air — tiny r makes drag enormous relative to weight.
The Key Equation
- \tau — Shear stress (Pa (N m⁻²)): The internal friction force per unit area acting between adjacent fluid layers.
- \eta — Dynamic viscosity (Pa·s): The fluid's intrinsic resistance to shear deformation; higher η means thicker, more resistive fluid.
- \frac{dv}{dy} — Velocity gradient (shear rate) (s⁻¹): The rate at which fluid speed changes with perpendicular distance y; the steeper this gradient, the larger the shear stress.
Worked Example
Problem. A steel ball bearing of radius r = 1.5 mm and density ρ_s = 7800 kg/m³ falls through glycerol (η = 1.49 Pa·s, ρ_f = 1260 kg/m³). Calculate the terminal velocity of the ball using Stokes' Law.
Answer. Terminal velocity ≈ 21.5 mm/s (about 2.15 cm/s). The ball sinks very slowly because glycerol is nearly 1500 times more viscous than water.
Origins · 1 minJean Léonard Marie Poiseuille and how this was discovered
Poiseuille was a physician and physicist who derived the law governing viscous flow through cylindrical tubes (Poiseuille's Law, 1838–1840). Motivated by wanting to understand blood circulation, he built precise glass capillary experiments and showed that flow rate is proportional to the fourth power of the tube radius and the pressure difference, and inversely proportional to viscosity and tube length. His work provided the first quantitative framework for viscous pipe flow. Poiseuille originally trained as a physician and measured blood pressure using a mercury manometer attached directly to a horse's artery — so his fluid dynamics work was inseparable from his medical research.
Related Concepts
- Bernoulli's Equation — Bernoulli's equation describes ideal (inviscid) flow; understanding viscosity reveals when and why real fluids deviate from Bernoulli's predictions.
- Poiseuille Flow — Poiseuille's Law is a direct application of viscosity to pipe flow, predicting volume flow rate as a function of viscosity, pressure, and geometry.
- Stokes' Drag and Terminal Velocity — Stokes' Law uses viscosity to calculate the drag force on a sphere moving through a fluid, leading directly to terminal velocity calculations.
- Reynolds Number and Turbulence — The Reynolds number Re = ρvL/η compares inertial to viscous forces, determining whether flow is laminar (smooth) or turbulent (chaotic).