Viscosity, Laminar Flow, and Surface Tension
Viscosity is the internal friction of a fluid that resists flow, laminar flow is smooth flow in parallel layers (at low Reynolds number Re = ρvd ÷ η), turbulent flow is chaotic, and surface tension is the tendency of a liquid surface to contract, which causes drops, capillary rise and the support of small insects. In brief: the drag on a small sphere in a viscous fluid is F = 6πηrv (Stokes’ law), and its terminal speed is v = 2r2(ρs − ρf)g ÷ 9η. Flow in a pipe is usually laminar below Re ≈ 2,000 and turbulent above about 4,000. In a narrow tube, a liquid rises by h = 2γ cos θ ÷ ρgr. Poiseuille’s law shows that the flow rate in a pipe is proportional to the fourth power of the radius.
Honey flows more slowly than water, blood flow depends on the width of vessels, and water striders walk on ponds. This guide explains viscosity and its measurement, the types of flow and the Reynolds number, surface tension and capillarity, compares fluids in a table, works through numerical examples and corrects common mistakes.
What is viscosity?
Viscosity arises from the forces between the molecules of a fluid and their exchange of momentum between moving layers. When a fluid flows, adjacent layers move at different speeds and exert a drag on each other. The dynamic viscosity η is defined through F ÷ A = η (Δv ÷ Δy): the shear force per unit area is proportional to the velocity gradient. The SI unit is the pascal-second (Pa·s); another unit is the poise (1 P = 0.1 Pa·s). The viscosity of liquids falls as temperature rises (honey flows easily when warm), whereas that of gases rises. Water at 20 °C has η = 1.0 × 10−3 Pa·s, air 1.8 × 10−5 Pa·s, motor oil about 0.1 to 0.3 Pa·s, and glycerol about 1.4 Pa·s. Stokes’ law: a sphere of radius r moving slowly with speed v through a fluid experiences a drag F = 6πηrv. A falling sphere reaches a terminal speed when the weight minus the buoyancy equals the drag: vt = 2r2(ρs − ρf)g ÷ 9η, which is the basis of the falling-ball viscometer.
What are laminar and turbulent flow?
In laminar flow, the fluid moves in smooth, parallel layers, with no mixing between them. In a pipe, the speed is greatest at the center and zero at the walls (parabolic profile). In turbulent flow, the motion is irregular, with eddies and mixing, and the drag is greater. The type of flow is predicted by the Reynolds number, a dimensionless quantity: Re = ρvd ÷ η, where d is a characteristic length, such as the pipe diameter. For pipes, Re < 2,000 usually means laminar flow and Re > 4,000 turbulent flow. The flow of blood is mostly laminar, while the flow of water in a tap is turbulent. For laminar flow in a pipe, Poiseuille’s law gives the volume flow rate Q = πr4Δp ÷ 8ηL: the flow rate is proportional to the fourth power of the radius and to the pressure drop, and inversely proportional to the length and viscosity. Halving the radius of an artery reduces the flow 16 times for the same pressure, so the heart must work much harder when arteries are narrowed by plaque.
What is surface tension?
Molecules in the bulk of a liquid are attracted by neighbors on all sides, but those at the surface are pulled only inward and sideways. The surface behaves like a stretched elastic skin that tends to minimize its area. The surface tension γ is the force per unit length along the surface (N/m), or equally the energy per unit area (J/m2). For water at 20 °C, γ = 0.0728 N/m; for mercury, 0.485 N/m; soap solutions reduce it to about 0.025 N/m. Effects:
- small drops and bubbles are spherical (the shape with the minimum surface area for a given volume); the excess pressure in a bubble is 2γ ÷ r for a drop (4γ ÷ r for a soap bubble with two surfaces);
- water striders and needles can stay on the surface;
- capillary action: in a narrow tube of radius r, a liquid that wets the glass (contact angle θ less than 90°) rises to a height h = 2γ cos θ ÷ ρgr, while mercury is depressed;
- detergents lower the surface tension, which allows water to penetrate fabrics.
Fluid properties compared
| Fluid | Viscosity η (Pa·s) | Surface tension γ (N/m) |
|---|---|---|
| Air | 1.8 × 10−5 | — |
| Water | 1.0 × 10−3 | 0.0728 |
| Blood (37 °C) | about 3 × 10−3 | about 0.058 |
| Glycerol | 1.4 | 0.063 |
| Mercury | 1.5 × 10−3 | 0.485 |
Worked example: Stokes, Reynolds and capillary rise
- Falling ball. A steel ball of radius 1.0 mm (ρ = 7,800 kg/m3) falls in glycerol (ρ = 1,260 kg/m3, η = 1.4 Pa·s). vt = 2r2(ρs − ρf)g ÷ 9η = 2 × (1.0 × 10−3)2 × 6,540 × 9.8 ÷ (9 × 1.4) = 0.128 ÷ 12.6 = 0.0102 m/s, or 1.0 cm/s. In water, the same ball would be much faster and the flow would not be laminar (Stokes’ law would not apply).
- Reynolds number. Water flows through a pipe of diameter 2.0 cm at 1.0 m/s. Re = ρvd ÷ η = 1,000 × 1.0 × 0.020 ÷ 1.0 × 10−3 = 20,000, so the flow is turbulent. For the same pipe with honey (η = 5 Pa·s, ρ = 1,400) at 0.10 m/s: Re = 1,400 × 0.10 × 0.020 ÷ 5 = 0.56, which is laminar.
- Poiseuille. If the radius of an artery is reduced by 20 percent (to 0.8 of the original), the flow at the same pressure falls to 0.84 = 0.41, a decrease of 59 percent.
- Capillary rise. Water in a clean glass tube of radius 0.50 mm (θ ≈ 0°): h = 2γ cos θ ÷ ρgr = 2 × 0.0728 × 1 ÷ (1,000 × 9.8 × 5.0 × 10−4) = 0.1456 ÷ 4.9 = 0.030 m = 3.0 cm. In a tube of radius 0.10 mm, the rise is 5 times higher, 15 cm.
- Pressure in a drop. A water droplet of radius 1.0 mm has an excess pressure of 2γ ÷ r = 2 × 0.0728 ÷ 0.0010 = 146 Pa.
How were these properties studied?
Isaac Newton described viscosity in the Principia. Jean-Louis-Marie Poiseuille studied blood flow in capillary tubes in the 1840s. George Stokes derived the drag law in 1851, and Osborne Reynolds demonstrated the transition to turbulence with dye in a pipe in 1883. Surface tension was explained by Thomas Young and Pierre-Simon Laplace in the early 1800s. Ludwig Prandtl introduced the boundary layer concept in 1904.
Which mistakes are common?
Mistake 1: Thinking that thick liquids are always denser
Viscosity and density are different. Oil is less dense than water but more viscous.
Mistake 2: Applying Stokes’ law at high Reynolds number
It is valid for slow, laminar flow around small spheres (Re < 1).
Mistake 3: Mixing up radius and diameter in Poiseuille’s law
The flow depends on r4, so errors are strongly amplified.
Mistake 4: Thinking that surface tension acts only on thin films
It acts at any liquid surface, but its effects are noticeable at small scales.