§14.6 — Viscosity and Stokes’ Law

Real fluids do not flow without resistance. Viscosity is the property that describes how a fluid opposes motion, analogous to friction in solids. It arises from intermolecular forces and the internal structure of the fluid. Viscosity causes energy dissipation, converts kinetic energy to heat, and produces a drag force on objects moving through fluids.

Viscosity and shear stress

When a layer of fluid slides past an adjacent layer, there is a tangential force resisting the relative motion—this is viscous drag. For simple Newtonian fluids (like water, oil, and air), the shear stress is proportional to the velocity gradient:

\[ \tau = \eta \frac{dv}{dy} \]

where:

• \( \tau \) is the shear stress (force per unit area parallel to the flow)
• \( \eta \) is the dynamic viscosity (SI units: Pa·s or kg/(m·s))
• \( \frac{dv}{dy} \) is the velocity gradient perpendicular to the flow direction

The larger the velocity gradient, the greater the shear stress. A more viscous fluid (larger \( \eta \)) exerts more resistance for the same gradient.

Common viscosity values

Fluid	Dynamic Viscosity (\( \eta \))	Temperature
Air	1.8 × 10⁻⁵ Pa·s	20°C
Water	1.0 × 10⁻³ Pa·s	20°C
Olive oil	0.08 Pa·s	20°C
Honey	2 Pa·s	20°C
Glycerin	1.5 Pa·s	20°C
Mercury	1.5 × 10⁻³ Pa·s	20°C

Water is about 100 times more viscous than air, and honey is roughly 1000 times more viscous than water.

Kinematic viscosity is the ratio of dynamic viscosity to density:

\[ \nu = \frac{\eta}{\rho} \]

(SI units: m²/s). This is sometimes more relevant for fluid dynamics because it combines inertial and viscous effects.

Stokes' law: drag on a sphere

One of the most important results in fluid dynamics is Stokes' law, which gives the drag force on a small sphere moving slowly through a viscous fluid:

\[ F_D = 6 \pi \eta r v \]

where:

• \( \eta \) is the dynamic viscosity
• \( r \) is the sphere's radius
• \( v \) is the velocity of the sphere relative to the fluid

This formula is valid for laminar flow at low Reynolds numbers (Re < 1, roughly \( v < 0.01 \) m/s for water around small objects).

Key insight: drag force is proportional to velocity (not velocity squared as in air resistance at high speeds). This linear dependence allows straightforward calculations and is why viscous effects dominate at small scales.

Worked example: Falling ball in oil

A steel sphere of radius \( r = 2 \) mm (density \( \rho_s = 7800 \) kg/m³) falls through oil (density \( \rho_o = 900 \) kg/m³, viscosity \( \eta = 0.1 \) Pa·s). What is the terminal velocity (constant velocity when drag balances gravity)?

Step 1: Calculate forces

Weight: \[ W = m_s g = \rho_s V_s g = 7800 \times \frac{4}{3}\pi (0.002)^3 \times 9.8 = 0.0164 \text{ N} \]

Buoyancy (from §14.3): \[ F_B = \rho_o V_s g = 900 \times \frac{4}{3}\pi (0.002)^3 \times 9.8 = 0.00188 \text{ N} \]

Net downward force before drag: \[ F_{\text{net}} = W - F_B = 0.0164 - 0.00188 = 0.0145 \text{ N} \]

Step 2: Apply Stokes' law at terminal velocity

At terminal velocity, the drag force equals the net downward force: \[ F_D = F_{\text{net}} \] \[ 6 \pi \eta r v_t = 0.0145 \] \[ 6 \pi \times 0.1 \times 0.002 \times v_t = 0.0145 \] \[ 0.00377 \, v_t = 0.0145 \] \[ v_t = 3.85 \text{ m/s} \]

The sphere reaches a terminal velocity of about 3.85 m/s. After this speed is attained, acceleration ceases and the ball falls at constant velocity.

Poiseuille's law: flow through a pipe

For laminar flow through a cylindrical pipe, Poiseuille's law gives the volume flow rate:

\[ Q = \frac{\pi r^4 \Delta P}{8 \eta L} \]

where:

• \( r \) is the pipe radius
• \( \Delta P \) is the pressure difference across the pipe
• \( L \) is the pipe length
• \( \eta \) is the fluid viscosity

Notice the fourth power of radius: halving the radius reduces flow rate by a factor of 16. This is why clogged arteries (reduced diameter) dramatically reduce blood flow. Also, flow rate is inversely proportional to viscosity: thicker fluids flow more slowly.

Reynolds number and the transition to turbulence

The Reynolds number is a dimensionless quantity that predicts whether flow is laminar (orderly, no mixing) or turbulent (chaotic, with vortices and mixing):

\[ \text{Re} = \frac{\rho v D}{\eta} = \frac{v D}{\nu} \]

where \( D \) is a characteristic length (e.g., pipe diameter).

• Re < 1: Stokes flow (linear drag)
• Re ≈ 1–1000: Laminar flow (nonlinear drag, but organized)
• Re > 1000: Transition and turbulent flow (quadratic drag, chaotic)

For example, a sphere falling through water at 1 m/s (D ≈ 0.01 m) has Re ≈ 10,000, deep in the turbulent regime. Stokes' law no longer applies; instead, drag follows \( F_D \propto v^2 \).

Energy dissipation in viscous flow

The power dissipated (energy per unit time) by viscous forces is:

\[ P = F_D \times v = \eta A \frac{dv}{dy} \times v \]

For Stokes flow: \[ P = 6 \pi \eta r v^2 \]

This energy goes into heat, warming the fluid. This is why stirring thick honey generates noticeable heat, and why internal friction in engines dissipates energy.

Real-world applications

Lubrication: Oil reduces friction between moving parts because its viscosity creates a thin film that prevents metal-on-metal contact.

Blood flow: Blood viscosity and artery diameter are critical; narrower vessels or thicker blood (high hematocrit) increase resistance, raising blood pressure.

Microfluidics: At very small scales (Re << 1), inertia is negligible and viscous effects dominate. Devices can use Stokes flow to separate particles or mix fluids with no turbulence.

Sedimentation: Particles settling in a fluid reach terminal velocity when their weight (minus buoyancy) equals viscous drag. Centrifuges speed up this process by increasing the effective "gravity."

What you should be able to do

• Define dynamic and kinematic viscosity
• Apply Stokes' law to find drag force on a sphere
• Calculate terminal velocity for objects falling through viscous media
• Use Poiseuille's law to relate pressure drop, flow rate, and viscosity
• Compute Reynolds number and predict flow regime (laminar vs turbulent)
• Explain energy dissipation and heat generation in viscous flow
• Recognize practical consequences in medicine, engineering, and everyday situations

Preview of §14.7

Turbulence and complex fluid flow: beyond Stokes' linear regime, chaos, vortices, and self-similar structures.

📚 See also: Halliday Vol 2, Ch 14, §14.6.

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