Unit 8: Fluid Mechanics · Lesson 7
Viscosity and Turbulence in Real Fluids
What this lesson is for
- Students will understand viscosity and turbulence in real fluids
- Students will see God's design in physical laws
Viscosity and Turbulence in Real Fluids
Bernoulli's equation applies to ideal fluids — non-viscous and perfectly smooth-flowing. Real fluids, however, have internal friction called viscosity and can transition from orderly (laminar) to chaotic (turbulent) flow. Understanding these effects is essential for engineering blood vessels, pipelines, and aerodynamic surfaces.
Viscosity Defined
Viscosity (η) is a fluid's resistance to flow — its "thickness." It arises from intermolecular attractions between fluid layers moving at different speeds. The SI unit is the Pascal-second (Pa·s) or equivalently N·s/m².
Typical viscosities at room temperature:
- Water: η ≈ 0.001 Pa·s (1 mPa·s = 1 centipoise)
- Blood: η ≈ 0.003 Pa·s
- Motor oil (SAE 30): η ≈ 0.3 Pa·s
- Honey: η ≈ 10 Pa·s
Honey flows far more slowly than water because its sugar molecules form strong attractions that resist the relative sliding of layers. God designed each fluid with viscosity suited to its purpose — blood thick enough to carry cells but thin enough to flow easily through capillaries.
Poiseuille's Law: Flow in Pipes
For laminar flow of a viscous fluid in a cylindrical pipe:
Q = (π r⁴ ΔP) / (8 η L)
Where:
- Q = volumetric flow rate (m³/s)
- r = pipe radius (m)
- ΔP = pressure difference end-to-end (Pa)
- η = dynamic viscosity (Pa·s)
- L = pipe length (m)
The r⁴ dependence is critical: doubling the pipe radius increases flow rate by 16 times. This is why arteries narrowed by even 30% (plaque buildup) dramatically reduce blood flow — a 30% reduction in radius means flow drops to (0.7)⁴ ≈ 24% of normal. Engineers use this law to size water pipes, irrigation lines, and medical catheters.
Reynolds Number: Laminar vs Turbulent
The Reynolds number predicts whether flow is smooth (laminar) or chaotic (turbulent):
Re = ρvd / η
Where:
- ρ = fluid density (kg/m³)
- v = average velocity (m/s)
- d = pipe diameter (m)
- η = dynamic viscosity (Pa·s)
Interpretation:
- Re < 2300: Laminar flow — smooth, parallel streamlines, Poiseuille's law applies
- 2300 < Re < 4000: Transitional — unstable, can switch between states
- Re > 4000: Turbulent flow — chaotic, energy-wasting eddies, greater resistance
Example: Blood Flow in an Artery
A major artery has diameter d = 0.01 m (1 cm), blood velocity v = 0.5 m/s, ρ_blood = 1060 kg/m³, η_blood = 0.003 Pa·s.
Re = ρvd / η = (1060 × 0.5 × 0.01) / 0.003 = 5.3 / 0.003 ≈ 1767
Re < 2300, so blood flow in major arteries is laminar — exactly as designed for efficient, non-destructive flow. During vigorous exercise, increased velocity can push Re toward turbulent levels, which is why doctors can sometimes hear turbulent blood flow as a "bruit" with a stethoscope.
Practical Sizing of Straws and Pipes
The r⁴ dependence explains why:
- Drinking straws are sized to the minimum radius that allows comfortable sipping (too narrow → very high resistance)
- Municipal water mains are large (reduces Re and friction losses)
- Nasal passages use branching architecture to keep Re low in smaller branches
God's Engineering in Blood Vessels
The cardiovascular system — branching from the aorta through arterioles to capillaries — is a masterpiece of fluid engineering. Vessel diameters and branching angles are tuned to minimize the work the heart must do while ensuring every cell receives oxygen. Murray's Law (branching optimization) matches what engineers derive mathematically, yet it was designed by God before any human understood the equations.
Homework
Complete assigned practice problems. Show all work with units.
The first three of this lesson's 10 questions
Viscosity is best defined as:
Show the answer
Answer: A fluid's internal resistance to flow due to intermolecular friction
In Poiseuille's Law, if pipe radius doubles while all else stays constant, flow rate changes by a factor of:
Show the answer
Answer: 16
Blood has a viscosity of approximately:
Show the answer
Answer: 0.003 Pa·s
Another 7 questions follow in the lesson itself, each one graded the moment it is answered.