Fluid Dynamics
High-Yield Summary
- Viscosity is a fluid's internal resistance to flow (Pa·s). Low viscosity flows freely (water); high viscosity resists motion (honey).
- Laminar flow: smooth, parallel layers, fastest at the center of a pipe. Governed by Poiseuille's Law — flow rate ∝ r⁴, ∝ pressure gradient, ∝ 1/viscosity, ∝ 1/length.
- Turbulent flow: chaotic mixing once fluid speed exceeds a critical speed, tied to the Reynolds number (low = laminar, high = turbulent).
- Continuity equation (A1v1 = A2v2): fluid speeds up when a pipe narrows, to keep mass flow rate constant.
- Bernoulli's principle (P + ½ρv² + ρgh = constant): faster-moving fluid has lower pressure — the basis of the Venturi effect.
Continuity and Bernoulli's Equation
A1v1 = A2v2 | P + ½ρv² + ρgh = constant
- A = Cross-sectional area of the pipe
- v = Fluid speed
- P = Pressure energy
- ½ρv² = Kinetic energy term
- ρgh = Potential energy term (height-dependent)
- Applies to incompressible, non-viscous fluid flowing along a streamline.
Poiseuille's Law and Critical Speed
Flow rate ∝ r⁴ · ΔP / (η · L) | vc = (NR · η) / (ρ · D)
- r = Pipe radius
- ΔP = Pressure gradient along the pipe
- η = Fluid viscosity
- L = Pipe length
- vc = Critical speed — threshold for turbulence onset
- NR = Reynolds number (dimensionless)
- D = Pipe/channel diameter
- Doubling pipe radius increases flow rate 16-fold (2⁴) — why vessel diameter matters so much clinically.
Laminar vs. Turbulent Flow
| Laminar Flow | Turbulent Flow |
|---|---|
| Smooth, parallel layers/streamlines | Chaotic, irregular, mixing streamlines |
| Parabolic velocity profile (fastest at center) | Irregular, unpredictable velocity profile |
| Low Reynolds number | High Reynolds number |
| Low viscosity fluid or slow speed, below critical speed | Higher velocity or lower viscosity, above critical speed |
Common MCAT Trap
- Poiseuille's Law's r⁴ dependence is extreme — a small vessel narrowing (e.g., plaque buildup) causes a disproportionately large drop in flow, not a proportional one.
- Fluid velocity is always tangent to a streamline — particles never cross from one streamline to another, even in a curving channel.
- Even in turbulent flow, a thin laminar boundary layer persists right at the pipe surface/obstacle.
Quick Recall
If a pipe's radius doubles, what happens to flow rate (Poiseuille's Law)?
Per Bernoulli's principle, what happens to pressure when fluid speeds up through a narrowing (Venturi effect)?