Fluid Dynamics
Fluid dynamics studies how liquids and gases flow under the action of forces. The PMDC MDCAT 2026 syllabus expects you to understand the equation of continuity, Bernoulli's principle, viscosity-driven drag, terminal velocity, and the difference between laminar and turbulent flow. Expect 1-2 MCQs from this chapter.
Equation of Continuity
For an incompressible, non-viscous fluid flowing steadily through a pipe of varying cross-section, the mass flowing per second past every point is the same. Mathematically:
A1v1 = A2v2 (Av = constant)
Where A is the cross-sectional area and v is the speed of the fluid. This is a direct consequence of conservation of mass. Wherever the pipe narrows, the fluid must speed up; wherever it widens, the fluid slows down. The product Av is called the volume flow rate (m³/s).
Worked example
Water flows through a pipe whose radius narrows from 4 cm to 2 cm. If v1 = 2 m/s, then v2 = (A1/A2)v1 = (16/4)(2) = 8 m/s. Speed quadruples because area scales with r².
Bernoulli's Equation
For an ideal (incompressible, non-viscous, streamline) fluid, the sum of pressure energy, kinetic energy and potential energy per unit volume is constant along a streamline:
P + ½ρv² + ρgh = constant
- P
- Static pressure of the fluid (Pa).
- ½ρv²
- Dynamic pressure due to fluid motion.
- ρgh
- Hydrostatic pressure due to height h above a reference level.
Direct consequence: where speed is high, pressure is low. This single statement explains aeroplane lift, the lift of a roof in a storm, the action of an atomiser/spray gun, the carburettor, and the Venturi meter.
Applications
- Aeroplane lift: aerofoil shape forces faster flow over the top → lower pressure on top → upward lift.
- Atomiser / paint spray: high-speed air across a tube tip lowers the pressure there, sucking liquid up the tube where it shears into droplets.
- Venturi meter: measures flow rate by reading the pressure drop across a constriction.
- Curving cricket ball / Magnus effect: spin produces unequal flow speeds on the two sides.
Fluid Flow (Laminar, Turbulent)
Two distinct flow regimes occur in real fluids:
| Property | Laminar / streamline | Turbulent |
|---|---|---|
| Motion | Smooth, parallel layers | Chaotic, eddies and vortices |
| Velocity at a point | Constant in time | Fluctuates rapidly |
| Mixing between layers | None | Strong |
| Energy loss | Low | High (dissipated as heat) |
| Reynolds number Re = ρvd/η | < 1000 (transition 1000–2000) | > 2000 |
| Bernoulli applies? | Yes (along a streamline) | No |
| Examples | Slow flow in narrow tubes, blood in capillaries | Smoke from chimney, water from open tap, blood in aorta |
Fluid Drag
When a body moves through a fluid, it experiences a retarding force called drag. For a small spherical object moving slowly through a viscous fluid, Stokes' law gives the drag:
F = 6πηrv
where η is the coefficient of viscosity (Pa·s), r is the radius of the sphere, and v is its speed relative to the fluid. Drag rises linearly with speed at low Re, but for fast/large objects (turbulent regime) drag rises with v².
Coefficient of viscosity (η)
Viscosity is the internal friction of a fluid — resistance to shear. SI unit: Pa·s (or N·s/m²). For liquids viscosity decreases with temperature (e.g. honey thins on heating); for gases it increases with temperature.
Terminal Velocity
A body falling through a viscous fluid eventually reaches a constant speed when the net force on it is zero. At this point the weight is balanced by buoyancy plus the viscous drag. For a sphere of radius r, density ρ in a fluid of density σ:
vt = 2r²(ρ − σ)g / (9η)
Key features:
- vt increases with r² — larger droplets fall much faster.
- vt increases with the density difference (ρ − σ).
- vt decreases as viscosity η increases.
- If ρ = σ the body floats; if ρ < σ it rises (e.g. air bubble in water).
A raindrop reaches terminal velocity within metres of falling. For a 2 mm drop, vt ≈ 6–9 m/s. Without air drag a raindrop falling from 1 km would strike at >140 m/s.
Worked MCQs
Five MCQs that capture the high-yield testing patterns for this chapter.
Q1. Water flows through a pipe whose radius reduces from 4 cm to 2 cm. If the speed in the wider section is 1 m/s, the speed in the narrow section is:
By continuity A1v1 = A2v2. Areas scale with r², so A1/A2 = (4/2)² = 4, giving v2 = 4 × 1 = 4 m/s.
Q2. Bernoulli's equation is a statement of conservation of:
P + ½ρv² + ρgh = constant expresses conservation of mechanical energy per unit volume of an ideal fluid along a streamline. Continuity, by contrast, is conservation of mass.
Q3. Stokes' drag on a small sphere of radius r moving slowly with speed v through a fluid of viscosity η is:
Stokes' law: F = 6πηrv. Linear in radius and in speed; valid only at low Reynolds numbers (laminar regime).
Q4. The terminal velocity of a sphere falling through a viscous medium is doubled if:
vt ∝ r². To double vt we need r² to double, i.e. r becomes √2 times its original value.
Q5. Which condition is NOT required for Bernoulli's equation to be applied?
Bernoulli's equation requires laminar (streamline) flow, not turbulent. The other three conditions are essential assumptions of the derivation.
Quick Recap
- Continuity: A1v1 = A2v2 (mass conservation).
- Bernoulli: P + ½ρv² + ρgh = constant (energy conservation).
- Higher speed → lower pressure (lift, atomiser, Venturi).
- Stokes' drag F = 6πηrv at low Re.
- Terminal velocity vt = 2r²(ρ − σ)g/(9η).
- Re < 1000 laminar; Re > 2000 turbulent.