Chapter Review
Fluid Dynamics
Viscosity, Fluid Drag and Terminal Velocity · Continuity Equation and Bernoulli Equation
Laminar vs Turbulent Flow
Fluid flow is classified as laminar (streamline) or turbulent based on the Reynolds number — a dimensionless quantity that predicts flow regime transitions.
Key Points
- •Laminar flow: smooth, parallel streamlines that never cross; Re < 2000
- •Turbulent flow: chaotic eddies and unpredictable particle paths; Re > 3000
- •Transition region (2000 < Re < 3000): flow is unstable and can switch regimes
- •Re = ρvD/η — all units cancel, making it dimensionless
- •Higher velocity or lower viscosity increases Re, promoting turbulence
Formula
$$R_e = \frac{\rho v D}{\eta}$$
Viscosity
Viscosity (η) is the internal friction between adjacent fluid layers — it quantifies resistance to flow.
Key Points
- •SI unit: Pa·s (equivalent to kg·m⁻¹·s⁻¹, N·s·m⁻²)
- •In liquids, viscosity decreases with temperature (molecules move apart)
- •In gases, viscosity increases with temperature (more molecular collisions)
- •Water: η ≈ 0.8 × 10⁻³ Pa·s; Glycerin: η ≈ 630 × 10⁻³ Pa·s
- •Ideal fluids are assumed non-viscous; real fluids always have some viscosity
Stokes' Law
Stokes' Law gives the viscous drag force on a small sphere moving slowly through a fluid under laminar conditions.
Key Points
- •F_d = 6πηrv — drag is linear in both radius and velocity (not r² or v²)
- •Valid only for laminar flow (low Reynolds number) around smooth spheres
- •Drag is directly proportional to viscosity: more viscous fluid → greater drag
- •At high speeds, Stokes' Law breaks down — drag becomes proportional to v²
- •Dimensional check: Pa·smm/s = N ✓
Formula
$$F_d = 6\pi \eta r v$$
Terminal Velocity
Terminal velocity is the constant speed reached when weight is exactly balanced by Stokes' drag plus buoyancy — net force and acceleration become zero.
Key Points
- •Derived from: W = F_d + F_b → ⁴⁄₃πr³ρ_sg = 6πηrv_t + ⁴⁄₃πr³ρ_fg
- •v_t ∝ r² — doubling radius quadruples terminal velocity
- •v_t ∝ (ρ_s − ρ_f) — denser sphere (relative to fluid) falls faster
- •v_t ∝ 1/η — more viscous fluid → slower terminal velocity
- •When ρ_s = ρ_f (neutral buoyancy): v_t = 0
- •Always convert radius to metres before substituting — errors scale by 10⁶
Formula
$$v_t = \frac{2r^2(\rho_s - \rho_f)g}{9\eta}$$
Equation of Continuity
The equation of continuity expresses conservation of mass for an incompressible fluid — the volume flow rate Av remains constant at every cross-section.
Key Points
- •A₁v₁ = A₂v₂ — product of area and velocity is constant throughout the pipe
- •v ∝ 1/A: smaller cross-section means faster flow
- •For circular pipes: A = πr², so v ∝ 1/r² (halving diameter quadruples speed)
- •A falling water stream narrows as gravity increases speed, so area must decrease
- •Applies only to incompressible, steady flow
Formula
$$A_1 v_1 = A_2 v_2$$
Bernoulli's Equation
Bernoulli's equation is a statement of energy conservation per unit volume along a streamline for ideal fluids — static pressure, dynamic pressure, and hydrostatic pressure sum to a constant.
Key Points
- •Three assumptions: incompressible, non-viscous, steady (laminar) flow
- •Each term has units of pressure (Pa) = energy per unit volume
- •P term = internal energy, ½ρv² = kinetic energy per unit volume, ρgh = gravitational PE per unit volume
- •Horizontal flow (h₁ = h₂): P + ½ρv² = constant — pressure and speed trade off inversely
- •Static fluid (v = 0): P + ρgh = constant — reduces to hydrostatic equation
- •Does NOT apply to viscous fluids or across streamlines
Formula
$$P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$$
Torricelli's Theorem
The speed of efflux from a tank orifice equals the free-fall velocity from the fluid surface to the hole depth — a direct consequence of Bernoulli's equation.
Key Points
- •v = √(2gh) — independent of the liquid's density (ρ cancels out)
- •Both surface and orifice are at atmospheric pressure P₀, so P₀ cancels
- •v ∝ √h: doubling depth increases speed by √2 ≈ 1.41, not 2
- •Pressurized tank (P_top > P_atm): v = √(2gh + 2(P_top − P_atm)/ρ)
- •Maximum horizontal range when hole is at half the tank height (y = H/2)
Formula
$$v = \sqrt{2gh}$$
Venturi Effect
When fluid flows through a constriction, continuity forces it to speed up and Bernoulli's equation causes pressure to drop — this is the Venturi effect.
Key Points
- •Pressure drop in throat: ΔP = ½ρ(v₂² − v₁²) where v₂ > v₁
- •Combined with continuity: v₂ = (A₁/A₂)v₁ to express everything in terms of v₁
- •For circular pipes: speed ratio = (d₁/d₂)² — diameter ratio squared
- •Used in Venturi meters, carburetors, aspirators, and spray bottles
- •Manometer between wide and narrow sections reads the pressure difference directly
Formula
$$P_1 - P_2 = \frac{1}{2}\rho(v_2^2 - v_1^2)$$
Dynamic Lift and Magnus Effect
Dynamic lift arises from asymmetric airflow over surfaces — faster flow over one side creates lower pressure, producing a net force. The Magnus effect is the lateral deflection of spinning objects due to the same principle.
Key Points
- •Aerofoil: curved upper surface → air moves faster above → lower pressure → net upward lift
- •F_lift ∝ (v_top² − v_bottom²) × A — depends on squared speed difference times wing area
- •Magnus effect: spinning ball drags air, adding speed on one side and subtracting on the other → lateral swing
- •Applications: airplane wings, chimney draft, atomizers/sprayers, spinning cricket balls
- •Pitot tube measures velocity from stagnation pressure vs static pressure: v = √(2ΔP/ρ)
Formula
$$F_{lift} = \frac{1}{2}\rho(v_{top}^2 - v_{bottom}^2) \times A$$
Formulas
Reynolds Number
Dimensionless predictor of laminar vs turbulent flow regime.
Formula
$R_e = \frac{\rho v D}{\eta}$
Stokes' Law
Viscous drag on a slow sphere through a fluid.
Formula
$F_d = 6\pi \eta r v$
Terminal Velocity
Constant falling speed when drag + buoyancy balance weight.
Formula
$v_t = \frac{2r^2(\rho_s - \rho_f)g}{9\eta}$
Equation of Continuity
Volume flow rate is constant for incompressible steady flow.
Formula
$A_1 v_1 = A_2 v_2$
Bernoulli's Equation
Energy conservation per unit volume along a streamline.
Formula
$P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$
Torricelli's Efflux
Exit speed from an orifice at depth h below the surface.
Formula
$v = \sqrt{2gh}$
Venturi Pressure Drop
Pressure difference between wide and narrow pipe sections.
Formula
$P_1 - P_2 = \frac{1}{2}\rho(v_2^2 - v_1^2)$
Dynamic Lift
Upward force from pressure difference across an aerofoil.
Formula
$F_{lift} = \frac{1}{2}\rho(v_{top}^2 - v_{bottom}^2) \times A$