Chapter Review
Fluid Mechanics
Fluid Flow and Viscosity · Bernoulli's Applications
Flow Types & Reynolds Number
Fluid flow is classified as laminar (smooth, parallel streamlines) or turbulent (chaotic eddies) based on the dimensionless Reynolds number.
Key Points
- •Laminar flow: Re < 2000 — particles follow stable, predictable paths
- •Turbulent flow: Re > 3000 — chaotic mixing with unpredictable particle trajectories
- •Transition region exists between Re 2000–3000
- •Ideal fluid assumptions: incompressible, non-viscous, irrotational, steady flow
- •Streamlines never cross in steady flow; crowded streamlines indicate higher speed
Formula
$$R_e = \frac{\rho v D}{\eta}$$
Equation of Continuity
For an incompressible fluid in steady flow, the volume flow rate (Av) is constant at every cross-section — a direct consequence of conservation of mass.
Key Points
- •Velocity is inversely proportional to cross-sectional area: $v \propto 1/A$
- •For circular pipes: $v \propto 1/r^2$, so halving the diameter quadruples the speed
- •A falling water stream narrows because gravity increases v, forcing A to decrease
- •Always convert diameter to area before applying continuity
Formula
$$A_1 v_1 = A_2 v_2$$
Bernoulli's Equation
An expression of energy conservation for ideal fluids — the sum of static pressure, dynamic pressure, and hydrostatic pressure remains constant along a streamline.
Key Points
- •Each term has units of pressure (Pa) and represents energy per unit volume
- •At constant height: higher speed → lower pressure (core insight for all applications)
- •At constant speed: reduces to hydrostatic equation $P + \rho gh = \text{const}$
- •Valid only along a streamline for incompressible, non-viscous, steady flow
- •Pressure difference for horizontal flow: $\Delta P = \frac{1}{2}\rho(v_2^2 - v_1^2)$
Formula
$$P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$$
Viscosity & Stokes' Law
Viscosity is internal friction between fluid layers. Stokes' Law gives the drag force on a small sphere moving slowly through a viscous fluid under laminar conditions.
Key Points
- •SI unit of viscosity: Pa·s (= kg·m⁻¹·s⁻¹)
- •Viscosity decreases with temperature in liquids but increases in gases
- •Stokes' drag is proportional to $r$ (not $r^2$) and to $v$ (not $v^2$)
- •Valid only at low Reynolds numbers (laminar regime around the sphere)
- •At high speeds, Stokes' Law breaks down — drag becomes proportional to $v^2$
Formula
$$F_d = 6\pi \eta r v$$
Terminal Velocity
The constant speed reached when the net force on a falling object is zero — weight exactly balanced by drag and buoyancy.
Key Points
- •$v_t \propto r^2$ — doubling radius quadruples terminal velocity
- •$v_t \propto (\rho_s - \rho_f)$ — denser sphere relative to fluid falls faster
- •$v_t \propto 1/\eta$ — more viscous fluid means slower terminal velocity
- •Neutral buoyancy ($\rho_s = \rho_f$) gives $v_t = 0$
- •Object asymptotically approaches $v_t$ — never truly reaches it exactly
- •Always convert radius to metres before substituting (mm → m common error)
Formula
$$v_t = \frac{2r^2(\rho_s - \rho_f)g}{9\eta}$$
Torricelli's Theorem
The speed of efflux from a hole in a tank equals the speed of an object falling freely from the fluid surface to the hole's depth.
Key Points
- •Efflux speed is independent of the liquid's density — $\rho$ cancels out
- •Atmospheric pressure cancels (both surface and exit are at $P_0$)
- •$v \propto \sqrt{h}$ — doubling depth increases speed by factor $\sqrt{2}$, not 2
- •For pressurized tanks: $v = \sqrt{2gh + 2(P_{top} - P_{atm})/\rho}$
- •Maximum horizontal range when orifice is at half the tank height
Formula
$$v = \sqrt{2gh}$$
Venturi Effect & Meter
Fluid flowing through a constriction speeds up (continuity) and its pressure drops (Bernoulli). The Venturi meter exploits this to measure flow rate.
Key Points
- •Speed ratio from continuity: $v_2/v_1 = (d_1/d_2)^2$ for circular pipes
- •Manometer between wide and narrow sections reads the pressure difference directly
- •Applications: carburetors (fuel suction), aspirators/filter pumps, atomizers
- •Common error: using diameter as radius when calculating area
Dynamic Lift & Magnus Effect
An aerofoil generates lift because air moves faster over its curved top surface (lower pressure) than its flatter bottom surface (higher pressure).
Key Points
- •Lift force: $F_L = \frac{1}{2}\rho(v_{top}^2 - v_{bottom}^2) \times A$
- •Magnus effect: a spinning ball creates asymmetric airflow, producing lateral 'swing'
- •Chimney draft enhanced by wind blowing across the top (low pressure)
- •Atomizers/sprayers use fast airflow over a tube to draw liquid upward
Formula
$$F_L = \frac{1}{2}\rho(v_{top}^2 - v_{bottom}^2) \times A$$
Pitot Tube & Blood Pressure
A Pitot tube measures flow velocity from the difference between stagnation pressure and static pressure. Blood pressure measurement similarly relies on detecting flow changes in a compressed artery.
Key Points
- •Stagnation point: where $v = 0$ and all kinetic energy converts to pressure
- •Pitot tubes on aircraft measure airspeed via ram pressure
- •Systolic pressure (~120 torr): peak during heartbeat; diastolic (~75–80 torr): minimum between beats
- •Blood slows dramatically in capillaries due to enormous total cross-sectional area (continuity)
Formula
$$v = \sqrt{\frac{2(P_{stag} - P_{static})}{\rho}}$$
Formulas
Reynolds Number
Dimensionless number predicting laminar vs turbulent flow regime.
Formula
$R_e = \frac{\rho v D}{\eta}$
Equation of Continuity
Volume flow rate is constant in steady incompressible flow.
Formula
$A_1 v_1 = A_2 v_2$
Bernoulli's Equation
Energy conservation per unit volume along a streamline.
Formula
$P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$
Stokes' Law
Viscous drag on a slow-moving sphere in laminar flow.
Formula
$F_d = 6\pi \eta r v$
Terminal Velocity
Constant falling speed when weight equals drag plus buoyancy.
Formula
$v_t = \frac{2r^2(\rho_s - \rho_f)g}{9\eta}$
Torricelli's Theorem
Efflux speed equals free-fall velocity from depth h.
Formula
$v = \sqrt{2gh}$
Dynamic Lift Force
Net upward force from asymmetric airflow over a wing.
Formula
$F_L = \frac{1}{2}\rho(v_{top}^2 - v_{bottom}^2) \times A$
Pitot Tube Velocity
Flow speed from stagnation vs static pressure difference.
Formula
$v = \sqrt{\frac{2\Delta P}{\rho}}$