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Fluid Mechanics Bernoulli, Continuity, Reynolds Cheat Sheet
A printable reference covering Bernoulli’s equation, continuity, Reynolds number, head loss, friction factor, and pipe-flow design for grades 11-12.
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Fluid mechanics connects conservation laws to real engineering systems such as pipes, pumps, nozzles, and water networks. This cheat sheet focuses on the equations students use most often: continuity, Bernoulli’s equation, Reynolds number, and head loss. It helps students organize when each equation applies and how to connect pressure, velocity, elevation, and energy loss. These tools are essential for analyzing flow systems in engineering design problems. Continuity states that mass flow is conserved, so incompressible flow often uses A1v1 = A2v2. Bernoulli’s equation tracks mechanical energy per unit weight through pressure head, velocity head, and elevation head. Reynolds number predicts whether flow is likely laminar, transitional, or turbulent. Head loss and friction factor are then used to model real viscous effects in pipes.
Key Facts
- For steady incompressible flow, the continuity equation is A1v1 = A2v2.
- Volumetric flow rate is Q = Av, where Q is flow rate, A is cross-sectional area, and v is average velocity.
- Mass flow rate is m_dot = rho Q = rho Av, where rho is fluid density.
- Bernoulli’s equation for ideal steady incompressible flow is P/(rho g) + v^2/(2g) + z = constant.
- The pressure form of Bernoulli’s equation is P + 1/2 rho v^2 + rho g z = constant.
- Reynolds number for pipe flow is Re = rho v D / mu = v D / nu.
- Typical pipe-flow ranges are laminar when Re < 2300, transitional when 2300 <= Re <= 4000, and turbulent when Re > 4000.
- Darcy-Weisbach head loss is h_f = f(L/D)(v^2/(2g)), where f is the Darcy friction factor.
Vocabulary
- Continuity equation
- A conservation of mass equation stating that flow rate remains consistent through a steady flow path unless fluid is added or removed.
- Bernoulli’s equation
- An energy conservation equation that relates pressure head, velocity head, and elevation head along a streamline for ideal flow.
- Reynolds number
- A dimensionless number comparing inertial forces to viscous forces to predict whether flow is laminar or turbulent.
- Head loss
- The loss of mechanical energy per unit weight of fluid caused by friction, fittings, valves, or other flow resistance.
- Friction factor
- A dimensionless value used in pipe-flow equations to represent the resistance caused by wall friction.
- Hydraulic diameter
- An effective diameter used for noncircular flow passages, defined as D_h = 4A/P_wetted.
Common Mistakes to Avoid
- Using Bernoulli’s equation without checking assumptions is wrong because the simple form ignores pumps, turbines, heat transfer, and major losses from viscosity.
- Mixing pressure and head units is wrong because P, 1/2 rho v^2, and rho g z are pressure terms, while P/(rho g), v^2/(2g), and z are head terms.
- Using diameter instead of area in continuity is wrong because flow rate depends on A = pi D^2/4, so doubling diameter quadruples area.
- Forgetting fluid viscosity in Reynolds number is wrong because Re = rho v D / mu depends directly on dynamic viscosity and cannot be found from velocity alone.
- Using the wrong friction factor type is wrong because Darcy friction factor and Fanning friction factor differ by a factor of 4.
Practice Questions
- 1 Water flows through a pipe with diameter 0.10 m at an average speed of 2.0 m/s. Find the volumetric flow rate Q using Q = Av.
- 2 A pipe narrows from area 0.040 m^2 to area 0.010 m^2. If the inlet velocity is 1.5 m/s, find the outlet velocity for steady incompressible flow.
- 3 Water with rho = 1000 kg/m^3 and mu = 0.001 Pa s flows in a 0.050 m diameter pipe at 1.2 m/s. Calculate Re and classify the flow as laminar, transitional, or turbulent.
- 4 Explain why real pipe-flow design often needs both Bernoulli’s equation and a head-loss equation instead of Bernoulli’s equation alone.