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.
Understanding Fluid Mechanics Bernoulli, Continuity, Reynolds
A flow problem becomes easier when you first choose the two locations being compared. Mark each location on a sketch and list pipe diameter, height above a reference level, pressure, and average speed. Then decide whether the flow is steady.
Steady means the conditions at a fixed point do not change with time. For water in a normal pipe, density changes so little that it can usually be treated as constant.
Air needs more care. If air speed is high or pressure changes greatly, compression can change density, so the simple incompressible method may not be accurate.
Pipe size has a strong effect because area depends on diameter squared. A pipe with twice the diameter has four times the flow area. For a fixed volume flow rate, that larger area gives a much lower average speed.
This is why wide supply pipes can carry large amounts of water without extreme friction loss. A nozzle does the opposite. It reduces area, raises speed, and converts some pressure energy into kinetic energy.
The pressure drop in a narrowing section does not mean energy has vanished. In an ideal model, the energy has changed form. Real fluids lose part of it to internal friction.
Bernoulli calculations require careful choices about what is known. Gauge pressure is often easiest because atmospheric pressure appears on both sides when both points are exposed to the same air. A tank open to the atmosphere is a common example.
The water surface has nearly zero gauge pressure and, if the tank is much wider than the outlet, nearly zero surface speed. Its height can then explain the outlet speed.
This idea appears in fountains, fuel systems, spray nozzles, and aircraft pitot tubes. Do not apply the ideal relation across a pump or turbine without adding the energy supplied or removed by that device.
Viscosity is the reason actual pipe systems need more than an ideal energy balance. Near a solid wall, fluid sticks to the surface and its speed falls to zero. A velocity gradient forms between the wall and the faster fluid near the pipe center.
In laminar flow, fluid moves in smooth layers and the profile is strongly curved. In turbulent flow, swirling motion mixes momentum across the pipe, creating much larger losses. Roughness matters most in turbulent flow because wall bumps disturb the moving fluid.
Engineers use pipe length, diameter, roughness, fittings, valves, and bends to estimate total loss. Minor losses from fittings can be important when a system has short pipes or many turns. Check units throughout, use the inside diameter, and remember that average speed is not the maximum speed inside the pipe.