ME 310 Chap.5 The Bernoulli Equation
The Bernoulli Equation
Five lectures on one equation
The calendar devotes five lectures in the second half of September to the Bernoulli equation, and it is the last topic before the first exam, which covers the first three textbook chapters. The equation comes from Newton's second law applied along a streamline to a particle of fluid, with pressure and gravity kept and viscous forces neglected, in steady flow of constant density.
It states that pressure plus half density times speed squared plus density times gravity times elevation is constant along the streamline.
Heads and the energy picture
Dividing by specific weight turns each term into a height: pressure head, velocity head and elevation head.
Their sum is the total head, and plotting it along a flow path gives the energy line, while subtracting the velocity head gives the hydraulic grade line.
Where the hydraulic grade line dips below a pipe, the pressure there is below atmospheric, a warning sign for siphons and suction lines.
Across streamlines
Newton's law normal to a curved streamline requires a pressure gradient to supply centripetal acceleration, so pressure rises away from the centre of curvature.
Bernoulli may connect different streamlines only in irrotational flow; otherwise each streamline has its own constant.
Applications
Bringing a flow to rest at a stagnation point raises pressure by the dynamic pressure, which is how Pitot-static probes measure airspeed. A jet leaving a large tank reaches the speed of free fall from the surface height.
Pairing Bernoulli with mass conservation gives flow meters such as the Venturi tube.
Knowing when to stop
The equation fails when its assumptions fail: losses in long pipes, shaft work from pumps and turbines, unsteady flow, rotational flow across streamlines, and gas flows fast enough for density to change.
Predicted pressures below the vapour pressure signal cavitation rather than a real answer.
A note on units
Bernoulli can be written per unit volume, in pascals, or per unit weight, in metres of head. Choose one form for a whole solution; mixing a pressure in pascals with a head in metres is one of the commonest numerical slips.
What this chapter covers
- 01
Newton's law along a streamline
- 02
Pressure, velocity and elevation heads
- 03
Energy line and hydraulic grade line
- 04
Pressure gradient across curved streamlines
- 05
Stagnation, static and dynamic pressure
- 06
Free jets, Pitot tubes and Venturi meters
- 07
Assumptions, cavitation and limits
Worked example · free
Jet from a pressurised tank
- 2Choose points: the water surface (p = 50 kPa gage, V ≈ 0, z = 2.0 m) and the jet (p = 0 gage, z = 0).
- 2Bernoulli: 50,000/1000 + 9.81 × 2.0 = V²/2, so V²/2 = 69.6 m²/s².
- 1V = √139.2 = 11.8 m/s.
- 1Without the air pressure the speed would be √(2 × 9.81 × 2.0) = 6.3 m/s.
Key terms
- Stagnation Pressure
- Stagnation pressure is the pressure reached when a moving fluid is brought to rest without losses; it equals static plus dynamic pressure.
- Dynamic Pressure
- Dynamic pressure is half the density times the square of the flow speed, the pressure equivalent of the flow's kinetic energy.
- Hydraulic Grade Line
- The hydraulic grade line is the plot of pressure head plus elevation along a flow path, showing where pressure falls below atmospheric.
- Streamline
- A streamline is a line drawn tangent to the velocity at every point of a flow at one instant.
- Cavitation
- Cavitation is the local boiling of a liquid where its pressure falls below the vapour pressure, followed by damaging bubble collapse.
- Venturi Meter
- A Venturi meter is a flow-measuring constriction that infers flow rate from the pressure drop between the inlet and the throat.
The Bernoulli Equation FAQ
What are the assumptions behind the Bernoulli equation?
Steady flow, negligible viscous effects, constant density and application along a single streamline, or between streamlines if the flow is irrotational. There must also be no pump, turbine or other shaft work between the two points.
Why does a Pitot-static tube measure speed?
Its nose brings the approaching flow to rest and senses stagnation pressure, while side ports sense static pressure. The difference is the dynamic pressure, so the speed follows from the square root of twice that difference over the fluid density.
Is Bernoulli on Exam I?
The Fall 2026 calendar places all five Bernoulli lectures before the in-class Exam I on 30 September, which covers textbook chapters 1 to 3, and Bernoulli is textbook chapter 3. Confirm the scope announced in class.
Why does pressure drop where a pipe narrows?
Mass conservation forces the liquid to move faster through the smaller area. Along a streamline with no losses and no height change, the extra kinetic energy must come from pressure, so the static pressure falls in the constriction.
Exam move
Before writing Bernoulli, pick two points where you already know two of the three terms, and state the four assumptions beside the equation.
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