Bernoulli's equation is one of the most useful energy ideas in fluid mechanics. It connects pressure, speed, and height for fluid moving steadily along a streamline. Engineers use it to estimate how fluids behave in pipes, nozzles, ducts, wings, and measuring devices.
The main idea is that mechanical energy can shift between pressure energy, kinetic energy, and gravitational potential energy.
Key Facts
- Bernoulli's equation: P + 1/2 rho v^2 + rho g h = constant along a streamline.
- Between two points: P1 + 1/2 rho v1^2 + rho g h1 = P2 + 1/2 rho v2^2 + rho g h2.
- Pressure term P represents fluid pressure energy per unit volume, measured in pascals.
- Kinetic term 1/2 rho v^2 increases when fluid speed increases.
- Elevation term rho g h increases when the fluid is higher in a gravitational field.
- Continuity for incompressible flow: A1 v1 = A2 v2, so a smaller pipe area usually means faster flow.
Vocabulary
- Pressure
- Pressure is the force exerted per unit area by a fluid on its surroundings.
- Streamline
- A streamline is a path that is everywhere tangent to the fluid velocity at each point.
- Incompressible flow
- Incompressible flow means the fluid density stays nearly constant as it moves.
- Dynamic pressure
- Dynamic pressure is the kinetic energy per unit volume of a moving fluid, equal to 1/2 rho v^2.
- Venturi effect
- The Venturi effect is the pressure drop that occurs when a fluid speeds up through a narrow section of a pipe.
Common Mistakes to Avoid
- Using Bernoulli's equation when flow is highly viscous or turbulent is wrong because energy is then lost to friction and mixing, so the ideal constant-energy form does not apply well.
- Forgetting the elevation term rho g h is wrong when the pipe changes height because gravitational potential energy affects pressure and speed.
- Assuming higher speed means higher pressure is wrong in ideal horizontal flow because increasing velocity usually corresponds to decreasing static pressure.
- Mixing gauge pressure and absolute pressure in the same equation is wrong because all pressure values must use the same reference level.
Practice Questions
- 1 Water flows through a horizontal pipe that narrows from area 0.020 m^2 to 0.0050 m^2. If the speed in the wide section is 1.5 m/s, what is the speed in the narrow section?
- 2 In a horizontal pipe, water with density 1000 kg/m^3 speeds up from 2.0 m/s to 6.0 m/s. If the pressure before the narrowing is 150000 Pa, what is the pressure in the narrow section, ignoring losses?
- 3 A fluid moves through a pipe that rises upward while also narrowing. Explain how pressure, speed, and height can trade energy according to Bernoulli's equation.