8.5 Fluid Dynamics: Bernoulli's Equation

Bernoulli's Equation

For an ideal, incompressible fluid in steady flow along a streamline, P + ½ρv² + ρgh = constant. This links pressure (P), kinetic energy density (½ρv²), and gravitational potential energy density (ρgh) at any two points in the flow. It's essentially conservation of energy applied to fluids, ignoring viscosity and turbulence.

Speed–Pressure Tradeoff

At the same height, Bernoulli's equation shows that where a fluid moves faster, pressure is lower, and where it moves slower, pressure is higher. This explains lift on airplane wings and why fluid speeds up through a narrow pipe section (from continuity) while its pressure drops there.

Applying the Equation

Combine Bernoulli's equation with the continuity equation (A₁v₁ = A₂v₂) to solve problems with changing pipe area. Common setups: fluid exiting a tank (Torricelli's law), flow through pipes of varying diameter, and comparing pressure/speed at two points connected by a streamline.

Key terms

Bernoulli's Equation

P + ½ρv² + ρgh = constant along a streamline; expresses conservation of energy for an ideal fluid.

Continuity Equation

A₁v₁ = A₂v₂; used with Bernoulli's equation to relate speed and pipe cross-sectional area.

Speed-Pressure Relationship

At constant height, faster fluid flow corresponds to lower pressure, and slower flow corresponds to higher pressure.

Ideal Fluid Assumptions

Incompressible, non-viscous, and steady (laminar) flow along a streamline — required for Bernoulli's equation to hold.

Application Example

Airplane wing lift: air moves faster over the curved top, lowering pressure there relative to below, creating net upward force.

Try an AP-style question

According to Bernoulli's principle, where fluid speed increases along a horizontal pipe, the pressure…

1. becomes zero
2. stays the same
3. decreases
4. increases