Bernoulli's equation
Summary
Bernoulli's equation is conservation of energy for an ideal fluid (incompressible, nonviscous, steady flow): P + ½ρv² + ρgh is constant along a streamline. At the same height, faster flow means lower pressure, as in a narrowed pipe.
Detail
Meaning: For an ideal fluid, pressure does work that changes the fluid's kinetic and gravitational potential energy. So the sum of pressure, kinetic energy per volume and potential energy per volume is the same at any two points on a streamline. With the continuity equation (A₁v₁ = A₂v₂), a narrow section has higher speed, so its pressure is lower if the height is unchanged.
Equation: P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂, where P is pressure (Pa), ρ is fluid density (kg/m³), v is speed (m/s), g is gravitational acceleration (9.8 m/s²), and h is height (m).
On the MCAT: Typical setups are a Venturi tube, a narrowed artery, an airplane wing, or fluid draining through a hole in a tank (v = √(2gh)). You compare pressure or speed at two points of different width or height. In a horizontal pipe, P + ½ρv² is constant. If v = 0, it reduces to P₂ = P₁ + ρgΔh for a static fluid.
Common confusion: Students expect a constriction to raise pressure because the fluid is squeezed. Speed rises and pressure falls, because energy shifts from pressure to kinetic energy. The equation also ignores viscosity. In real flow, Poiseuille's law applies, and pressure drops along a uniform tube even though speed stays constant.
Sources
- Physics (Cutnell and Johnson)
- Physics (Giancoli)
- University Physics (Young and Freedman)
Written for the MCAT and checked before it was published. For study reference only. Spotted an error? Let us know.