Bernoulli’s Equation Study Pack

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Last updated May 21, 2026

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Bernoulli’s Equation Study Guide

Unpack the physics behind fluid flow by working through Bernoulli's equation — P + ½ρv² + ρgh = constant — and understanding what each term reveals about pressure, kinetic energy, and gravitational potential energy per unit volume. This pack covers the continuity equation, the conditions for valid application, and real-world cases including Venturi meters, airfoil lift, and Torricelli's theorem.

Key Takeaways

  • •Bernoulli's equation expresses conservation of energy for a moving, incompressible, non-viscous fluid: P + ½ρv² + ρgh = constant along any streamline.
  • •When fluid speeds up through a constriction (as described by the continuity equation A₁v₁ = A₂v₂), its static pressure drops — a direct consequence of energy conservation.
  • •The three terms in Bernoulli's equation represent static pressure energy, kinetic energy per unit volume, and gravitational potential energy per unit volume, respectively.
  • •Bernoulli's equation applies only under specific conditions: steady (laminar) flow, an incompressible fluid, negligible viscosity, and flow confined to a single streamline.
  • •Practical applications include the Venturi meter (measuring flow speed via pressure differences), lift on airfoils, and the behavior of fluid ejected from a tank (Torricelli's theorem).
  • •Torricelli's theorem is a special case of Bernoulli's equation showing that fluid exits an opening at speed v = √(2gh), where h is the depth of the opening below the free surface.

Foundations: Energy Conservation in Flowing Fluids

Bernoulli's equation is not a new physical law — it is the principle of conservation of energy rewritten for a fluid in motion, and understanding where it comes from makes every application easier to interpret.

Energy in a Moving Fluid Element

  • •A small parcel of fluid carries three forms of mechanical energy: pressure energy (work done by surrounding fluid to push it along), kinetic energy from its velocity, and gravitational potential energy from its height.
  • •Dividing each energy term by volume converts them into energy densities: static pressure P (Pa), kinetic pressure ½ρv² (Pa), and hydrostatic pressure ρgh (Pa), where ρ is fluid density, v is speed, g is gravitational acceleration, and h is height above a reference level.
  • •Because no energy is added or lost in ideal flow, the sum of these three terms is the same at every point along a streamline: P + ½ρv² + ρgh = constant.

The Streamline Concept

  • •A streamline is an imaginary line drawn so that the velocity vector of the fluid is always tangent to it; in steady flow, fluid parcels follow the same path repeatedly.
  • •Bernoulli's equation holds along a single streamline — comparing conditions at two points on the same streamline, written as P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂.
  • •Applying the equation across different streamlines is only valid if the flow is irrotational (no swirling eddies), an additional assumption beyond the basic derivation.

Assumptions and Validity Conditions

Bernoulli's equation is a powerful but idealized tool; knowing its assumptions is essential for judging when it gives accurate predictions and when corrections are needed.

Steady Flow Requirement

  • •Steady (laminar) flow means fluid properties at any fixed point do not change over time — every parcel passing a given location has the same velocity, pressure, and density.
  • •Turbulent flow, which involves chaotic mixing and eddies, violates this condition; Bernoulli's equation cannot be applied directly to turbulent regions.

Incompressibility Assumption

  • •An incompressible fluid has constant density regardless of pressure changes; liquids such as water satisfy this well under ordinary conditions.
  • •Gases can be treated as incompressible only when flow speeds are well below the speed of sound (roughly below Mach 0.3), making Bernoulli applicable to slow air flows but not to high-speed aerodynamics.

Negligible Viscosity

  • •Viscosity is internal fluid friction; Bernoulli's equation ignores viscous energy losses entirely.
  • •In real pipes and channels, viscous dissipation reduces total mechanical energy along the flow path — engineers add a head-loss term (from the Darcy-Weisbach equation) to account for this.

About this Study Pack

Created by Kibin to help students review key concepts, prepare for exams, and study more effectively. This Study Pack was checked for accuracy and curriculum alignment using authoritative educational sources. See sources below.

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