--- title: "Continuity and Bernoulli's Equation for Ideal Flow" description: "For steady incompressible flow in an unbranched pipe, continuity links cross-section and speed. Applying the work-energy theorem to a fluid element relates pressure, speed and height along a streamlin" canonical: https://lightmysky.com/learn/science/continuity-and-bernoullis-equation-for-ideal-flow-mt_pKxlVYTy0Y source: https://lightmysky.com/learn/science/continuity-and-bernoullis-equation-for-ideal-flow-mt_pKxlVYTy0Y.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Continuity and Bernoulli's Equation for Ideal Flow For steady incompressible flow in an unbranched pipe, continuity links cross-section and speed. Applying the work-energy theorem to a fluid element relates pressure, speed and height along a streamline under ideal-flow assumptions. Subject: Science · Area: Forces & Motion · Ages 19 to 20 Page: https://lightmysky.com/learn/science/continuity-and-bernoullis-equation-for-ideal-flow-mt_pKxlVYTy0Y ## Ready when they can - Uses continuity to get the speed in one section of a pipe from the speed and area in another - Derives Bernoulli's equation as work done on a fluid element rather than quoting it - Explains lift, a Venturi meter and the flow from a hole in a tank with the same relation, and states where the ideal assumptions fail ## Lesson: One flow, one energy budget For an incompressible fluid in a sealed pipe, whatever flows in must flow out. The volume rate stays the same along the pipe, so speed times area is the constant linking the sections. Where the pipe narrows, the fluid must speed up. A thumb over a hose speeds the jet because the area shrinks, and a wide slow river quickens through a narrow gorge. *(drawing: Flow speed in m/s. Area 8 square metres at 3 m/s, then area 2 square metres at 12 m/s: both carry 24 cubic metres a second.)* Follow one parcel of fluid along a streamline. Pressure behind it does work, pressure ahead does negative work, and gravity does work as its height changes. The work-energy theorem sets the total equal to the change in kinetic energy. Every term carries the same parcel volume, so divide through by it. What is left is p + rho*v*v/2 + rho*g*z, equal at both points, where p is pressure, rho is density, v is speed, z is height and g is the gravitational field strength. At one height, a rise in v means a fall in p. This needs steady, incompressible, frictionless flow along one streamline. **Example.** One relation handles lift, meters, and draining tanks once you pick the right two points. Faster air over a wing means lower pressure above it, so the pressure below pushes up. A Venturi throat speeds flow and drops pressure in step, and the pressure drop reveals the flow rate. For a hole in a tank, the still high surface converts height into exit speed set by the depth. State the assumptions every time: steady, incompressible, frictionless flow along one streamline. Long pipes with sticky viscous losses, turbulent churning, or compressible gas flows need richer models. Part of mastering Bernoulli is knowing when to reach for it and when viscosity rules it out. **Recap.** Continuity sets the speed, Bernoulli spends the energy budget, and ideal assumptions set the boundary. ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Pressure in Liquids and the Atmosphere](https://lightmysky.com/learn/science/pressure-in-liquids-and-the-atmosphere-mt_eLGNkEJAZl) - [Work as a Line Integral Along a Path](https://lightmysky.com/learn/science/work-as-a-line-integral-along-a-path-mt_UFuQUPGAt7) ## Opens up - [Viscosity, Poiseuille Flow and the Reynolds Number](https://lightmysky.com/learn/science/viscosity-poiseuille-flow-and-the-reynolds-number-mt_-1sWgkRtMs)