--- title: "Continuity and Bernoulli's Equation for Ideal Flow" description: "Mass cannot pile up in a pipe, so a narrower section carries the same flow at a higher speed. Applying the work-energy theorem to a moving element of fluid then ties pressure, speed and height into on" 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: 2026-09-12 --- > **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 Mass cannot pile up in a pipe, so a narrower section carries the same flow at a higher speed. Applying the work-energy theorem to a moving element of fluid then ties pressure, speed and height into one statement that holds along a streamline. 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. If a pipe of area 6 at speed 2 narrows to area 3, the new speed is 4. Bernoulli's relation comes from tracking the work done on a small parcel of fluid as it moves along. Pressure pushes on both ends while gravity may raise or lower the parcel, and the net work shows up as a change in kinetic energy. Read it as an energy budget per unit volume: faster flow or greater height must be paid for by lower pressure. The derivation only needs work, energy, and an incompressible, nearly frictionless fluid. **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. **Tip.** 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)