Bernoulli's Principle: Velocity & Pressure

Interactive Bernoulli's Principle & Venturi Effect Simulation

An interactive fluid dynamics simulation visualizing the relationship between pipe geometry, fluid velocity, and pressure. Users can dynamically adjust the entrance, throat, and exit radii of a horizontal pipe to observe the Continuity Equation (A_1v_1 = A_2v_2) and Bernoulli's Principle in real-time. As the pipe narrows, the simulation demonstrates how fluid velocity increases quadratically to conserve mass, resulting in a corresponding pressure drop (the Venturi effect).

Key features include live piezometer standpipes visualizing pressure head, a tracked fluid parcel with a dynamic velocity vector, dual Cartesian graphs mapping velocity and pressure gradients along the pipe, and threshold warnings for fluid cavitation. Ideal for AP Physics, college engineering, and fluid mechanics education.

Why does fluid pressure drop where a pipe narrows?

In short: Fluid cannot pile up, so the volume flow rate Q = Av stays the same along the pipe (continuity). Where the pipe narrows, the fluid must speed up, and since A ∝ r², the speed v ∝ 1 ÷ r². The faster fluid has more kinetic energy, and Bernoulli’s principle says that for steady, horizontal flow of an ideal fluid P + ½ρv² stays constant along a streamline, so the pressure must fall.

Common doubts

• “How can the pressure be lower where the fluid moves faster?” A pressure difference is what speeds the fluid up. The pressure behind a fluid element is higher than the pressure ahead of it, so it accelerates into the narrow section, where the pressure is lower.
• “Does halving the radius double the speed?” No. The speed depends on the cross-sectional area, which goes as r². Halving the radius makes the fluid four times faster.
• “What is a streamline?” It is the path a small bit of fluid follows in steady flow. Bernoulli’s equation connects the pressure, speed and height at different points along one streamline.

Change the entrance, throat and exit radii in the simulation above, press FLOW, and watch the speed and pressure graphs respond.

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