---
title: "Continuity & Bernoulli Flow Bench Interactive Lab | Sandbox Physics"
description: "Where does the pressure go when liquid flows through a narrow part of a pipe? Into speed."
canonical_url: "https://sandboxphysics.com/labs/bernoulli-flow/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/bernoulli-flow/"
last_modified: "2026-10-27"
---

M093 · Fluid mechanics

# Continuity & Bernoulli Flow Bench

**Where does the pressure go when liquid flows through a narrow part of a pipe?** Into speed. The same volume has to pass every second, so the liquid speeds up where the pipe narrows: half the area, twice the speed. Bernoulli's equation says the pressure drops to pay for that speed. Pressure recovers downstream as the pipe widens, but what friction takes along the way is never paid back.

Shape a transparent Venturi tube, move a pressure probe and compare the energy grade line with the hydraulic grade line. Record raw pressure and speed readings, then use them to work out the flow and a Darcy friction factor.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The flow is steady, incompressible and one-dimensional, with a prescribed Darcy factor and a known diffuser loss. If the pressure anywhere drops below the vapor threshold, that single-phase solution is not used for flow or for readings.

Physics tutorial

## Track pressure and energy through a throat

**Background** Liquid flows steadily through a pipe under constant gravity, and the flow is treated as incompressible and one-dimensional. Centreline heights are measured upward from the laboratory datum.

**Why it matters** Before trusting a fitted curve, see how the shape of the pipe, the energy carried by the liquid and the simulated sensor readings relate to each other.

Start with the essentials

- **Focus question**: Can the pressure recover while energy is still lost?
- **One-sentence intuition**: In the throat the liquid trades pressure head for speed, and dissipative losses lower the total head.

### Core mathematical model

#### Continuity: the same flow through every section

$$
Q=A(s)v(s),\qquad A(s)=\pi r(s)^2
$$

Every cross-section carries the same prescribed volume flow, so a smaller area means a higher speed.

#### Energy and hydraulic grade lines

$$
\begin{aligned}H(s)&=z(s)+\frac{p(s)}{\rho g}+\frac{v(s)^2}{2g}\\H_{\mathrm{hyd}}(s)&=H(s)-\frac{v(s)^2}{2g}\end{aligned}
$$

Pressure here is gauge pressure, and the source sets the head at the inlet.

#### Head loss from local speed

$$
h_L(s)=f_D\!\int_0^s\!\frac{v(u)^2}{2gD(u)}\,du+K\chi(s)\frac{v_t^2}{2g}
$$

The known diffuser loss builds up after the throat. The Darcy loss uses the local diameter and speed at each point.

#### Estimating flow and friction from the readings

$$
\begin{aligned}\widehat Q&=\frac1n\sum_i A_i v_i\\\widetilde H_i&=b-f_D X_i\end{aligned}
$$

First correct the measured total head for the known diffuser loss. Then fit it against the wall-loss coordinate, which is computed from the estimated flow.

#### Check that the pressure stays above vapor pressure

$$
\min_s\!\left[p_a+p(s)\right]>p_v
$$

If the pressure crosses the threshold, the solution is rejected. The model does not compute cavitating flow.

### Common difficulties

#### Gauge and absolute pressure

**Typical misconception** A negative gauge pressure always means the liquid cannot flow.

**Better mental model** Add the ambient pressure first, then compare the result with the specified vapor pressure. The check covers the whole path.

#### The fitted friction factor is an estimate

**Typical misconception** The fitted coefficient is just the reference slider under another name.

**Better mental model** Clear the record and the estimate disappears. Noise can even give an unphysical negative coefficient: the fit is unconstrained, so the result is not clipped, and it is not a safety certificate.

### Run the experiment

1. 01

   #### Find the throat

   Move the gold probe handle through the narrow section.

   **What to observe:** The speed rises and the pressure drops. Without losses, the energy head stays flat.

2. 02

   #### Separate recovery from loss

   Choose With losses and compare the inlet, throat and outlet.

   **What to observe:** Pressure recovers downstream, but the energy grade line stays below its inlet value.

3. 03

   #### Record readings and estimate

   Set a Pressure noise bound, press Acquire pressure profile, then press Export raw readings.

   **What to observe:** The pressure and speed readings give estimates of flow and friction; the reference flow is never used in the fit.

4. 04

   #### Push past the limit of the model

   Choose Pressure boundary or lower the ambient pressure.

   **What to observe:** Even a smooth solution can drop below the vapor threshold, and then acquisition is disabled.

Physical conclusion

Pressure can recover downstream even when energy has been lost. The liquid must also stay above its vapor threshold along the whole pipe for the solution to hold.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)