The Continuity Equation

For an incompressible fluid (constant density), conservation of mass requires that the volume flow rate $Q$ is constant along a streamtube:

$A_1 v_1 = A_2 v_2 \implies Q = Av = \text{const}$

This is the continuity equation — what flows in must flow out.

Volume Flow Rate

The volume flow rate $Q$ is the volume of fluid passing a cross-section per unit time:

$Q = Av$

where $A$ is the cross-sectional area (m²) and $v$ is the flow velocity (m/s). Units: m³/s.

Exit Velocity

Rearranging the continuity equation gives the velocity at any cross-section:

$v_2 = \frac{A_1 v_1}{A_2}$

A narrowing pipe ($A_2 < A_1$) produces a higher velocity — this is why water speeds up at a garden hose nozzle. A widening pipe slows the flow.

Mass Flow Rate

The mass flow rate $\dot{m}$ accounts for fluid density $\rho$ (kg/m³):

$\dot{m} = \rho Q = \rho A v$

Units: kg/s. For an incompressible fluid, $\dot{m}$ is also conserved along a streamtube.

Your Task

Implement the following functions:

• flow_rate(A, v) — returns volume flow rate $Q = Av$ in m³/s
• exit_velocity(A1, v1, A2) — returns exit velocity $v_2 = A_1 v_1 / A_2$ in m/s
• mass_flow_rate(rho, A, v) — returns mass flow rate $\dot{m} = \rho A v$ in kg/s