Fluid mechanics

The continuity equation

Mass conservation makes the mass flow rate through every section of a steady stream equal.

In steady flow, fluid does not accumulate in a pipe segment: mass entering per second equals mass leaving. For an incompressible liquid, a narrower pipe entails a higher mean speed.

\dot m=\rho Av=\text{constant},\qquad A_1v_1=A_2v_2\quad(\rho=\text{constant})

Definition: Flow rate

Volume flow rate $Q=Av$ is measured in m³/s; mass flow rate is $\dot m=\rho Av$ in kg/s. $v$ is the mean velocity normal to the section. The general form also applies to compressible fluids whose density varies.

Observe faster, more closely spaced markers in the narrow section while the flow rate remains constant.

A narrowing pipe

At constant density, $Q=Av$ is conserved. Halving the pipe radius quarters its area and quadruples the mean speed. Closer streamlines indicate higher speed, not a larger flow rate.

Compressible flow

For a compressible gas, density can vary along a duct; the correct relation is then $\rho Av=\mathrm{constant}$ . The simpler $Av=\mathrm{constant}$ applies only when density can be treated as constant.

Example: Speed in a nozzle

Water flows from a 4 cm diameter pipe at 0.50 m/s into a 2 cm nozzle. Find its mean speed in the nozzle.

Solution

$A_1/A_2=(4/2)^2=4$ , so $v_2=v_1A_1/A_2=2.0\,\mathrm{m/s}$ .

Quick check

Example: Flow through a narrowing nozzle

Water passes from a pipe of diameter $4.0,mathrm{cm}$ into a nozzle of diameter $2.0,mathrm{cm}$ . Since area scales as diameter squared, $A_2=A_1/4$ . If the initial mean speed is $0.50,mathrm{m/s}$ , conservation of volume flow gives $v_2=(A_1/A_2)v_1=2.0,mathrm{m/s}$ . The flow rate stays fixed, $Q=A_1v_1=A_2v_2$ : the speed rises because the cross-section shrinks, not because fluid is created.

A circular incompressible pipe’s radius halves. How must mean speed change to keep flow rate constant?

Which quantity is conserved in steady compressible gas flow?

References

Frank M. White (2016). Fluid Mechanics