Equation of Continuity: MDCAT Physics notes

Ideal fluid flow

The equation of continuity is derived for an ideal fluid, which is:

• incompressible (density constant),
• non-viscous (no internal friction),
• in steady (streamline) flow, so velocity at any point does not change with time,
• irrotational (no swirling eddies).

The equation

Consider fluid entering a pipe of cross-section $A_1$ at speed $v_1$ and leaving a narrower section $A_2$ at speed $v_2$. In time $\Delta t$ the mass entering is $\rho A_1 v_1 \Delta t$ and the mass leaving is $\rho A_2 v_2 \Delta t$. Because mass can neither be created nor stored in the pipe (and the fluid is incompressible), these must be equal:

$$\rho A_1 v_1 = \rho A_2 v_2 \quad\Rightarrow\quad A_1 v_1 = A_2 v_2$$

So the mass flow rate $\rho A v$ and the volume flow rate $Av$ are constant along the pipe. The equation of continuity is really the law of conservation of mass for a moving fluid.

Consequences

• $v \propto \dfrac{1}{A}$: where the pipe is narrow, the fluid moves faster; where it is wide, it moves slower.
• For a circular pipe $A = \pi r^2$, so $v \propto \dfrac{1}{r^2}$ (or $1/d^2$). Halving the diameter makes the speed four times larger.
• Putting a thumb over the end of a garden hose makes water spurt faster.
• A river flows faster in its narrow parts than in wide, deep parts.

Change in pipe	Effect on speed
Area halved	Speed doubles
Radius halved	Speed becomes 4 times
Area doubled	Speed halves
Radius doubled	Speed becomes one-fourth

Worked examples

1. Oil flows at $2\ \mathrm{m\,s^{-1}}$ through a pipe of area $0.06\ \mathrm{m^2}$ that narrows to $0.015\ \mathrm{m^2}$. Then $v_2 = A_1v_1/A_2 = (0.06 \times 2)/0.015 = 8\ \mathrm{m\,s^{-1}}$.

2. A pipe of radius 4 cm carries water at $1\ \mathrm{m\,s^{-1}}$ into a pipe of radius 2 cm. Since $v \propto 1/r^2$, $v_2 = 1 \times (4/2)^2 = 4\ \mathrm{m\,s^{-1}}$.

3. Volume flow rate in example 1 is $Av = 0.06 \times 2 = 0.12\ \mathrm{m^3\,s^{-1}}$, the same in both sections.

Key formulas

• $A_1 v_1 = A_2 v_2$
• Mass flow rate $= \rho A v$ (kg s$^{-1}$)
• Volume flow rate $= A v$ (m$^3$ s$^{-1}$)
• $v_2 = v_1 (r_1/r_2)^2$ for circular pipes

Common MDCAT traps

• The speed rises to keep the mass flow rate constant, not the pressure or the energy.
• Smaller area means greater speed, not smaller.
• If radius or diameter is given, square the ratio; do not use it directly.
• Continuity comes from conservation of mass; Bernoulli's equation comes from conservation of energy.

Quick revision

• $Av$ = constant for an incompressible fluid in steady flow.
• Continuity = conservation of mass.
• Narrow section: high speed.
• $v \propto 1/r^2$ for a round pipe.
• Assumes ideal fluid: incompressible, non-viscous, steady.