Equation of Continuity

The physical idea

Consider a steady flow of an incompressible fluid through a tube whose cross section changes, with a wide section of area $A_{1}$ and a narrow section of area $A_{2}$. For a steady flow the velocity of a particle is constant at a given point but may vary from point to point. A particle has to move faster in the narrower section, where there is less space, to accommodate the particles behind it; entering a wider section it slows down because there is more space. Because the fluid is incompressible, the particles move faster through a narrow section and slow down through a wider section — if the fluid did not speed up in a narrow region it would have to be compressed to fit into the narrow space.

Diagram to draw

A tube of flow of varying cross section. Mark a wider section of area $A_{1}$ at point A, cut by the surface EFGH, and a narrower section of area $A_{2}$ at point B, cut by the surface PQRS. Show velocities $v_{1}$ and $v_{2}$ at the two sections. All the fluid is confined to the tube of flow, so no fluid can leave or enter through its walls.

Derivation

All the fluid that passes through a tube of flow must pass through any cross section that cuts the tube. Matter is neither created nor destroyed within the region between $A_{1}$ and $A_{2}$, so the mass of fluid in that region is constant over time: if a mass $m$ enters at $A_{1}$, an equal mass must leave at $A_{2}$.

Let the fluid cross the section EFGH at A with speed $v_{1}$ in a time interval $\Delta t$. The mass entering through this cross section is $\rho A_{1}v_{1}\Delta t$.

Let the fluid cross the section PQRS of area $A_{2}$ at B with speed $v_{2}$ in the same interval $\Delta t$. The mass leaving through this cross section is $\rho A_{2}v_{2}\Delta t$.

Since the fluid is incompressible, the mass entering at A equals the mass leaving at B:

$$\rho A_{1}v_{1}\Delta t = \rho A_{2}v_{2}\Delta t$$

$$A_{1}v_{1} = A_{2}v_{2} \qquad\text{or}\qquad Av = \text{constant}$$

This is the equation of continuity in fluid dynamics. It states that the volume rate of flow of an incompressible fluid in a steady flow is the same throughout the flow, and it is an expression of the conservation of mass.

Flux

• $Av = \dfrac{dV}{dt}$ is the volume rate of flow, the volume of fluid per unit time passing through any cross section of the tube of flow. It is called the volume flux.
• $\rho\dfrac{dV}{dt} = \dfrac{dm}{dt}$ is called the mass flux.
• The basic unit of measuring flow is the cusec, one cubic foot per second (28.317 litre per second).

Standard results

• $A_{1}v_{1} = A_{2}v_{2}$, i.e. $Av$ = constant, for a steady flow of an incompressible fluid.
• $v \propto \dfrac{1}{A}$; for a circular pipe $A = \pi d^{2}/4$, so $v \propto \dfrac{1}{d^{2}}$ and $d \propto \dfrac{1}{\sqrt{v}}$.
• Volume flux $= Av$; mass flux $= \rho A v$.