Continuity Equation in Fluid Flow

The study of fluids in motion is called fluid dynamics. A flowing fluid always has a pattern of flow; either steady or unsteady. In steady flow pattern the particles passing through a fixed point move in the same flow line or in other words the flow pattern does not change with time.

But in unsteady flow the flow pattern changes with time and becomes turbulent also called turbulent flow. In steady flow as shown in Figure 1 the fluid particles passing through the point $p_1$ must pass through the points $p_2$ and $p_3$.

Figure 1 In steady flow all particles passing through a given point move in the same flow line; it means the particles moving through a point must move through other points in the flow line.

The flow line in which a tangent at any point gives the direction of fluid flow at that point is called streamline. If the cross section of the flow tube varies, the flow speed along a flow line changes.

In Figure 1 all the particles which flow in the flow line shown have velocity $\vec v_1$ at the point $p_1$ and have velocity $\vec v_2$ at the point $p_2$ and so on.

In steady flow the streamlines coincide with the flow lines. The flow pattern remains steady until a certain velocity called critical velocity. If the velocity of the fluid becomes greater than the critical velocity, the flow line looses its steady nature and becomes irregular and chaotic.

The internal friction in a fluid is called viscosity. The motion of one layer of fluid against another is opposed by the viscosity of the fluid. The more viscous fluid is more sticky such as honey; it easily sticks when you touch it.

The viscous fluid sticks to the walls of the flow tube and therefore the fluid in contact with the walls of the flow tube remains at rest while the other layers move against each other.

Continuity Equation

We consider an incompressible fluid with zero viscosity whose flow is steady in a flow tube of varying cross-section as shown in Figure 2. But note that small viscosity is needed to insure that the flow is steady but we neglect it here.

Figure 2 The cross-section of the flow tube is not the same throughout its length but the mass entering the left end must leave the right end.

The cross-sectional area of the left end of the flow tube is $A_1$ and that of the right end is $A_2$. The fluid entering the left end at cross-section $A_1$ has speed $v_1$ and the fluid leaving the right end at cross-section $A_2$ has speed $v_2$.

The volume of the fluid which enters the left end in a very small interval of time $dt$ is $d{V_1} = {A_1}{v_1}dt$ while the volume of the fluid which leaves the right end in the same time interval is $d{V_2} = {A_2}{v_2}dt$.

If $\rho$ is the density of the fluid, the mass entering the left end in the time interval $dt$ is $\rho {A_1}{v_1}dt$ and the mass leaving the right end in the same time interval is $\rho {A_2}{v_2}dt$.

Since the fluid flow is ideal (steady flow without viscosity and compressibility), the mass which enters the left end in the time interval $dt$ must be equal to the mass that leaves the right end in the same time interval:

\[\begin{align*} \rho {A_1}{v_1}dt &= \rho {A_2}{v_2}dt\\ {\rm{or,}}\quad {A_1}{v_1} &= {A_2}{v_2} \tag{1} \label{1} \end{align*}\]

The Eq. \eqref{1} is called continuity equation. The quantity $A_1v_1$ is the volume flow rate at the left end and $A_2v_2$ is the volume flow rate at the right end.

The continuity equation shows that the volume flow rate at one cross-section is equal to the volume flow rate at another cross-section of the flow tube. Therefore, the quantity $Av$ is the volume flow rate which is constant for steady flow of incompressible fluid with zero viscosity.

Mechanics
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