Class 11 > Unit # 07: Fluid Dynamics > Equation of Continuity

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Fluid Dynamics - Equation of Continuity

Q. Derive equation of continuity.

Equation of Continuity

"Consider a steady, laminar flow of an incompressible fluid through an enclosed tube or pipe of varying cross-sectional area. The speed of the fluid varies inversely with the cross-sectional area of the tube in accordance with the law of conservation of mass."

Figure: flow of an incompressible fluid through an enclosed tube or pipe of varying cross-sectional area.

The mass flow rate is defined as the mass $\Delta m$ of fluid that passes a given point per unit time $\Delta t$.

• At cross-sectional area $A_1$, the volume of fluid passing through in time $\Delta t$ is $A_1 \Delta x_1$, where $\Delta x_1 = v_1 \Delta t$ is the distance the fluid moves in time $\Delta t$ with velocity $v_1$.
• The mass of this fluid element is $\Delta m_1 = \rho_1 (A_1 v_1 \Delta t)$, where $\rho_1$ is the fluid density.

Therefore, the mass flow rate through cross-section $A_1$ is:

$\frac{\Delta m_1}{\Delta t} = \rho_1 A_1 v_1$    --- (i)

Similarly, through cross-sectional area $A_2$ downstream, the mass flow rate is:

$\frac{\Delta m_2}{\Delta t} = \rho_2 A_2 v_2$    --- (ii)

Mathematical Derivation

Since no fluid leaks in or out through the impermeable walls of the tube, the law of conservation of mass dictates that the mass flow rate must remain constant throughout the pipe. Thus, the mass flow rates through $A_1$ and $A_2$ must be equal:

$\rho_1 A_1 v_1 = \rho_2 A_2 v_2$

For an incompressible fluid (where density remains constant, $\rho_1 = \rho_2 = \rho$), we can cancel $\rho$ from both sides:

$A_1 v_1 = A_2 v_2 \quad \text{or} \quad A v = \text{constant}$

Physical Significance & Volume Flow Rate:

• The equation $A v = \text{constant}$ is called the equation of continuity.
• The product $Av$ represents the volume rate of flow (volume of fluid passing a given cross-section per second, with SI unit $\text{m}^3/\text{s}$).
• Inverse Relationship: The product of area and velocity demonstrates that where the cross-sectional area is large ($A$), the fluid velocity is small ($v$); conversely, where the cross-sectional area is small, the fluid velocity is large.

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