Question

The equation of continuity is obtained as a result of which of the following laws?

• A. Law of conservation of momentum for liquid flow
• B. Law of conservation of energy
• C. Law of equipartition of energy
• D. Law of conservation of mass

Answer 1

The correct option is D Law of conservation of mass

Considering the steady flow of fluid through a duct (i.e the inlet and outlet flows do not vary with time). The velocity $V$ and density $\rho$ of the fluid will be constant over any cross-sectional area $A$, due to steady state flow.


Since the flow is steady, there is no mass accumulation, in the considered control volume (CV) i.e.the fluid volume considered for study. Which means that the incoming mass at cross-sectional area $A_1$ must be equal to the outgoing mass at cross-sectional area $A_2$ during a short time interval $\Delta t$.

Incoming volume flow at cross-sectional area $A_1=A_1{V}_1\Delta t$
Outgoing volume flow at cross-sectional area $A_2=A_2{V}_2\Delta t$

Incoming mass at cross-sectional area $A_1=\rho A_1{V}_1\Delta t$
Outgoing mass at cross-sectional area $A_2=\rho A_2{V}_2\Delta t$
Applying conservation mass,
$\rho A_1{V}_1\Delta t=\rho A_2{V}_2\Delta t$
$\therefore \rho A_1{V}_1=\rho A_2{V}_2...\left(i\right)$
Now considering ideal fluid $\rho =\text{constant}$, so Eq.$\left(i\right)$ becomes:
$A_1{V}_1=A_2{V}_2...\left(ii\right)$
Here, Eq.$\left(i\right)$ is general continuity equation, while Eq.$\left(ii\right)$ is continuity equation for ideal fluid (considering steady state flow)
$\Rightarrow$Option $\left(d\right)$ is correct.