Question

For compressible fluid, continuity equation is
A. \[{\rho _1}{A_1}{v_1} = {\rho _2}{A_2}{v_2}\]
B. \[{\rho _2}{A_1}{v_1} = {\rho _1}{A_2}{v_2}\]
C. \[{A_1}{v_1} = {A_2}{v_2}\]
D. \[\dfrac{{{\rho _1}}}{{{\rho _2}}} = \dfrac{{{A_1}}}{{{A_2}}} = \dfrac{{{v_1}}}{{{v_2}}}\]

Answer 1

Hint: Use the relation between density, mass and volume. Express the mass of compressible fluid at both incoming and outgoing sections of pipe. Use the law of conservation of mass at both sections.

Complete step by step answer:
We know that density of fluid is equal to mass per unit volume. Therefore, if the mass changes at a certain point, the density of fluid also changes.
We assume the compressible fluid is passing through a pipe whose entering section has cross sectional area is \[{A_1}\] outgoing section gas cross sectional area \[{A_2}\]. Also, the density of compressible fluid at the entering section is \[{\rho _1}\]and the density of fluid at section \[{A_2}\] is \[{\rho _2}\].
Now we express the volume of fluid entering the section \[{A_1}\] in unit time as follows,
\[{V_1} = {A_1}{v_1}\]
Here, \[{v_1}\] is the velocity of the fluid at \[{A_1}\].
We have to express the mass of the fluid entering the pipe as follows,
\[{m_1} = {\rho _1}{A_1}{v_1}\] …… (1)
Here, \[{\rho _1}\] is the density of the fluid at \[{A_1}\].
We can also express the volume of fluid coming out of the section \[{A_2}\] as follows,
\[{V_2} = {A_2}{v_2}\]
Here, \[{v_2}\] is the velocity of the fluid at \[{A_2}\].
Also, \[{m_2} = {\rho _2}{A_2}{v_2}\] . …… (2)
Here, \[{\rho _2}\] is the density of the fluid at \[{A_2}\].
We have from the law of conservation of mass, the mass of fluid entering the pipe is equal to the mass of fluid coming out of the pipe. Therefore, we can write,
\[{\rho _1}{A_1}{v_1} = {\rho _2}{A_2}{v_2}\]
This is the equation of continuity for compressible fluid.

So, the correct answer is “Option A”.

Note:
For a compressible fluid, the volume of the fluid decreases due to applied stress. Since the volume is inversely proportional to the density of the fluid, the density of the fluid increases. Therefore, the equation of continuity for compressible fluid involves density off fluid at both sections. In case of incompressible fluid, the density does not change at both the sections, therefore the equation of continuity for incompressible fluid is given as, \[{A_1}{v_1} = {A_2}{v_2}\].