Question

According to equation of continuity,

• A. $w_1 a_1 = w_2 a_2$
• B. $w_1 v_1 = w_2 v_2$
• C. $a_1 v_1 = a_2 v_2$
• D. $a_1/v_1 = a_2/v_2$

Hint:

When flow area decreases ($a_2 < a_1$), velocity must increase ($v_2 > v_1$) to keep volumetric flow rate constant.
This simple principle explains how nozzles work.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The continuity equation is a mathematical expression of the conservation of mass in a fluid flow system.
The question asks for the standard simplified form of this equation under typical flow conditions.

Step 2: Key Formula or Approach:
The fundamental conservation of mass for steady flow of a fluid through a conduit is written as:
\[ \dot{m}_1 = \dot{m}_2 \Rightarrow \rho_1 a_1 v_1 = \rho_2 a_2 v_2 \] Where:
$\rho$ is the fluid density.
$a$ is the cross-sectional area of the conduit.
$v$ is the average fluid velocity.

Step 3: Detailed Explanation:

• For an incompressible fluid (such as most liquids under ordinary operating conditions), the density remains constant throughout the flow.

• Therefore, we can set:
\[ \rho_1 = \rho_2 = \text{constant} \]
• Substituting this into the continuity equation allows us to cancel the density terms from both sides:
\[ a_1 v_1 = a_2 v_2 \]
• This simplified form represents the conservation of volumetric flow rate for an incompressible fluid flow.

Step 4: Final Answer:
According to the equation of continuity for incompressible flow, $a_1 v_1 = a_2 v_2$, which corresponds to option (C).