Question

The continuity equation for an incompressible fluid states that:

• A. The velocity is constant everywhere
• B. The volume flow rate is constant along a streamline
• C. The pressure is constant along a streamline
• D. The density changes with velocity

Hint:

Remember: Conservation of Mass translates to "Constant Volumetric Flow Rate" ($A_{1}V_{1} = A_{2}V_{2}$) ONLY if the fluid density is constant (i.e., the fluid is incompressible).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the fundamental statement of the continuity equation applied to an incompressible fluid flow.

Step 2: Key Formula or Approach: The continuity equation is a direct representation of the Law of Conservation of Mass in fluid dynamics.
For a steady-state fluid flow, the mass entering a control volume must equal the mass exiting.

Step 3: Detailed Explanation:

• The mass flow rate (\(\dot{m}\)) of a fluid passing through a cross-sectional area \(A\) with velocity \(V\) is defined as:
\[ \dot{m} = \rho A V \]
where \(\rho\) is the fluid density.

• For steady flow along a streamline or inside a channel, conservation of mass dictates that the mass flow rate is constant:
\[ \rho_{1} A_{1} V_{1} = \rho_{2} A_{2} V_{2} = \text{constant} \]

• An incompressible fluid is defined as a fluid whose density remains constant throughout the flow (\(\rho_{1} = \rho_{2} = \rho\)).

• Dividing the mass conservation equation by the constant density \(\rho\) yields:
\[ A_{1} V_{1} = A_{2} V_{2} = \text{constant} \]

• The product of cross-sectional area and fluid velocity (\(A V\)) represents the volumetric flow rate (or volume flow rate, \(Q\)).

• Therefore, for an incompressible fluid flow, the volumetric flow rate remains constant along a streamline.

Step 4: Final Answer:
The continuity equation for an incompressible fluid states that the volume flow rate is constant along a streamline.