The simplified equation of continuity is represented as
A1v1 = A2v2
A1v2 = A2v2
A1v1 = A1v2
A2v1 = A1v1
The correct answer and explanation is:
The correct equation of continuity is:
[
A_1 v_1 = A_2 v_2
]
This equation is a fundamental principle in fluid dynamics, derived from the conservation of mass. It states that for an incompressible, steady-flowing fluid, the product of the cross-sectional area ((A)) and velocity ((v)) remains constant along a streamline.
Explanation:
- Conservation of Mass:
The equation of continuity is based on the principle that mass cannot be created or destroyed in a fluid flow. This means that the amount of fluid entering a section of a pipe must be equal to the amount of fluid exiting. - Mathematical Derivation:
Consider a fluid flowing through a pipe with varying cross-sectional areas. If ( \rho ) is the fluid density, then the mass flow rate at any point is given by: [
\text{Mass Flow Rate} = \rho A v
] For an incompressible fluid (( \rho ) remains constant), the equation simplifies to: [
A_1 v_1 = A_2 v_2
] This implies that if the area of the pipe decreases, the velocity of the fluid must increase, and vice versa. - Applications:
- Aerodynamics: Used to analyze airflow over wings.
- Hydraulics: Determines flow rates in pipes.
- Medical Field: Blood flow analysis in arteries.
- Real-Life Example:
When you pinch a garden hose, the water speed increases due to the reduced cross-sectional area, demonstrating the equation of continuity.
Now, I will generate an image illustrating this concept.
Here is an educational diagram illustrating the equation of continuity in fluid dynamics. The image visually represents how velocity increases as the pipe narrows, following the principle ( A_1 v_1 = A_2 v_2 ). Let me know if you need any modifications or further explanations!
