Which of the following correctly depicts the continuity equation in its integral form ? (The options as printed in the paper)
- A∂ρ/∂x + ∇·(ρV) = 0
- B∂V/∂t − ∇·(ρV) = 0
- C∂ρ/∂t + ∇·(ρV) = 0Correct
- D∂ρ/∂t − ∇·(ρV) = 0
Solution
Concept
Continuity equation (conservation of mass)
Formula
∂ρ/∂t + ∇·(ρ V) = 0
Step-by-step solution
1. Conservation of mass: rate of change of mass in a control volume = net inflow. 2. In differential form: ∂ρ/∂t + ∇·(ρV) = 0. 3. Only option (c) has this form (with a plus sign and a time derivative of density).
Explanation
The continuity equation says that the rate of increase of density in a volume equals the net mass inflow. Option (c) is the standard form (the paper calls it "integral form").
Why the other options are wrong
(a) uses ∂/∂x instead of ∂/∂t, (b) uses ∂V/∂t, and (d) has the wrong sign for the divergence term.
Shortcut
∂ρ/∂t + ∇·(ρV) = 0; for incompressible flow ∇·V = 0.
Common mistake
Getting the sign of the divergence term wrong.
Correct answer: C) ∂ρ/∂t + ∇·(ρV) = 0