Questions: The Continuity Equation (Conservation of Mass)

Question 1

Water flows steadily through a pipe that narrows from a cross-sectional area of 0.04 m² to 0.01 m². If the inlet velocity is 2 m/s, what is the outlet velocity?

• A. 0.5 m/s
• B. 2 m/s
• C. 4 m/s
• D. 8 m/s

Answer: (D)

For incompressible steady flow, A₁V₁ = A₂V₂. So V₂ = (A₁/A₂)V₁ = (0.04/0.01)(2) = 8 m/s. The velocity must increase by the area ratio because the same volume of fluid must pass through a smaller opening per unit time.

Question 2

Incompressibility (∇·V = 0) is a property of the fluid material itself, meaning mainly liquids like water can satisfy it.

• T. True
• F. False

Answer: False

Incompressibility is a flow assumption, not a material property. Gases — which are physically compressible — can be treated as incompressible when the Mach number is low (roughly Ma < 0.3). At low speeds, density changes in a gas are negligible, and ∇·V = 0 is an excellent approximation. The assumption describes the flow regime, not the fluid.

Question 3

What does the condition ∇·V = 0 physically mean for an incompressible velocity field?

Model answer: ∇·V = 0 means the velocity field has zero divergence everywhere: there are no sources or sinks of fluid volume. Any fluid element that flows into a control volume must be matched by an equal volume flowing out. Volume is neither created nor destroyed within the flow domain.

The divergence of a vector field measures the net outward flux per unit volume. For incompressible flow, this is zero everywhere, enforcing that mass (and volume, since density is constant) is conserved locally at every point — not just globally across a duct. This constraint severely restricts what velocity fields are physically realizable.