Subtopic: Physical interpretations – line, surface and volume integrals
Question Statement:
The physical interpretation of the surface integral $\iint_S \vec{v} \cdot \hat{n} \, dS$, where $\vec{v}$ is the fluid velocity vector field and $\hat{n}$ is the unit outward normal, represents the:
Options:
(A) Total volumetric flow rate (flux) of fluid crossing surface $S$
(B) Total kinetic energy of fluid enclosed by $S$
(C) Vorticity of fluid along the boundary of $S$
(D) Static pressure gradient acting across $S$
Answer Key:
A
Step-by-Step Solution & Explanation
The dot product $\vec{v} \cdot \hat{n}$ gives the normal component of fluid velocity across the surface element $dS$. Integrating over $S$ computes the net volumetric discharge rate $\iint_S \vec{v} \cdot \hat{n} \, dS$ in $\text{m}^3/\text{s}$ passing through surface $S$.