Surface Integral/Examples/Fluid in Motion
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Example of Surface Integral
Let $\mathbf v$ be the velocity within a body of fluid $B$ as a point-function.
Let $S$ be a surface through which $B$ is in motion.
Let $\d S$ be a small element of $S$ whose center is at a point $P$.
Then the flow rate of $B$ through $S$ is given by the surface integral:
- $\ds \iint_S \mathbf v \cdot \mathbf {\hat n} \rd S$
where $\mathbf {\hat n}$ denotes the unit normal to $S$ at $\d S$ in the direction of flow of $B$.
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {II}$: The Products of Vectors: $3$. Line and Surface Integrals