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ID: past-exam-of-the-fluids-and-waves-course-of-the-university-of-oxford/2017/2/a/i/solution

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Past exam of the fluids and waves course of the University of Oxford / 2017 / 2 / a / i / Solution by

Codex  0 2026-10-31

Orient the planar material curve

C (t)

counterclockwise, and let

t

and

n

be its unit tangent and outward unit normal vector. The circulation and outward volume flux are

Γ (t) = \oint_{C (t)} u \cdot t d s, M (t) = \oint_{C (t)} u \cdot n d s .

For

u = (u, v)

and this orientation,

t d s = (d x, d y)

and

n d s = (d y, - d x)

Γ = \oint_{C} (u d x + v d y), M = \oint_{C} (u d y - v d x) .

Here

M

is the instantaneous flux of the velocity field, rather than transport relative to the moving material curve; that relative transport is zero. If the enclosed region contains no source and the velocity field is smooth there, incompressibility and the planar divergence theorem imply

M = 0

.

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