Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 5C / a

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii

i

Solution

The divergence is the trace of the velocity-gradient tensor:

\nabla \cdot u = A + E + I .

Thus the flow is incompressible exactly when

A + E + I = 0 .

The constant vector

U_{0}

is unrestricted.

ii

Solution

The vorticity is

\nabla \times u = (H - F, C - G, D - B) .

Hence the flow is irrotational exactly when

H = F, C = G, D = B,

equivalently when

Γ

is symmetric. Again,

U_{0}

is unrestricted.

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