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 Incoming links: Insulated steady heat balance

Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics V / 2 / d / ii / Solution 2026-10-25

Integrate the steady state equation over

R

and use the divergence theorem:

0 = \iint_{\partial R} k \nabla T \cdot n d A + ρ ∭_{R} S (x) d V .

The zero heat flux condition makes the boundary term vanish. But

ρ > 0

and

S > 0

throughout a region of positive volume, so

ρ ∭_{R} S d V > 0,

a contradiction. Thus no steady-state solution exists. This is the insulated steady heat balance: positive internal heat generation continually increases the stored thermal energy when no heat can escape.

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