Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 37A / a

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- Solution a

Solution

Let

u

be the Stokes flow and let

v

be any other admissible incompressible flow with the same no-slip boundary condition on the sphere and the same decay at infinity. Put

w = v - u, e (q) = \frac{1}{2} (\nabla q + \nabla q^{T}) .

Then

\nabla \cdot w = 0

and

w = 0

on the sphere. The viscous dissipation satisfies

D [v] = 2 μ \int_{D} e (u + w) : e (u + w) d V = D [u] + 4 μ \int_{D} e (u) : e (w) d V + 2 μ \int_{D} e (w) : e (w) d V .

For the Stokes stress

σ = - p I + 2 μ e (u),

incompressibility of

w

gives

2 μ e (u) : e (w) = σ : \nabla w .

The divergence theorem, the homogeneous boundary data, decay at infinity, and

\nabla \cdot σ = 0

give

\int_{D} σ : \nabla w d V = \int_{\partial D} w \cdot σ n d S - \int_{D} w \cdot (\nabla \cdot σ) d V = 0.

Therefore

D [v] - D [u] = 2 μ \int_{D} e (w) : e (w) d V \geq 0.

This proves the Minimum-dissipation theorem for Stokes flow.

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