Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 54 / 2 / b

Table of contents
- Solution b

Solution

Apply the continuity equation and integrate by parts, with the stated vanishing boundary terms and finite moments. The second mass moment tensor satisfies

\dot{I}_{ij} = \int ρ (u_{i} x_{j} + x_{i} u_{j}) d τ,

and differentiation again gives

\ddot{I}_{ij} = 2 \int ρ u_{i} u_{j} d τ + \int ρ (x_{i} D_{t} u_{j} + x_{j} D_{t} u_{i}) d τ .

Insert the stress-divergence equation from part (a). The two force integrals become

\int (x_{i} \partial_{k} T_{jk} + x_{j} \partial_{k} T_{ik}) d τ = - \int (T_{ji} + T_{ij}) d τ = - 2 T_{ij} .

The first term is

4 K_{ij}

. Hence the magnetized-fluid tensor virial theorem is

\frac{1}{2} \ddot{I}_{ij} = 2 K_{ij} - T_{ij} .

Here

T

is the volume-integrated stress, whose sign differs from some gravitational potential-energy tensor conventions. The derivation also requires the advective mass-moment surface terms to vanish; this is automatic for an isolated sufficiently decaying configuration.

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