Question 34.3: Derive an expression for κ in terms of the kinetic coefficient......
Derive an expression for κ in terms of the kinetic coefficients.
Substitution of eqn 34.45 into eqn 34.44 yields
\epsilon =-\frac{\mathcal{L} _{\varepsilon T} }{\mathcal{L} _{\varepsilon \varepsilon } }. (34.45)
\pmb{\varepsilon} =\epsilon ∇T (34.44)
\pmb{\varepsilon} = -\frac{\mathcal{L} _{\varepsilon T} }{\mathcal{L} _{\varepsilon \varepsilon} } ∇T. (34.49)
Putting this into eqn 34.39 implies that
\pmb {J}_{Q} = \mathcal{L} _{T\varepsilon} \pmb{\varepsilon} + \mathcal{L} _{TT} ∇T. (34.39)
\pmb{J}_{Q} =\left(\frac{\mathcal{L} _{\varepsilon \varepsilon } \mathcal{L} _{TT} −\mathcal{L} _{T\varepsilon } \mathcal{L} _{\varepsilon T} }{\mathcal{L} _{\varepsilon \varepsilon } } \right)∇T, (34.50)
and hence comparison with eqn 34.43 yields
\pmb{J}_{Q} = −κ∇T, (34.43)
κ = -\left[\frac{\mathcal{L} _{\varepsilon \varepsilon } \mathcal{L} _{TT} −\mathcal{L} _{T\varepsilon } \mathcal{L} _{\varepsilon T} }{\mathcal{L} _{\varepsilon \varepsilon } } \right]. (34.51)