= Critical Rayleigh number for an isothermal layer
{title2=$\operatorname{Ra}_c=27\pi^4/4$}
For fixed <temperatures> and stress-free impermeable boundaries, linear marginal <Rayleigh-Bénard convection> with dimensionless horizontal <wavenumber> $q$ obeys $\operatorname{Ra}(q)=(q^2+\pi^2)^3/q^2$. <Differentiation> gives $q_c=\pi/\sqrt2$ and the displayed minimum. The threshold depends on the mechanical <boundary conditions>: two rigid no-slip boundaries instead give approximately $1708$.
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