Determine the minimum critical Rayleigh number for the onset of convection in
a layer of porous material heated from below with an isothermal and
impermeable lower boundary and an isothermal constant pressure upper boundary.
This boundary condition corresponds to a permeable boundary between a
saturated porous layer and an overlying fluid. What is the horizontal
wavelength that corresponds to the minimum value of
\(\mathrm{Ra}_{\mathrm{cr}}\) ? Take the layer thickness to be \(b\), and let the
upper boundary, \(y=0\), have temperature \(T=T_{0}\) and the lower boundary,
\(y=b\), have temperature \(T=T_{1}\). Assume that at the onset of convection
\(T^{\prime}\) has the form
$$
T^{\prime}=T_{0}^{\prime} \sin \frac{2 \pi x}{\lambda} Y(y)
$$
and show that \(Y(y)\) is a solution of
$$
\frac{d^{4} Y}{d \bar{y}^{-4}}-2 a^{2} \frac{d^{2} Y}{d
y^{-2}}+Y\left(a^{4}-a^{2} R a\right)=0
$$
where
$$
a \equiv \frac{2 \pi b}{\lambda} \quad \bar{y} \equiv \frac{y}{b}
$$
Show that the general solution of Equation (9.135) can be written as
$$
Y=c_{1} e^{\gamma \bar{y}}+c_{2} e^{-\gamma \bar{y}}+c_{3} \sin \delta
\bar{y}+c_{4} \cos \delta \bar{y}
$$
where \(c_{1}, c_{2}, c_{3},\) and \(c_{4}\) are constants of integration and
$$
\begin{array}{l}
\gamma^{2}=a^{2}+a \sqrt{\mathrm{Ra}} \\
\delta^{2}=a \sqrt{\mathrm{Ra}}-a^{2}
\end{array}
$$
Show that the boundary conditions are
$$
\begin{aligned}
Y=0 & \text { on } \bar{y}=0 \text { and } 1 \\
\frac{d^{2} Y}{d y^{2}}=0 & \text { on } \bar{y}=1 \\
\frac{d}{d \bar{y}}\left(\frac{d^{2} Y}{d \bar{y}^{2}}-a^{2} Y\right)=0 &
\text { on } \bar{y}=0 .
\end{aligned}
$$
Substitute Equation (9.137) into each of these boundary conditions to obtain
four homogeneous equations for the four unknown constants \(c_{1}, c_{2},\)
\(c_{3},\) and \(c_{4} .\) Show that a nontrivial solution of these equations
requires
$$
\gamma \tan \delta+\delta \tanh \gamma=0
$$
This transcendental equation is an eigenvalue equation that implicitly gives
\(\mathrm{Ra}_{\mathrm{cr}}\) as a function of \(a\), since both \(\gamma\) and
\(\delta\) are defined in terms of \(\mathrm{Ra}\) and \(a\) in Equations (9.138)
and (9.139) . The critical Rayleigh number can be found by numerically solving
Equations (9.138),(9.139) , and (9.143) . The value of min
\(\left(\mathrm{Ra}_{\mathrm{cr}}\right)\) turns out to be \(27.1 .\) One way of
proceeding is to choose a value of \(a\) (there exists an
\(\mathrm{Ra}_{\mathrm{cr}}\) for each
a). Then try a value of \(\delta\). Compute \(\gamma\) from \(\gamma^{2}=2 a^{2}+\)
\(\delta^{2} .\) Then compute \(\tan \delta / \delta\) and \(-\tanh \gamma /
\gamma\). Iterate on
\(\delta\) until these ratios are equal. With \(\delta\) determined
\(\mathrm{Ra}_{\mathrm{cr}}\) follows from Equation \((9.139) .\) Repeat the
process for different values of \(a\) until \(\min \left(\operatorname{Ra}_{c
r}\right)\) is found.
