5.3.1. Square lattice

With $\boldsymbol{k}_{1}=k_{c}\boldsymbol{e}_{x}$ and $\boldsymbol{k}_{2}=k_{c}\boldsymbol{e}_{y}$ , modes generated at order $A_{1}A_{2}$ are

(5.8) $$\begin{eqnarray}A_{1}A_{2}\left[\begin{array}{@{}c@{}}F_{A_{1}A_{2}}(z)\\ G_{A_{1}A_{2}}(z)\end{array}\right]\exp \left[\text{i}(\boldsymbol{k}_{1}+\boldsymbol{k}_{2})\boldsymbol{\cdot }\boldsymbol{r}\right]+\text{c.c.},\end{eqnarray}$$

and

(5.9) $$\begin{eqnarray}A_{1}A_{2}\left[\begin{array}{@{}c@{}}\bar{F}_{A_{1}A_{2}}(z)\\ \bar{G}_{A_{1}A_{2}}(z)\end{array}\right]\exp \left[\text{i}(\boldsymbol{k}_{1}-\boldsymbol{k}_{2})\boldsymbol{\cdot }\boldsymbol{r}\right]+\text{c.c.}\end{eqnarray}$$

Since $F_{11}$ and $G_{11}$ are real, $(F_{A_{1}A_{2}},G_{A_{1}A_{2}})$ and $(\bar{F}_{A_{1}A_{2}},\bar{G}_{A_{1}A_{2}})$ are identical and real. They satisfy

(5.10) $$\begin{eqnarray}\displaystyle & & \displaystyle [(D^{2}-2k_{c}^{2})^{2}-2(s/\mathit{Pr})(D^{2}-2k_{c}^{2})]F_{A_{1}A_{2}}-2k_{c}^{2}\mathit{Ra}G_{A_{1}A_{2}}\nonumber\\ \displaystyle & & \displaystyle \quad =(2/\mathit{Pr})[F_{11}(D^{3}F_{11})+(DF_{11})(D^{2}F_{11})]-(4/\mathit{Pr})k_{c}^{2}F_{11}(DF_{11}],\end{eqnarray}$$

(5.11) $$\begin{eqnarray}(D^{2}-2k_{c}^{2}-2s)G_{A_{1}A_{2}}+F_{A_{1}A_{2}}=2F_{11}(DG_{11}).\end{eqnarray}$$

Boundary conditions on $(F_{A_{1}A_{2}},G_{A_{1}A_{2}})$ are the same as those on $(F_{11},G_{11})$ . The functions $F_{A_{1}A_{2}}$ and $G_{A_{1}A_{2}}$ are shown in figure 7 for different values of $L_{s}$ . The coupling between the modes $\exp (\text{i}k_{c}x)$ and $\exp (\text{i}k_{c}y)$ is significant and it is more important in the situation of wall adhesion than for SFBCs. In this latter case, at the critical conditions,

(5.12) $$\begin{eqnarray}\displaystyle & \displaystyle F_{A_{1}A_{2}}(z)=-\frac{27}{100}{\rm\pi}^{2}\left(\frac{{\rm\pi}}{\mathit{Pr}}+1\right)\sin (2{\rm\pi}z), & \displaystyle\end{eqnarray}$$

(5.13) $$\begin{eqnarray}\displaystyle & \displaystyle G_{A_{1}A_{2}}=-{\rm\pi}\frac{27+150\mathit{Pr}}{473\mathit{Pr}}\sin (2{\rm\pi}z). & \displaystyle\end{eqnarray}$$

$A_{1}A_{2}$

${\it\zeta}$

Figure 7.

Modes generated in a square lattice at order $A_{1}A_{2}$ , at the critical conditions, for $\mathit{Pr}=10$ , and different values of $L_{s}$ : (1) $L_{s}=0$ (NSBC); (2) $L_{s}=0.1$ ; and (3) $L_{s}=10^{4}$ very close to SFBC.

5.3.2. Hexagonal lattice

With $\boldsymbol{k}_{1}=k_{c}\boldsymbol{e}_{x},\boldsymbol{k}_{2}=k_{c}(-\boldsymbol{e}_{x}/2+(\sqrt{3}/2)\boldsymbol{e}_{y})$ and $\boldsymbol{k}_{3}=k_{c}(-\boldsymbol{e}_{x}/2-(\sqrt{3}/2)\boldsymbol{e}_{y})$ , modes generated at order $A_{p}A_{q}$ with $1\leqslant p\leqslant 3,~1\leqslant q\leqslant 3$ and $p\neq q$ are

(5.14) $$\begin{eqnarray}A_{p}A_{q}\left[\begin{array}{@{}c@{}}F_{A_{p}A_{q}}(z)\\ G_{A_{p}A_{q}}(z)\end{array}\right]\exp \left[\text{i}(\boldsymbol{k}_{p}+\boldsymbol{k}_{q})\boldsymbol{\cdot }\boldsymbol{r}\right]+\text{c.c.},\end{eqnarray}$$

and

(5.15) $$\begin{eqnarray}A_{p}A_{q}\left[\begin{array}{@{}c@{}}\bar{F}_{A_{p}A_{q}}(z)\\ \bar{G}_{A_{p}A_{q}}(z)\end{array}\right]\exp \left[\text{i}(\boldsymbol{k}_{p}-\boldsymbol{k}_{q})\boldsymbol{\cdot }\boldsymbol{r}\right]+\text{c.c.}\end{eqnarray}$$

The functions $F_{ApAq}$ and $G_{ApAq}$ satisfy

(5.16) $$\begin{eqnarray}\displaystyle & & \displaystyle [(D^{2}-k_{c}^{2})^{2}-2(s/\mathit{Pr})(D^{2}-k_{c}^{2})]F_{A_{p}A_{q}}-k_{c}^{2}\mathit{Ra}G_{A_{p}A_{q}}\nonumber\\ \displaystyle & & \displaystyle \quad =(1/\mathit{Pr})\left[F_{11}D^{3}F_{11}+2\left(DF_{11}\right)\left(D^{2}F_{11}\right)-3k_{c}^{2}F_{11}\left(DF_{11}\right)\right]\!,\end{eqnarray}$$

(5.17) $$\begin{eqnarray}\displaystyle & & \displaystyle \left(D^{2}-k_{c}^{2}-2s\right)G_{A_{p}A_{q}}+F_{A_{p}A_{q}}=2F_{11}\left(DG_{11}\right)+G_{11}\left(DF_{11}\right)\!.\end{eqnarray}$$

$\bar{F}_{ApAq}$

$\bar{G}_{ApAq}$

(5.18) $$\begin{eqnarray}\displaystyle & & \displaystyle [(D^{2}-3k_{c}^{2})^{2}-(2s/\mathit{Pr})(D^{2}-3k_{c}^{2})]\bar{F}_{A_{p}A_{q}}-3k_{c}^{2}\mathit{Ra}\bar{G}_{A_{p}A_{q}}\nonumber\\ \displaystyle & & \displaystyle \quad =(3/\mathit{Pr})\left[F_{11}\left(D^{3}F_{11}\right)-k_{c}^{2}F_{11}\left(DF_{11}\right)\right]\!,\end{eqnarray}$$

(5.19) $$\begin{eqnarray}\displaystyle & & \displaystyle \left(D^{2}-3k_{c}^{2}-2s\right)\bar{G}_{A_{p}A_{q}}+\bar{F}_{A_{p}A_{q}}=2F_{11}\left(DG_{11}\right)-G_{11}\left(DF_{11}\right)\!.\end{eqnarray}$$

$(F_{A_{p}A_{q}},G_{A_{p}A_{q}})$

$(\bar{F}_{A_{p}A_{q}},\bar{G}_{A_{p}A_{q}})$

$(F_{11},G_{11})$

$F_{A_{p}Aq},G_{ApAq},\bar{F}_{ApAq}$

$\bar{G}_{ApA_{q}}$

$\mathit{Pr}=10$

(5.20a,b ) $$\begin{eqnarray}F_{A_{p}A_{q}}=-\frac{9{\rm\pi}^{3}}{104}\left(1+\frac{3}{\mathit{Pr}}\right)\sin (2{\rm\pi}z),\quad G_{A_{p}A_{q}}=-\frac{3{\rm\pi}}{52}\left(9+\frac{1}{\mathit{Pr}}\right)\sin (2{\rm\pi}z),\end{eqnarray}$$

(5.21a,b ) $$\begin{eqnarray}\bar{F}_{A_{p}A_{q}}=-\frac{81{\rm\pi}^{3}}{5000}\left(3+\frac{11}{\mathit{Pr}}\right)\sin (2{\rm\pi}z),\quad \bar{G}_{A_{p}A_{q}}=-\frac{3{\rm\pi}}{2500}\left(121+\frac{27}{\mathit{Pr}}\right)\sin (2{\rm\pi}z).\end{eqnarray}$$

As for squares, the amplitude of modes arising from the quadratic coupling between modes with vector $\boldsymbol{k}_{p}$ and $\boldsymbol{k}_{q}$ is more important than that of the first harmonic.

Figure 8.

Modes generated in an hexagonal lattice at the critical conditions for $\mathit{Pr}=10$ : factor of $A_{p}A_{q}\exp [\text{i}(\boldsymbol{k}_{p}+\boldsymbol{k}_{q})\boldsymbol{\cdot }\boldsymbol{r}]$ in (a,b) and $A_{p}A_{q}\exp [\text{i}(\boldsymbol{k}_{p}-\boldsymbol{k}_{q})\boldsymbol{\cdot }\boldsymbol{r}]$ in (c,d), with (1) $L_{s}=0$ , (2) $L_{s}=0.1$ and (3) $L_{s}=10^{4}$ .

5.4.1. Bifurcation to rolls

The modification of the fundamental at order $A^{3}$ is governed by (4.18) and (4.19) with $m=n=1$ , i.e. 

(5.23) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\mathscr{S}_{1}^{2}-3\frac{s}{\mathit{Pr}}\mathscr{S}_{1}\right)F_{13}-k_{c}^{2}\mathit{Ra}G_{13}=\frac{g_{1}}{\mathit{Pr}}\mathscr{S}_{1}F_{11}-\left[NI_{w}\right]_{E^{1}A^{3}}-\left[NV\right]_{E^{1}A^{3}}\!, & \displaystyle\end{eqnarray}$$

(5.24) $$\begin{eqnarray}\displaystyle & \left(\mathscr{S}_{1}-3s\right)G_{13}+F_{13}=g_{1}G_{11}-\left[NI_{{\it\theta}}\right]_{E^{1}A^{3}}\!. & \displaystyle\end{eqnarray}$$

(5.25a ) $$\begin{eqnarray}\displaystyle & \displaystyle F_{13}=0,\quad DF_{13}-L_{s}D^{2}F_{13}={\it\alpha}L_{s}\frac{\partial }{\partial x}\left({\it\Gamma}\frac{\partial u}{\partial z}\right)_{E^{1}A^{3}}\!,\quad G_{13}=0\quad \text{at}~z=0, & \displaystyle\end{eqnarray}$$

(5.25b ) $$\begin{eqnarray}\displaystyle & \displaystyle F_{13}=0,\quad DF_{13}+L_{s}D^{2}F_{13}=-{\it\alpha}L_{s}\frac{\partial }{\partial x}\left({\it\Gamma}\frac{\partial u}{\partial z}\right)_{E^{1}A^{3}}\!,\quad G_{13}=0\quad \text{at}~z=1.~\ & \displaystyle\end{eqnarray}$$

(5.26) $$\begin{eqnarray}\displaystyle & F_{13}=DF_{13}=0\quad \text{at}~z=0,1~\text{for NSBC}, & \displaystyle\end{eqnarray}$$

(5.27) $$\begin{eqnarray}\displaystyle & \displaystyle F_{13}=D^{2}F_{13}=0\quad \text{at}~z=0,1~\text{for SFBC}. & \displaystyle\end{eqnarray}$$

(5.28) $$\begin{eqnarray}\boldsymbol{L}\boldsymbol{\cdot }\boldsymbol{X}_{13}=g_{1}\boldsymbol{M}\boldsymbol{\cdot }\boldsymbol{X}_{11}-\boldsymbol{NI}-\boldsymbol{NV}+3s\boldsymbol{M}\boldsymbol{\cdot }\boldsymbol{X}_{13}\quad \text{with}~\boldsymbol{X}_{13}=\left(F_{13},G_{13}\right)\!.\end{eqnarray}$$

The nature of the bifurcation is determined at the critical conditions, i.e.  $s=0$ .

In order to calculate the first Landau coefficient $g_{1}$ from the Fredholm solvability condition, i.e. orthogonality of the inhomogeneous part of (5.28) to the null-space of the adjoint operator of $\boldsymbol{L}$ , we decompose $\boldsymbol{X}_{13}$ into a homogeneous $\boldsymbol{X}_{13H}$ and inhomogeneous $\boldsymbol{X}_{13NH}$ parts:

(5.29) $$\begin{eqnarray}\boldsymbol{X}_{13}=\boldsymbol{X}_{13H}+\boldsymbol{X}_{13NH}.\end{eqnarray}$$

Here $\boldsymbol{X}_{13H}$ satisfies

(5.30) $$\begin{eqnarray}\boldsymbol{L}\boldsymbol{\cdot }\boldsymbol{X}_{13H}=g_{1H}\boldsymbol{M}\boldsymbol{\cdot }\boldsymbol{X}_{11}-\boldsymbol{N}\boldsymbol{I}-\boldsymbol{N}\boldsymbol{V},\end{eqnarray}$$

with homogeneous boundary conditions, i.e.  $F_{13H}=DF_{13H}-L_{s}D^{2}F_{13H}=0$ at $z=0$ and similarly at $z=1$ . By applying the solvability condition to (5.30), we obtain

(5.31a,b ) $$\begin{eqnarray}g_{1H}=\langle \boldsymbol{NI}+\boldsymbol{NV},\tilde{\boldsymbol{X}}_{ad}\rangle ,\quad \tilde{\boldsymbol{X}}_{ad}=\frac{\boldsymbol{X}_{ad}}{\left\langle \boldsymbol{M}\boldsymbol{\cdot }\boldsymbol{X}_{11},\boldsymbol{X}_{ad}\right\rangle }.\end{eqnarray}$$

Then $\boldsymbol{X}_{13H}$ is then determined by solving (5.30). Here, $\boldsymbol{X}_{13NH}$ is a correction term that accounts for the non-homogeneity of the boundary conditions. Substituting (5.29) into (5.28), we obtain at the critical conditions

(5.32) $$\begin{eqnarray}\boldsymbol{L}\boldsymbol{\cdot }\boldsymbol{X}_{13H}=g_{1}\boldsymbol{M}\boldsymbol{\cdot }\boldsymbol{X}_{11}-\boldsymbol{NI}-\boldsymbol{NV}-\boldsymbol{L}\boldsymbol{\cdot }\boldsymbol{X}_{13NH}.\end{eqnarray}$$

By applying the solvability condition to (5.32), we get

(5.33) $$\begin{eqnarray}g_{1}=g_{1H}+\langle \boldsymbol{L}\boldsymbol{\cdot }\boldsymbol{X}_{13NH},\tilde{\boldsymbol{X}}_{ad}\rangle .\end{eqnarray}$$

The technique of solution adopted is to iterate a few times between (5.28) and (5.33). At the start, $\boldsymbol{X}_{13NH}$ is assumed to be identically zero in (5.33). A first approximation to $g_{1}$ is then obtained: $g_{1}^{(1)}=g_{1H}$ . This is put into (5.28), which is solved at the critical conditions, with non-homogeneous boundary conditions, to obtain a first approximation of $\boldsymbol{X}_{13}$ . Using (5.29), a first approximation of $\boldsymbol{X}_{13NH}$ is deduced. Then $\boldsymbol{X}_{13NH}$ is put into (5.33). This process is repeated until it converges to a desired level of accuracy. Note that (5.28) and (5.30) are solved with an additional condition

(5.34) $$\begin{eqnarray}\boldsymbol{X}_{13}=\boldsymbol{X}_{13H}=0\quad \text{at}~z=1/2,\end{eqnarray}$$

as suggested by Herbert (Reference Herbert1980, Reference Herbert1983), Sen & Venkateswarlu (Reference Sen and Venkateswarlu1983) and Generalis & Fujimura (Reference Generalis and Fujimura2009). Without this normalization, $\boldsymbol{X}_{13}$ is defined up to an arbitrary multiple of the solution $\boldsymbol{X}_{11}$ of the linear problem (3.8). Finally, $g_{1}$ can be written as the sum of three contributions. The first one arises from the nonlinear inertial terms $\boldsymbol{NI}$ , the second from the nonlinear viscous terms $\boldsymbol{NV}$ and the third from the inhomogeneity of the boundary conditions,

(5.35) $$\begin{eqnarray}g_{1}=g_{1}^{I}+g_{1}^{V}+g_{1}^{NH}\end{eqnarray}$$

with

(5.36a−c ) $$\begin{eqnarray}g_{1}^{I}=\langle \boldsymbol{NI},\tilde{\boldsymbol{X}}_{ad}\rangle ,\quad g_{1}^{V}=\langle \boldsymbol{NV},\tilde{\boldsymbol{X}}_{ad}\rangle ,\quad g_{1}^{NH}=\langle \boldsymbol{L}\boldsymbol{\cdot }\boldsymbol{X}_{13NH},\tilde{\boldsymbol{X}}_{ad}\rangle .\end{eqnarray}$$

Using (5.22), the contribution of the nonlinear viscous term $g_{1}^{V}$ can be written as

(5.37) $$\begin{eqnarray}g_{1}^{V}=-{\it\alpha}g_{1}^{NN},\end{eqnarray}$$

with ${\it\alpha}$ defined by (2.11) and $g_{1}^{NN}$ that does not depend on the rheological parameters. Similarly, it can be shown that $X_{13NH}=-{\it\alpha}\tilde{X}_{13NH}$ , where $\tilde{X}_{13NH}$ does not depend on the rheological parameters. Hence,

(5.38) $$\begin{eqnarray}g_{1}^{NH}=-{\it\alpha}g_{1}^{BC}\quad \text{with}~g_{1}^{BC}=\langle \boldsymbol{L}\boldsymbol{\cdot }\tilde{\boldsymbol{X}}_{13NH},\tilde{\boldsymbol{X}}_{ad}\rangle .\end{eqnarray}$$

Note that, for NSBCs and SFBCs,

(5.39) $$\begin{eqnarray}\boldsymbol{X}_{13NH}=0\quad \text{thus}~g_{1}^{BC}=0.\end{eqnarray}$$

The first Landau constant $g_{1}$ as well as the different contributions $g_{1}^{I},g_{1}^{V}$ and $g_{1}^{NH}$ are determined for different critical sets $(\mathit{Ra}_{c},k_{c},L_{s},\mathit{Pr})$ and different values of the degree of shear-thinning ${\it\alpha}$ . The integrals in (5.36) are evaluated numerically by means of Clenshaw and Curtis method.

In figure 9(a), $g_{1}^{I}$ is plotted as a function of $L_{s}$ . As expected, $g_{1}^{I}$ is negative, i.e. the bifurcation is supercritical for a Newtonian fluid. The absolute value of $g_{1}^{I}$ decreases with increasing $L_{s}$ , i.e. slip promotes the development of the convection. For SFBCs, an analytical expression of $g_{1}^{I}$ is obtained,

(5.40) $$\begin{eqnarray}g_{1}^{I}=-\frac{9{\rm\pi}^{4}}{8}\frac{\mathit{Pr}}{1+\mathit{Pr}}.\end{eqnarray}$$

The contribution of the nonlinear inertial terms gathers the feedback from thermal advection terms and velocity terms:

(5.41) $$\begin{eqnarray}\displaystyle g_{1}^{I} & = & \displaystyle \langle NI_{w}(F_{22}|F_{11}),\tilde{F}_{ad}\rangle +\langle NI_{{\it\theta}}(G_{22}|F_{11}),\tilde{G}_{ad}\rangle \nonumber\\ \displaystyle & & \displaystyle +\,\langle NI_{{\it\theta}}(F_{22}|G_{11}),\tilde{G}_{ad}\rangle +\langle NI_{{\it\theta}}(G_{02}|F_{11}),\tilde{G}_{ad}\rangle .\end{eqnarray}$$

For $\mathit{Pr}\geqslant 1$ , the analysis of the different terms shows that $g_{1}^{I}$ is dominated by the nonlinear thermal convection terms, more precisely the term involving $G_{02}$ , the modification of the conductive temperature. The contributions of the other nonlinear inertial term is practically negligible.

Figure 9.

Contribution of the nonlinear inertial terms (a) and nonlinear viscous terms (b) to the first cubic Landau constant, versus the slip parameter $L_{s}$ for different values of the Prandtl number: (1) $\mathit{Pr}=1$ , (2) $\mathit{Pr}=5$ , (3) $\mathit{Pr}=10$ .

In figure 9(b), we plot $g_{1}^{NN}$ as a function of $L_{s}$ for different values of $\mathit{Pr}$ . For SFBCs, an analytical expression is obtained,

(5.42) $$\begin{eqnarray}g_{1}^{NN}=-\frac{1803{\rm\pi}^{8}}{64}\frac{\mathit{Pr}}{1+\mathit{Pr}}.\end{eqnarray}$$

As can be observed, $g_{1}^{NN}$ is negative, and $g_{1}^{V}=-{\it\alpha}g_{1}^{NN}>0$ . Therefore, shear-thinning effects promote a subcritical bifurcation, which is understandable since the viscosity, which damps convection, is reduced.

Concerning $g_{1}^{NH}$ , its absolute value does not exceed $g_{1}/100$ . The maximum value is reached at $L_{s}\approx 0.1$ .

In figure 10 we plot $g_{1}$ as a function of $L_{s}$ at $\mathit{Pr}=10$ and different values of ${\it\alpha}$ . For low shear-thinning effects, $g_{1}<0$ and the bifurcation is supercritical, while for sufficiently high shear-thinning effects, $g_{1}>0$ and the bifurcation is subcritical. Using (5.35) and (5.37), the critical degree of shear-thinning ${\it\alpha}_{c}$ above which the bifurcation changes from supercritical to subcritical is given by

(5.43) $$\begin{eqnarray}{\it\alpha}_{c}=\frac{g_{1}^{I}}{g_{1}^{NN}+g_{1}^{BC}}.\end{eqnarray}$$

Figure 10.

Cubic Landau constant versus $L_{s}$ at $\mathit{Pr}=10$ and different values of ${\it\alpha}$ : (1) ${\it\alpha}=0$ , i.e. Newtonian case; (2) ${\it\alpha}=6.25\times 10^{-6}$ ; (3) ${\it\alpha}=10^{-4}$ ; (4) ${\it\alpha}=2.25\times 10^{-4}$ ; (5) ${\it\alpha}=3.0625\times 10^{-4}$ ; (6) ${\it\alpha}=4\times 10^{-4}$ .

Figure 11 shows the variation of ${\it\alpha}_{c}$ as a function of $\mathit{Pr}$ for NSBCs and SFBCs. For this latter case, we have

(5.44) $$\begin{eqnarray}{\it\alpha}_{c}=\frac{24}{601{\rm\pi}^{4}}\approx 4.1\times 10^{-4}.\end{eqnarray}$$

This result agrees with Balmforth & Rust (Reference Balmforth and Rust2009) but disagrees with the value ${\it\alpha}_{c}=14\times 10^{-4}$ found by Albaalbaki & Khayat (Reference Albaalbaki and Khayat2011). For NSBCs, and $\mathit{Pr}\geqslant 1$ , ${\it\alpha}_{c}=2.15\times 10^{-4}$ . The plot of ${\it\alpha}_{c}$ versus $L_{s}$ for $\mathit{Pr}=10$ is visible in figure 14, which also displays some results for two other patterns.

Figure 11.

Critical value of the degree of shear-thinning ${\it\alpha}_{c}$ above which the bifurcation to rolls becomes subcritical, versus Prandtl number.

5.4.2. Bifurcation to squares

As compared with rolls, for square patterns, the modification of the fundamental comprises an additional term, which arises from the interaction of modes with wavevectors $\boldsymbol{k}_{1}=k_{c}\boldsymbol{e}_{x}$ and $\boldsymbol{k}_{2}=k_{c}\boldsymbol{e}_{y}$ . Hence, the modification of the fundamental

(5.45) $$\begin{eqnarray}w=F_{13}(z)\left(A_{1}^{3}\text{e}^{\text{i}k_{c}x}+A_{2}^{3}\text{e}^{\text{i}k_{c}y}+\text{c.c.}\right)+\tilde{F}_{13}(z)\left(A_{2}^{2}A_{1}\text{e}^{\text{i}k_{c}x}+A_{1}^{2}A_{2}\text{e}^{\text{i}k_{c}y}+\text{c.c.}\right)\!,\end{eqnarray}$$

and similarly for ${\it\theta}$ by replacing $F$ by $G$ . The amplitude equations for $A_{1}$ and $A_{2}$ are obtained using symmetries introduced by the square lattice. Employing crystallographic terminology (Golubitsky et al. Reference Golubitsky, Swift and Knoblock1984; McKenzie Reference McKenzie1988; Silber & Knoblock Reference Silber and Knoblock1988), these are the symmetries of a square $D_{4}$ in addition to the two-torus $T_{2}$ of translation in the two horizontal directions. Requiring equivariance with respect to the group $D_{4}\times T_{2}$ leads to the amplitude equations

(5.46) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}A_{1}}{\text{d}t}=sA_{1}+\left(g_{1}A_{1}^{2}+{\it\lambda}_{1}A_{2}^{2}\right)A_{1}, & \displaystyle\end{eqnarray}$$

(5.47) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}A_{2}}{\text{d}t}=sA_{2}+\left(g_{1}A_{2}^{2}+{\it\lambda}_{1}A_{1}^{2}\right)A_{2}. & \displaystyle\end{eqnarray}$$

$g_{1}$

${\it\lambda}_{1}$

Equations satisfied by $\tilde{F}_{13}(z)$ and $\tilde{G}_{13}(z)$ are similar to that satisfied by $F_{13}(z)$ and $G_{13}(z)$ , i.e. (5.23) and (5.24), respectively, by replacing $g_{1}$ by ${\it\lambda}_{1}$ and evaluating the nonlinear forcing terms that are factor of $A_{2}^{2}A_{1}\exp (\text{i}k_{c}x)$ . At criticality, $s=0$ , the coupling coefficient ${\it\lambda}_{1}$ is obtained with a solvability condition. As for $g_{1}$ , it can be written

(5.48) $$\begin{eqnarray}{\it\lambda}_{1}={\it\lambda}_{1}^{I}-{\it\alpha}{\it\lambda}_{1}^{NN}-{\it\alpha}{\it\lambda}_{1}^{BC}.\end{eqnarray}$$

For SFBCs,

(5.49a−c ) $$\begin{eqnarray}{\it\lambda}_{1}^{I}=-\frac{9{\rm\pi}^{4}\left(120+72\mathit{Pr}+673\mathit{Pr}^{2}\right)}{3784\mathit{Pr}\left(1+\mathit{Pr}\right)},\quad {\it\lambda}_{1}^{NN}=-\frac{921{\rm\pi}^{8}\mathit{Pr}}{32(1+\mathit{Pr})},\quad {\it\lambda}_{1}^{BC}=0.\end{eqnarray}$$

The critical value of ${\it\alpha}$ above which the bifurcation to squares becomes subcritical,

(5.50) $$\begin{eqnarray}{\it\alpha}_{c}=\frac{g_{1}^{I}+{\it\lambda}_{1}^{I}}{\left(g_{1}^{NN}+{\it\lambda}_{1}^{NN}\right)+\left(g_{1}^{BC}+{\it\lambda}_{1}^{BC}\right)}.\end{eqnarray}$$

Figure 12 shows the variation of ${\it\alpha}_{c}$ as a function of $\mathit{Pr}$ for NSBC and SFBC. In this latter case,

(5.51) $$\begin{eqnarray}{\it\alpha}_{c}=\frac{16\left(20+12\mathit{Pr}+191\mathit{Pr}^{2}\right)}{63\,855{\rm\pi}^{4}\mathit{Pr}^{2}}.\end{eqnarray}$$

Figure 12.

Critical value of the degree of shear-thinning ${\it\alpha}_{c}$ above which the bifurcation to squares becomes subcritical, versus Prandtl number.

5.4.3. Bifurcation to hexagons

For hexagonal patterns, the modification of the fundamental modes reads

(5.52) $$\begin{eqnarray}\displaystyle w & = & \displaystyle F_{13}(z)\left(A_{1}^{3}\text{e}^{\text{i}\boldsymbol{k}_{1}\boldsymbol{\cdot }\boldsymbol{r}}+A_{2}^{3}\text{e}^{\text{i}\boldsymbol{k}_{2}\boldsymbol{\cdot }\boldsymbol{r}}+A_{3}^{3}\text{e}^{\text{i}\boldsymbol{k}_{3}\boldsymbol{\cdot }\boldsymbol{r}}+\text{c.c.}\right)+\tilde{\tilde{F}}_{13}(z)\left[\left(A_{2}^{2}+A_{3}^{2}\right)A_{1}\text{e}^{\text{i}\boldsymbol{k}_{1}\boldsymbol{\cdot }\boldsymbol{r}}\right.\nonumber\\ \displaystyle & & \displaystyle +\,\left.\left(A_{3}^{2}+A_{1}^{2}\right)A_{2}\text{e}^{\text{i}\boldsymbol{k}_{2}\boldsymbol{\cdot }\boldsymbol{r}}+\left(A_{1}^{2}+A_{2}^{2}\right)A_{3}\text{e}^{\text{i}\boldsymbol{k}_{3}\boldsymbol{\cdot }\boldsymbol{r}}+\text{c.c.}\right]\!,\end{eqnarray}$$

and similarly for ${\it\theta}$ by replacing $F$ by $G$ . The amplitude equations for $A_{1},A_{2}$ and $A_{3}$ are obtained using symmetries introduced by the hexagonal lattice. The symmetry group that leaves the hexagonal lattice invariant is $D_{6}\times T_{2}$ ; $D_{6}$ represents the rotational ( $2{\rm\pi}/3$ ) and reflection symmetries. The amplitude equations are (Golubitsky et al. Reference Golubitsky, Swift and Knoblock1984)

(5.53) $$\begin{eqnarray}\frac{\text{d}A_{1}}{\text{d}t}=sA_{1}+\left[g_{1}A_{1}^{2}+{\it\delta}_{1}\left(A_{2}^{2}+A_{3}^{2}\right)\right]A_{1}.\end{eqnarray}$$

Equations for $\text{d}A_{2}/\text{d}t,~\text{d}A_{3}/\text{d}t$ are obtained by cyclic permutation of $A_{1},A_{2}$ and $A_{3}$ . The quadratic term proportional to $A_{2}A_{3}$ in (5.53) is equivariant with respect to the group $D_{6}\times T_{2}$ , but vanishes here because of the up and down reflection symmetry. Equations satisfied by $\tilde{\tilde{F}}_{13}(z)$ and $\tilde{\tilde{G}}_{13}(z)$ are similar to that satisfied by $F_{13}(z)$ and $G_{13}(z)$ , i.e. (5.23) and (5.24), respectively, by replacing $g_{1}$ by ${\it\delta}_{1}$ and evaluating the nonlinear forcing terms that are factor of $A_{2}^{2}A_{1}\exp (\text{i}k_{c}x)$ . The determination of ${\it\delta}_{1}$ follows the same procedure as for $g_{1}$ and ${\it\lambda}_{1}$ . As previously, the coupling coefficient ${\it\delta}_{1}$ can be written

(5.54) $$\begin{eqnarray}{\it\delta}_{1}={\it\delta}_{1}^{I}-{\it\alpha}{\it\delta}_{1}^{NN}-{\it\alpha}{\it\delta}_{1}^{BC}.\end{eqnarray}$$

For SFBCs

(5.55a−c ) $$\begin{eqnarray}{\it\delta}_{1}^{I}=-\frac{9{\rm\pi}^{4}\left(1728+1113\mathit{Pr}+1273\mathit{Pr}^{2}\right)}{65\,000\mathit{Pr}\left(1+\mathit{Pr}\right)},\quad {\it\delta}_{1}^{NN}=-\frac{2175{\rm\pi}^{8}\mathit{Pr}}{64\left(1+\mathit{Pr}\right)},\quad {\it\delta}_{1}^{BC}=0.\end{eqnarray}$$

The critical value of the degree of shear-thinning ${\it\alpha}$ above which the bifurcation to hexagons becomes subcritical

(5.56) $$\begin{eqnarray}{\it\alpha}_{c}=\frac{g_{1}^{I}+2{\it\delta}_{1}^{I}}{\left(g_{1}^{NN}+2{\it\delta}_{1}^{NN}\right)+\left(g_{1}^{NH}+2{\it\delta}_{1}^{BC}\right)}.\end{eqnarray}$$

The variation of ${\it\alpha}_{c}$ as a function of $\mathit{Pr}$ is plotted in figure 13 for NSBC and SFBC. In this latter case,

(5.57) $$\begin{eqnarray}{\it\alpha}_{c}=\frac{24\left(3456+2226\mathit{Pr}+33\,599\mathit{Pr}^{2}\right)}{16\,664\,375{\rm\pi}^{4}\mathit{Pr}^{2}}.\end{eqnarray}$$

Finally, figure 14 shows ${\it\alpha}_{c}$ versus $L_{s}$ for rolls, squares and hexagons, at $\mathit{Pr}=10$ .

Figure 13.

Critical value of the degree of shear-thinning ${\it\alpha}_{c}$ above which the bifurcation to hexagons becomes subcritical, versus Prandtl number.

Figure 14.

For rolls (R), squares (S) and hexagons (H), critical value of the degree of shear-thinning ${\it\alpha}_{c}$ , versus the slip parameter $L_{s}$ for $\mathit{Pr}=10$ .

From now on, for the sake of simplicity, in order to not overload this article, we restrict ourselves to the two limiting cases $L_{s}=0$ , NSBCs, or $L_{s}=+\infty$ , SFBCs.

5.5.1. Saturation and coupling coefficients outside the critical conditions

Departing sufficiently from the critical conditions, we can no longer assume $s=0$ in the equations that describe the modification of the fundamental at cubic order. Hence, these equations become unconditionally solvable. To calculate $g_{1}$ , ${\it\lambda}_{1}$ or ${\it\delta}_{1}$ , an iterative process is used as suggested by Sen & Venkateswarlu (Reference Sen and Venkateswarlu1983). Details of the procedure are given in appendix B.

Figure 15 shows the variation of the ratios $g_{1}^{I}({\it\epsilon})/g_{1}^{I}({\it\epsilon}=0)$ and $g_{1}^{NN}({\it\epsilon})/g_{1}^{NN}({\it\epsilon}=0)$ versus the reduced Rayleigh number ${\it\epsilon}=\mathit{Ra}/\mathit{Ra}_{c}-1$ . The contribution to $g_{1}$ of the non-Newtonian viscous terms increases more rapidly than that of the inertial terms. At ${\it\epsilon}=0$ , the saturation coefficient $g_{1}=g_{1}^{I}-{\it\alpha}g_{1}^{NN}$ changes sign for ${\it\alpha}={\it\alpha}_{c}$ . It is clear from figure 15 that, for ${\it\epsilon}\neq 0$ , $g_{1}$ changes sign for ${\it\alpha}<{\it\alpha}_{c}$ . This result will be used when determining the domain of existence of the stationary solutions of the amplitude equations at cubic order.

Figure 15.

For rolls, ratios $g_{1}^{I}/g_{1}^{I}~({\it\epsilon}=0)$ and $g_{1}^{NN}/g_{1}^{NN}~({\it\epsilon}=0)$ versus the reduced Rayleigh number, ${\it\epsilon}$ : (1) $g_{1}^{I}/g_{1}^{I}~({\it\epsilon}=0)$ for SFBC; (2) $g_{1}^{I}/g_{1}^{I}~({\it\epsilon}=0)$ for NSBC; (3) $g_{1}^{NN}/g_{1}^{NN}~({\it\epsilon}=0)$ for SFBC; and (4) $g_{1}^{NN}/g_{1}^{NN}~({\it\epsilon}=0)$ for NSBC.

Ratios between the coupling and saturation coefficients play a fundamental role in the pattern selection. They are shown in figure 16 versus the reduced Rayleigh number, for different values of ${\it\alpha}$ , $\mathit{Pr}=10$ and NSBCs; similar curves are obtained for SFBCs. For ${\it\alpha}\leqslant 0.9{\it\alpha}_{c}$ , one has: (i) ${\it\lambda}_{1},{\it\delta}_{1}$ and $g_{1}$ are of the same, negative, sign; (ii) $|{\it\lambda}_{1}|>|g_{1}|$ and $|{\it\delta}_{1}|>|g_{1}|$ ; and (iii) ${\it\lambda}_{1}/g_{1}$ and ${\it\delta}_{1}/g_{1}$ increase with increasing shear-thinning effects.

Figure 16.

For squares and hexagons, ratios (a) ${\it\lambda}_{1}/g_{1}$ and (b) ${\it\delta}_{1}/g_{1}$ versus the reduced Rayleigh number, at $\mathit{Pr}=10$ and different values of ${\it\alpha}$ in the case of NSBC: (1) ${\it\alpha}=0$ , Newtonian fluid; (2) ${\it\alpha}=0.1{\it\alpha}_{c}$ ; (3) ${\it\alpha}=0.2{\it\alpha}_{c}$ ; (4) ${\it\alpha}=0.3{\it\alpha}_{c}$ ;(5) ${\it\alpha}=0.4{\it\alpha}_{c}$ ; (6) ${\it\alpha}=0.5{\it\alpha}_{c}$ ; (7) ${\it\alpha}=0.6{\it\alpha}_{c}$ ; and (8) ${\it\alpha}=0.7{\it\alpha}_{c}$ .

5.5.2. Pattern selection on a square lattice

The amplitude equations satisfying the symmetry requirements of a square lattice are (5.46) and (5.47). The coefficients $s$ , $g_{1}$ and ${\it\lambda}_{1}$ in these equations depend on ${\it\alpha}$ and on the reduced Rayleigh number ${\it\epsilon}$ . The stationary solutions are obtained by setting $f_{i}(A_{1},A_{2})=0$ , where $f_{i}$ is the right-hand side of the amplitude equations. Their stability is determined by the sign of the eigenvalues ${\it\chi}_{i}$ of the Jacobian matrix $\unicode[STIX]{x1D611}_{ij}=\partial f_{i}/\partial A_{j}$ evaluated at steady states. In the following, the stability of the stationary solutions is examined in detail.

(i) Conduction state, $A_{1}=A_{2}=0$ . The stability eigenvalues associated to this state are ${\it\chi}_{1}={\it\chi}_{2}=s$ . The conductive state is stable if ${\it\epsilon}<0$ and unstable if ${\it\epsilon}>0$ .

(ii) Steady convection with rolls, $A_{1}=\sqrt{-s/g_{1}},A_{2}=0$ or $A_{1}=0,A_{2}=\sqrt{-s/g_{1}}$ . The associated eigenvalues are: ${\it\chi}_{1}=-2s$ and ${\it\chi}_{2}=s((g_{1}-{\it\lambda}_{1})/g_{1})$ . In the supercritical regime, according to figures 10 and 16(a), $g_{1}$ and ${\it\lambda}_{1}$ are negative and $|{\it\lambda}_{1}|>|g_{1}|$ . Thus, stable rolls appear at a supercritical bifurcation from ${\it\epsilon}=0$ . In the neighbourhood of the points ( ${\it\epsilon},{\it\alpha}$ ) where $g_{1}$ changes sign, the stability of rolls cannot be determined and a higher-order analysis is required. In the subcritical regime, roll solutions exist but are unstable.

(iii) Steady convection with square patterns, $A_{1}=A_{2}=\sqrt{-s/(g_{1}+{\it\lambda}_{1})}$ . The associated eigenvalues are ${\it\chi}_{1}=-2s$ and ${\it\chi}_{2}=(2s({\it\lambda}_{1}-g_{1}))/({\it\lambda}_{1}+g_{1})$ . In the supercritical regime, $g_{1}$ and ${\it\lambda}_{1}$ are negative and $|{\it\lambda}_{1}|>|g_{1}|$ , therefore ${\it\chi}_{2}>0$ and squares are unstable. As explained by Fauve (Reference Fauve, Huerre and Rossi1998), when $|{\it\lambda}_{1}|>|g_{1}|$ , the interaction between the two sets of rolls is too strong and one of the two sets of rolls nonlinearly damps out the other. As indicated previously, this interaction is more stronger with increasing shear-thinning effects (figure 16 a). At cubic order, the stability of the squares cannot be determined in the region where $g_{1}+{\it\lambda}_{1}$ changes sign. In the subcritical regime, square solutions exist but are unstable.

5.5.3. Pattern selection on a hexagonal lattice

The system of amplitude equations satisfying the symmetry requirements of a hexagonal lattice is given by (5.53). The stationary solutions are determined and their linear stability is investigated.

(i) Steady convection with rolls, $A_{1}=\sqrt{-s/g_{1}},A_{2}=A_{3}=0$ . The eigenvalues associated to this state are ${\it\chi}_{1}=-2s$ and ${\it\chi}_{2}={\it\chi}_{3}=s((g_{1}-{\it\delta}_{1})/g_{1})$ . In the supercritical regime, according to figures 10 and 16(b), ${\it\delta}_{1}$ and $g_{1}$ are negative and $|{\it\delta}_{1}|>|g_{1}|$ , thus ${\it\chi}_{3}<0$ . As on the square lattice, stable rolls appear supercritically from ${\it\epsilon}=0$ . This analysis is not valid in the vicinity of the points ( ${\it\epsilon},{\it\alpha}$ ) where $g_{1}$ changes sign.

(ii) Steady convection with hexagons, $A_{1}=A_{2}=A_{3}=\sqrt{-s/(g_{1}+2{\it\delta}_{1})}$ . The associated eigenvalues are ${\it\chi}_{1}=-2s,{\it\chi}_{2}={\it\chi}_{3}=2s(({\it\delta}_{1}-g_{1})/(g_{1}+2{\it\delta}_{1}))$ . In the supercritical regime, $g_{1}$ and ${\it\delta}_{1}$ are negative and $|{\it\delta}_{1}|>|g_{1}|$ , thus ${\it\chi}_{3}>0$ and hexagons are unstable. This analysis is not valid in the vicinity of the points $({\it\alpha},{\it\epsilon})$ where $g_{1}+2{\it\delta}_{1}$ changes sign. In the subcritical regime, hexagon solutions exist but are unstable.

5.5.4. Comparison with Albaalbaki & Khayat (Reference Albaalbaki and Khayat2011)

Finally, our results show that only rolls are stable near critical conditions. Furthermore, shear-thinning effects reinforce convection in the form of rolls. This contradicts Albaalbaki & Khayat (Reference Albaalbaki and Khayat2011) findings, where stable squares and hexagons were found near critical conditions. For an hexagonal lattice, Albaalbaki & Khayat (Reference Albaalbaki and Khayat2011) wrote the amplitude equations up to $O(A^{4})$ , where we denote $O(A)$ the common order of magnitude of the amplitudes $A_{1},~A_{2},~A_{3}$ of all modes. They indicated that the terms of $O(A^{2})$ vanish, but that terms of $O(A^{4})$ , the terms proportional to $a_{5},a_{6},a_{7}$ in their equation (3.14) or to $a_{5}+a_{6}+2a_{7}$ in their equation (5.6), play an important role, because ‘non-Newtonian effects are symmetry breaking’ (their § 5.2 p. 527).

In fact, non-Newtonian effects do not break the midplane reflection symmetry. Therefore, all of the even terms in the amplitude equations vanish. In the appendix D, detailed calculations are provided to show that the coefficients $a_{5},a_{6},a_{7}$ in equation (3.14) of Albaalbaki & Khayat (Reference Albaalbaki and Khayat2011) vanish.