Study of a periodically forced magnetohydrodynamic system using Floquet analysis and nonlinear Galerkin modelling

Abstract

We present a detailed study of Rayleigh–Bénard magnetoconvection with periodic gravity modulation and uniform vertical magnetic field. Linear stability analysis is carried out using Floquet theory to construct the stability boundaries in order to estimate the magnitude of forcing amplitude $\epsilon$ required for having convection in the system for a fixed Rayleigh number Ra, wave number k and modulating frequency $\Omega$. The effects of varying Prandtl number Pr and Chandrasekhar number Q on the threshold of convection are also investigated. A higher Pr value reduces the value of the threshold, whereas a higher Q value increases it. Bicritical states are also observed at which the minimum forcing amplitude needed for convection to begin occurs at two different k values in harmonic and sub-harmonic regions, respectively. We also construct a nonlinear Galerkin model and compare the results with those obtained from linear stability analysis. Two-dimensional (2D) oscillatory convection is observed at the onset, while quasiperiodic and chaotic behaviours are found at higher Ra values. 2D as well as nonlinear convective flow patterns are observed for primary and higher-order instabilities, respectively. Bifurcation diagrams with respect to different parameters such as $\epsilon$, Ra and Q are provided for thorough understanding of the forced nonlinear system.

Appendices

Appendix I: The magnetohydrodynamic equations

The properties of the concerned magnetohydrodynamic system are as described in the first paragraph of Sect. 2. Let us have a look at how we arrive at the governing Eqs. 3–6 of our system. The equation of continuity is given by

$$\begin{aligned} \frac{\partial \rho }{\partial t} + {{\varvec{\nabla }}\cdot }(\rho {\varvec{v}})=0 \end{aligned}$$

(38)

while the equation of momentum transfer is given by

$$\begin{aligned} \rho \left[ \frac{\partial {\varvec{v}}}{\partial t} +({\varvec{v}}{\cdot {\varvec{\nabla }}}){\varvec{v}}\right]= & {} -\,{{\varvec{\nabla }}} P + \rho g {{\varvec{e_3}}} + {\varvec{f}}_L \nonumber \\&+\,\zeta \nabla ^2{{\varvec{v}}} + \frac{\zeta }{3}{{\varvec{\nabla }}}({\varvec{\nabla }}\cdot {\varvec{v}}) \end{aligned}$$

(39)

where P is the non-magnetic pressure, $\rho$ is the density and $\zeta$ is the dynamic viscosity of the fluid. The term $\rho g {{\varvec{e_3}}}$ is the buoyant force in the system due to the constant temperature gradient across the layer of fluid. The term ${\varvec{f}}_L$ represents the Lorentz force which arises due to the application of the external magnetic field and is given by

$$\begin{aligned} {\varvec{f}}_L = {\varvec{J}}\times {\varvec{B}} \end{aligned}$$

(40)

Using Maxwell’s equations, the above can be written as

$$\begin{aligned} {\varvec{J}}\times {\varvec{B}} = -{\varvec{\nabla }}\left( \frac{B^2}{2\mu }\right) + \frac{1}{\mu } ({\varvec{B}}{\cdot {\varvec{\nabla }}}){\varvec{B}} \end{aligned}$$

(41)

Substituting this, Eq. 39 can be written as

$$\begin{aligned} \rho \left[ \frac{\partial {\varvec{v}}}{\partial t} +({\varvec{v}}{\cdot {\varvec{\nabla }}}){\varvec{v}}\right]= & {} -{{\varvec{\nabla }}} \left( P+\frac{B^2}{2\mu }\right) + \rho g {{\varvec{e_3}}} \nonumber \\&+ \frac{1}{\mu } ({\varvec{B}}{\cdot {\varvec{\nabla }}}){\varvec{B}} + \zeta \nabla ^2{{\varvec{v}}} \nonumber \\&+ \frac{\zeta }{3}{{\varvec{\nabla }}}({\varvec{\nabla }}\cdot {\varvec{v}}) \end{aligned}$$

(42)

The magnetic induction equation is given by

$$\begin{aligned} \frac{\partial {\varvec{B}}}{\partial t} = {\varvec{\nabla }}\times ({\varvec{v}}\times {\varvec{B}}) + \frac{1}{\mu \sigma } \nabla ^2{{\varvec{B}}} \end{aligned}$$

(43)

which can be simplified as

$$\begin{aligned} \frac{\partial {\varvec{B}}}{\partial t} +({\varvec{v}}{\cdot {\varvec{\nabla }}}){\varvec{B}} = ({\varvec{B}}{\cdot {\varvec{\nabla }}}){\varvec{v}} + \frac{1}{\mu \sigma } \nabla ^2{{\varvec{B}}} \end{aligned}$$

(44)

Here, $\sigma$ is the electrical conductivity of the fluid and relates to the magnetic diffusivity $\lambda$ as $\lambda =1/(\mu \sigma )$. The heat diffusion equation is given by

$$\begin{aligned}&\rho \left[ \frac{\partial }{\partial t}(C_V T) +({\varvec{v}}{\cdot {\varvec{\nabla }}})(C_V T)\right] \nonumber \\&\quad = {{\varvec{\nabla }}} (k_T {{\varvec{\nabla }}} T) +\chi - P({{\varvec{\nabla }}}\cdot {\varvec{v}}) \end{aligned}$$

(45)

where $C_V$, T, $k_T$, and $\chi$ are the specific heat at constant volume, temperature of the fluid, thermal conductivity of the fluid, and heat dissipation term in the fluid, respectively.

At this point, the Boussinesq approximation [1] comes in which is based on the smallness of the volume expansion coefficient $\alpha$. Since the range of $\alpha$ lies between $10^{-3}$ and $10^{-4}$ for most of the fluids, thus for small variation in temperature ($<10^\circ$), the variation in density is at the most 1$\%$. Consequently, the variations of specific heat, thermal conductivity, viscosity, etc., are also of the same order. So, the variations of these quantities with the temperature are small, and they are considered to be constant except in the buoyancy term of the equation for momentum transfer. The order of magnitude of the buoyancy force is comparable with that of the inertial term and hence cannot be neglected. Under this approximation, the velocity field becomes solenoidal (from Eq. 38),

$$\begin{aligned} {\varvec{\nabla }}\cdot {\varvec{v}}=0 \end{aligned}$$

(46)

and the momentum and heat transfer equations become

$$\begin{aligned} \frac{\partial {\varvec{v}}}{\partial t} +({\varvec{v}}{\cdot {\varvec{\nabla }}}){\varvec{v}}= & {} -\frac{1}{\rho _0} {{\varvec{\nabla }}} \left( P+\frac{B^2}{2\mu }\right) +\left( \frac{\rho }{\rho _0}\right) g {{\varvec{e_3}}} \nonumber \\&\quad +\frac{1}{\mu \rho _0} ({\varvec{B}}{\cdot {\varvec{\nabla }}}){\varvec{B}} + \nu \nabla ^2{{\varvec{v}}} \end{aligned}$$

(47)

and

$$\begin{aligned} \frac{\partial T}{\partial t} +({\varvec{v}}\cdot {{\varvec{\nabla }}})T = \kappa \nabla ^2 T \end{aligned}$$

(48)

respectively, where $\nu =\zeta /\rho _0$ denotes the kinematic viscosity, $\rho _0$ is the fluid density at the temperature $T_0$ of the lower plate, $\kappa =k_T/(\rho _0 C_V)$ is the thermal diffusivity and $\delta \rho =-\rho _0 \alpha (T-T_0)$ is the change in density due to the change in temperature. In the conduction state, the velocity is zero and consequently, there are no currents in the fluid. The temperature $T_s(z)$ in the conduction state of the fluid is independent of time and is given by

$$\begin{aligned} T_s(z) = T_0- \beta z \end{aligned}$$

(49)

where the adverse temperature gradient across the fluid layer is given by $\beta =\Delta T/d=(T_0-T_1)/d$. $T_1$ is the temperature of the upper plate. The variation of fluid density in the vertical direction is given by

$$\begin{aligned} \rho _s(z)=\rho _0(1+\alpha \beta z) \end{aligned}$$

(50)

and the pressure distribution by

$$\begin{aligned} P_s(z)= P_0 + \rho _0 g\left[ (d-z)+\frac{\alpha \beta }{2} (d^2-z^2)\right] -\frac{B_0^2}{2\mu } \end{aligned}$$

(51)

where $P_s(z)$ is the pressure of the fluid in the conduction state, $P_0$ is a constant pressure at the upper plate located at $z=d$, and $B_0$ is the magnitude of the applied external magnetic field. As soon as the convective state sets in, the fluid velocity becomes finite and the other quantities are modified as

$$\begin{aligned}&T_{s}(z) \rightarrow T(x,y,z,t)= T_{s}(z)+\theta (x,y,z,t) \end{aligned}$$

(52)

$$\begin{aligned}&\rho _{s}(z) \rightarrow \rho (x,y,z,t)= \rho _{s}(z) + \delta \rho (x,y,z,t)\end{aligned}$$

(53)

$$\begin{aligned}&P_{s}(z) \rightarrow P(x,y,z,t)= P_{s}(z) + p(x,y,z,t)\end{aligned}$$

(54)

$$\begin{aligned}&{\varvec{B_0}} \rightarrow {\varvec{B}}(x,y,z,t) = {\varvec{B_0}} + {\varvec{b}}(x,y,z,t) \end{aligned}$$

(55)

where $\delta \rho (x,y,z,t)$, $\theta (x,y,z,t)$, p(x, y, z, t) and ${\varvec{b}}(x,y,z,t)$ denote the changes in density, temperature, pressure and the uniform magnetic field, respectively, due to the onset of convection. Please note that the convective pressure p also includes the change due to the induced magnetic field. For Boussinesq fluids, the change in density is given by $\delta \rho = \rho _0 \alpha \theta$. Considering an external uniform magnetic field which is applied in the vertical direction ${\varvec{B}} = B_0{\varvec{e_3}} + {\varvec{b}}(x,y,z,t)$ and using the scaling factors as given in Sect. 2 for non-dimensionalization, the set of dimensionless equations come out to be

$$\begin{aligned}&\frac{\partial {\varvec{v}}}{\partial t}+({\varvec{v}}\cdot {\varvec{\nabla )}}{\varvec{v}}=-{\varvec{\nabla }} p + \nabla ^2{\varvec{v}} \nonumber \\&\quad +\,Q\left[ \frac{\partial {\varvec{b}}}{\partial z}+Pm({\varvec{b}}{\cdot {\varvec{\nabla }}}){\varvec{b}} \right] \nonumber \\&\quad +\, Ra\theta {\varvec{e_3}} \end{aligned}$$

(56)

$$\begin{aligned}&Pm\left[ \frac{\partial {\varvec{b}}}{\partial t} + ({\varvec{v}}\cdot {\varvec{\nabla )}}{\varvec{b}}\right] = \frac{\partial {\varvec{v}}}{\partial z}+ \nabla ^2{\varvec{b}} + Pm({\varvec{b}}{\cdot {\varvec{\nabla }}}){\varvec{v}} \end{aligned}$$

(57)

$$\begin{aligned}&Pr\left[ \frac{\partial \theta }{\partial t} + ({\varvec{v}}\cdot {\varvec{\nabla )}}\theta \right] =\nabla ^2\theta +v_3 \end{aligned}$$

(58)

$$\begin{aligned}&{\varvec{\nabla }}\cdot {\varvec{v}}=0 \end{aligned}$$

(59)

$$\begin{aligned}&{\varvec{\nabla }}\cdot {\varvec{b}}=0 \end{aligned}$$

(60)

The four dimensionless control parameters as mentioned in Sect. 2 are thermal Prandtl number Pr, magnetic Prandtl number Pm, Chandrasekhar number Q and Rayleigh number Ra. In the limit $Pm\rightarrow 0$ for the case of terrestrial fluids and modulating gravity, the above equations narrow down to the set of Eqs. 3–6.

Appendix II: The model

The 16-mode low-dimensional Galerkin model is presented below.

$$\begin{aligned} {\dot{W}}_{101}= & {} a_1 [a_2 W_{101} + a_3f(t) T_{101} + a_4 W_{011}W_{112}\nonumber \\&\quad +\,a_5 W_{211}W_{112} + a_6 W_{011}Z_{112} + a_7 W_{011}Z_{110}\nonumber \\&\quad +\, a_8 W_{211}Z_{112} + a_9 W_{121}Z_{220} + a_{10} W_{112}Z_{211}\nonumber \\&\quad +\, a_{11} W_{211}Z_{110} + a_{12} Z_{121}Z_{220} + a_{13} Z_{112}Z_{211}\nonumber \\&\quad +\, a_{14} Z_{211}Z_{110}] \end{aligned}$$

(61)

$$\begin{aligned} {\dot{W}}_{011}= & {} a_1 [a_2 W_{011} + a_3f(t) T_{011} + a_4 W_{101}W_{112}\nonumber \\&\quad +\, a_5 W_{121}W_{112} - a_6 W_{101}Z_{112} - a_7 W_{101}Z_{110}\nonumber \\&\quad -\, a_8 W_{121}Z_{112} - a_9 W_{211}Z_{220} - a_{10} W_{112}Z_{121}\nonumber \\&\quad -\, a_{11} W_{121}Z_{110} + a_{12} Z_{211}Z_{220} + a_{13} Z_{112}Z_{121}\nonumber \\&\quad +\, a_{14} Z_{121}Z_{110}] \end{aligned}$$

(62)

$$\begin{aligned} {\dot{W}}_{121}= & {} b_1 [b_2 W_{121} + b_3f(t) T_{121} + b_4 W_{011}W_{112}\nonumber \\&\quad +\, b_5 W_{211}W_{112} + b_6 W_{211}Z_{112} + b_7 W_{112}Z_{211}\nonumber \\&\quad +\, b_8 W_{211}Z_{110} + b_9 W_{011}Z_{110} + b_{10} W_{101}Z_{220}\nonumber \\&\quad +\, b_{11} W_{011}Z_{112} + b_{12} Z_{112}Z_{211} + b_{13} Z_{211}Z_{110}] \end{aligned}$$

(63)

$$\begin{aligned} {\dot{W}}_{211}= & {} b_1 [b_2 W_{211} + b_3f(t) T_{211} + b_4 W_{101}W_{112}\nonumber \\&\quad +\, b_5 W_{121}W_{112} - b_6 W_{121}Z_{112} - b_7 W_{112}Z_{121}\nonumber \\&\quad -\, b_8 W_{121}Z_{110} - b_9 W_{101}Z_{110} - b_{10} W_{011}Z_{220}\nonumber \\&\quad -\, b_{11} W_{101}Z_{112} + b_{12} Z_{112}Z_{121} + b_{13} Z_{121}Z_{110}] \end{aligned}$$

(64)

$$\begin{aligned} {\dot{W}}_{112}= & {} c_1 [c_2 W_{112} + c_3f(t) T_{112} + c_4 W_{101}W_{211}\nonumber \\&\quad +\, c_5 W_{011}W_{101} + c_6 W_{121}W_{211} + c_7 W_{011}W_{121}\nonumber \\&\quad +\, c_8 W_{011}Z_{121} + c_9 W_{101}Z_{211} + c_{10} W_{211}Z_{121}\nonumber \\&\quad +\, c_{11} W_{121}Z_{211} + c_{12} Z_{211}Z_{121}] \end{aligned}$$

(65)

$$\begin{aligned} {\dot{Z}}_{121}= & {} d_1 [d_2 Z_{121} + d_3 W_{211}W_{112} + d_4 W_{011}W_{112}\nonumber \\&\quad +\, d_5 W_{011}Z_{112} + d_6 W_{101}Z_{220} + d_7 W_{112}Z_{211}\nonumber \\&\quad +\, d_8 W_{211}Z_{110} + d_9 W_{211}Z_{112} + d_{10} W_{011}Z_{110}\nonumber \\&\quad +\, d_{11} Z_{112}Z_{211} + d_{12} Z_{211}Z_{110}] \end{aligned}$$

(66)

$$\begin{aligned} {\dot{Z}}_{211}= & {} d_1 [d_2 Z_{211} - d_3 W_{121}W_{112} - d_4 W_{101}W_{112}\nonumber \\&\quad +\, d_5 W_{101}Z_{112} + d_6 W_{011}Z_{220} + d_7 W_{112}Z_{121}\nonumber \\&\quad +\, d_8 W_{121}Z_{110} + d_9 W_{121}Z_{112} + d_{10} W_{101}Z_{110}\nonumber \\&\quad -\, d_{11} Z_{112}Z_{121} - d_{12} Z_{121}Z_{110}] \end{aligned}$$

(67)

$$\begin{aligned} {\dot{Z}}_{112}= & {} e_1 [e_2 Z_{112} + e_3 W_{101}W_{211} + e_4 W_{011}W_{121}\nonumber \\&\quad + e_5 W_{011}Z_{121} + e_6 W_{101}Z_{211} + e_7 W_{112}Z_{220}] \end{aligned}$$

(68)

$$\begin{aligned} {\dot{Z}}_{110}= & {} f_1 Z_{110} + f_2 W_{101}W_{211} + f_3 W_{011}W_{121}\nonumber \\&\quad +\, f_4 W_{211}Z_{121} + f_5 W_{121}Z_{211} + f_6 W_{101}Z_{211}\nonumber \\&\quad +\, f_7 W_{011}Z_{121} \end{aligned}$$

(69)

$$\begin{aligned} {\dot{Z}}_{220}= & {} g_1 Z_{220} + g_2 W_{011}W_{211} + g_3 W_{101}W_{121}\nonumber \\&\quad + g_4 W_{011}Z_{211} + g_5 W_{101}Z_{121} + g_6 W_{112}Z_{112} \end{aligned}$$

(70)

$$\begin{aligned} {\dot{T}}_{101}= & {} h_1 [h_2 T_{101} + h_3 W_{101} + h_4 W_{112}T_{211}\nonumber \\&\quad +\, h_5 W_{101}T_{002} + h_6 W_{011}T_{112} + h_7 W_{211}T_{112}\nonumber \\&\quad +\, h_8 Z_{211}T_{112} + h_9 Z_{110}T_{011} + h_{10} Z_{110}T_{211}\nonumber \\&\quad +\, h_{11} Z_{112}T_{011} + h_{12} Z_{112}T_{211} + h_{13} Z_{220}T_{121}] \end{aligned}$$

(71)

$$\begin{aligned} {\dot{T}}_{011}= & {} h_1 [h_2 T_{011} + h_3 W_{011} + h_4 W_{112}T_{121}\nonumber \\&\quad +\, h_5 W_{011}T_{002} + h_6 W_{101}T_{112} + h_7 W_{121}T_{112}\nonumber \\&\quad -\, h_8 Z_{121}T_{112} - h_9 Z_{110}T_{101} - h_{10} Z_{110}T_{121}\nonumber \\&\quad -\, h_{11} Z_{112}T_{101} - h_{12} Z_{112}T_{121} - h_{13} Z_{220}T_{211}] \end{aligned}$$

(72)

$$\begin{aligned} {\dot{T}}_{121}= & {} i_1 [i_2 T_{121} + i_3 W_{121} + i_4 W_{011}T_{112}\nonumber \\&\quad +\, i_5 W_{112}T_{011} + i_6 W_{211}T_{112} + i_7 W_{121}T_{002}\nonumber \\&\quad +\, i_8 Z_{211}T_{112} + i_9 Z_{110}T_{011} + i_{10} Z_{110}T_{211}\nonumber \\&\quad +\, i_{11} Z_{112}T_{011} + i_{12} Z_{112}T_{211} + i_{13} Z_{220}T_{101}] \end{aligned}$$

(73)

$$\begin{aligned} {\dot{T}}_{211}= & {} i_1 [i_2 T_{211} + i_3 W_{211} + i_4 W_{101}T_{112}\nonumber \\&\quad +\, i_5 W_{112}T_{101} + i_6 W_{121}T_{112} + i_7 W_{211}T_{002}\nonumber \\&\quad -\, i_8 Z_{121}T_{112} - i_9 Z_{110}T_{101} - i_{10} Z_{110}T_{121}\nonumber \\&\quad -\, i_{11} Z_{112}T_{101} - i_{12} Z_{112}T_{121} - i_{13} Z_{220}T_{011}] \end{aligned}$$

(74)

$$\begin{aligned} {\dot{T}}_{112}= & {} j_1 [j_2 T_{112} + j_3 W_{112} + j_4 W_{101}T_{211}\nonumber \\&\quad +\, j_5 W_{121}T_{011} + j_6 W_{011}T_{121} + j_7 W_{011}T_{101}\nonumber \\&\quad +\, j_8 W_{211}T_{101} + j_9 W_{121}T_{211} + j_{10} W_{101}T_{011}\nonumber \\&\quad +\, j_{11} W_{211}T_{121} + j_{12} Z_{211}T_{121} + j_{13} Z_{121}T_{011}\nonumber \\&\quad +\, j_{14} Z_{121}T_{211} + j_{15} Z_{211}T_{101}] \end{aligned}$$

(75)

$$\begin{aligned} {\dot{T}}_{002}= & {} k_1 [k_2 T_{002} + k_3 W_{011}T_{011} + k_4 W_{101}T_{101}\nonumber \\&\quad +\, k_5 W_{211}T_{211} + k_6 W_{121}T_{121}] \end{aligned}$$

(76)

The values of the coefficients are given below.

$a_1=\frac{1}{80(\pi ^2+k^2)}$, $a_2=-80[(\pi ^2+k^2)^2+\pi ^2Q]$, $a_3=80k^2Ra$, $a_4=20\pi (\pi ^2+k^2)$, $a_5=2\pi (17k^2-11\pi ^2)$, $a_6=10(k^2-\pi ^2)$, $a_7=20(\pi ^2-k^2)$, $a_8=9\pi ^2-5k^2$, $a_9=7\pi ^2-5k^2$, $a_{10}=2(3\pi ^2-k^2)$, $a_{11}=2(5k^2-\pi ^2)$, $a_{12}=-4\pi$, $a_{13}=-2\pi$, and $a_{14}=-4\pi$.

$b_1=\frac{1}{80(\pi ^2+5k^2)}$, $b_2=-80[(\pi ^2+5k^2)^2+\pi ^2Q]$, $b_3=400k^2Ra$, $b_4=20\pi (5k^2+13\pi ^2)$, $b_5=18\pi (5k^2+\pi ^2)$, $b_6=3(13\pi ^2-25k^2)$, $b_7=6(5k^2+\pi ^2)$, $b_8=6(25k^2+3\pi ^2)$, $b_9=20(3\pi ^2+5k^2)$, $b_{10}=10(5k^2-3\pi ^2)$, $b_{11}=50(\pi ^2-k^2)$, $b_{12}=18\pi$, and $b_{13}=36\pi$.

$c_1=\frac{1}{200(2\pi ^2+k^2)}$, $c_2=-400[(2\pi ^2+k^2)^2+\pi ^2Q]$, $c_3=200k^2Ra$, $c_4=-20\pi (\pi ^2+11k^2)$, $c_5=-200\pi (\pi ^2+k^2)$, $c_6=-18\pi (5k^2+\pi ^2)$, $c_7=-20\pi (\pi ^2+11k^2)$, $c_8=10(4\pi ^2-k^2)$, $c_9=10(k^2-4\pi ^2)$, $c_{10}=3(5k^2-8\pi ^2)$, $c_{11}=3(8\pi ^2-5k^2)$, and $c_{12}=-18\pi$.

$d_1=\frac{1}{80(\pi ^2+5k^2)}$, $d_2=-80[(\pi ^2+5k^2)^2+\pi ^2Q]$, $d_3=54\pi ^2(\pi ^2+5k^2)$, $d_4=-20\pi ^2(\pi ^2+5k^2)$, $d_5=100\pi (\pi ^2+5k^2)$, $d_6=-40\pi (\pi ^2+5k^2)$, $d_7=18\pi (\pi ^2+5k^2)$, $d_8=-16\pi (\pi ^2+5k^2)$, $d_9=-18\pi (\pi ^2+5k^2)$, $d_{10}=80\pi (\pi ^2+5k^2)$, $d_{11}=9(\pi ^2+5k^2)$, and $d_{12}=18(\pi ^2+5k^2)$.

$e_1=\frac{1}{10(2\pi ^2+k^2)}$, $e_2=-20[(2\pi ^2+k^2)^2+\pi ^2Q]$, $e_3=-2\pi ^2(2\pi ^2+k^2)$, $e_4=2\pi ^2(2\pi ^2+k^2)$, $e_5=-4\pi (2\pi ^2+k^2)$, $e_6=-4\pi (2\pi ^2+k^2)$, and $e_7=-5\pi (2\pi ^2+k^2)$.

$f_1=-2k^2$, $f_2=\frac{\pi ^2}{5}$, $f_3=-\frac{\pi ^2}{5}$, $f_4=-\frac{\pi }{20}$, $f_5=-\frac{\pi }{20}$, $f_6=-\frac{\pi }{10}$, and $f_7=-\frac{\pi }{10}$.

$g_1=-8k^2$, $g_2=\frac{2}{5}\pi ^2$, $g_3=-\frac{2}{5}\pi ^2$, $g_4=\frac{4}{5}\pi$, $g_5=\frac{4}{5}\pi$, and $g_6=\pi$.

$h_1=\frac{1}{80Pr}$, $h_2=-80(\pi ^2+k^2)$, $h_3=80$, $h_4=20\pi Pr$, $h_5=80\pi Pr$, $h_6=20\pi Pr$, $h_7=14\pi Pr$, $h_8=-2Pr$, $h_9=-20Pr$, $h_{10}=10Pr$, $h_{11}=10Pr$, $h_{12}=-5Pr$, and $h_{13}=-5Pr$.

$i_1=\frac{1}{80Pr}$, $i_2=-80(\pi ^2+5k^2)$, $i_3=80$, $i_4=60\pi Pr$, $i_5=-40\pi Pr$, $i_6=18\pi Pr$, $i_7=80\pi Pr$, $i_8=6Pr$, $i_9=20Pr$, $i_{10}=30Pr$, $i_{11}=-10Pr$, $i_{12}=-15Pr$, and $i_{13}=10Pr$.

$j_1=\frac{1}{40Pr}$, $j_2=-80(2\pi ^2+k^2)$, $j_3=40$, $j_4=-30\pi Pr$, $j_5=-14\pi Pr$, $j_6=-30\pi Pr$, $j_7=-20\pi Pr$, $j_8=-14\pi Pr$, $j_9=-9\pi Pr$, $j_{10}=-20\pi Pr$, $j_{11}=-9\pi Pr$, $j_{12}=-3Pr$, $j_{13}=-2Pr$, $j_{14}=3Pr$, and $j_{15}=2Pr$.

$k_1=\frac{\pi }{4Pr}$, $k_2=-16\pi$, $k_3=-2Pr$, $k_4=-2Pr$, $k_5=-Pr$, and $k_6=-Pr$.