## 3 Linearized System  
The linearized equations about the motionless base state (4) for a two-dimensional disturbance can be written in terms of a streamfunction \(\psi\), defined in the usual way for incompressible flow. In dimensionless form, the streamfunction-vorticity equation is given by:  
\[\frac{1}{\Pr}\left(\psi_{zzt}+\psi_{xxt}\right)=\psi_{xxxx}+2\psi_{xxzz}+\psi_{ zzzz}. \tag{5}\]  
Subscripting by coordinates refers to partial differentiation by the respective coordinate. The pressure, which is needed for the normal stress boundary condition, can be found from  
\[p_{x}=\psi_{xxz}+\psi_{zzz}-\frac{1}{\Pr}\psi_{zt}. \tag{6}\]  
The mathematical analysis leading to Equations (5) and (6) is outlined in Davis and Homsy [11]. The linear stability analysis is a standard mathematical approach having linearized the equations about the base state. Heat transport couples temperature convection and diffusion. The linearized version is  
\[T_{t}=T_{xx}+T_{zz}+\psi_{x}. \tag{7}\]  
We apply seven boundary conditions at the upper and lower surfaces. At \(z=0\), material surface (no penetration):  
\[\psi=0; \tag{8}\]  
no slip:  
\[\psi_{z}=0; \tag{9}\]  
fixed temperature:  
\[T=0. \tag{10}\]  
At \(z=1\), normal stress:  
\[p=-2\psi_{xz}+Gh-K^{-1}h_{xx}; \tag{11}\]  
tangential stress:  
\[0=\psi_{zz}-\psi_{xx}+M\left(T_{x}+h_{x}\right); \tag{12}\]  
material surface (kinematic condition):  
\[h_{t}=-\psi_{x}; \tag{13}\]  
no heat flux:  
\[T_{z}=0. \tag{14}\]  
We presume separation of variables with a factor that is a wave of real wavenumber \(k\) and (complex) phase velocity \(c\) by normal mode expansion of the field variables:  
\[p_{x}=\psi_{xxz}+\psi_{zzz}-\frac{1}{\Pr}\psi_{zt}. \tag{6}\]0 \[p_{x}=\psi_{xxz}+\psi_{zzz}-\frac{1}{\Pr}\psi_{zt}. \tag{6}\]1 \[p_{x}=\psi_{xxz}+\psi_{zzz}-\frac{1}{\Pr}\psi_{zt}. \tag{6}\]2 \[p_{x}=\psi_{xxz}+\psi_{zzz}-\frac{1}{\Pr}\psi_{zt}. \tag{6}\]3The wave system is thus reduced to a two-point boundary value problem of ordinary differential equations (ODEs) in \(P\left(z\right),\Psi\left(z\right),\Theta\left(z\right)\) and \(H\), with algebraic constraints: Bulk equations,  
\[p_{x}=\psi_{xxz}+\psi_{zzz}-\frac{1}{\Pr}\psi_{zt}. \tag{6}\]4  
\[p_{x}=\psi_{xxz}+\psi_{zzz}-\frac{1}{\Pr}\psi_{zt}. \tag{6}\]5  
\[p_{x}=\psi_{xxz}+\psi_{zzz}-\frac{1}{\Pr}\psi_{zt}. \tag{6}\]6  
Boundary conditions, at \(z=0\),  
\[p_{x}=\psi_{xxz}+\psi_{zzz}-\frac{1}{\Pr}\psi_{zt}. \tag{6}\]7 \[p_{x}=\psi_{xxz}+\psi_{zzz}-\frac{1}{\Pr}\psi_{zt}. \tag{6}\]8 \[p_{x}=\psi_{xxz}+\psi_{zzz}-\frac{1}{\Pr}\psi_{zt}. \tag{6}\]9  
At \(z=1\),  
\[T_{t}=T_{xx}+T_{zz}+\psi_{x}. \tag{7}\]0 \[T_{t}=T_{xx}+T_{zz}+\psi_{x}. \tag{7}\]1 \[T_{t}=T_{xx}+T_{zz}+\psi_{x}. \tag{7}\]2 \[T_{t}=T_{xx}+T_{zz}+\psi_{x}. \tag{7}\]3  
The algebraic manipulation package Mathematica 5.2 (Wolfram, Champaign, IL, USA) was used to manipulate the governing equations for this system. As the bulk equations are linear ODEs with constant coefficients, the general solutions can be represented by linear combinations of complex exponentials given by the characteristic roots, \(\pm k\pm\lambda=\sqrt{k^{2}-\frac{ikc}{\text{Pr}}}\) and \(\pm\Lambda=\sqrt{k^{2}-ikc}\), yielding  
\[T_{t}=T_{xx}+T_{zz}+\psi_{x}. \tag{7}\]4 \[T_{t}=T_{xx}+T_{zz}+\psi_{x}. \tag{7}\]5 \[T_{t}=T_{xx}+T_{zz}+\psi_{x}. \tag{7}\]6  
The seven boundary conditions (19) and (20) reduce to a matrix equation in the seven unknowns \(\alpha=\left[A,B,C,D,F,J,H\right]^{T}\), \(\mathbf{Z}\alpha=0\), where the matrix \(\mathbf{Z}=\mathbf{Z_{stick}}\) is given by  
\[\mathbf{Z_{stick}}=\left[