Appendix 2: Linearized equations

Let $W=W_{0}+\hat{W}, \phi=\phi_{0}+\hat{\phi}, \tau_{11}=\tau_{11}^{0}+\hat{\tau_{11}}, \tau_{33}=\tau_{33}^{0}+\hat{\tau_{33}}$ where $\hat{W}$ etc. are assumed to be small perturbations. Linearizing and dropping the hats we obtain the equations given below.  

$$3(1-\beta)\frac{\partial^2 W}{\partial \xi^{2}}+a(t) \frac{\... ...}} {\partial \xi}-\frac{\partial \tau_{11}}{\partial \xi}=0$$ (236)

 

$$\frac{\partial \phi}{\partial t}=-\frac{1}{2}\frac{\partial W}{\partial \xi}$$ (237)

 

$${\rm De} \frac{\partial \tau_{11}}{\partial t}+(1+De)\tau_{1... ...beta+{\rm De}\tau_{11}^{0}) \frac{\partial W}{\partial \xi}$$ (238)

 

$${\rm De}\frac{\partial \tau_{33}}{\partial t}+(1-2{\rm De})\... ...beta+{\rm De}\tau_{33}^{0}) \frac{\partial W}{\partial \xi}$$ (239)

Here

$$a(t)=6(1-\beta)+ Ca^{-1}e^{-\phi_{0}}+2(\tau_{33}^{0}-\tau_{11}^{0}).$$

Equations (7.12)-(7.15) must be solved together with prescribed initial conditions and the boundary conditions

$$W=0 \: \mbox{on} \: \xi =0 , 1.$$ (240)