Appendix 2: Linearized equations

 

$$\rho \frac{\partial u}{\partial \rho}-\frac{\partial v}{\partial \psi}=0;$$ (216)

 

$$\rho\frac{\partial p}{\partial \rho} =\rho\frac{\partial \Si... ...al \zeta}{\partial \psi} -\Delta +(1-\beta) \nabla^{2}u,$$ (217)

 

$$-\frac{\partial p}{\partial \psi}= \rho \frac{\partial \zeta... ...-\frac{\partial \Gamma}{\partial \psi}+(1-\beta)\nabla^{2}v,$$ (218)

 

$$0=\rho\frac{\partial \gamma}{\partial \rho} -\frac{\partial \Pi}{\partial \psi}+(1-\beta)\nabla^{2}w,$$ (219)

 

$${\rm De}\frac{\partial \Sigma}{\partial t}+\Sigma=2\beta\rho\frac{\partial u} {\partial \rho}$$ (220)

 

$${\rm De}\frac{\partial \zeta}{\partial t}+\zeta= \beta (\rh... ...partial v}{\partial \rho}-\frac{\partial u}{\partial \psi}),$$ (221)

 

$${\rm De}\frac{\partial \gamma}{\partial t}+\gamma=-\mbox{We}... ...artial \psi})+\beta \rho \frac{\partial w} {\partial \rho},$$ (222)

 

$${\rm De}\frac{\partial \Gamma}{\partial t}+\Gamma=-2\beta \frac{\partial v} {\partial \psi},$$ (223)

 

$${\rm De}\frac{\partial \Pi}{\partial t}+\Pi=-\mbox{We}(\Gamm... ...l v}{\partial \psi})-\beta \frac{\partial w}{\partial \psi},$$ (224)

 

$${\rm De}\frac{\partial \Delta}{\partial t}+\Delta=-2{\rm De}( \Pi-\beta \frac{\partial w}{\partial \psi}).$$ (225)