Appendix 1: Governing equations

For axisymmetric, creeping flow the leading order equations (for $\alpha<<1$), after dropping the subscripts, are [3]:  

$$\rho\frac{\partial u}{\partial \rho}- \frac{\partial v}{\partial \psi}=0,$$ (209)

 

$$\rho\frac{\partial p}{\partial \rho} =\rho\frac{\partial \Si... ...al \zeta}{\partial \psi} -\Delta +(1-\beta) \nabla^{2}u,$$ (210)

 

$$-\frac{\partial p}{\partial \psi}= \rho \frac{\partial \zeta... ...-\frac{\partial \Gamma}{\partial \psi}+(1-\beta)\nabla^{2}v,$$ (211)

 

$$0=\rho\frac{\partial \gamma}{\partial \rho} -\frac{\partial \Pi}{\partial \psi}+(1-\beta)\nabla^{2}w,$$ (212)

   $$\begin{eqnarray} \Sigma+De\frac{\partial \Sigma}{\partial t}&=& -\mbox{We} (u... ...\ & & +2\beta\rho\frac{\partial u}{\partial \rho}, \nonumber \end{eqnarray}$$

   $$\begin{eqnarray} \zeta+De\frac{\partial \zeta}{\partial t}&=& -\mbox{We}(u\rho... ...partial \rho}- \frac{\partial u}{\partial \psi}), \nonumber \end{eqnarray}$$

   $$\begin{eqnarray} \gamma + {\rm De}\frac{\partial \gamma}{\partial t}&=& -\mbox{... ...si} ) +\beta \rho \frac{\partial w}{\partial \rho}, \nonumber \end{eqnarray}$$

 

$$\Gamma+{\rm De}\frac{\partial \Gamma}{\partial t} = - \mbo... ...{\partial \psi}) -2\beta\frac{\partial v}{\partial \psi},$$ (213)

   $$\begin{eqnarray} \Pi+{\rm De}\frac{\partial \Pi}{\partial t}&=& -\mbox{We} (u\... ...tial \psi}) -\beta \frac{\partial w}{\partial \psi},\nonumber \end{eqnarray}$$

and  

$$\Delta+{\rm De} \frac{\partial \Delta}{\partial t}= -\mbox... ... -\gamma \rho \frac{\partial w}{\partial \rho}). \nonumber$$

In these equation the Deborah number is defined based on the rotation rate ${\rm De}=\lambda\Omega$, while the Weissenberg number is based on the shear rate ${\rm We}=\lambda\dot\gamma=\lambda\Omega/\alpha$.The no-slip boundary conditions in this case are  

$$u=v=w=0, \; \mbox{on} \; \psi=0,$$ (214)

and  

$$u=v=0, \; w=1, \; \mbox{on} \;\psi=1.$$ (215)

In addition, we require u, v, and w to be bounded at $\rho =0$ and as $\rho \rightarrow \infty$. The Laplacian is defined as

$$\nabla^{2} \equiv \rho^{2}\frac{\partial^{2}}{\partial \rho^... ...artial}{\partial \rho}+\frac{\partial^{2}}{\partial \psi^{2}}.$$