Appendix 1: Governing equations

For (h/a) <<1, and assuming that inertia is negligible, one can show [4,7] that for an Oldroyd-B fluid, the dimensionless equations (after eliminating pressure) for axisymmetric flow in cylindrical coordinates are as follows:

Continuity equation  

$$\frac{\partial u}{\partial r}+\frac{u}{r}+\frac{\partial w}{\partial z}=0.$$ (184)

Momentum equations

 

$$(1-\beta)\frac{\partial^{3} u}{\partial z^{3}}+\frac{\partia... ...\Sigma}{\partial z}-\frac{\partial \Delta} {\partial z})=0,$$ (185)

 

$$(1-\beta)\frac{\partial^{2} v}{\partial z^{2}}+\frac{\partia... ...artial r}+\frac{2}{r}\zeta+\frac{\partial\Pi}{\partial z}=0.$$ (186)

Constitutive equations

 

$${\rm De}{\partial\Sigma\over\partial t}+ \Sigma =-{\rm De} [... ...ial u}{\partial r} -2\gamma \frac{\partial u}{\partial z}],$$ (187)

$$\begin{eqnarray} {\rm De}{\partial\zeta\over\partial t}+ \zeta &= & - {\rm De} [... ...}{\partial z}-\gamma \frac{\partial v}{\partial z} ], \nonumber \end{eqnarray}$$

   $$\begin{eqnarray} {\rm De}{\partial\gamma\over\partial t}+ \gamma &=& - {\rm De} ... ... {\partial r}] + \beta \frac{\partial u}{\partial z}, \nonumber \end{eqnarray}$$

   $$\begin{eqnarray} {\rm De}{\partial\Delta\over\partial t}+ \Delta &=& -{\rm De} [... ...frac{u}{r}\Delta-2\Pi \frac{\partial v}{\partial z}], \nonumber \end{eqnarray}$$

   $$\begin{eqnarray} {\rm De}{\partial\Pi\over\partial t}+ \Pi &=& -{\rm De} [u\frac... ...\partial r}] +\beta \frac{\partial v}{\partial z}, \nonumber \end{eqnarray}$$

   $$\begin{eqnarray} {\rm De}{\partial\Gamma\over\partial t}+ \Gamma & =& -{\rm De} ... ... z} ]\\ & & +2\beta \frac{\partial w}{\partial z}. \nonumber \end{eqnarray}$$

Boundary conditions  

$$u=w=0, \hspace{.2in}v=0 \hspace{.2in} \mbox{on} \hspace{.2in} z=0,$$ (188)

 

$$u=w=0, \hspace{.2in} v=r, \hspace{.2in} \mbox{on} \hspace{.1in} z=1.$$ (189)

For this model the domain is taken to be the infinite region $r \ge 0$.