Problem formulation

In what follows we use cylindrical coordinates with the dimensionless velocity given by

$${\bf v}=(u,v,w)$$

(i.e. u is the radial, v the azimuthal and w the axial velocity) and the extra stress tensor

$$\mbox{\boldmath$\tau$}=\left ( \begin{array} {ccc} \Sigma &... ... \zeta&\Delta&\Pi \\ \gamma&\Pi&\Gamma \end{array} \right ).$$

One solution of this problem is the torsional flow given by

$${\bf v}=(0,rz,0)$$ (152)

and

$$\tau =\left ( \begin{array} {ccc} 0&0&0\\ 0&2r^{2}\beta{\rm De}&r\beta \\ 0&r\beta &0 \end{array} \right ).$$ (153)

Our goal is to determine the stability of this solution and to find the solutions which bifurcate from it when stability is lost. For ease of analysis we shall consider a class of similarity solutions which include the viscometric torsional solution. Introduce the following similarity variables

$$u=r\frac{d f}{d z}, \hspace{.2in} w=-2f(z), \hspace{.2in} v=rg(z),$$

$$\Sigma=r^{2}\hat{\Sigma}(z),\hspace{.2in} \zeta=r^{2}\hat{\zeta}(z), \hspace{.2in} \gamma=r\hat{\gamma}(z),$$

$$\Delta=r^{2}\hat{\Delta}(z), \hspace{.2in} \Pi=r\hat{\Pi}(z), \hspace{.2in} \Gamma=\hat{\Gamma}(z),$$

Substituting in the governing equations (Appendix 1) and dropping the hats we obtain the following.  

$$(1-\beta)\frac{\partial^{4} f}{\partial z^{4}}+ \frac{\part... ...l\Sigma} {\partial z}-\frac{\partial \Delta }{\partial z}=0$$ (154)

$$(1-\beta)\frac{\partial^2 g}{\partial z^{2}}+\frac{\partial \Pi} {\partial z}+4\zeta =0$$ (155)

$${\rm De} \frac{\partial \Sigma}{\partial t}+\Sigma=2{\rm De}... ...a} {\partial z}+\gamma \frac{\partial^{2} f}{\partial z^{2}})$$ (156)

$${\rm De} \frac{\partial \zeta}{\partial t}+\zeta={\rm De}(2f... ...al g}{\partial z}+\Pi \frac{\partial^{2} f} {\partial z^{2}})$$ (157)

$${\rm De} \frac{\partial \gamma}{\partial t}+\gamma-\beta \fr... ...f}{\partial z}+\Gamma \frac{\partial^{2} f} {\partial z^{2}})$$ (158)

$${\rm De} \frac{\partial \Delta}{\partial t}+\Delta=2{\rm De}... ...rtial \Delta} {\partial z}+\Pi \frac{\partial g}{\partial z})$$ (159)

$${\rm De} \frac{\partial \Pi}{\partial t}+\Pi-\beta \frac{\pa... ...artial f} {\partial z}+\Gamma \frac{\partial g}{\partial z}).$$ (160)

 

$${\rm De}\frac{\partial \Gamma}{\partial t}+\Gamma+4\beta \fr... ...\Gamma}{\partial z} -2\Gamma \frac{\partial f}{\partial z})$$ (161)

The boundary conditions become

$$f=f^{'}=g=0, \: \mbox{for} \: z=0$$ (162)

and

$$f=f^{'}=0, \: g=1 \: \mbox{for}\: z=1.$$ (163)