Stability of torsional flow

To determine the stability of this solution we linearize equations (5.3)-(5.10) about the given solution to obtain  

$$(1-\beta)\frac{\partial^{4} f}{\partial z^{4}}+ \frac{\part... ...l\Sigma} {\partial z}-\frac{\partial \Delta }{\partial z}=0$$ (164)

$$(1-\beta)\frac{\partial^2 g}{\partial z^{2}}+\frac{\partial \Pi} {\partial z}+4\zeta =0$$ (165)

$${\rm De}\frac{\partial \Sigma}{\partial t}+\Sigma=0$$ (166)

$${\rm De}\frac{\partial \zeta}{\partial t}+\zeta-{\rm De}(\gamma +\beta\frac{\partial^{2} f}{\partial z^{2}})=0$$ (167)

$${\rm De}\frac{\partial \gamma}{\partial t}+\gamma -\beta \frac{\partial^{2} f}{\partial z^{2}}=0$$ (168)

$${\rm De}\frac{\partial \Delta}{\partial t}+\Delta -2{\rm De}(\Pi +\beta \frac{\partial g}{\partial z})=0$$ (169)

$${\rm De}\frac{\partial \Pi}{\partial t}+\Pi -{\rm De}(\Gamm... ...artial f}{\partial z}) -\beta \frac{\partial g}{\partial z}=0$$ (170)

 

$${\rm De}\frac{\partial \Gamma}{\partial t}+\Gamma +4\beta \frac{\partial f}{\partial z}=0$$ (171)

The boundary conditions are  

$$f=f{'}=g=0, \: \mbox{for} \: z=0,1.$$ (172)

We look for normal mode solutions of the form

$$(f,g,\Sigma,\zeta,\gamma, \Delta, \Pi, \Gamma )= e^{\sigma ... ...mma_{1}, \zeta_{1}, \gamma_{1}, \Delta_{1},\Pi_{1},\Gamma_{1})$$

Substituting into equation (5.13)-(5.20) and simplifying we obtain the following eigenvalue problem.  

$$\frac{\partial^{4} f_{1}}{\partial z^{4}} +\Lambda^{2} \frac{\partial^2 f_{1}}{\partial z^{2}}=0$$ (173)

and the boundary conditions  

$$f_{1}=\frac{\partial f_{1}}{\partial z}=0, \: \mbox{for} \: z=0,1.$$ (174)

where  

$$\Lambda^{2}=4\frac{\beta {\rm De}^{2}(\sigma^{2}+4\sigma+3+2\beta)} {(\sigma+1)^{2}[1+\sigma(1-\beta)]^{2}}$$ (175)

This is a standard Sturm-Louiville problem. It is straightforward to show that the eigenvalues are given by $\Lambda=2\kappa$ where $\kappa$ is a root of the transcendental equations  

$$\sin \kappa=0$$ (176)

and  

$$\tan \kappa =\kappa.$$ (177)

There are infinitely many eigenvalues the smallest of which is given by $\kappa =\pi$. Substituting in (5.24), we obtain a quartic equation for $\sigma$. One can then solve for the eigenvalue $\sigma$. The most dangerous mode corresponds to the smallest root $\kappa =\pi$. For $\beta=1$ the equation for the critical eigenvalue is given by

$$({\rm De}^{2}-\pi^{2})\sigma^{2}+2(2{\rm De}^{2}-\pi^{2})\sigma +(5{\rm De}^{2}-\pi^{2})=0.$$ (178)

The base viscometric flow is stable if $\Re(\sigma) <0$ and unstable if $\Re(\sigma) \gt$ The sign of $\sigma$ depends on the parameters ${\rm De}$ and $\beta$. If ${\rm De}$ is small all the eigenvalues have negative real parts and the flow is stable. As ${\rm De}$ increases past a critical value ${\rm De}_{c}$, the critical $\sigma$, which is real, changes sign from negative to positive and the flow becomes unstable. The critical value of ${\rm De}$ which can be obtained explicitly from (5.24) by setting $\sigma=0$ and solving for ${\rm De}$ is

$${\rm De}_{c}=\frac{\pi}{\sqrt{\beta (3+2\beta)}}$$ (179)

This result was first obtained by Phan-Thien [7]. The eigenfunction is given by  

$$f_{1}=1-\cos(2\pi z)$$ (180)

The other functions $g_{1},\Sigma_{1} \cdots$ can be obtained by backward substitution.