Bifurcations

We saw in the last section that as the Deborah number increases past the value ${\rm De}_{c}$ the base torsional flow loses stability as one real eigenvalue crosses into the right half-plane. This is precisely the scenario for bifurcation to another steady state. Let

$${\rm De}={\rm De}_{c}+\lambda$$

Then as $\lambda$ increases from negative to positive the flow loses stability. In order to determine the bifurcations that occur at $\lambda=0$ we will use the center manifold theory as expounded in the previous part. Denote by $\phi$ the eigenfunction corresponding to the critical mode. Then the solution of the nonlinear problem is  

$${\bf q}=x\phi+y, \: <y,\phi^{*}\gt=0$$ (181)

where $\phi^{*}$ is the solution of the adjoint problem to (5.13)- (5.20) and (5.21). Due to the symmetry under reflection across the midplane, the equation on the center manifold has the form  

$$\frac{\partial x}{\partial t}=\sigma(\lambda)x+c(\lambda)x^{3}$$ (182)

and the center manifold is of the form  

$$y=x^{2}\psi(x,\lambda)+O(x^{3}).$$ (183)

The coefficient $\sigma$ is given by (5.24) with $\Lambda=2\pi$. In order to determine $\psi$ and c we substitute (5.30) into the governing equations (perturbed around the base solution) and simplify using (5.31) and (5.32). The resulting equations are then solved recursively by equating coefficients of the same powers of x. This calculation is tedious but can be done using a Computer Algebra System such as Maple or Mathematica. We give results for $\beta=1$ and $\beta= 1/2$ in the table 5.1.

 

$$\beta \sigma$$ c

$$\frac{\sqrt{2}\lambda}{\pi}+O(\lambda^{2}) -\frac{669\pi^{4}}{128} +O(\lambda)$$ 0.5

$$\frac{5\sqrt{5}\lambda}{3\pi}+O(\lambda^{2}) \frac{728\pi^{4}}{75} +O(\lambda)$$ 1.0

From this table, we see that for the Maxwell fluid ($\beta=1$) there is a subcritical pitchfork bifurcation at $\lambda=0$. The bifurcating solutions are therefore unstable. This results agrees with numerical results obtained by Walsh [11]. On the other hand for $\beta=0.5$ the bifurcation is supercritical and stable. This also agrees with the analysis of Crewther et al. [1]. In fact it can be shown [4]) that there is a value of $\beta= \beta_{c}$ such that for $\beta <\beta_{c}$ the bifurcation is supercritical and for $\beta\gt \beta_{c}$ it is subcritical. This value of $\beta_{c}$ is approximately 0.59.