Magnetorotational-type Instability in Couette-Taylor Flow of a Viscoelastic Polymer Liquid

We describe an instability of viscoelastic Couette–Taylor flow that is directly analogous to the magnetorotational instability (MRI) in astrophysical magnetohydrodynamics, with polymer molecules playing the role of magnetic field lines. By determining the conditions required for the onset of instability and the properties of the preferred modes, we distinguish it from the centrifugal and elastic instabilities studied previously. Experimental demonstration and investigation should be much easier for the viscoelastic instability than for the MRI in a liquid metal. The analogy holds with the case of a predominantly toroidal magnetic field such as is expected in an accretion disk, and it may be possible to access a turbulent regime in which many modes are unstable.

Figure 1

Contours of the growth rate $\mathrm{Re}(s)/{\Omega }_1$, optimized with respect to $m$ and $k$, for case 1 ($p=-3/2$). Contour values increase from 0 to 0.35 in steps of 0.025 starting from the thick line (negative contours are not plotted). The preferred azimuthal mode number $m$ is indicated by numerals.

Figure 2

Eigenfunction of the preferred mode in case 1 at the onset of instability for $(\mathrm{De},\mathrm{Re})=(3.3,40)$, where $(m,k)=(11,95)$. The azimuthal velocity perturbation is indicated by color/grayscale. One vertical wavelength is shown.

Figure 3

Growth rate for case 2 ($p=+3/2$). Contour values increase from 0 to 0.07 in steps of 0.005. The preferred value of $m$ is 2 throughout. Note the different horizontal scale.

Figure 4

Growth rate for case 3 ($p=-3$). Contour values increase from 0 to 0.7 in steps of 0.05. Note the different vertical scale.