Don Huynh, Stanislav Boldyrev , Vladimir Pariev University of Wisconsin - Madison

1. Analog of Astrophysical Magnetorotational Instability in a Couette-Taylor Flow of Polymer Fluids Don Huynh, StanislavBoldyrev, Vladimir Pariev University of Wisconsin - Madison

2. Acknowledgements Mark Anderson RiccardoBonazza Cary Forest Michael Graham Daniel Klingenberg

3. Mechanical Analog of MRI • Two particles in different orbital radii connected by a weak spring • The particle at the smaller radius is moving at a faster velocity than the particle at the larger radius • This causes the spring to stretch • Since the spring wants to restore equilibrium it slows the particle at the smaller radius down while speeds up the particle at larger radius • The particle at smaller radius falls into a lower orbit and the particle at larger radius moves into a higher orbit, which further stretches the spring • Leads to instability

4. Magnetorotational Instability • Two fluid elements connected by magnetic field lines • Magnetic field lines act as the spring • Elastic polymer act as magnetic field

5. Comparison of MHD and Viscoelastic Fluid Equations MHD Momentum Equation Viscoelastic Fluid Momentum Equation

6. Comparison of MHD and Polymer Solution Equations From the induction equation and the Oldroyd-B constitutive equation, Tm and Tpsatisfy the following equations, If η → 0 and τ → ∞, one can neglect the dissipation terms

7. Ogilvie and Proctor (2003) Narrow Gap Solution • In the limit of a narrow gap (ΔR/R << 1) cylindrical Couette flow is equivalent to a plane Couette flow (linear shear flow) in a rotating channel

8. Ogilvie and Proctor (2003) Basic Flow The basic flow is the plane Couette flow: The polymeric stress is and can be represented using three auxiliary fields

9. Linear Perturbations

10. “Elasto-Rotational” Instability Consider unsheared (“axisymmetic”) modes (ky = 0) and WKB approximation with solutions of the form uxα sin(k xx) Instability first appears at a stationary bifurcation (s = 0) For a Keplerian profile ( ), the Rossby number ( , where is Oort’s first constant ) is ¾ => A = ¾Ω

11. “Elasto-Rotational” Instability In the limit of τ→∞ with kz2 >> kx2 the dispersion relation for the onset of instability is If we identify , the critical angular velocity is identical to the ideal MHD case for a magnetic field along the z-axis.

12. Experimental Setup • Two concentric cylinders are attached to a motor on different gear ratios to rotate at different angular velocities • Filled between the two cylinders is a polymer solution

13. Experiment continued • Want to see how the polymer behaves under Keplerian angular velocity profile • Reflective particles added to visualize the fluid flow • Instability can be easily seen m = 0 mode

14. Phys. Rev. E 80, 066310 (2009) Results • onset of instability agrees qualitatively with computations • when sign of ∂Ω/∂r was reversed no instability detected • suggests instability observed is different from purely elastic instability • Keplerian profile => r12Ω1 < r22Ω2 Rayleigh’s inertial instability requires r12Ω1 > r22Ω2 so observed instability is different

15. Numerical Simulations • Work in progress • Try to use results to find an ideal polymer to use

16. Conclusion There is a close analogy between an electrically conducting fluid containing a magnetic field and a viscoelastic fluid The instability observed are different from purely elastic instability and Rayleigh’s inertial instability This instability is analogous to the MRI of a vertical magnetic field and can be used to study MRI in a lab setting.