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

Acknowledgements Mark Anderson Mark Anderson Riccardo Bonazza Riccardo Bonazza Cary Forest Cary Forest Michael Graham Michael Graham Daniel Klingenberg Daniel Klingenberg 2

Mechanical Analog of MRI Two particles in different orbital radii connected by a weak spring 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 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 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 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 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 Leads to instability 3

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

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

Comparison of MHD and Polymer Solution Equations From the induction equation and the Oldroyd-B constitutive equation, T m and T p satisfy the following equations, 6 If η 0 and τ, one can neglect the dissipation terms

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 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 7Ogilvie and Proctor (2003)

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

Linear Perturbations 9

Elasto-Rotational Instability Consider unsheared (axisymmetic) modes (k y = 0) and WKB approximation with solutions of the form u x α sin(k x x) Instability first appears at a stationary bifurcation (s = 0) For a Keplerian profile ( ), the Rossby number (, where is Oorts first constant ) is ¾ => A = ¾Ω 10

Elasto-Rotational Instability In the limit of τ with k z 2 >> k x 2 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. 11

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

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

Results onset of instability agrees qualitatively with computations onset of instability agrees qualitatively with computations when sign of Ω/r was reversed no instability detected when sign of Ω/r was reversed no instability detected suggests instability observed is different from purely elastic instability suggests instability observed is different from purely elastic instability Keplerian profile => r 1 2 Ω 1 r 2 2 Ω 2 so observed instability is different Keplerian profile => r 1 2 Ω 1 r 2 2 Ω 2 so observed instability is different 14Phys. Rev. E 80, (2009)

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

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