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
 Riccardo Bonazza
 Cary Forest
 Michael Graham
 Daniel Klingenberg

2
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
3
Magnetorotational Instability



Two fluid elements
connected by magnetic
field lines
Magnetic field lines act as
the spring
Elastic polymer act as
magnetic field
4
Comparison of MHD and
Viscoelastic Fluid Equations
MHD Momentum Equation
(
u
t
 u  u )   pM    TM   2 u
TM 
BB
0

B2
2 0
I
Viscoelastic Fluid Momentum Equation
(
u
t
 u  u )   pP    TP   2 u
5
Comparison of MHD and
Polymer Solution Equations
From the induction equation and the Oldroyd-B constitutive
equation, Tm and Tp satisfy the following equations,
Tm
 u  Tm  ( u )  Tm  Tm   u  
T
t
Tp
 u  Tp  ( u )  Tp  Tp   u  
T
t

0
1

[ B B  ( B ) B ]
(Tp 
2
2
p

I)
If η → 0 and τ → ∞, one can neglect the dissipation terms
6
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
Ogilvie and Proctor (2003)
7
Basic Flow
The basic flow is the plane Couette flow: u  2 Axy
 De
0
 1

The polymeric stress is Tp  p   De 2 De 2  1 0 
 
0
1 
 0
and can be represented using three auxiliary fields
Tp  B1 B1  B2 B2  B3 B3
1


0

1 
1
0  p 12 


0 p 2  
B1,2  (
) De  ( De 2  1) 2  , B3  (
) 0




1
0


 


Ogilvie and Proctor (2003)
8
Linear Perturbations
u(r , t )  u0 ( x)  Re[u( x)est ikyyikzz ]
T  T0  Re[t ( x)e st ikyy ikzz ]
(1) u x ' ik y u y  ik z u z  0
(2) [( s  2iAxk y )u x  2u y   p ' t xx ' ik y t xy  ik z t xz   (u x " k 2u x )
(3) [( s  2iAxk y )u y  2(  A)u x  ik y p  t xy ' ik y t yy  ik z t yz   (u y " k 2u y )
(4) [( s  2iAxk y )u z  ik z p  t xz ' ik y t yz  ik z t zz   (u z " k 2u z )
(5) t xx   [( s  2iAxk y )t xx  2iT0 xy k y u x ]  2  p u x '
(6) t xy   [( s  2iAxk y )t xy  2 At xx  T0 xy (u x ' ik y u y )  iT0 yy k y u x ]   p (u y ' ik y u x )
(7) t xz   [( s  2iAxk y )t xz  iT0 xy k y u z ]   p (u z ' ik z u x )
(8) t yy   [( s  2iAxk y )t yy  4 At xy  2T0 xy u y ' 2iT0 yy k y u y ]  2i  p k y u y
(9) t yz   [( s  2iAxk y )t yz  2 At xz  T0 xy u z ' iT0 yy k y u z ]  i  p (k z u z  k z u y )
(10) t zz   [( s  2iAxk y )t zz ]  2i  p k z u z
9
“Elasto-Rotational” Instability
Consider unsheared (“axisymmetic”) modes (ky = 0) and WKB approximation
with solutions of the form ux α sin(k xx)
 [qs  (q  p )(kx2  kz2 )]2 (kx2  kz2 )  4( A)q2kz2  4A pkz2 (kx2  kz2 )  0
Instability first appears at a stationary bifurcation (s = 0)
(  p )2 (kx2  kz2 )3  4( A)kz2  4A p kz2 (kx2  kz2 )  0
r d
A
A


For a Keplerian profile (   r ), the Rossby number ( Ro  , where
2 dr


3
2
is Oort’s first constant ) is ¾ => A = ¾Ω
(  p )2 (kx2  kz2 )3  c2kz2  3c2 p kz2 (kx2  kz2 )  0
 c2 
(  p )2 (k x2  k z2 )3
k z2 [3 p (k x2  k z2 )  1]
10
“Elasto-Rotational” Instability
In the limit of τ→∞ with kz2 >> kx2 the dispersion relation for the onset of
2 4
2
instability is  p kz  4( A)  4A p kz  0
0  p
If we identify B 
, the critical angular velocity is identical to the

2
z
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
Filled between the two
cylinders is a polymer
solution
12
Experiment continued


m = 0 mode

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
13
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
Phys. Rev. E 80, 066310 (2009)
14
Numerical Simulations

Work in progress

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
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.
16