Analog of Analog of Astrophysical Astrophysical

Magnetorotational Magnetorotational Instability in a Instability in a

Couette-Taylor Flow Couette-Taylor Flow of Polymer Fluidsof Polymer Fluids

Don Huynh, Stanislav Boldyrev, Vladimir Don Huynh, Stanislav Boldyrev, Vladimir ParievPariev

University of Wisconsin - MadisonUniversity of Wisconsin - Madison

AcknowledgementsAcknowledgements

Mark AndersonMark Anderson Riccardo BonazzaRiccardo Bonazza Cary ForestCary Forest Michael GrahamMichael Graham Daniel KlingenbergDaniel Klingenberg

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Mechanical Analog of Mechanical Analog of MRIMRI

Two particles in different orbital Two particles in different orbital radii connected by a weak springradii connected by a weak spring

The particle at the smaller radius The particle at the smaller radius is moving at a faster velocity than is moving at a faster velocity than the particle at the larger radiusthe particle at the larger radius

This causes the spring to stretch This causes the spring to stretch Since the spring wants to restore Since the spring wants to restore

equilibrium it slows the particle equilibrium it slows the particle at the smaller radius down while at the smaller radius down while speeds up the particle at larger speeds up the particle at larger radiusradius

The particle at smaller radius The particle at smaller radius falls into a lower orbit and the falls into a lower orbit and the particle at larger radius moves particle at larger radius moves into a higher orbit, which further into a higher orbit, which further stretches the springstretches the spring

Leads to instability Leads to instability

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Magnetorotational Magnetorotational InstabilityInstability

Two fluid elements Two fluid elements connected by connected by magnetic field linesmagnetic field lines

Magnetic field lines Magnetic field lines act as the springact as the spring

Elastic polymer act Elastic polymer act as magnetic fieldas magnetic field

4

Comparison of MHD and Comparison of MHD and Viscoelastic Fluid Viscoelastic Fluid

EquationsEquations MHD Momentum EquationMHD Momentum Equation

Viscoelastic Fluid Momentum Viscoelastic Fluid Momentum EquationEquation

5

2( ) MM

uu u T u

tp

����������������������������������������������������������������� �����

2( ) PP

uu u T u

tp

����������������������������������������������������������������� �����

2

0 02M

BB BT I

Comparison of MHD and Comparison of MHD and Polymer Solution Polymer Solution

EquationsEquations

2 2

0

( ) [ ( ) ]

1( ) ( )

mT

m m m

pT p

p p p p

Tu T u T T u B B B B

t

T

u T u T T u T It

From the induction equation and the Oldroyd-B From the induction equation and the Oldroyd-B constitutive equation, Tconstitutive equation, Tm m and Tand Tp p satisfy the satisfy the following equations,following equations,

6

If If ηη → 0 and → 0 and τ τ → ∞, one can neglect the → ∞, one can neglect the dissipation terms dissipation terms

Narrow Gap SolutionNarrow Gap Solution

In the limit of a narrow gap (In the limit of a narrow gap (ΔΔR/R << 1) R/R << 1) cylindrical Couette flow is equivalent to a cylindrical Couette flow is equivalent to a plane Couette flow (linear shear flow) in a plane Couette flow (linear shear flow) in a rotating channelrotating channel

7Ogilvie and Proctor (2003)

Basic FlowBasic Flow

2

1 0

2 1 0

0 0 1

pp

De

T De De

1 1 10 022 2 2

1,2 3

10

( ) ( 1) , ( ) 0

0 1

p pB De De B

����������������������������

The polymeric stress isThe polymeric stress is

and can be represented using three auxiliary fieldsand can be represented using three auxiliary fields

8

2u Axy

The basic flow is the plane The basic flow is the plane Couette flow: Couette flow:

1 1 2 2 3 3pT B B B B B B ������������������������������������������������������������������������������������

Ogilvie and Proctor (2003)

Linear PerturbationsLinear Perturbations0( , ) ( ) Re[ ( ) ]y zst ik y ik zu r t u x u x e

0 Re[ ( ) ]y zst ik y ik zT T t x e

9

2

2

2

(1) ' 0

(2) [( 2 ) 2 ' ' ( " )

(3) [( 2 ) 2( ) ' ( " )

(4) [( 2 ) ' ( " )

(5)

x y y z z

y x y xx y xy z xz x x

y y x y xy y yy z yz y y

y z z xz y yz z zz z z

xx

u ik u ik u

s iAxk u u p t ik t ik t u k u

s iAxk u A u ik p t ik t ik t u k u

s iAxk u ik p t ik t ik t u k u

t

0

0 0

0

0 0

[( 2 ) 2 ] 2 '

(6) [( 2 ) 2 ( ' ) ] ( ' )

(7) [( 2 ) ] ( ' )

(8) [( 2 ) 4 2 ' 2 ]

y xx xy y x p x

xy y xy xx xy x y y yy y x p y y x

xz y xz xy y z p z z x

yy y yy xy xy y yy y y

s iAxk t iT k u u

t s iAxk t At T u ik u iT k u u ik u

t s iAxk t iT k u u ik u

t s iAxk t At T u iT k u

0 0

2

(9) [( 2 ) 2 ' ] ( )

(10) [( 2 ) ] 2

p y y

yz y yz xz xy z yy y z p z z z y

zz y zz p z z

i k u

t s iAxk t At T u iT k u i k u k u

t s iAxk t i k u

““Elasto-Rotational” Elasto-Rotational” InstabilityInstability

Consider unsheared (“axisymmetic”) modes (kConsider unsheared (“axisymmetic”) modes (kyy = 0) and WKB = 0) and WKB approximation with solutions of the form uapproximation with solutions of the form uxx αα sin(k sin(k xxx)x)

Instability first appears at a stationary bifurcation (s = 0)Instability first appears at a stationary bifurcation (s = 0)

For a Keplerian profile ( ), the Rossby number For a Keplerian profile ( ), the Rossby number ( , where ( , where

is Oort’s first constant ) is ¾ => A = ¾is Oort’s first constant ) is ¾ => A = ¾ΩΩ

2 2 2 2 2 2 2 2 2 2[ ( )( )] ( ) 4 ( ) 4 ( ) 0p x z x z z p z x zqs q k k k k A q k A k k k

2 2 2 3 2 2 2 2( ) ( ) 4 ( ) 4 ( ) 0p x z z p z x zk k A k A k k k

2 2 2 3 2 2 2 2 2 2( ) ( ) 3 ( ) 0p x z c z c p z x zk k k k k k 2 2 2 3

22 2 2

( ) ( )

[3 ( ) 1]p x z

cz p x z

k k

k k k

10

ARo

2

r dA

dr

3

2r

““Elasto-Rotational” Elasto-Rotational” InstabilityInstability

In the limit of In the limit of τ→∞τ→∞ with k with kzz22 >> k >> kxx

22 the dispersion relation for the dispersion relation for the onset of the onset of

instability isinstability is

If we identify , the critical angular velocity is If we identify , the critical angular velocity is identical to the identical to the

ideal MHD case for a magnetic field along the z-axis.ideal MHD case for a magnetic field along the z-axis.

2 4 24 ( ) 4 0p z p zk A A k

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02 pzB

Experimental SetupExperimental Setup Two concentric Two concentric

cylinders are cylinders are attached to a attached to a motor on different motor on different gear ratios to gear ratios to rotate at different rotate at different angular velocitiesangular velocities

Filled between the Filled between the two cylinders is a two cylinders is a polymer solutionpolymer solution

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Experiment continuedExperiment continued

Want to see how Want to see how the polymer the polymer behaves under behaves under Keplerian angular Keplerian angular velocity profilevelocity profile

Reflective particles Reflective particles added to visualize added to visualize the fluid flowthe fluid flow

Instability can be Instability can be easily seeneasily seen

m = 0 m = 0 modemode

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ResultsResults

onset of instability agrees qualitatively with onset of instability agrees qualitatively with computationscomputations

when sign of ∂when sign of ∂ΩΩ//∂∂r was reversed no r was reversed no instability detectedinstability detected

suggests instability observed is different suggests instability observed is different from purely elastic instabilityfrom purely elastic instability

Keplerian profile => rKeplerian profile => r1122ΩΩ11 < r < r22

22ΩΩ22 Rayleigh’s inertial instability requires rRayleigh’s inertial instability requires r11

22ΩΩ11 > r> r22

22ΩΩ22 so observed instability is different so observed instability is different

14Phys. Rev. E 80, 066310 (2009)

Numerical SimulationsNumerical Simulations

Work in progressWork in progress Try to use results to find an ideal polymer Try to use results to find an ideal polymer

to useto use

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ConclusionConclusion

There is a close analogy between an There is a close analogy between an electrically conducting fluid containing electrically conducting fluid containing a magnetic field and a viscoelastic fluid a magnetic field and a viscoelastic fluid

The instability observed are different The instability observed are different from purely elastic instability and from purely elastic instability and Rayleigh’s inertial instabilityRayleigh’s inertial instability

This instability is analogous to the MRI This instability is analogous to the MRI of a vertical magnetic field and can be of a vertical magnetic field and can be used to study MRI in a lab setting.used to study MRI in a lab setting.

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