THE EVOLUTION OF LINEARIZED PERTURBATIONS IN A MAGNETOHYDRODYNAMIC BAROCLINIC
COUETTE FLOW
Vijayalakshmi A.R.
Department of Mathematics, Maharani’s Science College for Women, Bangalore – 560 001, India, 08088854962.
ABSTRACT
By using the initial value problem approach the evolution of linearized perturbations in a magnetohydrodynamic shear flow is studied. Here the resulting equation in time posed by using Fourier transform is solved for the Fourier amplitudes for baroclinic couette flow with a point source of the field of transverse velocity and magnetic field as the initial disturbance. Solutions are obtained for small values of Alfven velocity. The velocity and magnetic field plots are drawn for different values of Alfven velocity.
Key words : Baroclinic couette flow, initial value problem, Fourier transform, Alfven velocity.
INTRODUCTION
The stability of electrically conducting shear flow has various applications in geophysics and astrophysics.
The stability of plane Poiseuille flow was investigated by [1] with the assumption that the mean magnetic field is every where constant. In this case the stability equation is similar to that of Orr-Sommerfeld equation with only one new term. [2] derived the general stability equation for small magnetic Reynolds number and obtained numerical results for the case where the initial perturbations of the magnetic field vanish. [3] proved that, when a uniform magnetic field is parallel to the flow and sufficiently large, the wave number vector of the most unstable disturbance is not, in general parallel to the flow i,e., it is a three – dimensional disturbance. By using the method of separation of variables [4] studied the stability of dissipative magnetohydrodynamic shear flow in a parallel magnetic field for unbounded plane Couette flow.
[5] have shown that the addition of small diffusivity, dissipation is strongly stabilising and there is eventual collapse of all the modes. Using energy method, stability criterion was obtained when the fluid is bounded by horizontal planes. [6] investigated the linear stability of an ideal, plane – parallel magnetohydrodynamic shear flow and found a new sufficient condition for the stability of the flow. [7] studied the effects of shear flow and Alfven waves on two – dimensional magnetohydrodynamic turbulence and found numerically the underlying physical mechanisms
for the reduction of turbulent transport and turbulence level by shear flow and magnetic field.
[8] investigated the evolution of magnetohydrodynamic shear flows with non-constant tranverse magnetic field both near the plate and far from it by analysing the possibility of reverse flow and the instability of the solutions.
[9] has investigated the evolution of linearized perturbations in a magnetohydrodynamic boundary layer and [10] studied the evolution of linearized perturbations in a baroclinic stratified couette flow. In the present paper, we have investigated for magnetohydrodynamic baroclinic couette flow, with unit pulse of velocity and magnetic field as initial conditions. Here the disturbances are resolved into rotational and irrotational components. The rotational solution is the solution for the hypothetical initial-value problem for which the mean flow is unbounded but coincides with the actual flow in the layer. The irrotational solution in each layer is specified uniquely by satisfying the interfacial conditions and boundary conditions.
MATHEMATICAL FORMULATION
The basic equations of motion for inviscid, incompressible, magnetohydrodynamic Boussinesq fluid ignoring gravity are
0 q .
, (1) 0
H
.
, (2)
q. q - P μm
H. Ht
ρ q
, (3)
q. H
H. qt
H
, (4)
where
H2 μm p
P , is the total pressure,
μmis the magnetic permeability.
The basic state of the system is (U
y σy,0,0) q0 , ,0,0)
(H0 H0
, (y)
P0
P , where
is the shear of the mean flow where and H0are assumed to be constants. To study the linear stability, we superimpose a small wave like perturbation upon the mean flow i,e., q
q0 q
, 0 H
H H
, P
P0
P where q , H
,P are the perturbed quantities of velocity, magnetic field and pressure respectively.
By employing (i) moving co-ordinates transformation, defined by T t, ξ x - σ y t, η y, ζz (ii) three - dimensional Fourier transformation given by
i αξ βη γζ dξ dηdζe T ζ; η; ξ; u T
γ; β; α;
uˆ
with similar expressions for
vˆ,wˆ , Hˆ x,
Hˆ y,Hˆz and Pˆ(iii) Squire transformation defined by α
α2γ2
12 and φ = arctan
γα , the velocity and magnetic field components in the α and φ directions are givenby α
wˆ γ αuˆ
u
,
α wˆ α γuˆ w
,
,
α Hˆz x γ Hˆ α
Hx
α
Hˆz x α Hˆ γ
Hz
.
The resulting equations are
2 K2 Hˆ y 0 VAiα vˆ K2 dT
d (5)
αvˆ T i
d Hˆy
d . (6)
where
ρ0 2 H0 μm 2
VA , VAis the Alfven velocity, ρ
0 is the equilibrium density.
β σαT
2α2
K2 and α2
α2γ2
.The pressure amplitude Pˆ is obtained by taking the divergence of the momentum equations. It is found that
K2 αvˆ σ
Pˆ i2 when K2 0.
From equation (5) we obtain two sets of solutions for vˆ, (i) for K20, when the disturbance is rotational and (ii) for K2 0, when the disturbance is irrotational, since the vanishing of the product k2vˆ is equivalent to Laplace equation 2vˆ0 in real space. But for
Hˆy solution exists for only K2 0, since for K2 0, the resulting equation implies that H is force free magnetic field i,e., there is no magnetic field. Hence 0
Hˆ y
2
which corresponds to irrotational solution, is not taken into account.
Now considering the case K2 0, we assume perturbation solution for
vˆR, the rotational component of vˆ for small values of VA2
Alfven velocity
in the form
α,β,γ,T
vˆ0
α,β,γ,T
VA2 vˆ1
α,β,γ,T
VA2 2 vˆ2
α,β,γ,T
...vˆR (7)
2 2Hˆy2
α,β,γ,T
...VA γ,T
β, 1α, Hˆy 2 VA γ,T β, 0 α, Hˆy γ,T
β, y α,
Hˆ (8)
We find that,
β σαT
2α2
γ β, 0 α, Ωˆ vˆ0
, (9)
α
σαT 1 β
tan 3 α
σαT β 3 1 α2
σαT2 2 β
log α 3 1 α
σαT 1 β
α tan σαT β σ
α3 Ωˆ0 vˆ1
α2
β σαT
23 1 α σαT β 3 1 α
σαT β σ
Ωˆ1 α3 2 i α
σαT 1 β
6 tan 1
, (10)
2
α σαT β α2
σαT 2 2 β
log α 3α
18σ Ωˆ0 α5 7 α
σαT 1 β
tan 3 α
σαT β vˆ2
3α 6σ
Ωˆ0 α5 4 α
σαT β 3α
10σ Ωˆ0 α5 α2
σαT 2 2 β
log α 5 α
σαT β 3α
36σ Ωˆ0 α5 5
9σ2 Ωˆ1 α5 8iα 3α
σ Ωˆ0 α5 2 α
σαT β 27σ2
Ωˆ1 α5 5iα 3α
3σ Ωˆ0 α5 3 2 α
σαT β
3α 3σ
Ωˆ0 α5 2 α
σαT 1 β
α tan σαT β 9σ2
Ωˆ1 α5 8iα 3α
σ Ωˆ0 α5 2 α
σαT 1 β
tan
α2 σαT 2 2 β
log α 3 α
σαT β α
σαT β σ2
Ωˆ1 α5 5 iα α
σαT β
β σαT
2α2
1 3σ2
Ωˆ1 α5 α 2i
, (11)
The solution for K2 0 is found by considering the perturbation equations where two – dimensional Fourier transform is used instead of the full three – dimensional decomposition.
Using moving co-ordinate transformation , K2vˆ0 corresponds to
α2 σ2α2T2
vI 0η vI αT 2 2iσ
η vI 2
, (12) where
iαξ γζ dξdζ
e ς,T η, I ξ, v γ;T
η, I α, I v v
, (13) is the irrotational part of v. The solution of equation (2.12) is found to be
T e αη iσαTη B
T eαη iσαTηI A
v
. (14) where A (T) and B(T) are constants of integration .
In order to combine vˆR and vI
to obtain the complete solution and satisfy the matching condition vˆRmust be inverted once to obtain vR
α,η,γ;T
i,e.,
-iβη dβe γ;T β, R α, vˆ 2π T 1
γ;
η, R α,
v
. (15)
With initial velocity and initial magnetic field given by
x,y,z,0
V0δ
x-x0 δy-y0 δz-z0v . (16)
Hy
x,y,z,0
H0δ
x-x0 δ y-y0 δz-z0 (17) In terms of moving co-ordinates and the three-dimensional Fourier transform is
γz0 βy0αx0 i 0e γ V β, 0 α, Ω γ β, 0 α, v
. (18)
γz0 βy0αx0 i 0e H~ γ β, 1α, Ω γ β, α, y0 H
(19)
vR
is found to be
3α 18σ
α5 4 5VA 3σ
α3 2 2VA V0 η e α 3α 36σ
α5 4 VA σα - 6
α3 2 VA - 0 1 V η 0 σαT 0 γz
αx i R e v
η η d
η α η η e α 3α
36σ α5 4 5VA 3σ
α3 2 VA V0 η η d
η
η α η η e α η
9σ2
H~0 α5 8iα 3α
σ V0 α5 η 2
e α 2
η i 27σ2
H~0 α3 α 46i 3α
3σ V0 α5 4 3σ
H~0 α3 2i
η η
dη . 2η α η η- α e- 2 i
9σ H~0 α5 iα 4 3α 10σ
V0 α5 η η d
η α η η- α e-
i
(20)
Now the complete solution will be
vI vR
v . (21) vR
and vI
are given by equations (20) and (14).
MAGNETOHYDRODYNAMIC BAROCLINIC PLANE COUETTE FLOW
The problem of magnetohydrodynamic baroclinic plane couette flow arises in geophysical fluid dynamics. The boundary condition for the problem is the pressure p must be constant for η H(Fig.1). By considering the Fourier transforms of the momentum and magnetic induction equations and solving for P
α,γ,η,T
and using the pressure matching condition i,e., the pressure is continuous across the interface , we obtain the relations
αH iσ αTH αH iσ αTH αH i σαTH
α e A α e B iσ α 1 αH e A
αH iσ αTH
iσ α 1 αH e B = F T
1
(22)
αH iσ αTH αH iσ αTH αH iσ αTH
α e A α e B iσ α 1 αH e A
αH iσ αTH
iσ α 1 αH e B F T
2
(23)
Fig. 1. Sketch of bounded magnetohydrodynamic Baroclinic plane Couette flow Where
2v v 2 Hy 2 2 iσ αTHF T iσ αT 2i σα v iα V α V σT H e
1 η T T A η A y
η H
(24)
2v v 2 Hy 2 2 - iσ αTHF T iσ αT 2i σα v iα V α V σT H e
2 η T T A η A y
η H
. (25)
Equations (22) and (23) can be written as
αH αH
αH αH 1 αH e 1 αH e F
αe αe A iσ α A 1
αH αH
αH αH B 1 αH e 1 αH e B F
αe αe 2
. (26)
Multiplying equation (26) by the inverse of the matrix on the left-hand side, the above system can be represented as
B A αH
2 coth αH
2 H 2
α 2 sinh
αH 2
αH 2 sinh
αH H 2
α 2 coth αH 2 2 2 cosφ iσ B
A
y ση σyU
ηy
ηH
,0,0) (H0 H0
ξ
ηH
αH αH
e F e F
1 2 1
αH αH
2α sinh 2α H e F e F
1 2
, (27)
with α αcosφ. With the solutions proportional to eωT, the values for eigen frequencies are
2αH
coth
2αH
2αH
2 124 2 4
φ cos
ω iσ . (28)
The eigenvectors for A and B are found to be
1T eω ν1
1T eω
&
2T ω 2e ν
2T ω
e (29)
respectively, where
2
2αH
coth
2αH
ν0
αH 2
αH 2 sinh
ν1,2 (30)
12 . αH 2 coth αH
2 2 4 αH 2 0 4
ν
(31)
The forced solution will result from the integration of
ω T αH αH αH αH ω T
ω T1 2 T e ν e F ν e e F e 1
e e 2 1 2 2
AB ν e1 ω T1 ν e2 ω T2 0 eαH ν e1 αH F1 ν e1 αH e αH F e2 ω T2 dT .
(32)
where A and B are found to be
αH αH αH αH ω T1
e ν e F ν e e F e
3 4
2 2
αH αH αH αH ω T2
e ν e1 F5 ν e1 e F e6 A
B eαH ν e αH F ν eαH e αH F ν eω T1
3 4 1
2 2
αH αH αH αH ω T2
e ν e F ν e e F ν e
5
1 1 6 2
. (33)
The values of the coefficients are given in APPENDIX.
RESULTS AND DISCUSSIONS
In this problem, we have studied the evolution of linearized perturbations of a basic flow of an inviscid magnetohydrodynamic baroclinic plane couette flow using piecewise linear velocity profiles by using unit pulse for velocity and magnetic field as initial distributions.
In these broken line (piecewise linear) profiles, we have resolved the perturbations into rotational and irrotational components. Plots are drawn to observe the variation of amplitude of rotational velocity
vˆR with time. Figs..2 (a)-(b) are plots of
vˆR Vs T for different values of VA(
VA= 0, 0.2, 0.5) and φ (φ 0 , 45 0 0). As time increases there is decay in vˆR . Figs..3 (a)- (b) are plots of ˆHy Vs T for different values of
VA(
VA= 0, 0.2, 0.5) and φ (φ 0 , 45 0 0). As time increases there is decay in ˆHy .
REFERENCES
1. J.T. Stuart, On the stability of viscous flow between parallel planes in the presence of co – planar magnetic field, Proc. R. Soc. Lond. A 221, (1954) 189.
2. F.D. Hains, Stability diagrams for magnetogasdynamic channel flow, Phys. Fluids, 8(11),(1965) 2014.
3. J.C.R. Hunt, On the stability of parallel flows with parallel magnetic field, Proc. R. Soc.
Lond. A, 293, (1966)342.
4. J. Lerner and E. Knobloch, The stability of dissipative magnetohydro-dynamic shear flow in a parallel magnetic field, Geophysics and Astrophysics, Fluid Dynamics, 33,(1985) 295.
5. M. Venkatachalappa and A.M. Soward, The stability of stratified conducting shear flow in an aligned magnetic field, Geophysics and Astrophysics, Fluid Dynamics, 54, (1990)109.
6. D.W. Hughes and S.M. Tobias, On the stability of magnetohyrodynamic shear flows, Proc. R. .Soc. London A, 457,(2001)1365
7. J.I. Douglas, Eun-jin Kim and A. Thyagaraja, Effects of flow shear and Alfven waves on two-dimensional magnetohydrodynamic turbulence, Physics of Plasma,15,(2008)23.
8. Nunez and Manuel, MHD shear flows with non-constant transverse magnetic field, Physics Letters A, 376, 19 (2012) 1624.
9. A. R. Vijayalakshmi and P.M. Balagondar, The Evolution of Linearized Perturbations in a Magnetohydrodynamic Boundary layer, Int J of Applied Mechanics and Engineering,19,2,(2014)397.
10. A.R. Vijayalakshmi and P.M. Balagondar, The Evolution of Linearized Perturbations in a Baroclinic stratified couette layer”, Int J of Multidispl.Research & Advcs in Engg, 6, III, (2014)1.
Fig. 2 Curves of
vˆR versus T for (a)
φ 0 o
and (b)φ 45 o
for different values of VA0 10 20 30 40 50
0 2 4 6 8 10 12
14 V
A
0 0.2 0.5
(b)
T
0 10 20 30 40 50
0 1 2 3 4 5 6
7 V
A
0 0.2 0.5
(a)
T
vˆR
vˆR
Fig. 3 Curves of versus T for (a) different
φ 0 o
and (b)φ 45 o
for different values ofVA APPENDIX
F1F|η H, F2F|η H ω T2
e 1 iσ αTH
F F e
3 1 iσ αy ω
0 2
,
ω T2
e 1 iσ αTH
F F e
4 2 iσ αy ω
0 2
ω T1
e 1 -iσ αTH
F F e
5 1 iσ αy ω
0 2
,
ω T1
e 1 -iσ αTH
F F e
6 2 iσ αy ω
0 2
52 3 4 5 α - y 8α α H
V α V α
i αx γz σαTy
0 0 0 A A 0 0
F e V 1- - e 1 α - y
0 6σ α 36σ α3 0 9σ2
α -y η α η 2 3
2 3 4 5 5 0 1- α - y 2V α V
V αA 5V αA 2iα e 0 A 0
V dη
0 3σ 18σ α3 σ α3 η - y η 3σ
0
0 10 20 30 40 50
0.0 0.2 0.4 0.6 0.8 1.0
1.2 V
A
0 0.2 0.5
(a)
T 0.00 10 20 30 40 50
0.2 0.4 0.6 0.8 1.0
VA
0 0.2 0.5
(b)
T
Hˆy Hˆy
Hˆy
5 5 -α - y - η α η 1 α - y
α V 2αα H
5 i 0 0 η e 0 0 1 dη
3 2 2
18 20 σ α 9σ α - y0 η - y0 η
2 2
3 5 5 -α - y - η 2V α α H
α H 2i α V 23αα H
0
0 0 0 e α y iσ α A 0
σ 3σ α3 9σ2 0 3σ
4
2 2 4 4 α y 4 4 4α V
2V α iV α
1 A A 0 2i α 2i α 0 4 2
V - - e H V iα V
0 σ 3 σ α 90 σ α2 4 2 0 9σ3 276 σ2 15 σ α4 2 A A
4 4 4
4i α V 6i α H iα V α - y η α η
α 1 V4 0 0 0 e 0
2 A 4 2 2 4 2
y0 y0 20σ α 9σ 3σ
α 1
i α VA2 y0 η y0 η 2 dη
.
