On the Hydromagnetic Stability of a Rotating Fluid Annulus

The linear stability of a rotating fluid in the annulus between two concentric cylinders is investigated in the presence of a magnetic field which is azimuthal as well as in axial direction. Several results of MHD stability have been derived by using the inner product method. It is shown that when the swirl velocity component is large, the hydromagnetic effects become small compared with those due to swirl. The presence of a velocity field and imposed magnetic field will lead to the basic state to more stability.


Introduction
Howard and Gupta [1] investigated the stability of inviscid flows between two concentric cylinders which have an axial velocity component depending only on r in addition to a swirl velocity component in the direction of increasing azimuthal angle θ.Acheson [2] studied the hydromagnetic instability of a uniformly rotating fluid in the annular region between two concentric infinitely long cylinders.Following that analysis, a detailed hydromagnetic instability arising out of such a configuration has been simplified to yield a series of stability conditions.It has been shown that regardless of the magnetic field profiles, any unstable disturbance must have the ratio between angular velocity with the phase velocity in the azimuthal direction as negative and hence must propagate against the basic rotation.Zhang and Busse [3] investigated the instability of an electrically conducting fluid of magnetic diffusivity and viscosity in a rapidly rotating sphere when toroidal magnetic field is present.Liu et al. [4] examined the stability of an azimuthal base flow for both axisymmetric and plane-polar disturbances for an electrically conducting fluid confined between stationary, concentric, infinitely-long cylinders.When an axial magnetic field is applied, the interaction between the radial electric current and the magnetic field gives rise to an azimuthal electromagnetic body force which drives an azimuthal velocity and infinitesimal axisymmetric disturbances to an instability in the flow.Goodman and Ji [5] investigated axisymmetric stability of viscous resistive magnetized Couette flow, with emphasis on flow that would be hydrodynamically stable according to Rayleigh's criterion: opposing gradient of angular velocity and specific angular momentum.In this regime, magnetorotational instabilities may occur.In studies of magnetic Taylor-Couette flow in the presence of an imposed axial magnetic field, Willis and Barenghi [6] find that values of the imposed magnetic field which alters only slightly, the transition from circular-Couette flow to Taylor-vortex can shift the transition from Taylor-vortex flow to wavy modes by a substantial amount.Deka and Gupta [7] have analyzed linear stability of wide-gap MHD dissipative Couette flow of an incompressible electrically conducting fluid between two rotating concentric circular cylinders when a uniform axial magnetic field is present.Rajaee and Shoki [8] considered the case when a transition layer exists between two fluids, and both density and magnetic field change across this layer.The numerical calculations show that while the increase of the Mach number and compressibility have a destabilizing influence, the increases in magnetic field strength and density provide a stabilizing effect.Jasmine [9] investigated stability of radial flow subjected to a radial magnetic field.The stability condition derived is shown to remain valid even when the local velocity is not entirely radial, and that the magnetic field exerts a stabilizing effect on the flow.We have extended this work when both velocity components and magnetic field components are azimuthal as well as axial.
In this presentation, we consider a non-dissipative fluid rotating uniformly in the annular region between two infinitely long cylinders.The objective is to investigate the stability of MHD flow with velocity components for an incompressible fluid permeated by a magnetic field, where the components are along (r, θ, z) directions in cylindrical polar coordinate system, n is positive and A, B are constants.The magnetic lines of force are in general twisted by non-axisymmetric disturbances to the basic flow, whereas the axisymmetric disturbances only bend but do not twist the lines of force, and when the swirl velocity component r is large, the hydromagnetic effects become small compared with those due to swirl.These ideas have led us to investigate the MHD stability with respect to non-axisymmetric disturbances to the basic flow.The presence of velocity field in the basic state may cause more stability.Several results of MHD stability have been derived by using the inner product method.

Mathematical Formulation
Let us consider the basic flow (0, V θ, V z ) of an incompressible, inviscid and perfectly conducting fluid between two concentric cylinders of radii R 1 and R 2 permeated by a magnetic field (0, A/r n , B/r n ).The governing equations are: where V is the velocity vector, t, P, ρ, μ, and r H r represent time, pressure, density, magnetic permeability, and magnetic field, respectively.
The perturbed velocity is . The perturbed magnetic field and the total presure (hydrodynamic and hydromagnetic) are respectively taken as (2 ) The linearized momentum equations are The linearized magnetic induction equations are , 0 where In terms of the variables , and , the Eqs.(4a-4c) become Substituting Eq. ( 6) an (8c) d Eq. (8a-8c) in Eq. (3a-3c), we get ( ) Clearly the induced magnetic field ) , , ( given by (8a-8c) satisfies by irtue of (7).o co-axial c , we m e radial component of the velocity vanishes for these value of r, thus Eqs.(9a If the fluid is contained between tw ylinders of radii R 1 and R 2 ust require that th -9c) must be considered together with boundary conditions We rewrite equations (9a-9c) in the form (11a-11c) can be written in the matrix form as where , ) , , ( , and Here M, iG, and H are indepe c .According to Barston [1 r product can be defined as ndent of 1], the inne , ) ( , where ξ is the complex conjugate of ξ .Taking inner product with ξ The above equation leads to (15) where is a quadratic equation in c with real co-efficients.Its roots are where .

Co usion
e investigated the stability of MHD flow when both velocity components and magnetic s are azimuthal as well as axial.The observations are: We thus demonstrate that for a velocity field and imposed magnetic field as mentioned abo mor 106, 24 (1985).son, J. Fluid Mech.52, 529 (1972).doi:10.1017/S0022112072001570ve, the basic state will lead to e stability.
total pressure.Analysing the disturbances into normal modes, we seek solution of the foregoing equations whose dependence on t, θ , z is given by , where c = complex number, m = an integer, and k = real number.We linearize the equtions in the usual way and seek solutions in which all perturbation quantities 0 D is a positive real quantity.The motion is accordingly oscillatory.Again if ( then D is also real and the motion is oscillatory.(b)But, if the inequality does not hold then