This paper investigates the stability of a thin electrically conductive fluid under an applied uniform magnetic filed during spin coating. A generalized nonlinear kinematic model is derived by the long-wave perturbation method to represent the physical system. After linearizing the nonlinear evolution equation, the method of normal mode is applied to study the linear stability. Weakly nonlinear dynamics of film flow is studied by the multiple scales method. The Ginzburg-Landau equation is determined to discuss the necessary conditions of the various critical flow states, namely, subcritical stability, subcritical instability, supercritical stability, and supercritical explosion. The study reveals that the rotation number and the radius of the rotating circular disk generate similar destabilizing effects but the Hartmann number gives a stabilizing effect. Moreover, the optimum conditions can be found to alter stability of the film flow by controlling the applied magnetic field.

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Weakly Nonlinear Stability Analysis of a Thin Magnetic Fluid during Spin Coating

Stability of narrow-gap Taylor-Dean flow with radial heating: Stationary critical modes

Analysis of a conducting fluid in a thin annulus with rotating insulated walls under radial magnetic effect

A linear stability analysis has been carried out for hydromagnetic current-induced flow. A viscous electrically conducting fluid between concentric cylinders is driven electromagnetically by the interaction ofa superimposed radial current and a uniform axial magnetic field. The assumption of small-gap approximation is made and the governing equations with respect to nonaxisymmetric disturbances are derived and solved by a direct numerical procedure. Both of the two different types of boundary conditions, namely ideally conducting and weakly conducting walls, are considered. For 0 ≤ Q ≤ 5000, where Q is the Hartmann number, which represents the strength of magnetic field in the axial direction, it is found that the instability sets in as a steady secondary flow for the case of weakly conducting walls but not for ideally conducting walls. For ideally conducting walls, it is demonstrated that the onset mode is due to nonaxisymmetric rather than axisymmetric disturbances as Q exceeds a certain critical value. The transition of the onset of instability from axisymmetric modes to nonaxisymmetric modes is discussed in detail and the possibility of axisymmetric oscillatory modes is examined

Hydromagnetic stability of current-induced flow in a small gap between concentric cylinders

The effect of air shear on the hydromagnetic instability is studied through (i) linear stability, (ii) weakly nonlinear theory, (iii) sideband stability of the filtered wave, and (iv) numerical integration of the nonlinear equation. Additionally, a discussion on the equilibria of a truncated bimodal dynamical system is performed. While the linear and weakly nonlinear analyses demonstrate the stabilizing (destabilizing) tendency of the uphill (downhill) shear, the numerics confirm the stability predictions. They show that (a) the downhill shear destabilizes the flow, (b) the time taken for the amplitudes corresponding to the uphill shear to be dominated by the one corresponding to the zero shear increases with magnetic fields strength, and (c) among the uphill shear-induced flows, it takes a long time for the wave amplitude corresponding to small shear values to become smaller than the one corresponding to large shear values when the magnetic field intensity increases. Simulations show that the streamwise and transverse velocities increase when the downhill shear acts in favor of inertial force to destabilize the flow mechanism. However, the uphill shear acts oppositely. It supports the hydrostatic pressure and magnetic field in enhancing films stability. Consequently, reduced constant flow rates and uniform velocities are observed.

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Air-Aided Shear on a Thin Film Subjected to a Transverse Magnetic Field of Constant Strength: Stability and Dynamics

The stability analysis of the flow of a viscous electrically conducting fluid between concentric rotating cylinders in the presence of an axial magnetic field is extended to the case where the primary flow includes a pressure gradient acting in the azimuthal direction. The pressure gradient is produced electromagnetically by the interaction of a superimposed radial current and the uniform axial magnetic field. The assumption of small gap approximation is made and the governing equations with respect to both axisymmetric and non–axisymmetric three–dimensional disturbances are derived and solved by a direct numerical procedure. A parametric study covering wide ranges of Q , the Hartmann number which represents the strength of axial magnetic field, and β, a parameter characterizing the ratio of current induced and rotation velocities, is conducted for weakly conducting cylinders and the situation of practical interest where the outer cylinder is stationary and the inner cylinder is rotating. The area where the onset mode is non–axisymmetric is shown in the plane β, Q ). It is found that the most stable state occurs approximately along a critical curve (β + 4.3) Q 2 + 56250(β + 3.75) = 0 and the critical axial wavenumber always has discontinuity when the parameters Q and β cross this curve. The critical mode transition of the onset of instability will be demonstrated in detail and results for the critical wavenumber and the critical Taylor number are presented. The corresponding values of the radial current density required for the appearance of secondary flow are also determined.

Ch'o–Kuang Chen

Papers

Hydromagnetic stability of current-induced flow in a small gap between concentric rotating cylinders