Hydrodynamic and hydromagnetic stability

Flow of an electrically conducting fluid in a toroidal duct of square cross-section is analysed. The flow is driven by the azimuthal Lorentz force resulting from the interaction between the radial electric currents created by the difference of electric potential maintained between the cylinder walls and the strong magnetic field imposed in the axial direction. The flow geometry and the value of the Hartmann number correspond to the experiment of Moresco & Alboussiere (J. Fluid Mech., vol. 504, 2004, pp. 167–181). The purpose of the analysis is to reveal the flow features at Reynolds numbers below the threshold of transition to turbulence in Hartmann layers. We find that the flow experiences a complex evolution. The laminar base flow experiences the first instability at the Reynolds number significantly smaller than that of the threshold. The instability is axisymmetric and oscillatory. Turbulence appears at a slightly higher Reynolds number. Right up to the Hartmann layer instability, the turbulence remains localized in a layer near the outer cylinder wall. It is demonstrated that the turbulence may affect the transition in the Hartmann layers via unsteady forcing of the outer flow.

Instabilities and turbulence in magnetohydrodynamic flow in a toroidal duct prior to transition in Hartmann layers

Magnetohydrodynamic mixing of two fluids in an annular microchannel is modelled as a two-dimensional laminar convection-diffusion problem and examined using asymptotic analysis and numerical simulation. The time T required for mixing of a plug of solute depends on the Peclet number Pe and on the geometry of the annulus. Three scaling regimes are identified: purely diffusive, Taylor-dispersive, and convection-dominated; each has a characteristic power-law dependence of T upon Pe. Consequences of these results for optimal micromixer design are discussed.

Modelling annular micromixers

Modelling Annular Micromixers

Instability of a flow of an electrically conducting fluid in an annular channel is analyzed. Strong constant magnetic field is imposed in the axial direction. Similarly to toroidal duct experiments, the flow is driven by the azimuthal Lorentz force resulting from the interaction between the magnetic field and the radial electric currents created by a difference of electric potential imposed between the cylinders. The instability of the base flow, while clearly of centrifugal nature, is significantly different from the Dean instability detected earlier in hydrodynamic systems and similar MHD systems at low and moderate magnetic fields. Growing perturbations are oscillating and axisymmetric and consist of counter-rotating toroidal vortices arranged side by side in the radial direction and having meridional cross-sections in the form of elongated ellipses oriented slightly obliquely to the axial direction. Simulations of the secondary flow show an interesting feature of periodic transitions between two symmetric solutions.

Instability of magnetohydrodynamic flow in an annular channel at high Hartmann number

Electromagnetically driven flows between infinite conducting cylinders subjected to a constant axial magnetic field B0 are considered. The linear stability problem for axisymmetric stationary disturbances is treated. In contrast with Chandrasekhar’s conjecture, the present numerical results indicate that the critical wave‐number tends to infinity with the magnetic field. The nonlinear state of the flow, slighly above transition, is investigated by means of Stuart’s energy approach. Further theoretical results are compared with experiment. At criticality, excellent agreement is found between linear stability theory and experiment. Above transition, only the shape of the experimental curves is correctly predicted by Stuart’s energy method.

P. Tabeling

Papers

Magnetohydrodynamic Taylor vortex flow under a transverse pressure gradient