Instabilities of rotational flows in azimuthal magnetic fields of arbitrary radial dependence

Using the WKB approximation we perform a linear stability analysis for a rotational flow of a viscous and electrically conducting fluid in an external azimuthal magnetic field that has an arbitrary radial profile B_{phi}(R). In the inductionless approximation, we find the growth rate of the three-dimensional perturbation in a closed form and demonstrate in particular that it can be positive when the velocity profile is Keplerian and the magnetic field profile is slightly shallower than R^{-1}.Comment: 15 pages, 2 figures, slightly extended, the case of finite Rm treated, results were partially presented at the IUTAM Symposium on Vortex Dynamics, Fukuoka, Japan, March 10 201

Mathematical models for turbulent round jets based on “ideal” and “lossy” conservation of mass and energy

[EN] We propose mathematical models for turbulent round atomized liquid jets that describe its dynamics in a simple but comprehensive manner with the apex angle of the cone being the main disposable parameter. The basic assumptions are that (i) the jet is statistically stationary and that (ii) it can be approximated by a mixture of two fluid with the phases in local dynamic equilibrium, or so-called locally homogeneous flow (LHF). The models differ in their particular balance of explanatory capability and precision. To derive them we impose partial conservation of the initial mass and energy fluxes, introducing loss factors again as disposable parameters. Depending on each model, the equations admit explicit or implicit analytical solutions or a numerical solution in the discretized model case. The described variables are the the two-phase fluid’s composite density and velocity, both as functions of the distance from the nozzle, from which the dynamic pressure is calculated.FFM was in part supported by the Mexican Council of Science and Technology (CONACYT), the Bank of Mexico (BANXICO)’s FIDERH program and KUMIAY InterFranco, F.; Fukumoto, Y. (2017). Mathematical models for turbulent round jets based on “ideal” and “lossy” conservation of mass and energy. En Ilass Europe. 28th european conference on Liquid Atomization and Spray Systems. Editorial Universitat Politècnica de València. 456-463. https://doi.org/10.4995/ILASS2017.2017.4778OCS45646

Mass entrainment rate of an ideal momentum turbulent round jet

We propose a two-phase-fluid model for a full-cone turbulent round jet that describes its dynamics in a simple but comprehensive manner with only the apex angle of the cone being a disposable parameter. The basic assumptions are that (i) the jet is statistically stationary and that (ii) it can be approximated by a mixture of two fluids with their phases in dynamic equilibrium. To derive the model, we impose conservation of the initial volume and total momentum fluxes. Our model equations admit analytical solutions for the composite density and velocity of the two-phase fluid, both as functions of the distance from the nozzle, from which the dynamic pressure and the mass entrainment rate are calculated. Assuming a far-field approximation, we theoretically derive a constant entrainment rate coefficient solely in terms of the cone angle. Moreover, we carry out experiments for a single-phase turbulent air jet and show that the predictions of our model compare well with this and other experimental data of atomizing liquid jets.Comment: 17 pages, 10 figure

Short-wavelength analysis of magnetorotational instability of resistive MHD flows

Local stability analysis is made of axisymmetric rotating flows of a perfectly conducting fluid and resistive flows with viscosity, subjected to external azimuthal magnetic field to non-axisymmetric as well as axisymmetric perturbations. For perfectly conducting fluid (ideal MHD), we use the Hain-Lüst equation, capable of dealing with perturbations over a wide range of the axial wavenumber k to take short wavelength approximation. When the magnetic field is sufficiently weak, the maximum growth rate is given by the Oort A-value. As the magnetic field is increased, the Keplerian flow becomes unstable to waves of short axial wavelength. We also incorporate the effect of the viscosity and the electric resistivity and apply the WKB method in the same way as we do to the perfectly conducting fluid. In the inductionless limit, i.e. when the magnetic diffusivity is much larger than the viscosity, Keplerian-rotation flow of arbitrary distributions of the magnetic field, including the Liu limit, becomes unstable

A unifying picture of helical and azimuthal magnetrotational instability, and the universal significance of the Liu limit

The magnetorotational instability (MRI) plays a key role for cosmic structure formation by triggering turbulence in the rotating flows of accretion disks that would be otherwise hydrodynamically stable. In the limit of small magnetic Prandtl number, the helical and the azimuthal versions of MRI are known to be governed by a quite different scaling behavior than the standard MRI with a vertical applied magnetic field. Using the short-wavelength approximation for an incompressible, resistive, and viscous rotating fluid, we present a unified description of helical and azimuthal MRI, and we identify the universal character of the Liu limit for the critical Rossby number. From this universal behavior we are also led to the prediction that the instability will be governed by a mode with an azimuthal wavenumber that is proportional to the ratio of axial to azimuthal applied magnetic field, when this ratio becomes large and the Rossby number is close to the Liu limit

楕円回転流の弱非線形安定性のためのオイラー・ラグランジュ混合法

九州大学応用力学研究所研究集会報告 No.22AO-S8 「非線形波動研究の新たな展開 : 現象とモデル化」Report of RIAM Symposium No.22AO-S8 Development in Nonlinear Wave: Phenomena and Modeling定常剛体回転流は軸対称性と並進対称性のおかげで中立安定であるが，対称性を破る摂動を加えると不安定化する．楕円形にひずんだ流線をもつ回転流の線形不安定性は縮退する2個の3次元Kelvin波のパラメータ共鳴として普遍的にとらえることができる．これらは，ハミルトニアンHopf (あるいはピッチフォーク) 分岐を起こして新たな状態に移行するが，非線形段階を記述する数学的道具が欠如している；通常のオイラー的記述の枠組みでは波の非線形相互作用によって誘起される平均流ですら正しく計算できていない．最近，われわれは、ラグランジュ的記述によって平均流の計算を系統的に進める糸口を見つけた．従来のオイラー的扱いの不備を指摘し，弱非線形振幅方程式の係数をすべて決定する方法を紹介する

Analysis of azimuthal magnetorotational instability of rotating MHD flows and Tayler instability via an extended Hain-Lust equation

We consider a differentially rotating flow of an incompressible electrically conducting and viscous fluid subject to an external axial magnetic field and to an azimuthal magnetic field that is allowed to be generated by a combination of an axial electric current external to the fluid and electrical currents in the fluid itself. In this setting we derive an extended version of the celebrated Hain-Lust differential equation for the radial Lagrangian displacement that incorporates the effects of the axial and azimuthal magnetic fields, differential rotation, viscosity, and electrical resistivity. We apply the Wentzel-Kramers-Brillouin method to the extended Hain-L¨ust equation and derive a new comprehensive dispersion relation for the local stability analysis of the flow to three-dimensional disturbances. We confirm that in the limit of low magnetic Prandtl numbers, in which the ratio of the viscosity to the magnetic diffusivity is vanishing, the rotating flows with radial distributions of the angular velocity beyond the Liu limit, become unstable subject to a wide variety of the azimuthal magnetic fields, and so is the Keplerian flow. In the analysis of the dispersion relation we find evidence of a new long-wavelength instability which is caught also by the numerical solution of the boundary value problem for a magnetized Taylor-Couette flow

Weakly nonlinear saturation of stationary resonance of a rotating flow in an elliptic cylinder

Abstract. We address weakly nonlinear stability of a uniformly rotating flow confined in a cylinder of elliptic cross-section to three-dimensional disturbances. A Lagrangian approach is developed to derive unambiguously the drift current induced by nonlinear interaction of isovortical disturbances. This approach rescues the insufficiency inherent in the Eulerian approach and provides a direct path to reach the amplitude equations in the Hamiltonian normal form. The nonlinear effect saturates the stationary instability mode, and asymptotic form of its saturation amplitude is gained, in a tidy form, in the short-wavelength regime