Multiple-diffusive instabilities in rotating complex fluids

Self-sustained shear driven Hall MHD dynamos

K. Deguchi

kengo.deguchi@monash.edu

Monash University

The Hall effect on an MHD dynamo driven by shear is studied. As with many turbulent transitions in purely hydrodynamic shear flows, whether a dynamo is generated or not must be treated in the framework of a subcritical transition problem, for which dynamical systems theory is useful. We focus on the steady-state solution that seems to be at the edge of basin of attraction of dynamo turbulence, and derive its behavior at high Reynolds numbers by matched asymptotic expansion. The structure of the solution is described by the interaction between the mean field and current sheets. Its overall framework is somewhat similar to that of mean field theory, but it does not require any artificial assumptions for closure.

Oleg Kirillov

Double-diffusive instabilities in rotating hydrodynamic and magnetohydrynamic flows

Kirillov, Oleg

oleg.kirillov@northumbria.ac.uk

Northumbria University

The Prandtl number, i.e. the ratio of the fluid viscosity to a diffusivity parameter of other physical nature such as thermal diffusivity or ohmic dissipation, plays a decisive part for the onset of instabilities in hydrodynamic and magnetohydrodynamic flows. The studies of many particular cases suggest a significant difference in stability criteria obtained for the Prandtl number equal to 1 from those for the Prandtl number deviating from 1. We demonstrate this for a circular Couette flow with a radial temperature gradient and for a differentially rotating viscous flow of electrically conducting incompressible fluid subject to an external azimuthal magnetic field. Furthermore, in the latter case we find that the local dispersion relation is governed by a pseudo-Hermitian matrix both in the case when the magnetic Prandtl number, Pm, is Pm=1 and in the case when Pm=-1. This implies that the complete neutral stability surface contains three Whitney umbrella singular points and two mutually orthogonal intervals of self-intersection. At these singularities the double-diffusive system reduces to a marginally stable G-Hamiltonian system. The role of double complex eigenvalues (exceptional points) stemming from the singular points in exchange of stability between modes is demonstrated.

Joris Labarbe

Instability windows of Chandrasekhar-Friedman-Schutz instability

J. Labarbe and O.N. Kirillov

joris.labarbe@northumbria.ac.uk

Northumbria University

Rotating, self-gravitating mass of incompressible ideal fluid possesses an axially symmetric equilibrium configuration known as the Maclaurin spheroid. Fluid viscosity causes dissipation-induced instability of this equilibrium. Chandrasekhar discovered that radiation reaction force due to emission of gravitational waves could lead to a radiative instability of the Maclaurin spheroids. In the presence of both viscosity and resistivity, the ratio of these two dissipative forces plays a crucial role, determining the instability window for the Chandrasekhar-Friedman-Schutz (CFS) instability. The CFS instability is commonly accepted nowadays as one of the main triggers of gravitational radiation from single neutron stars that is the next goal for the existing (LIGO, Virgo) and planned (LISA) detectors of gravitational waves. Perturbation theory for eigenvalues in combination with numerical methods was proposed for calculation of instability windows of the CFS instability already in previous works. We are extending this approach to the multiple-parameter case by adopting established methodology to get better approximations and to put the CFS to the general context of the dissipation-induced instability theory.

Manikandan Mathur

Diffusive and curvature effects on symmetric instability in stratified vortices

Manikandan Mathur, Suraj Singh

manims@ae.iitm.ac.in

Department of Aerospace Engineering, Indian Institute of Technology Madras, Chennai - 600036, India.

We present a local stability analysis of an idealized model of stratified vortices that appear in geophysical settings. The base flow comprises an axisymmetric vortex with background rotation and an out-of-plane stable stratification, and a radial stratification in thermal wind balance with the out-of-plane momentum gradient. Solving the local stability equations along fluid particle trajectories in the base flow, the dependence of short-wavelength instabilities on Schmidt number Sc is studied, in the presence of curvature effects. In the diffusion-free limit, the well-known symmetric instability is recovered. In the viscous, double-diffusive regime, instability characteristics are shown to depend on three non-dimensional parameters (including Sc), and two different instabilities are identified: 1. a monotonic instability (same as symmetric instability at Sc = 1), and 2. an oscillatory instability (absent at Sc = 1). Separating the base flow and perturbation characteristics, two each of base flow and perturbation parameters (apart from Sc) are identified, and the entire parameter space is explored for the aforementioned instabilities. In comparison to Sc = 1, monotonic and oscillatory instabilities are shown to significantly expand the instability region in the space of base flow parameters as Sc moves away from unity. Neutral stability boundaries on the plane of Sc and a modified gradient Richardson number are then identified for both these instabilities. We conclude with a discussion of curvature effects, and the likelihood of monotonic and oscillatory instabilities in typical oceanic settings.

Jerome Mougel

Effect of the free surface on the stability and energy harvesting efficiency of a tensioned membrane in a uniform current.

N. Achour, J. Mougel, D. Fabre

jerome.mougel@imft.fr

IMFT

Flexible structures have recently been considered as alternative ways to extract energy from ocean waves (Alam 2012, Desmars et al 2018) or tidal currents (Trasch et al 2018), with the objective to find devices with complementary working characteristics compared to non-flexible energy harvesters. We investigate the dynamics of a finite length tensioned membrane with a localized linear damper to mimic energy extraction, which is placed in a uniform current parallel to a free surface. Such configuration resembles the so called Nemtsov’s membrane (Nemtsov 1985), recently studied in details and generalized to finite depth cases by Labarbe & Kirillov (2020, 2021), or to the infinite flag configuration close to a free surface studied recently by Mougel & Michelin (2020). The above studies reveal the importance of the free surface on the stability of the system, due to interactions between surface waves and structural waves when a current is present. In the present study, focus is placed on both forcing by incident waves (as already reported by Achour et al 2020 for weak currents) and stability analysis in order to investigate the role of the current on wave energy extraction by a flexible membrane, and shed additional light on the possible instability mechanism. In this objective, a linear potential flow model coupled to a tensioned beam is considered, and numerical results computed with the finite elements code FreeFEM++ (Hecht 2012) interfaced by StabFEM solver (Fabre et al 2019) are presented for a large range of physical parameters covering both subcritical and supercritical regimes.

Suraj Singh

Double-diffusive effects in the local instabilities of an elliptical vortex

Suraj Singh, Manikandan Mathur

surajsingh108talk@gmail.com

Department of Aerospace Engineering, Indian Institute of Technology Madras, Chennai - 600036, India.

Small-scale instabilities, which are local compared to the systems they exist in, play an instrumental role in understanding the mechanisms that lead to complex and often three-dimensional flow features. Vortices, for example, are ubiquitous in a turbulent flow and understanding their instabilities helps in a dynamical understanding of turbulence. A local stability approach, which calculates the inviscid evolution of relatively short-wavelength perturbations on a given base flow, has been instrumental in understanding various instabilities in vortical flows. In this study, we explore the effects of Schmidt number (Sc), which is the ratio between momentum and density diffusion coefficients, on the small-scale instabilities in an elliptical vortex with a stable stratification along its vortical axis. While the momentum and density diffusion coefficients are individually assumed to be small, their ratio Sc is allowed to be of arbitrary magnitude. The inviscid elliptical instability gets greatly modified due to the presence of a stable stratification. For Sc = 1, diffusion is shown to serve only as a suppressant of existing diffusion-free instabilities. We discover, however, that due to the presence of a stable stratification and a non-unity Sc, the vortex can be unstable in regimes which are stable based on a diffusion-free analysis. We characterize these non-unity Sc instabilities in detail, study how their growth rates depend on various base flow and perturbation parameters, and explore their potential connections with the diffusion-free instabilities

Laurette Tuckerman

Double-diffusive convection via 2 by 2 matrices

Laurette Tuckerman

laurette@pmmh.espci.fr

PMMH (CNRS, ESPCI, Sorbonne)

Convection due to competing or cooperating mechanisms, displays a variety of dynamical phenomena. One mechanism is usually a thermal gradient; typical examples of the second are rotation, a magnetic field, or a concentration gradient. The transition from conduction to convection is via a steady or a Hopf bifurcation; the point separating them is the best known codimension-two point. The steady bifurcation may be super or sub-critical, and the amplitude may undergo a qualitative transition from weak to strong.

All these features -- linear and nonlinear -- can be explained as manifestations of the behavior of the eigenvalues of a generic 2 x 2 matrix near the point where the eigenvalue branches intersect. For the stability problem, the eigenvalues have the conventional interpretation as growth rates, while for the nonlinear steady-state problems, they can be interpreted as the energy of steady states. Thus, there is a strong analogy between the stability problem and the bifurcation diagram.

Harunori Yoshikawa

Instabilities in nonisothermal Taylor-Couette flows in radial electric fields

H. Yoshikawa, A. Meyer, I. Mutabazi

Harunori.Yoshikawa@univ-cotedazur.fr

Université Côte d'Azur

In non-isothermal dielectric fluids subject to electric fields, the coupling between electric and temperature fields through the thermal variation of permittivity gives rise to an electrohydrodynamic force and can generate convective motion in fluids [see, e.g., Yoshikawa et al., 87, 043003, 2013]. The convection driven by this thermo-electrohydrodynamic effect is of importance in active control of heat and mass transport in small fluid systems, e.g., for manipulation of particles to modify locally their concentration [Kumar et al. Langmuir, 26(7), 5262-5272, 2010]. By the linear stability theory, we investigate instabilities provoked by the TEHD force in Taylor-Couette (TC) flow systems subject to radial temperature gradient and electric field. Different TC systems of different gap widths in different gravitational environments, i.e., in microgravity and on the Earth are considered. Depending on the radius ration of inner to outer cylinders and on the gravitational condition, different instabilities are observed. We elucidate the driving effects of different instabilities by an energetic analysis. We also examine effects of dielectric loss on the instabilities.