This research is focused on linear analysis of a plane-parallel flow stability in a transverse magnetic field (Hartmann flow) within a convective approximation. We derive and solve equations describing the perturbation growth. Perturbation modes and their nonexcitation conditions have been determined. An equation for the instability increment has been derived and it is shown that the equation has an unstable root. Additionally, we show that the resulting instabilities qualitatively agree with the experimental data.
We analyse numerically the linear stability of the fully developed liquid metal flow in a square duct with insulating side walls and thin electrically conducting horizontal walls with the wall conductance ratio $c=0.01\cdots 1$ subject to a vertical magnetic field with the Hartmann numbers up to $Ha=10^{4}.$ In a sufficiently strong magnetic field, the flow consists of two jets at the side walls walls and a near-stagnant core with the relative velocity $\sim(cHa)^{-1}.$ We find that for $Ha\gtrsim300,$ the effect of wall conductivity on the stability of the flow is mainly determined by the eff…
Direct numerical simulations and linear stability analysis are carried out to study mixed convection in a horizontal duct with constant-rate heating applied at the bottom and imposed transverse horizontal magnetic field. A two-dimensional approximation corresponding to the asymptotic limit of very strong magnetic field effect is validated and applied, together with full three-dimensional analysis, to investigate the flow's behaviour in the previously unexplored range of control parameters corresponding to typical conditions of a liquid metal blanket of a nuclear fusion reactor (Hartmann number…
In geo- and astrophysics, low Prandtl number convective flows often interact with magnetic fields. Although a static magnetic field acts as a stabilizing force on such flow fields, we find that self-organized convective flow structures reach an optimal state where the heat transport significantly increases and convective velocities reach the theoretical free-fall limit, i.e. the maximum possible velocity a fluid parcel can achieve when its potential buoyant energy is fully converted into kinetic energy. Our measurements show that the application of a static magnetic field leads to an anisotrop…
We study the linear stability of the flow of a viscous electrically conducting capillary fluid on a planar fixed plate in the presence of gravity and a uniform magnetic field. We first confirm that the Squire transformation for MHD is compatible with the stress and insulating boundary conditions at the free surface, but argue that unless the flow is driven at fixed Galilei and capillary numbers, the critical mode is not necessarily two-dimensional. We then investigate numerically how a flow-normal magnetic field, and the associated Hartmann steady state, affect the soft and hard instability mo…
The stability of the flow under the magnetic force is one of the classical problems in fluid mechanics. In this paper, the flow in a rectangular duct with different Hartmann (Ha) number is simulated. The finite volume method and the SIMPLE algorithm are used to solve a system of equations and the energy gradient theory is then used to study the (associated) stability of magnetohydrodynamics (MHD). The flow stability of MHD flow for different Hartmann (Ha) number, from Ha=1 to 40, at the fixed Reynolds number, Re=190 are investigated. The simulation is validated firstly against the simulation i…
The effects of a vertical static magnetic field on the flow structure and global transport properties of momentum and heat in liquid metal Rayleigh-Bénard convection are investigated. Experiments are conducted in a cylindrical convection cell of unity aspect ratio, filled with the alloy GaInSn at a low Prandtl number of $\mathit{Pr}=0.029$. Changes of the large-scale velocity structure with increasing magnetic field strength are probed systematically using multiple ultrasound Doppler velocimetry sensors and thermocouples for a parameter range that is spanned by Rayleigh numbers of $10^6 \le \m…
In this article the stability loss of the Hartmann flow are investigated by applying the equations for disturbances. The velocity and electric potential quasi-static MHD model is used. The equations allow us to calculate time-dependent disturbance fields using a base flow and an initial disturbance. Two type of initial perturbations are considered: the eigenfunction of the linearized MHD equations and a fluid injection into the flow. These two approaches lead to identical stability results. However, we found a significant difference in the practical implementation of the two approaches. Dealin…
We study the linear stability with respect to lateral perturbations of free surface films of polymer mixtures on solid substrates. The study focuses on the stability properties of the stratified and homogeneous steady film states studied in Part I [U. Thiele, S. Madruga and L. Frastia, Phys. Fluids 19, 122106 (2007)]. To this aim, the linearized bulk equations and boundary equations are solved using continuation techniques for several different cases of energetic bias at the surfaces, corresponding to linear and quadratic solutal Marangoni effects. For purely diffusive transport, an increase i…
Buoyancy-thermocapillary convection in a layer of volatile liquid driven by a horizontal temperature gradient arises in a variety of situations. Recent studies have shown that the composition of the gas phase, which is typically a mixture of vapour and air, has a noticeable effect on the critical Marangoni number describing the onset of convection as well as on the observed convection pattern. Specifically, as the total pressure or, equivalently, the average concentration of air is decreased, the threshold of the instability leading to the emergence of convective rolls is found to increase rat…
This study considers the linear stability of Poiseuille-Rayleigh-Bénard flows, subjected to a transverse magnetic field to understand the instabilities that arise from the complex interaction between the effects of shear, thermal stratification and magnetic damping. This fundamental study is motivated in part by the desire to enhance heat transfer in the blanket ducts of nuclear fusion reactors. In pure MHD flows, the imposed transverse magnetic field causes the flow to become quasi-2D and exhibit disturbances that are localised to the horizontal walls. However, the vertical temperature strati…
We study magnetic effects induced by rigidly rotating plates enclosing a cylindrical MHD Taylor-Couette flow at the finite aspect ratio $H/D=10$. The fluid confined between the cylinders is assumed to be liquid metal characterized by small magnetic Prandtl number, the cylinders are perfectly conducting, an axial magnetic field is imposed $\Ha \approx 10$, the rotation rates correspond to $\Rey$ of order $10^2-10^3$. We show that the end-plates introduce, besides the well known Ekman circulation, similar magnetic effects which arise for infinite, rotating plates, horizontally unbounded by any w…
It is demonstrated that the azimuthal magnetorotational instability (AMRI) also works with radially increasing rotation rates contrary to the standard magnetorotational instability for axial fields which requires negative shear. The stability against nonaxisymmetric perturbations of a conducting Taylor-Couette flow with positive shear under the influence of a toroidal magnetic field is considered if the background field between the cylinders is current-free. For small magnetic Prandtl number Pm-->0 the curves of neutral stability converge in the Hartmann number/Reynolds number plane approximat…
This study is concerned with numerical linear stability analysis of liquid metal flow in a square duct with thin electrically conducting walls subject to a uniform transverse magnetic field. We derive an asymptotic solution for the base flow which is valid not only for high but also moderate magnetic fields. This solution shows that for low wall conductance ratios $c\ll1,$ an extremely strong magnetic field with the Hartmann number $Ha\sim c^{-4}$ is required to attain the asymptotic flow regime considered in the previous studies. We use a vector stream function/vorticity formulation and a Che…
When representing convective instability mechanisms with the streamwise BiGlobal stability approach, results suffer from a sensitivity to the streamwise domain truncation length and boundary conditions. The methodology proposed in this paper resolves this sensitivity by considering a moving frame of reference. In that frame, the spectrum features discrete eigenvalues whose corresponding eigenfunctions decay exponentially in both the up- and downstream directions. Therefore, the truncation boundaries can be placed far enough that both variations in the domain length and artificial boundary cond…
The convective stability associated with carbon sequestration is usually investigated by adopting an unsteady diffusive basic profile. The method of normal modes is not applicable due to the time dependence of the nonlinear base profile. Therefore, the instability is quantified either in terms of critical times at which the boundary layer instability sets in or in terms of long time evolution of initial disturbances. This paper adopts an unstably stratified basic profile having a step function density with top heavy carbon saturated layer (boundary layer) overlying a lighter carbon free layer …
I consider the problem of weakly nonlinear stability of three-dimensional convective magnetohydrodynamic systems, where there is no alpha-effect or it is insignificant, to perturbations involving large scales. I assume that the convective MHD state (steady or evolutionary), the stability of which I investigate, does not involve large spatio-temporal scales, and it is stable to perturbations involving the same small spatial scales, as the perturbed state. Mean-field equations, which I derive for the perturbation using asymptotic techniques for multiscale systems, are a generalization of the equ…
The vectoral wind structure of daytime atmospheric boundary layer (ABL) turbulence is strongly dependent on the balance between shear-driven turbulence production of horizontal fluctuations (driven by winds at the mesoscale), and buoyancy-driven turbulence production of vertical velocity fluctuations (driven by solar heating), characterized by the global instability state parameter $-z_i/L > 0$. In the fully shear-driven neutral limit $-z_i/L \rightarrow 0$, the surface layer is dominated by coherent streamwise-elongated concentrations of negative streamwise fluctuating velocity (low-speed str…
Inspired by the experiment from Moresco \& Alboussière (2004, J. Fluid Mech.), we study the stability of a liquid metal flow in a rectangular, electrically insulating duct with a steady homogeneous transverse magnetic field. The Lorentz force tends to eliminate velocity variations along the magnetic field, leading to a quasi-two dimensional base flow with Hartmann boundary layers near the walls perpendicular to the magnetic field, and Shercliff layers near the walls parallel to the field. Since the Lorentz force strongly opposes the growth of perturbations with a dependence along the magnetic …
Magnetic fields vary in complexity for different stars. The stability of dipolar magnetic fields is known to depend on different quantities, e.g., the stellar rotation, the stratification, and the intensity of convective motions. Here, we study the dipolar stability in a system with an inner stable zone. We present preliminary results of dynamo simulations using the Rayleigh number as a control parameter. The stiffness of the stable zone is accordingly varied to keep a constant ratio of the Brunt-Vaisala frequency to the angular velocity. Similarly to the completely convective spherical shell,…
The diffusion of a solute from a concentrated source into a horizontal, stationary, fluid-saturated porous medium can lead to a convective motion when a gravitationally unstable density stratification evolves. In an inclined porous medium, the convective flow becomes intricate as it originates from a combination of diffusion and lateral flow, which is dominant near the source of the solute. Here, we investigate the role of inclination on the onset of convective instability by linear stability analyses of Darcy's law and mass conservation for the flow and the concentration field. We find that t…
We consider stability of steady convective flows in a horizontal layer with stress-free boundaries, heated below and rotating about the vertical axis, in the Boussinesq approximation (the Rayleigh-Benard convection). The flows under consideration are convective rolls or square cells, the latter being asymptotically equal to the sum of two orthogonal rolls of the same wave number k. We assume, that the Rayleigh number R is close to the critical one, R_c(k), for the onset of convective flows of this wave number: R=R_c(k)+epsilon^2; the amplitude of the flows is of the order of epsilon. We show t…
We study the stability of cylindrical Taylor-Couette flow in the presence of azimuthal magnetic fields, and show that one obtains non-axisymmetric magnetorotational instabilities, having azimuthal wavenumber m=1. For Omega_o/Omega_i only slightly greater than the Rayleigh value (r_i/r_o)^2, the critical Reynolds and Hartmann numbers are Re_c ~ 10^3 and Ha_c ~ 10^2, independent of the magnetic Prandtl number Pm. These values are sufficiently small that it should be possible to obtain these instabilities in the PROMISE experimental facility.
We consider arbitrary, possibly turbulent, Boussinesq flow which is smooth below a dissipative scale $l_d$. It is demonstrated that the stability of the flow with respect to growth of fluctuations with scale smaller than $l_d$ leads to a non-trivial constraint. That involves the dimensionless strength $Fl$ of fluctuations of the gradients of the scalar in the direction of gravity and the Rayleigh scale $L$ depending on the Rayleigh number $Ra$, the Nusselt number $Nu$ and $l_d$. The constraint implies that the stratified fluid at rest, which is linearly stable, develops instability in the limi…
Interaction of particles of dust with vortex convective flows is under theoretical consideration. It is assumed that the volume fraction of solid phase is small, variations of density due to nonuniform distribution of particles and those caused by temperature nonisothermality of medium are comparable. Equations for the description of thermal buoyancy convection of a dusty medium are developed in the framework of the generalized Boussinesq approximation taking into account finite velocity of particle sedimentation. The capture of a cloud of dust particles by a vortex convective flow is consider…
The present study is concerned with the stability of a flow of viscous conducting liquid driven by pressure gradient in the channel between two parallel walls subject to a transverse magnetic field. Although the magnetic field has a strong stabilizing effect, this flow, similarly to its hydrodynamic counterpart -- plane Poiseuille flow, is known to become turbulent significantly below the threshold predicted by linear stability theory. We investigate the effect of the magnetic field on 2D nonlinear travelling-wave states which are found at substantially subcritical Reynolds numbers starting fr…
This work addresses the effect of travelling thermal waves applied at the fluid layer surface, on the formation of global flow structures in 2D and 3D convective systems. For a broad range of Rayleigh numbers ($10^3\leq Ra \leq 10^7$) and thermal wave frequencies ($10^{-4}\leq Ω\leq 10^{0}$), we investigate flows with and without imposed mean temperature gradients. Our results confirm that the travelling thermal waves can cause zonal flows, i.e. strong mean horizontal flows. We show that the zonal flows in diffusion dominated regimes are driven purely by the Reynolds stresses, always travellin…
Rotating waves (RW) bifurcating from the axisymmetric basic magnetized spherical Couette (MSC) flow are computed by means of Newton-Krylov continuation techniques for periodic orbits. In addition, their stability is analysed in the framework of Floquet theory. The inner sphere rotates whilst the outer is kept at rest and the fluid is subjected to an axial magnetic field. For a moderate Reynolds number ${\rm Re}=10^3$ (measuring inner rotation) the effect of increasing the magnetic field strength (measured by the Hartmann number ${\rm Ha}$) is addressed in the range ${\rm Ha}\in(0,80)$ correspo…
In most fluid dynamics problems, the governing equations are nonlinear because of the presence of convective terms. Nevertheless, existence of solutions can be shown by direct sum provided one identifies, in the relevant Banach space of solutions, particular subspaces which are invariant under time evolution. As an example, we consider classical convection problems and show how symmetry arguments can help in identifying such subspaces, as well as the role it can play in some stability issues.
Zonal flows are often found in rotating convective systems. Not only are these jet-flows driven by the convection, they can also have a profound effect on the nature of the convection. In this work the cylindrical annulus geometry is exploited in order to perform nonlinear simulations seeking to produce strong zonal flows and multiple jets. The parameter regime is extended to Prandtl numbers that are not unity. Multiple jets are found to be spaced according to a Rhines scaling based on the zonal flow speed, not the convective velocity speed. Under certain conditions the nonlinear convection ap…
The stability of two-dimensional Poiseuille flow and plane Couette flow for concentrated suspensions is investigated. Linear stability analysis of the two-phase flow model for both flow geometries shows the existence of a convectively driven instability with increasing growth rates of the unstable modes as the particle volume fraction of the suspension increases. In addition it is shown that there exists a bound for the particle phase viscosity below which the two-phase flow model may become ill-posed as the particle phase approaches its maximum packing fraction. The case of two-dimensional Po…
Ortiz-Pérez and Dávalos investigated the stability of parallel convective flow. From the article it seems that this equations of flow obtained by the authors. Meanwhile, it is well studied theoretically and experimentally(exact solution obtained more than half a century ago.
We analyze both numerically and experimentally the stability of the steady jetting tip streaming produced by focusing a liquid stream with another liquid current when they coflow through the orifice of an axisymmetric nozzle. We calculate the global eigenmodes characterizing the response of this configuration to small-amplitude perturbations. In this way, the critical conditions leading to the instability of the steady jetting tip streaming are determined. The unstable perturbations are classified according to their oscillatory character and to the region where they are originated (convective …
We study the convective and absolute forms of azimuthal magnetorotational instability (AMRI) in a Taylor-Couette (TC) flow with an imposed azimuthal magnetic field. We show that the domain of the convective AMRI is wider than that of the absolute AMRI. Actually, it is the absolute instability which is the most relevant and important for magnetic TC flow experiments. The absolute AMRI, unlike the convective one, stays in the device, displaying a sustained growth that can be experimentally detected. We also study the global AMRI in a TC flow of finite height using DNS and find that its emerging …
The convective instability in a plane liquid layer with time-dependent temperature profile is investigated by means of a general method suitable for linear stability analysis of an unsteady basic flow. The method is based on a non-normal approach, and predicts the onset of instability, critical wave number and time. The method is applied to transient Rayleigh-Benard-Marangoni convection due to cooling by evaporation. Numerical results as well as theoretical scalings for the critical parameters as function of the Biot number are presented for the limiting cases of purely buoyancy-driven and pur…
We study two-phase stratified flow where the bottom layer is a thin laminar liquid and the upper layer is a fully-developed gas flow. The gas flow can be laminar or turbulent. To determine the boundary between convective and absolute instability, we use Orr--Sommerfeld stability theory, and a combination of linear modal analysis and ray analysis. For turbulent gas flow, and for the density ratio r=1000, we find large regions of parameter space that produce absolute instability. These parameter regimes involve viscosity ratios of direct relevance to oil/gas flows. If, instead, the gas layer is …
We consider a classical problem of linear stability of convective rolls in a plane layer with stress-free horizontal boundaries near the onset of convection. The problem has been studied by a number of authors, who have shown that rolls of wave number $k$ are unstable with respect to perturbations of different types, if some inequalities relating $k$ and the Rayleigh number $R$ are satisfied. The perturbations involve a large-scale mode. Certain asymptotic dependencies between wave numbers of the mode and overcriticality are always assumed in the available proofs of instability. We analyse the…
We analyze numerically the magnetorotational instability of a Taylor-Couette flow in a helical magnetic field (HMRI) using the inductionless approximation defined by a zero magnetic Prandtl number (Pm=0). The Chebyshev collocation method is used to calculate the eigenvalue spectrum for small amplitude perturbations. First, we carry out a detailed conventional linear stability analysis with respect to perturbations in the form of Fourier modes that corresponds to the convective instability which is not in general self-sustained. The helical magnetic field is found to extend the instability to a…
We analyze weakly nonlinear stability of a flow of viscous conducting liquid driven by pressure gradient in the channel between two parallel walls subject to a transverse magnetic field. Using a non-standard numerical approach, we compute the linear growth rate correction and the first Landau coefficient, which in a sufficiently strong magnetic field vary with the Hartmann number as $μ_{1}\sim(0.814-\mathrm{i}19.8)\times10^{-3}\textit{Ha}$ and $μ_{2}\sim(2.73-\mathrm{i}1.50)\times10^{-5}\textit{Ha}^{-4}$. These coefficients describe a subcritical transverse velocity perturbation with the equil…
Large-scale weakly nonlinear perturbations of convective magnetic dynamos in a rotating layer
Institute of Aeronautical Technology（航空技术研究所）；Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Ac. Sci.（俄罗斯科学院地震预测理论与数学地球物理研究所）
This article presents a modelling of the formation of spanwise vorticity in the turbulent streaks of the oblique bands and spots of transitional plane Couette flow. A functional model is designed to mimic the coherent flow in the streaks. The control parameters of the model are extracted from Direct Numerical Simulations (DNS) statistical data. A Reynolds stress is proposed to study the effect on the instability of this additional force maintaining the baseflow. Local (quasi-parallel) temporal stability analysis is performed on that model to investigate the linear development of the spanwise v…
We report the temporal and spatio-temporal stability analyses of anti-symmetric, free shear, viscoelastic flows obeying the Oldroyd-B constitutive equation in the limit of low to moderate Reynolds number and Weissenberg number. The resulting fourth order Orr-Sommerfeld equation is reduced to a set of six auxiliary equations which are numerically integrated starting from the rescaled far-field conditions, i.~e., via the Compound Matrix Method. Numerical results indicate that with increasing Weissenberg number: (a) the peak of the maximal growth rate (i. e., the maximum value of the imaginary co…
We analyze the stability of biological membrane tubes, with and without a base flow of lipids. Membrane dynamics are completely specified by two dimensionless numbers: the well-known Föppl--von Kármán number $Γ$ and the recently introduced Scriven--Love number $SL$, respectively quantifying the base tension and base flow speed. For unstable tubes, the growth rate of a local perturbation depends only on $Γ$, whereas $SL$ governs the absolute or convective nature of the instability. Furthermore, nonlinear simulations of unstable tubes reveal an initially localized disturbance results in propagat…
We report a novel form of convection in suspensions of the bioluminiscent marine bacterium $Photobacterium~phosphoreum$. Suspensions of these bacteria placed in a chamber open to the air create persistent luminiscent plumes most easily visible when observed in the dark. These flows are strikingly similar to the classical bioconvection pattern of aerotactic swimming bacteria, which create an unstable stratification by swimming upwards to an air-water interface, but they are a puzzle since the strain of $P.~phosphoreum$ used does not express flagella and therefore cannot swim. Systematic experim…
We investigate the stability of the steady convective flow in a plane tilted layer with ideal thermal conductivity of solid boundaries in the presence of uniform longitudinal temperature gradient. Analytically found the stability boundary with respect to the long-wave perturbations, find the critical Grashof number for the most dangerous among them of even spiral perturbation.
Fluid motion driven by thermal effects, such as that due to buoyancy in differentially heated three-dimensional (3D) enclosures, arise in several natural settings and engineering applications. It is represented by the solutions of the Navier-Stokes equations (NSE) in conjunction with the thermal energy transport equation represented as a convection-diffusion equation (CDE) for the temperature field. In this study, we develop new 3D lattice Boltzmann (LB) methods based on central moments and using multiple relaxation times for the three-dimensional, fifteen velocity (D3Q15) lattice, as well as …
