Marginal Stability Curves Research Articles

Convection in the Presence of Inertial Effects in an Active Mushy Layer

The present work, deals with the analytical investigation of convection in a mushy layer that results due to the morphological instability of the interface during the solidification of a binary or a multicomponent alloy. The dynamical behavior of a mushy layer strongly depends on the complex interactions between the convection and the transfer of heat and solute which can remarkably modify the shape, structure and rate of the crystal growth. Accordingly, the distribution and the dissolution of the dendrites in the layer, induced by the convective process eventually lead to the alteration of the permeability or the solid matrix of the mushy layer under consideration. The specific interest with which the study is carried out is to identify a model and the corresponding parametric values which could suppress the formation of chimney convection and annihilate the formation of freckles which cause imperfections in the resulting solid. The asymptotic limits considered here are a near-eutectic approximations, inertial effects, large far-field temperature and variable permeability. The consideration of large Stefan number incorporates a key balance for the existence of compositional convection. The important results of the present study are, (i) An active mushy layer is more stable than a passive mushy layer (ii) The far-field temperature has a destabilising effect on the marginal stability curves as expected and (iii) The influence of the governing parameters is remarkable on the vertical velocity component, temperature and local solid fraction profiles. Finally it is concluded that, through an analytical approach it is possible to determine the accurate solutions which could control or supress the chimney formation during the solidification process which is a burning problem in the areas like metal casting, sea dynamics etc. It is found that the results of the present study are very much closer to the experimental results.

Hydromagnetic Convection in a Mushy Layer with Variable Permeability – An Analytical Study

The problem of controlling the formation of freckles which cause imperfection during the solidification of a binary or multicomponent alloy is a challenging one. The present analytical study considers a hydromagnetic model with variable permeability near eutectic –approximation and large far-field temperature and the dynamics of the mushy layer is studied by using a modified perturbation technique. Our results suggest that although the effect of variable permeability is of stabilizing type, by a proper choice of the magnetic and expansion parameters it is possible to have an optimal control over the formation of chimney convection so that the freckles causing imperfection in the resulting solid could be drastically avoided. The profiles of marginal stability curves, vertical velocity, magnetic field, temperature and local solid-fraction are presented, which clearly predict the behavior of the system in an effective way. The results are in excellent agreement with the available results in the limiting cases.

Verifying effects of instability enhanced ion–ion Coulomb collisions on ion velocity distribution functions near the sheath edge in low temperature plasmas

Experiments have shown that ion velocity distribution functions (ivdfs) with non-Maxwellian tails created by ion–neutral collisions and ionizations along pre-sheaths in weakly collisional plasmas can be thermalized into Maxwellian distributions near the sheath edge. A recent theory suggests that ion–ion collisions enhanced by ion-acoustic instabilities can rapidly thermalize ions near the sheath edge into Maxwellian distributions and that increasing either electron temperature or neutral pressure of a plasma suppresses the growth of instabilities and eliminates the thermalization process. Measurements of ivdfs by laser induced fluorescence showed qualitative agreement between experimental data and a marginal stability curve inferred from the new theory.

A review of rotating cylinder wake transitions

Recent work on the flow past a rotating cylinder is reviewed and further investigated at low Reynolds numbers. The various two- and three-dimensional transitions that occur as the rotation rate is increased are detailed. Two steady states, steady state I and steady state II, are identified based on the physical characteristics of the wake and the drag force on the body. Steady state I occurs at lower rotation rates, while state steady state II occurs at higher rotation rates. Linear stability analysis shows that two three-dimensional modes become unstable on steady state I and steady state II. Floquet stability analysis of the unsteady base flows that occur at very low rotation rates shows the presence of five three-dimensional modes. The curves of marginal stability are presented, followed by a comparison of numerical simulations to their experimentally obtained counterparts. Furthermore, the spatio-temporal characteristics of each mode and the likely underlying physical mechanisms are briefly discussed.

Minimal model dynamics for twelvefold quasipatterns.

A dynamical model of the Swift-Hohenberg type is proposed to describe the formation of twelvefold quasipattern as observed, for instance, in optical systems. The model incorporates the general mechanisms leading to quasipattern formation and does not need external forcing to generate them. Besides quadratic nonlinearities, the model takes into account an angular dependence of the nonlinear couplings between spatial modes with different orientations. Furthermore, the marginal stability curve presents other local minima than the one corresponding to critical modes, as usual in optical systems. Quasipatterns form when one of these secondary minima may be associated with harmonics built on pairs of critical modes. The model is analyzed numerically and in the framework of amplitude equations. The results confirm the importance of harmonics to stabilize quasipatterns and assess the applicability of the model to other systems with similar generic properties.

Numerical simulations of thermal convection under the influence of an inclined magnetic field by using solenoidal bases

AbstractThe effect of an inclined homogeneous magnetic field on thermal convection between rigid plates heated from below under the influence of gravity is numerically simulated in a computational domain with periodic horizontal extent. The numerical technique is based on solenoidal (divergence‐free) basis functions satisfying the boundary conditions for both the velocity and the induced magnetic field. Thus, the divergence‐free conditions for both velocity and magnetic field are satisfied exactly. The expansion bases for the thermal field are also constructed to satisfy the boundary conditions. The governing partial differential equations are reduced to a system of ordinary differential equations under Galerkin projection and subsequently integrated in time numerically. The projection is performed by using a dual solenoidal bases set such that the pressure term is eliminated in the process. The quasi‐steady relationship between the velocity and the induced magnetic field corresponding to the liquid metals or melts is used to generate the solenoidal bases for the magnetic field from those for the velocity field. The technique is validated in the linear case for both oblique and vertical case by reproducing the marginal stability curves for varying Chandrasekhar number. Some numerical simulations are performed for either case in the nonlinear regime for Prandtl numbers Pr = 0.05 and Pr = 0.1. Copyright © 2014 John Wiley & Sons, Ltd.

Benard-Marangoni Convection with Free Slip Bottom and Mixed Thermal Boundary Conditions

The onset of cellular convection induced by surface tension gradients in a horizontal liquid layer heated from below is examined by making use of linear stability analysis for mixed thermal boundary conditions with free-slip condition at the lower boundary. We use a combination of analytical and numerical techniques to obtain a detailed description of marginal stability curves. It is established numerically that ‘the principle of exchange of stabilities’ is valid. The numerical results are presented for a wide range of the values of the parameters characterizing the nature of thermal boundary conditions. We investigate for the first time, a situation wherein value of the parameter characterizing the thermal condition at the upper boundary varies inversely to that characterizing the thermal condition at the lower boundary, and obtained distinct ranges in which increasing values of the parameter of the lower boundary lead to formation of convection cells of increasing or decreasing size at the onset of convection.

Torsion instability of soft solid cylinders

The application of pure torsion to a long and thin cylindrical rod is known to provoke a twisting instability, evolving from an initial kink to a knot. In the torsional parallel-plate rheometry of stubby cylinders, the geometrical constraints impose zero displacement of the axis of the cylinder, preventing the occurrence of such twisting instability. Under these experimental conditions, wrinkles occur on the cylinder's surface at a given critical angle of torsion. Here we investigate this subclass of elastic instability--which we call torsion instability--of soft cylinders subject to a combined finite axial stretch and torsion, by applying the theory of incremental elastic deformation superimposed on finite strains. We formulate the incremental boundary elastic problem in the Stroh differential form, and use the surface impedance method to build a robust numerical procedure for deriving the marginal stability curves. We present the results for a Mooney-Rivlin material and study the influence of the material parameters on the elastic bifurcation.

Stability of columnar convection in a porous medium

AbstractConvection in a porous medium at high Rayleigh number $\mathit{Ra}$ exhibits a striking quasisteady columnar structure with a well-defined and $\mathit{Ra}$-dependent horizontal scale. The mechanism that controls this scale is not currently understood. Motivated by this problem, the stability of a density-driven ‘heat-exchanger’ flow in a porous medium is investigated. The dimensionless flow comprises interleaving columns of horizontal wavenumber $k$ and amplitude $\widehat{A}$ that are driven by a steady balance between vertical advection of a background linear density stratification and horizontal diffusion between the columns. Stability is governed by the parameter $A= \widehat{A}\mathit{Ra}/ k$. A Floquet analysis of the linear-stability problem in an unbounded two-dimensional domain shows that the flow is always unstable, and that the marginal-stability curve is independent of $A$. The growth rate of the most unstable mode scales with ${A}^{4/ 9} $ for $A\gg 1$, and the corresponding perturbation takes the form of vertically propagating pulses on the background columns. The physical mechanism behind the instability is investigated by an asymptotic analysis of the linear-stability problem. Direct numerical simulations show that nonlinear evolution of the instability ultimately results in a reduction of the horizontal wavenumber of the background flow. The results of the stability analysis are applied to the columnar flow in a porous Rayleigh–Bénard (Rayleigh–Darcy) cell at high $\mathit{Ra}$, and a balance of the time scales for growth and propagation suggests that the flow is unstable for horizontal wavenumbers $k$ greater than $k\sim {\mathit{Ra}}^{5/ 14} $ as $\mathit{Ra}\rightarrow \infty $. This stability criterion is consistent with hitherto unexplained numerical measurements of $k$ in a Rayleigh–Darcy cell.

The stability of a static liquid column pulled out of an infinite pool

We study the stability of a static liquid column rising from an infinite pool, with its top attached to a horizontal plate suspended at a certain height above the pool's surface. Two different models are employed for the column's contact line. Model 1 assumes that the contact angle always equals Young's equilibrium value. Model 2 assumes a functional dependence between the contact angle and the velocity of the contact line, and we argue that, if this dependence involves a hysteresis interval, linear perturbations cannot move the contact line. It is shown that, within the framework of Model 1, all liquid columns are unstable. In Model 2, both stable and unstable columns exist (the former have larger contact angles θ and/or larger heights H). For Model 2, the marginal stability curve on the (θ, H)-plane is computed. The mathematical results obtained imply that, if the plate to which a stable liquid column is attached is slowly lifted up, the column's contact line remains pinned while the contact angle is decreasing. Once it reaches the lower boundary of the hysteresis interval, the column breaks down.

Time-dependent Marangoni-Bénard instability of an evaporating binary-liquid layer including gas transients

We are here concerned with Bénard instabilities in a horizontal layer of a binary liquid, considering as a working example the case of an aqueous solution of ethanol with a mass fraction of 0.1. Both the solvent and the solute evaporate into air (the latter being insoluble in the liquid). The system is externally constrained by imposing fixed “ambient” pressure, humidity, and temperature values at a certain effective transfer distance above the liquid-gas interface, while the ambient temperature is also imposed at the impermeable rigid bottom of the liquid layer. Fully transient and horizontally homogeneous solutions for the reference state, resulting from an instantaneous exposure of the liquid layer to ambient air, are first calculated. Then, the linear stability of these solutions is studied using the frozen-time approach, leading to critical (monotonic marginal stability) curves in the parameter plane spanned by the liquid layer thickness and the elapsed time after initial contact. This is achieved for different ratios of the liquid and gas thicknesses, and in particular yields critical times after which instability sets in (for given thicknesses of both phases). Conversely, the analysis also predicts a critical thickness of the liquid layer below which no instability ever occurs. The nature of such critical thickness is explained in detail in terms of mass fraction profiles in both phases, as it indeed appears that the most important mechanism for instability onset is the solutal Marangoni one. Importantly, as compared to the result obtained previously under the quasi-steady assumption in the gas phase [H. Machrafi, A. Rednikov, P. Colinet, and P. C. Dauby, Eur. Phys. J. Spec. Top. 192, 71 (2011)]10.1140/epjst/e2011-01361-y, it is shown that relaxing this assumption may yield essentially lower values of the critical liquid thickness, especially for large gas-to-liquid thickness ratios. A good-working analytical model is developed for the description of such delicate transient effects in the gas. The analysis reveals that the system considered in this paper is generally highly unstable, the instability setting in even for very small times and liquid thicknesses.

Three-dimensionality in the wake of a rapidly rotating cylinder in uniform flow

AbstractThe flow around an isolated cylinder spinning at high rotation rates in free stream is investigated. The existence of two steady two-dimensional states is confirmed, as is the existence of a secondary mode of vortex shedding. The stability of the two steady states to three-dimensional perturbations is established using linear stability analysis. At lower rotation rates on the first steady state, two three-dimensional modes are confirmed, and their structure and curves of marginal stability as a function of rotation rate and Reynolds number are determined. One mode (named mode $E$) appears consistent with a hyperbolic instability in the wake, while the second (named mode $F$) appears to be a centrifugal instability of the flow very close to the cylinder surface. At higher rotation rates on the second steady state, a single three-dimensional mode due to centrifugal instability (named mode ${F}^{\prime } $) is found. This mode becomes increasingly difficult to excite as the rotation rate is increased.

Three-dimensionality in the wake of a rotating cylinder in a uniform flow

AbstractThe wake of a rotating circular cylinder in a free stream is investigated for Reynolds numbers $\mathit{Re}\leqslant 400$ and non-dimensional rotation rates of $\alpha \leqslant 2. 5$. Two aspects are considered. The first is the transition from a steady flow to unsteady flow characterized by periodic vortex shedding. The two-dimensional computations show that the onset of unsteady flow is delayed to higher Reynolds numbers as the rotation rate is increased, and vortex shedding is suppressed for $\alpha \geqslant 2. 1$ for all Reynolds numbers in the parameter space investigated. The second aspect investigated is the transition from two-dimensional to three-dimensional flow using linear stability analysis. It is shown that at low rotation rates of $\alpha \leqslant 1$, the three-dimensional transition scenario is similar to that of the non-rotating cylinder. However, at higher rotation rates, the three-dimensional scenario becomes increasingly complex, with three new modes identified that bifurcate from the unsteady flow, and two modes that bifurcate from the steady flow. Curves of marginal stability for all of the modes are presented in a parameter space map, the defining characteristics for each mode presented, and the physical mechanisms of instability are discussed.

Thermo-electro-hydrodynamic instabilities in a dielectric liquid under microgravity

Linear stability analysis of a dielectric fluid confined in a cylindrical annulus of infinite length is performed under microgravity conditions. A radial temperature gradient and a high alternating electric field imposed over the gap induce an effective gravity that can lead to a thermal convection even in the absence of the terrestrial gravity. The linearized governing equations are discretized using a spectral collocation method on Chebyshev polynomials to compute marginal stability curves and the critical parameters of instability. The critical parameters are independent of the Prandtl number, but they depend on the curvature of the system. The critical modes are non-axisymmetric and are made of stationary helices.

Chebyshev collocation approach to stability of blood flows in a large artery

In this study, treating the large artery as a rigid channel with uniform width and the blood as an incompressible Newtonian fluid with variable viscosity due to transverse variation in hematocrit ratio, the basic flow structure and its temporal stability to small disturbances were studied. A fourth-order Eigenvalues problem which reduces to the well known Orr–Sommerfeld equation in some limiting cases was obtained and solved numerically by a spectral collocation technique with expansions in Chebyshev polynomials implemented in MATLAB. Graphical results for the basic flow axial velocity, disturbance growth rate and marginal stability curve are presented and discussed. It is worth pointing out that a transverse increase in the blood hematocrit ratio towards the central region of the artery had a stabilizing effect on the flow. Key words: Arterial blood flow, hematocrit ratio, variable viscosity, temporal stability, Chebyshev spectral collocation technique.

Electrokinetic instability in microchannels

The effect of geometric confinement on electroconvective instability due to nonequilibrium electro-osmotic slip at the interface of an electrolytic fluid and charge-selective solid is studied. It is shown that the topology of the marginal stability curves and the behavior of the critical parameters depend strongly on both channel geometry and dimensionless Debye length at low voltages for sufficiently deep channels, corresponding to the Rubinstein-Zaltzman instability mechanism, but that stability is governed almost entirely by channel depth for narrow channels at higher voltages. For shallow channels, it is shown that above a transition threshold, determined by both channel depth and Debye length, the low-voltage instability is completely suppressed.

Effect of vertical quasiperiodic vibrations on the stability of the free surface of an inviscid liquid layer

The aim of the present paper is to examine the effect of the vertical quasiperiodic oscillations on the stability of the free surface of an ideal horizontal liquid layer. The quasiperiodic motion considered here is characterized by two incommensurate frequencies ω1 and ω2. The governing system of equations is reduced to a quasiperiodic Mathieu equation. In this situation, using the harmonic balance method developed by Rand et al. [10, 11] and Hill’s determinants, we determine the marginal stability curves. We show that the quasiperiodic excitation produces a stabilizing or a estabilizing effect and is strongly depending on the ratio of the frequencies.

Pattern formation in fiber-reinforced tubular tissues: Folding and segmentation during epithelial growth

Constrained growth processes in living materials result in a complex distribution of residual strains, which in certain geometries may induce a bifurcation in the elastic stability. In this work, we investigate the combined effects of growth and material anisotropy in the epithelial pattern formation of tubular tissues. In order to represent the structural organization of most organs, we adopt a strain energy density which accounts for the presence of a nonlinear reinforcement made of cross-ply fibers distributed inside a ground matrix. Using a canonical transformation in mixed polar coordinates, we transform the nonlinear elastic boundary value problem into a variational formulation, performing a straightforward derivation of the Euler–Lagrange equations for perturbations in circumferential and longitudinal directions. The corresponding curves of marginal stability are obtained numerically: the results demonstrate that both the three-dimensional distribution of residual strains and the mechanical properties of fiber reinforcements within the tissue are fundamental to determine the emergence of a specific instability pattern. In particular, different proportions of axial and circumferential residual strains can model the epithelial formation of mucosal folds in the esophagus and of plicae circulares in the small intestine. The theoretical predictions are compared with morphological data for embryonic intestinal tissues, suggesting that the volumetric growth of the epithelium can also drive the early stages of villi morphogenesis.

Phase-locked solutions and their stability in the presence of propagation delays

We investigate phase-locked solutions of a continuum field of nonlocally coupled identical phase oscillators with distance-dependent propagation delays. Equilibrium relations for both synchronous and traveling wave solutions in the parameter space characterizing the non-locality and time delay are delineated. For the synchronous states a comprehensive stability diagram is presented that provides a heuristic synchronization condition as well as an analytic relation for the marginal stability curve. The relation yields simple stability expressions in the limiting cases of local and global coupling of the phase oscillators.

BPS spectrum of supersymmetricCP(N−1)theory withZNtwisted masses

We revisit the BPS spectrum of the supersymmetric CP(N-1) two-dimensional model with Z_N-symmetric twisted masses m_l (l=0,1, ..., N-1). A related issue we address is that of the curves of marginal stability (CMS) in this theory. Previous analyses were incomplete. We close the gap by exploiting a number of consistency conditions. In particular, we amend the Dorey formula for the BPS spectrum. Our analysis is based on the exact Veneziano--Yankielowicz-type superpotential and on the strong-coupling spectrum of the theory found from the mirror representation at small masses, |m_l| << \Lambda . We show that at weak coupling the spectrum, with necessity, must include N-1 BPS towers of states, instead of just one, as was thought before. Only one of the towers is seen in the quasiclassical limit. We find the corresponding CMS for these towers, and argue that in the large-N limit they become circles, filling out a band on the plane of a single mass parameter of the model at hand. Inside the CMS, N-1 towers collapse into N stable states.