Linear Stability Research Articles

On the validity of the exchange principle in rotatory electrothermoconvection

Abstract The electrothermoconvection in a rotating dielectric fluid layer heated from below (or above) is studied analytically using linear stability theory. First, we derive the necessary conditions for oscillatory motion when the fluid layer is heated from below and from above. Then as a consequence, sufficient conditions for the validity of the exchange principle are derived for the two configurations in terms of the parameters of the systems alone. The results hold for free boundaries for all wave numbers and for rigid boundaries with some restrictions. Further, the results for electrothermoconvection without rotation, rotatory Rayleigh–Bénard convection and Rayleigh–Bénard convection are also obtained as special cases which validate the existing results. To the best of author’s knowledge such results are not reported in the literature as far as the domain of electrothermoconvection studies is concerned.

On the stability of a particular class of one-dimensional states of dynamic equilibrium of the Vlasov–Poisson electron gas

   The one-dimensional problem of the linear stability of dynamic states of local thermodynamic equilibria with respect to small perturbations was studied for the case when the Vlasov–Poisson electron gas contains electrons with a stationary distribution function that is constant in physical space and variable in a continuum of velocities. The absolute instability of all considered one-dimensional dynamic states of any local thermodynamic equilibrium was proved using the direct Lyapunov method. The scope of applicability of the Newcomb–Gardner–Rosenbluth sufficient condition for linear stability was outlined. An a priori exponential estimation was obtained for the rise of small one-dimensional perturbations from below. Analytic counterexamples to the spectral Newсomb–Gardner theorem and the Penrose criterion were constructed. Earnshaw’s theorem was extended from classical mechanics tostatistical one.

Electromagnetically driven flow in unsupported electrolyte layers: lubrication theory and linear stability of annular flow

We consider a thin horizontal layer of a non-magnetic electrolyte containing a bulk solution of salt and carrying an electric current. The layer is bounded by two deformable free surfaces loaded with an insoluble surfactant and is placed in a vertical magnetic field. The arising Lorentz force drives the electrolyte in the plane of the layer. We employ the long-wave approximation to derive general two-dimensional hydrodynamic equations describing symmetric pinching-type deformations of the free surfaces. These equations are used to study the azimuthal flow in an annular film spanning the gap between two coaxial cylindrical electrodes. In weakly deformed films, the base azimuthal flow and its linear stability with respect to azimuthally invariant perturbations are studied analytically. For relatively thick layers and weak magnetic fields, the leading mode with the smallest decay rate is found to correspond to a monotonic azimuthal velocity perturbation. The Marangoni effect leads to further stabilisation of the flow while perturbations of the solute concentration in the bulk of the fluid have no influence on the flow stability. In strongly deformed films in the diffusion-dominated regime, the azimuthal flow becomes linearly unstable with respect to an oscillatory mixed mode characterised by the combination of radial and azimuthal velocity perturbations when the voltage applied between electrodes exceeds the critical value.

Probe particles in odd active viscoelastic fluids: How activity and dissipation determine linear stability

Probe particles in odd active viscoelastic fluids: How activity and dissipation determine linear stability

Emergence of a hexagonal pattern in shear-thickening suspensions under orbital oscillations

A dense particle suspension under shear may lose its uniform state to large local density and stress fluctuations, which challenge the mean-field description of the system. Here, we explore the novel dynamics of a non-Brownian suspension under orbital oscillations, where localized density waves along the flow direction appear beyond an excitation frequency threshold and self-organize into a hexagonal pattern across the system. The spontaneous occurrence of the inhomogeneity pattern arises from a coupling between particle advection and the shear-thickening nature of the suspension. Through linear stability analysis, we show that they overcome the stabilizing effects of particle pressure at sufficient particle volume fraction and oscillation frequency. In addition, the long-standing density waves degenerate into random fluctuations when replacing the free surface with rigid confinement. It indicates that the shear-thickened state is intrinsically heterogeneous, and the boundary conditions are crucial for developing local disturbance.

From a vortex gas to a vortex crystal in instability-driven two-dimensional turbulence

We study structure formation in two-dimensional turbulence driven by an external force, interpolating between linear instability forcing and random stirring, subject to nonlinear damping. Using extensive direct numerical simulations, we uncover a rich parameter space featuring four distinct branches of stationary solutions: large-scale vortices, hybrid states with embedded shielded vortices (SVs) of either sign, and two states composed of many similar SVs. Of the latter, the first is a dense vortex gas where all SVs have the same sign and diffuse across the domain. The second is a hexagonal vortex crystal forming from this gas when the linear instability is sufficiently weak. These solutions coexist stably over a wide parameter range. The late-time evolution of the system from small-amplitude initial conditions is nearly self-similar, involving three phases: initial inverse cascade, random nucleation of SVs from turbulence and, once a critical number of vortices is reached, a phase of explosive nucleation of SVs, leading to a statistically stationary state. The vortex gas is continued in the forcing parameter, revealing a sharp transition towards the crystal state as the forcing strength decreases. This transition is analysed in terms of the diffusivity of individual vortices using ideas from statistical physics. The crystal can also decay via an inverse cascade resulting from the breakdown of shielding or insufficient nonlinear damping acting on SVs. Our study highlights the importance of the forcing details in two-dimensional turbulence and reveals the presence of non-trivial SV states in this system, specifically the emergence and melting of a vortex crystal.

METHOD DEVELOPMENT AND VALIDATION OF RAPID ISOCRATIC RP-HPLC METHOD FOR SIMULTANEOUS ESTIMATION OF PARACETAMOL, CAFFEINE, AND PROPYPHENAZONE IN PHARMACEUTICAL FORMULATION

Objective: The objective is to develop a novel, rapid, simple, precise, accurate, and reproducible RP-HPLC method for simultaneous estimation of paracetamol, caffeine, and propyphenazone in bulk and a pharmaceutical dosage form. Method: Optimized chromatographic conditions were used for isocratic elution with Shimadzu C18 (4.6×250 mm, 5 μm), methanol, and 20mM phosphate buffer (60:40, v/v, pH 2.5) as mobile phase, flow rate 1.2 mL/min, and UV detector at λ max 272 nm. The method was validated for specificity, precision, linearity, accuracy, robustness, and solution stability as per the International Council for Harmonization (ICH) guidelines. Result: It was discovered that the retention time for paracetamol, caffeine, and propyphenazone was found to be 2.6 min, 3.0 min, and 7.5 min. The method proved to be rapid, simple, linear (R2>0.999), precise {relative standard deviation (RSD<2.0%)}, accurate (recovery 98–102%), sensitive, and robust. Conclusion: The proposed novel isocratic RP-HPLC method was found to be rapid (short run time of 10 min), highly selective, accurate, and sensitive. The method has been successfully applied to the simultaneous analysis of paracetamol, caffeine, and propyphenazone in a pharmaceutical formulation.

Linear stability analysis of MHD mixed convection flow of a radiating nanofluid in porous channel in presence of viscous dissipation

Purpose This paper aims to investigate the hydrodynamic instability properties of a mixed convection flow of nanofluid in a porous channel. Design/methodology/approach The treated single-phase nanofluid is a suspension consisting of water as the working fluid and alumina as a nanoparticle. The anisotropy of the porous medium and the effects of the inclination of the magnetic field are highlighted. The effects of viscous dissipation and thermal radiation are incorporated into the energy equation. The eigenvalue equation system resulting from the stability analysis is processed numerically by the spectral collocation method. Findings Analysis of the results in terms of growth rate reveals that increasing the volume fraction of nanoparticles increases the critical Reynolds number. Parameters such as the mechanical anisotropy parameter and Richardson number have a destabilizing effect. The Hartmann number, permeability parameter, magnetic field inclination, Prandtl number, wave number and thermal radiation parameter showed a stabilizing effect. The Eckert number has a negligible effect on the growth rate of the disturbances. Originality/value Linear stability analysis of Magnetohydrodynamics (MHD) mixed convection flow of a radiating nanofluid in porous channel in presence of viscous dissipation.

Turing Instability and Spatial Pattern Formation in a Model of Urban Crime

A nonlinear crime model is generalized by introducing self- and cross-diffusion terms. The effect of diffusion on the stability of non-negative constant steady states is applied. In particular, the cross-diffusion-driven instability, called Turing instability, is analyzed by linear stability analysis, and several Turing patterns driven by the cross-diffusion are studied through numerical investigations. When the Turing–Hopf conditions are satisfied, the type of instability highlighted in the ODE model persists in the PDE system, still showing an oscillatory behavior.

A-stable spectral deferred correction method for nonlinear Allen-Cahn model

This paper presents a type of A-stable spectral deferred correction (SDC) method. The scheme is initiated by the first-order backward Euler method. We adopt the linear stabilization approach for the Allen-Cahn model to get the linear semi-implicit SDC scheme. This is done by the addition and subtraction of the linear stabilization operators that have been provided for the Allen-Cahn problem. The ranges of the stabilization factors are then given. The aim is to achieve A-stability and error estimates for the implicit part of the scheme. Finally, the semi-implicit SDC scheme coupled with the Fourier spectral method is used to simulate the Allen-Cahn problem, which is also suitable for the highly nonlinear problem. Numerical experiments are given to numerically demonstrate the high-order accuracy and the energy decay property of the scheme with A-stability parameters.

Dynamical analysis of coupled curvature-matter scenario in viable f(R) dark energy models at de Sitter phase

We explore the interaction between dark matter and curvature-driven dark energy within viable f(R) gravity models, employing the phase-space analysis approach of linear stability theory. By incorporating an interacting term, denoted as Q=αHρ~m(κ23H2ρcurv+1) , into the continuity equations of both sectors, we examine dynamics of two f(R) gravity models that adhere to local gravity constraints and fulfill cosmological viability criteria. In de Sitter phase, our investigation reveals modifications of critical points compared to the conventional form, attributed to the introduced interaction. Through a comprehensive phase-space analysis, we illustrate the trajectories near critical points and outline the constraints on the phase space based on cosmological and cosmographic parameters. Furthermore, we can also observe stable behavior of this interacting system during the de Sitter phase of the Universe.

Elimination of lock-in phenomenon in vortex-induced vibration by passive modal control

Theoretical analysis and numerical results have shown that frequency lock-in in vortex-induced vibration (VIV) is caused by the instability of the structural mode rather than a resonant response to external excitations. However, there is a lack of experimental evidence supporting relevant theoretical research findings. This study investigates VIV suppression with a passive modal controller (PMC) for a circular cylinder at Reynolds numbers $Re = 60$ and $Re = 40$ , using experiments to distinguish the effects of stable and unstable wake modes. Comparative analysis before and after the implementation of the PMC reveals significant reduction in the vibration amplitude and the disappearance of the lock-in phenomenon at $Re = 60$ . The vibration frequency closely follows the vortex shedding frequency after control, while dynamic mode decomposition of the flow field indicates that the wake mode is dominant. For $Re = 40$ , the vibration is eliminated and the flow becomes steady. Additionally, the root loci of the coupled system are investigated before and after the PMC implementation via linear stability analysis. The results indicate that the PMC can alter the dynamic characteristics of the original system, causing the structural mode and PMC mode to couple when approaching the PMC frequency. Then, the interaction typically improves the stability of the structural mode. Finally, a parametric study is conducted in the experiment, as well as a linear stability analysis. The study provides experimental evidence that stability control of the structural mode is the key to suppressing VIV and eliminating the lock-in phenomenon.

The application of the "inverse problem" method for constructing confining potentials that make N-soliton waveforms exact solutions in the Gross-Pitaevskii equation.

In this work, we discuss an application of the "inverse problem" method to find the external trapping potential, which has particular N trapped soliton-like solutions of the Gross-Pitaevskii equation (GPE) also known as the cubic nonlinear Schrödinger equation (NLSE). This inverse method assumes particular forms for the trapped soliton wave function, which then determines the (unique) external (confining) potential. The latter renders these assumed waveforms exact solutions of the GPE (NLSE) for both attractive (g<0) and repulsive (g>0) self-interactions. For both signs of g, we discuss the stability with respect to self-similar deformations and translations. For g<0, a critical mass Mc or equivalently the number of particles for instabilities to arise can often be found analytically. On the other hand, for the case with g>0 corresponding to repulsive self-interactions which is often discussed in the atomic physics realm of Bose-Einstein condensates, the bound solutions are found to be always stable. For g<0, we also determine the critical mass numerically by using linear stability or Bogoliubov-de Gennes analysis, and compare these results with our analytic estimates. Various analytic forms for the trapped N-soliton solutions in one, two, and three spatial dimensions are discussed, including sums of Gaussians or higher-order eigenfunctions of the harmonic oscillator Hamiltonian.

On the magnetohydrodynamic stability of channel flow of Navier–Stokes–Voigt fluid

The linear stability of a pressure-driven channel flow of an electrically conducting Navier–Stokes–Voigt type of the viscoelastic fluid subject to a transverse magnetic field is investigated. The validity of Squire's theorem is proved, and the generalized eigenvalue problem for two-dimensional modes is obtained by adopting the Galerkin method, which is subsequently solved using the QZ-algorithm. Although the base flow retains its Newtonian fluid characteristics, the noticeable influence of the Kelvin–Voigt parameter in conjunction with the Hartmann number on the stability of fluid flow is perceived. Instability is exclusively identified within a specific range of the Kelvin–Voigt parameter, markedly affected by the Hartmann number. Furthermore, closed neutral stability curves arise, indicating the requirement of two values of the Reynolds number to completely assess the linear stability criteria, in contrast to the typical single value observed in the case of Newtonian fluids.

On viscous stratified Darcy–Forchheimer flow in a horizontal porous layer with thermal anisotropy and variable permeability

This study analyzes the effect of anisotropy and the internal heat source in a Darcy–Forchheimer porous layer. It is well known that the variations in viscosity can be attributed to the temperature. Therefore, in the present problem, we consider a linear variation in viscosity with temperature for simplicity. We first derived the linear instability theory and then established global stability using the energy functional approach. In the global stability analysis, we show that working with the L2 norm fails to give a sufficient condition for global stability by exhibiting that the associated maximization problem is unbounded in the underlying stability measure space. Then, we show that a conditional stability bound can be achieved by restricting the internal heat source parameter Q with higher-order norms. The eigenvalue problems obtained in linear and nonlinear theories were integrated numerically. The linear and nonlinear instability thresholds are then compared to identify the potential regions of sub-critical instabilities. It is observed that the system is stabilized when the horizontal component of thermal diffusivity dominates and is unstable when the vertical component of thermal diffusivity dominates. We also found that increasing the variable permeability parameter λ destabilized the system. It is observed that increasing viscosity stabilizes the system, and decreasing viscosity encourages the start of convection. It is also interesting that, in the presence of an internal heat source, the region of subcritical instability increases with increasing viscosity effect but reduces with increasing vertical permeability λ.

Comparison of commonly used and new methods to determine small molecule non-specific binding to human liver microsomes

Accurate measurement of non-specific binding of a drug candidate to human liver microsomes (HLM) can be critical for the accurate determination of key enzyme kinetic parameters such as Michaelis-Menton (Km), reversible inhibition (Ki), or inactivation (KI) constants. Several methods have been developed to determine non-specific binding of small molecules to HLM, such as rapid equilibrium dialysis (RED), ultrafiltration (UF), HLM bound to magnetizable beads (HLM-beads), ultracentrifugation (UC), the linear extrapolation stability assay (LESA), and the Transil™ system. Despite various differences in methodology between these methods, it is generally presumed that similar free fraction values (fu,mic) should be generated. To evaluate this hypothesis, a test set of 9 compounds were selected, representing low (high fu,mic value) and significant (low fu,mic value) HLM binding, respectively, across HLM concentrations tested in this manuscript. The fu,mic values were determined using a single compound concentration (1.0 µM) and three HLM concentrations (0.025, 0.50, and 1.0 mg/mL). When the HLM non-specific binding event is not extensive resulting in high fu,mic values, all methods generated similar fu,mic values. However, fu,mic values varied markedly across assay formats when high binding to HLM occurred, where fu,mic values differed by up to 33-fold depending on the method used. Potential causes for such discrepancies across the various methods employed, practical implications related to conduct the different assays, and implications to clinical drug-drug interaction (DDI) predictions are discussed.

Stability of penetrative convective currents in local thermal non-equilibrium

The aim of this paper is to investigate the onset of penetrative convection in a Darcy–Brinkman porous medium under the hypothesis of local thermal non-equilibrium. For the problem at stake, the strong form of the principle of exchange of stabilities has been proved, i.e. convective motions can occur only through secondary stationary motions. We perform linear and nonlinear stability analyses of the basic state, with particular regard to the behaviour of stability thresholds with respect to the relevant physical parameters characterizing the problem. The Chebyshev- τ method and the shooting method are employed and accurately implemented to solve the differential eigenvalue problems arising from linear and nonlinear analyses to determine critical Rayleigh numbers. Via numerical simulations, the stabilizing effect of the upper bounding plane temperature, of the Darcy number and of the interaction coefficient for the heat exchange, is demonstrated.

A highly sensitive, long-time stable Ag/AgCl ultra-micro sensor for in situ monitoring chloride ions inside the crevice using SECM

Tracking the variation of Cl− timely within the crevice is of great significance for comprehending the dynamic mechanism of crevice corrosion. The reported chloride ion selective electrodes are difficult to realize the long-time Cl− detection inside the confined crevice, due to their millimeter size or a relative limited lifespan. For this purpose, an Ag/AgCl ultra-micro sensor (UMS) with a radius of 12.5 μm was fabricated and optimized using laser drawing and electrodeposition techniques. Results show the AgCl film's structure is significantly impacted by the deposited current density, and further affects the linear response, life span and stability of Ag/AgCl UMS. The UMS prepared at current density of 0.1 mA/cm2 for 2 h shows a rapid response (several seconds), excellent stability and reproducibility, strong acid/alkali tolerance, sufficient linearity (R2 > 0.99), and long lifespan (86 days). Moreover, combined with the potentiometric mode of scanning electrochemical microscope (SECM), the Ag/AgCl UMS was successfully applied to monitor the in-situ radial Cl− concentration in micro-regions inside a 100 μm gap of stainless steel. The findings demonstrated that there was obvious radial difference in Cl− concentration inside the crevice, where the fastest rise in Cl− concentration was at the opening. The proposed method which combines the UMS with SECM has attractive practical applications for microzone Cl− monitoring in real time inside crevice. It may further promote the study of other localized corrosion mechanism and the development of microzone ions detection method.

Dual Solutions of Magnetized radiative Flow of Casson Nanofluid over a Stretching/Shrinking Cylinder: Stability Analysis

The enhanced thermal efficiency exhibited by Casson nanofluids offers significant practical applications across various industrial and engineering sectors. This study focuses on the mathematical investigation of the steady magnetohydrodynamic (MHD) boundary layer flow of Casson nanofluid through a stretched/shrinking cylinder, taking into account the effects of suction and thermal radiation. The governing partial differential equations (PDEs) have been subjected to a similarity transformation, resulting in a set of nonlinear ordinary differential equations (ODEs). These ODEs were solved numerically utilizing the code of bvp4c in the software of Matlab which offers high accuracy (4th order). The employed nanofluid model incorporates the effects of Brownian motion and thermophoresis. The present study illustrates the graphical depictions of the impacts of different governing parameters, namely Hartmann () number, curvature () parameter, Brownian motion () parameter, mass suction () parameter, thermal radiation () parameter, and thermophoresis () parameter, on heat transfer, flow, and mass transfer characteristics. Comprehensive determination and visual presentation of the coefficient of skin friction, local Nusselt number, and local Sherwood number were conducted for a range of estimates of applied parameters. Based on our examination, it has been determined that dual similarity solutions are present within a specific range of mass suction parameters. The relationship between the Casson parameter and various fluid dynamic properties, such as skin friction coefficient, heat transfer rate, and mass transfer rates, has been found to exhibit a decreasing trend. Furthermore, the stability analysis discovered that the first solution exhibits linear stability, whereas the second solution displays linear instability. Additionally, the motivation behind this study is to enhance industrial performance through the optimization of thermal power generation systems, thereby increasing their overall efficiency.

Adaptive cruise control design for collision risk avoidance

With advanced technologies such as perception and automatic control, automated vehicles (AVs) are gradually replacing human-driven vehicles (HDVs). They will become a significant component of transportation systems. As the crucial longitudinal control model for AVs, the existing adaptive cruise control (ACC) model faces challenges such as poor linear stability and compromised safety. To solve the problem, this paper proposes an adaptive cruise control model to avoid collision risk. First, the delay-considerate collision risk control strategy (DCRCS) is developed to consider vehicles' perception and control delay. Then, a proposed adaptive cruise control (PACC) model is designed by incorporating DCRCS into the ACC model based on the control gain coefficient. Finally, the effectiveness of the PACC model is discussed and analyzed in terms of stability, safety, efficiency, energy consumption, and pollutant emissions through a combined approach of theoretical analysis and simulation experiments. The result shows that (1) compared to the classical ACC model, the PACC model demonstrates significant improvements in system stability, reduced safety risks, enhanced efficiency, and decreased energy consumption and pollutant emissions. These enhancements are observed when the control gain coefficient is set to a positive value. (2) The equilibrium speed minimally influences the effectiveness of the PACC model. Under varying speed conditions, the PACC model consistently maintains stable performance. (3) The effectiveness of the PACC model is closely related to the selection of the control gain coefficient. Considering the effects of the control gain coefficient on stability, safety, efficiency, energy consumption, and pollutant emissions of the PACC model, the optimal range for the control gain coefficient is in the range of [0.8, 1.0]. In summary, compared to the classical ACC model, the PACC model exhibits outstanding performance in various aspects. It provides theoretical support for the longitudinal control of AVs in the future.