Linear Stability Results Research Articles

Stability analysis for viscoelastic fluid with thermorheological effects: Linear and nonlinear approaches

This study investigates the stability analysis of Rayleigh-Bénard configuration for a viscoelastic fluid subject to thermorheological effects, using the D2-Chebyshev-τ method. The fluid is modeled as a third-order viscoelastic fluid. This study accentuates how salting the fluid layer affects the thresholds for the onset of instability in a fluid of third order encompassing physically realistic rigid boundaries. The dynamic model incorporates advection-diffusion of temperature and solute concentration and a modified Navier–Stokes equation. We determine instability thresholds for the complex non-Newtonian fluid by analyzing the linear stability of the steady-state conduction solution. Our analysis proves the strong form of the principle of exchange of stabilities, demonstrating that convective motions can only occur through stationary motion. Additionally, a nonlinear stability analysis using the energy method is performed, deriving an unconditional nonlinear stability criterion. The results provide a comprehensive understanding of how variable viscosity and viscoelasticity impact system stability. Both the viscosity parameter and the third-grade fluid parameter exhibit stabilizing effects. Notably, we observe a discrepancy between the linear and global nonlinear stability results, indicating the presence of a subcritical instability region. This study contributes to the understanding of complex fluid dynamics in non-linear mechanical systems, with potential applications in various industrial and natural processes.

Linear and nonlinear stability of double diffusive convection in a micropolar fluid saturated porous layer with magnetic field and throughflow effects

The linear and nonlinear stability of double-diffusive convection in a porous layer saturated with micropolar fluid is examined. A transverse magnetic field is applied to the flow together with vertical throughflow. The normal mode technique is employed for linear stability analysis, whereas the energy method is used for nonlinear stability analysis. The resulting eigenvalue problems corresponding to linear and nonlinear stability theories are solved numerically by employing the bvp4c routine in MATLAB 2022(b). The critical thermal Rayleigh numbers for both linear and nonlinear analyses are computed for the different values of the governing parameters and presented graphically. A comparison is made between linear and nonlinear stability results. It is observed that the flow is more stable whenever a magnetic field is added to the flow, although the subcritical instability region also slightly increases. Increasing the Darcy number, Lewis number, coupling number, and absolute value of the throughflow parameter destabilizes the flow. On the other hand, raising the porosity of the medium and micropolar parameters stabilizes the flow. Furthermore, there is no subcritical gap in the absence of the throughflow effect, which is a good agreement between the linear and nonlinear thresholds.

Well-posedness and stability of a stochastic neural field in the form of a partial differential equation

A system of partial differential equations representing stochastic neural fields was recently proposed with the aim of modelling the activity of noisy grid cells when a mammal travels through physical space. The system was rigorously derived from a stochastic particle system and its noise-driven pattern-forming bifurcations have been characterised. However, due to its nonlinear and non-local nature, standard well-posedness theory for smooth time-dependent solutions of parabolic equations does not apply. In this article, we transform the problem through a suitable change of variables into a nonlinear Stefan-like free boundary problem and use its representation formulae to construct local-in-time smooth solutions under mild hypotheses. Then, we prove that fast-decaying initial conditions and globally Lipschitz modulation functions imply that solutions are global-in-time. Last, we improve previous linear stability results by showing nonlinear asymptotic stability of stationary solutions under suitable assumptions.

Stability of a viscous liquid film flowing down an inclined plane with respect to three-dimensional disturbances

An analysis is presented for the stability of a viscous liquid film flowing down an inclined plane with respect to three-dimensional disturbances under the action of gravity and surface tension. Using momentum-integral method, the nonlinear free surface evolution equation is derived by introducing the self-similar semiparabolic velocity profiles along the flow (x- and y-axis) directions. A normal mode technique and the method of multiple scales are used to obtain the theoretical (linear and nonlinear stability) results of this flow problem, which conceive the physical parameters: Reynolds number Re, Weber number We, angle of inclination of the plane θ and the angle of propagation of the interfacial disturbances ϕ. The temporal growth rate ωi+ and second Landau constant J2, based on which various (explosive, supercritical, unconditional, subcritical) stability zones of this flow problem are categorized, contain the shape factors B and β owing to the non-zero steady basic flow along the y-axis direction. A novel result which emerges from the linear stability analysis is that for any given value of Re, We and θ, any stability that arises in two-dimensional disturbances (ϕ = 0) must also be present in three-dimensional disturbances. For ϕ = 0, there exists a second explosive unstable zone (instead of unconditional stable zone) after a certain value of Re (or θ) due to the involvement of B and β in the expression of J2. This explosive unstable zone vanishes after a certain value of ϕ depending upon the values of Re, We and θ, which confirms the stabilizing influence of ϕ on the thin film flow dynamics irrespective of the values of Re, We and θ.

Non-Oberbeck–Boussinesq effects on the linear stability of a vertical natural convection boundary layer

The non-Oberbeck–Boussinesq effects on the stability of a vertical natural convection boundary layer are investigated using the linearised disturbance equations for air flows up to a temperature difference of $\Delta T=100\,{\rm K}$ . Based on the linear stability results, the neutral curve is shown to be sensitive to the choice of reference temperature. When evaluated using the film temperature $T_f$ , a lower film Grashof number is required to trigger the linear instability for larger $\Delta T$ . The relative contributions of shear and buoyant production to the perturbation kinetic energy budget reveals that the marginally unstable modes are amplified based on different mechanisms: for lower wavenumbers at relatively small Grashof number, the instability is driven by buoyancy; whereas for higher wavenumbers and larger Grashof number, the flow becomes unstable due to a shear instability. The use of reference temperature is found to scale the shear- and buoyant-driven instabilities differently so that no single reference temperature definition would collapse the neutral curves. The linear stability result further demonstrates that at a given Grashof number a higher temperature difference would give a larger amplification rate of the perturbation, which then leads to an earlier onset of the nonlinearities when evaluated at $T_f$ . Finally, by comparing the amplification rates obtained from direct numerical simulation and the linear stability results, the extent of the linear regime is determined for $\Delta T = 100\,{\rm K}$ .

Stability of low-pressure turbine boundary layers under variable Reynolds number and pressure gradient

The free-stream turbulence induced transition occurring under typical low-pressure turbine flow conditions is investigated by comparing linear stability theory with wind tunnel measurements acquired over a flat plate subjected to high turbulence intensity. The analysis was carried out, accounting for three different Reynolds numbers and four different adverse pressure gradients. First, a non-similarity-based boundary layer (BL) solver was used to compute base flows and validated against pressure taps and particle image velocimetry (PIV) measurements. Successively, the optimal disturbances and their spatial transient growth were calculated by coupling classical linear stability theory and a direct-adjoint optimization procedure on all flow conditions considered. Linear stability results were compared with experimental particle image velocimetry measurements on both wall-normal and wall-parallel planes. Finally, the sensitivity of the disturbance spatial transient growth to the spanwise wavenumber of perturbations, the receptivity position, and the location where disturbance energy is maximized were investigated via the built numerical model. Overall, the optimal perturbations computed by linear stability theory show good agreement with the streaky structures surveyed in experiments. Interestingly, the energy growth of disturbances was found to be maximum for all the flow conditions examined, when perturbations entered the boundary layer close to the position where minimum pressure occurs.

The dependent coordinates in the linearization of constrained multibody systems: Handling and elimination

In this paper, the handling of the dependent coordinates in the linearization of constrained multibody systems is clarified. To this end, a brief but comprehensive overview of linearization approaches used in the field of Multibody System Dynamics is presented. These procedures are illustrated by using a common notation and classified into four groups, depending on the initial form of the nonlinear equations of motion and the selection of the generalized coordinates (a redundant or a minimal set). The handling of the dependent coordinates in each of the procedures is discussed. The linearized equations of motion of constrained multibody systems are of great importance in several applications, such as linear stability and modal analyses, the design of linear state feedback controllers or the building of state and input estimators, like Kalman filters. When modeling a constrained multibody system, a number of coordinates greater than the number of degrees of freedom of the system is usually used. While some linearization approaches make use of the complete set of coordinates, several procedures are based on a coordinate partition in terms of independent and dependent coordinates to reduce the number of linearized equations of motion. In this latter scenario, the role of the dependent coordinates in the linearization is important and, in a general case, these dependent coordinates cannot be ignored to obtain the correct linearized equations of motion. This aspect, which may seem obvious, is not straightforward and is addressed in this work. The importance of considering the set of dependent coordinates in the linearization is demonstrated with a well-acknowledged bicycle benchmark multibody model and an electric kickscooter multibody model with rear and front suspensions. To this end, the Jacobian matrix and the linear stability results of these case studies are computed by considering different choices of independent and dependent coordinates.

Role of interfacial rheology on fingering instabilities in lifting Hele-Shaw flows.

The lifting Hele-Shaw cell setup is a popular modification of the classic, fixed-gap, radial viscous fingering problem. In the lifting cell configuration, the upper cell plate is lifted such that a more viscous inner fluid is invaded by an inward-moving outer fluid. As the fluid-fluid interface contracts, one observes the rising of distinctive patterns in which penetrating fingers having rounded tips compete among themselves, reaching different lengths. Despite the scholarly and practical relevance of this confined lifting flow problem, the impact of interfacial rheology effects on its pattern-forming dynamics has been overlooked. Authors of recent studies on the traditional injection-induced radial Hele-Shaw flow and its centrifugally driven variant have shown that, if the fluid-fluid interface is structured (i.e., laden with surfactants, particles, proteins, or other surface-active entities), surface rheological stresses start to act, influencing the development of the viscous fingering patterns. In this paper, we investigate how interfacial rheology affects the stability as well as the shape of the emerging fingered structures in lifting Hele-Shaw flows, at linear and early nonlinear dynamic stages. We tackle the problem by utilizing the Boussinesq-Scriven model to describe the interface and by employing a perturbative mode-coupling scheme. Our linear stability results show that interfacial rheology effects destabilize the interface. Furthermore, our second-order findings indicate that interfacial rheology significantly alters intrinsically nonlinear morphological features of the shrinking interface, inducing the formation of narrow sharp-tip penetrating fingers and favoring enhanced competition among them.

Motion of a parametrically driven damped coplanar double pendulum.

We present the results of linear stability of a damped coplanar double pendulum and its nonlinear motion, when the point of suspension is vibrated sinusoidally in the vertical direction with amplitude a and frequency ω. A double pendulum has two pairs of Floquet multipliers, which have been calculated for various driving parameters. We have considered the stability of a double pendulum when it is in any of its possible stationary states: (i) both pendulums are either vertically downward or upward and (ii) one pendulum is downward and other is upward. The damping is considered to be velocity-dependent, and the driving frequency is taken in a wide range. A double pendulum excited from its stable state shows both periodic and chaotic motion. The periodic motion about its pivot may be either oscillatory or rotational. The periodic swings of a driven double pendulum may be either harmonic or subharmonic for lower values of a. The limit cycles corresponding to the normal mode oscillations of a double pendulum of two equal masses are squeezed into a line in its configuration space. For unequal masses, the pendulum shows multi-period swings for smaller values of a and damping, while chaotic swings or rotational motion at relatively higher values of a. The parametric driving may lead to stabilization of a partially or fully inverted double pendulum.

Non-Oberbeck-Boussinesq effects and sub-critical primary bifurcations in porous media convection

Large temperature variations and highly unstable density stratification may exist between the cold and hot boundaries of the porous non-vacuum gas filled insulation layers of cryogenic storage tanks. For example, the density can vary by about a factor 3 in N2 filled liquefied natural gas (LNG) tank insulation layers and factor 15 in H2 filled liquid hydrogen (LH2) tank insulation layers. In such cases, the bifurcation of the convective states from the conduction state could be sub-critical leading to sudden jumps and hysteresis effects in heat transfer and boil-off rates. This work examines the impact of large physical property variation and non-Oberbeck-Boussinesq (non-OB) effects on the nature of bifurcation of convective states in gas filled porous insulation layers. Specifically, we show that the well-known super-critical (pitchfork) bifurcation of convective states that appear under the OB approximation changes to sub-critical bifurcation when the temperature difference exceeds a critical value. Further, under such large temperature variations, the emerging convective solutions could be highly asymmetric with center of circulation much closer to the cold boundary. We present linear stability results, local bifurcation analysis as well as numerical simulations of the full model equations for the two selected LNG and LH2 cases.

Nonlinear Stability and Existence of Two-Dimensional Compressible Current-Vortex Sheets

We are concerned with the nonlinear stability and existence of two-dimensional current-vortex sheets in ideal compressible magnetohydrodynamics. This is a nonlinear hyperbolic initial-boundary value problem with a characteristic free boundary. It is well-known that current-vortex sheets may be at most weakly (neutrally) stable due to the existence of surface waves solutions that yield a loss of derivatives in the energy estimate of the solution with respect to the source terms. We first identify a sufficient condition ensuring the weak stability of the linearized current-vortex sheets problem. Under this stability condition for the background state, we show that the linearized problem obeys an energy estimate in anisotropic weighted Sobolev spaces with a loss of derivatives. Based on the weakly linear stability results, we then establish the local-in-time existence and nonlinear stability of current-vortex sheets by a suitable Nash–Moser iteration, provided that the stability condition is satisfied at each point of the initial discontinuity. This result gives a new confirmation of the stabilizing effect of sufficiently strong magnetic fields on Kelvin–Helmholtz instabilities.

Higher-order complex cubic quintic Ginzburg-Landau equation: Chirped solitary waves

Propagation characteristics of the chirped dissipative solitary waves are investigated within the framework of higher-order complex cubic quintic Ginzburg-Landau equation. A potentially rich set of exact chirped dissipative pulses, such as, bright, dark, grey, antidark, kink, antikink is derived in the presence of the self-steepening, self-frequency shift and nonlinear gain/loss. The linear stability results are corroborated by the direct numerical simulations. The effect of the variation of model parameters on physical quantities, like the speed, amplitude and chirping, is explored.

The Interaction Between Frictional Slip and Viscous Fault Root Produces Slow Slip Events

AbstractWe consider a model where an unstable frictional region, governed by rate and state friction, interacts with a viscous zone with Newtonian rheology. The system is loaded at distance with a constant velocity. Pore pressure variations are considered and we show that the model of Segall and Rice (1995, https://doi.org/10.1029/95jb02403) relating porosity changes to variations of the state variable could be derived considering viscoplastic deformation of a population of identical asperities. We perform a linear stability analysis in the case of a constant pore pressure in agreement with the full numerical results. For a given value of the viscosity of the viscous region, stable slip is promoted at low normal stress and unstable slip at high normal stress. Near the transition from stable to unstable slip, modest acceleration of slip, resembling slow slip events (SSE) are observed. We show that our model can reproduce real SSE sequences in the Guerrero subduction zone which are the largest worldwide. The best fit parameters suggest that SSEs happen in areas of low effective normal stress (for the frictional region) and low viscosity (for the viscous region). In our model, SSEs happen in a regime where the viscous region is able to counteract the instability of the frictional one. We show that considering a rate strengthening rheology or a non newtonian one leads to the same linear stability results. Our work shows that a simple model with homogeneous spatial properties can lead to complex dynamics, covering a wide range of observed sliding modes, from steady‐state creep to seismic slip and SSEs.

Thermorheological effect on Rayleigh–Bénard magnetoconvection in a biviscous Bingham fluid with rough boundary condition on velocity and Robin boundary condition on temperature

AbstractThe thermorheological effect on the onset of Rayleigh–Bénard convection in a biviscous Bingham fluid in the presence of a horizontal magnetic field is investigated considering rough boundary conditions on velocity and Robin boundary conditions on temperature. The viscosity of the electrically conducting fluid is assumed to be sensitive to temperature variation. Linear and global nonlinear stability analyses are performed using the Chebyshev pseudospectral method to determine the existence of instability or otherwise. A general interpretation is made from the results to show the effects of the magnetic field and the variable viscosity on the system's stability. The biviscous Bingham parameter and the Chandrasekhar number are shown to have a delay in the onset of convection, while the effect of temperature sensitivity is to advance the onset. It is found that the results of linear and global nonlinear stability are not in agreement, so the region of subcritical instability exists. Also, the results obtained for Rayleigh–Bénard convection agree pretty well with those of Platten and Legros and Siddheshwar et al. for the limiting cases.

A nonlinear analytical formulation for the 1D modelling of a flexible beam in channel flow

The dynamics of flexible beams in confined flows has been a subject of research for many years, as its fundamental behaviour is found in many applications, from energy harvesters to musical instruments. Most studies are concerned solely with the conditions for linear stability and do not explore the ensuing nonlinear behaviour of the system. This is particularly delicate as fluttering beams in confined flows are known to often result in dynamics with intermittent impacts between the beam and the side-walls. Here we present a nonlinear analytical resolution to a simplified 1-D model, based on a modal beam and bulk-flow equations. The model accounts for dissipation through distributed frictional and localized​ head-loss terms. The latter are imposed at the boundary conditions and aim to describe the complex effects occurring outside the domain (turbulence, vortex shedding, etc.). The present analytical resolution leads to a compact system for linear stability analysis, but also to a nonlinear formulation of the fluid–structure interaction. The inclusion of a regularized contact model allows for the computation of the full nonlinear dynamics, including intermittent impacts. Linear stability results are compared to previously published results using 2-D CFD models, and the relative merits of the model are discussed. A variety of limit cycles, (1) with and without impacts, (2) in symmetric and asymmetric configurations and (3) with impacts both at the beam tip and along its length, are shown to illustrate the diversity of dynamics encountered. Moreover, we show that, at large flow velocities, particular model configurations can lead to aperiodic dynamics, a phenomenon reported in several experimental observations. To the authors knowledge, the proposed formulation presents, for the first time, a framework for the comprehensive understanding of the nonlinear dynamics associated with flexible beams in confined axial flow.

Dynamical analysis on a size-structured population model of Daphnia with delayed birth process

In this paper, we investigate a size-structured population model of Daphnia coupled with an unstructured algal food source. We introduce a delay into the boundary condition of this model, where the delay describes the effect of competition for finding food or resource in short supply juvenile period. Following the semigroup theory, we transform our model into the abstract boundary delay problem and obtain the existence and uniqueness of a positive stationary solution. By means of the spectral analysis and the characteristic equation technique, the instability and linear stability results of stationary solutions are formulated in terms of the basic reproduction number ℛ(F). Some numerical simulation examples are presented to illustrate the feasibility of our main results.

A Reduced and Linearized High Fidelity Waveboard Multibody Model for Stability Analysis

AbstractIn this paper, the robustness of a recently validated linearization approach is demonstrated with the linear stability analysis of a waveboard, a human-propelled two-wheeled vehicle consisting in two rotatable platforms, joined by a torsion bar and supported on two caster wheels. A multibody model with holonomic and nonholonomic constraints is used to describe the system. The nonlinear equations of motion, which constitute a differential-algebraic system of equations (DAE system), are linearized along the steady forward motion. With this approach, the minimal set of linearized equations of motion of the waveboard multibody model with toroidal wheels is derived. The procedure enables the generation of the Jacobian matrix in terms of the geometric and dynamic parameters of the multibody system, and the eigenvalues of the system are parameterized in terms of the design parameters. The resulting minimum set of linear equations leads to the elimination of null eigenvalues, while retaining all the stability information in spite of the reduction of the Jacobian matrix. The linear stability results of the waveboard obtained in previous work are validated with this approach. The procedure shows an excellent computational efficiency with the waveboard, its utilization being highly advisable to linearize the equations of motion of complex constrained multibody systems.

Linear and Nonlinear Stability Analyses of Double-Diffusive Convection in a Vertical Brinkman Porous Enclosure under Soret and Dufour Effects

Analytical and numerical investigations were performed to study the influence of the Soret and Dufour effects on double-diffusive convection in a vertical porous layer filled with a binary mixture and subject to horizontal thermal and solute gradients. In particular, the study was focused on the effect of Soret and Dufour diffusion on bifurcation types from the rest state toward steady convective state, and then toward oscillatory convective state. The Brinkman-extended Darcy model and the Boussinesq approximation were employed to model the convective flow within the porous layer. Following past laboratory experiments, the investigations dealt with the particular situation where the solutal and thermal buoyancy forces were equal but acting in opposite direction to favor the possible occurrence of the rest state condition. For this situation, the onset of convection could be either supercritical or subcritical and occurred at given thresholds and following various bifurcation routes. The analytical investigation was based on the parallel flow approximation, which was valid only for a tall porous layer. A numerical linear stability analysis of the diffusive and convective states was performed on the basis of the finite element method. The thresholds of supercritical, RTCsup, and overstable, RTCover, convection were computed. In addition, the stability of the established convective flow, predicted by the parallel flow approximation, was studied numerically to predict the onset of Hopf’s bifurcation, RTCHopf, which marked the transition point from steady toward unsteady convective flows; a route towards the chaos. To support the analytical analyses of the convective flows and the numerical stability methodology and results, nonlinear numerical solutions of the full governing equations were obtained using a second-order finite difference method. Overall, the Soret and Dufour effects were seen to affect significantly the thresholds of stationary, overstable and oscillatory convection. The Hopf bifurcation was marked by secondary convective flows consisting of superposed vertical layers of opposite traveling waves. A good agreement was found between the predictions of the parallel flow approximation, the numerical solution and the linear stability results.

The effect of incoming boundary layer thickness and Mach number on linear and nonlinear Rossiter modes in open cavity flows

The Rossiter modes of an open cavity were studied using global linear analysis and nonlinear numerical simulations. The length over depth ratio was two and the Reynolds numbers based on cavity depth were close to 1000. The effect of Mach on such cavities was studied. The global analysis revealed that, in the Mach range 0.1 to 0.9, for thick boundary layers, only R1 and R2 modes could become unstable, whereas for thin boundary layer up to R4 could be unstable. Compressibility was very destabilizing at low Mach. At moderate Mach the instability either saturated with Mach or had an irregular dependence. Analysis of the Rossiter mode eigenfunctions indicated that the acoustic feedback scaled to Ma^3, and explained the strong destabilizing effect of compressibility. The irregular dependence was associated with resonances between Rossiter modes and acoustic cavity modes. The analysis explained why the irregular Mach dependence occurred only for higher order Rossiter modes. In this parameter region three dimensional modes are stable or marginally unstable. Two-dimensional simulations were performed to evaluate how much of the nonlinear regime could be captured by the linear stability results. As the flow became more unstable, an increasing more complex final stage was reached. Yet the spectra present distinct tones that are close to linear predictions. However, a closer look reveals the dominant frequency in the saturated state corresponds to R1 while linear theory predicts R2 as the most unstable mode. This is associated with a nonlinear thickening of the mixing layer in the nonlinear regime. Moreover, the frequencies are much closer to Rossiter empirical predictions than the linear instability ones. Even at these conditions close to critical, the spectra are well described by the mode R1 and a cascade of nonlinearly generated harmonics, with no reminiscence of the linear instability.

Stability and Bifurcation Analysis of a Prey–Predator Model

The purpose of the present paper is to study the stability of a prey–predator model using KCC theory. The KCC theory is based on the assumption that the second-order dynamical system and geodesics equation, in associated Finsler space, are topologically equivalent. The stability (Jacobi stability) based on KCC theory and linear stability of the model are discussed in detail. Further, the effect of parameters on stability and the presence of chaos in the model are investigated. The critical values of bifurcation parameters are found and their effects on the model are investigated. The numerical examples of particular interest are compared to the results of Jacobi stability and linear stability and it is found that Jacobi stability on the basis of KCC theory is global than the linear stability.