Multiple Steady-state Solutions Research Articles

A critical state μ(I)-rheology model for cohesive granular flows

The dynamic behaviour of granular media can be observed widely in nature and in many industrial processes. Yet, the modelling of such media remains challenging as they may act with solid-like and fluid-like properties depending on the rate of the flow and can display a varying apparent friction, cohesion and compressibility. Over the last two decades, the $\mu (I)$ -rheology has become well established for modelling granular liquids in a fluid mechanics framework where the apparent friction $\mu$ depends on the inertial number $I$ . In the geo-mechanics community, modelling the deformation of granular solids typically relies on concepts from critical state soil mechanics. Along the lines of recent attempts to combine critical state and the $\mu (I)$ -rheology, we develop a continuum model based on modified cam-clay in an elastoplastic framework which recovers the $\mu (I)$ -rheology under flow. This model permits a treatment of plastic compressibility in systems with or without cohesion, where the cohesion is assumed to be the result of persistent inter-granular attractive forces. Implemented in a two- and three-dimensional material point method, it allows for the trivial treatment of the free surface. The proposed model approximately reproduces analytical solutions of steady-state cohesionless flow and is further compared with previous cohesive and cohesionless experiments. In particular, satisfactory agreements with several experiments of granular collapse are demonstrated, albeit with shear bands which can affect the smoothness of the surface. Finally, the model is able to qualitatively reproduce the multiple steady-state solutions of granular flow recently observed in experiments of flow over obstacles.

Coexistence and bistability of a modified Leslie–Gower predator–prey model with mixed movement for the predator and Holling-type II schemes

This paper centers on the analysis of the dynamics of a modified Leslie–Gower predator–prey model employing Holling-type II schemes, with the prey exhibiting pure random diffusion and the predator undergoing a mixed form of movement. The extinction of species and uniform persistence of this system are explored, and several conditions for the stability, uniqueness and multiplicity of positive steady-state solutions are derived. In contrast to the specialist and generalist predator–prey systems in open advective environments, the dynamics of this system are more intricate. It emerges that multiple positive steady-state solutions and the bistable phenomenon exist for this system when a small advection rate and a moderate predation rate are imposed. Numerical simulations reveal that the increase of diffusion rate for prey disadvantages the survival of itself and has no impact on predator invasion, while the increase of diffusion rate for predators favors the invasion of itself.

Rayleigh–Bénard Convection With Multiple Solutions in Trapezoidal Closed Cavities

Abstract Rayleigh–Bénard convection (RBC) in symmetric trapezoidal closed cavities with cavity angle ϕ=70°−110°, filled with air, is studied using numerical simulations where inclined side walls are adiabatic. In contrast to rectangular cavities, where no flow exists below a threshold value, there is a weak convection even at a low Rayleigh number (Ra) due to the fact that there is a component of thermal gradient in the horizontal direction in these cavities. Interestingly, these cavities show sudden and significant jumps in the convection, similar to square cavities (Rac = 2585.02 for ϕ=90°), as Ra increases beyond a critical value (Rac). It is noted here that these Rac represent symmetry-breaking pitchfork bifurcations. These bifurcations are seen in both acute (Rac = 8000 for ϕ=70°) and obtuse (Rac = 2300 for ϕ=110°) angle trapezoidal cavities. Moreover, it is observed that multiple steady-state solutions (MSSS) exist as Ra is further increased. A forward and backward continuation approach for numerical simulations is used to track the co-existence of MSSS. These steady-states have co-existing one-roll and two-roll convective patterns beyond another threshold value of Ra. Here, two types of critical Ra have been identified for different cavity angles; one shows the sudden jump in the convection, and the other is the one beyond which MSSS co-exist. Furthermore, a codimension two bifurcation analysis is carried out with Ra and ϕ as two parameters. The bifurcation analysis divides the parameter space into different regions based on the multiplicity of the solutions.

Flame stabilization by a highly conductive cylinder: Multiple steady-state solutions and dynamics

This study examines stabilization of a premixed flame by a circular cylinder placed perpendicularly to the uniform flow of the reacting mixture. It is assumed that the cylinder has a high thermal conductivity leading to an effectively uniform surface temperature. This temperature is not fixed, but it is determined by a thermal flame-cylinder balance. The numerical investigation is carried out on the basis of low Mach number Navier–Stokes equations coupled with the conservation equations for the energy and the fuel mass. Two models for transport coefficients are compared. The first model assumes they have constant values, and the second takes into account their change with temperature.Numerical modeling shows that within the specific range of parameters of this study the system can have several steady-state solutions corresponding to different cylinder temperatures. Moreover, at least two solutions are shown to be stable, corresponding to the hottest and coldest cylinder temperatures. The actual occurrence of one or another regime depends on the initial conditions.

A Non-Linear Non-Planar Coupling Mechanism of Suspended Cables in Thermal Conditions

Slight variations induced by thermal effects may bring unexpected discrepancies in both the system’s linear and non-linear responses. The present study investigates the temperature effects on the non-linear coupled motions of suspended cables subject to one-to-one internal resonances between the in-plane and out-of-plane modes. The classical non-linear flexible system is excited by a uniform distributed harmonic excitation with the primary resonance. Introducing a two-mode expansion and applying the multiple scale method, the polar and Cartesian forms of modulation equations are obtained. Several parametric investigations have highlighted the qualitative and quantitative discrepancies induced by temperature through the curves of force/frequency-response amplitude, time history diagrams, phase portraits, frequency spectrum, and Poincaré sections. Based on the bifurcation and stability analyses, temperature effects on the multiple steady-state solutions, as well as static and dynamic bifurcations, it is observed that the periodic motions may be bifurcated into the chaotic motions in thermal environments. The saddle-node, pitch-fork, and Hopf bifurcations are sensitive to temperature changes. Finally, our perturbation solutions are confirmed by directly integrating the governing differential equations, which yield excellent agreement with our results and validate our approach.

Experimental and analytical study on the stability of a low aspect ratio single-phase natural circulation loop coupled to a parabolic trough collector

Use of passive heat transport systems in renewable energy eliminates operator and external power requirements, thereby reducing the cost. Hence, a horizontal heater and horizontal cooler rectangular natural circulation loop (NCL) of low L/D (110) and aspect ratio (0.54) is coupled to a parabolic trough in the present work. Experimental investigations of the loop's steady state and stability characteristics were carried out at various powers corresponding to the solar irradiances using water as the heat transfer fluid. SolTrace was used to determine the heat loads at different direct normal irradiance (DNI) for the selected loop and trough dimensions. The upper threshold for stability was experimentally observed for the first time and reported here. Stability analysis carried out for different loop diameters revealed that the power at which the upper threshold of instability occurs reduces significantly with increase in loop diameter, enabling its observation in the low-pressure low temperature loop employed in the present study. Stabilizing techniques like orifice insertion and loop inclinations were also studied in detail. The results showed stability at higher heat loads (> 350 DNI), while the instability at lower heat loads can be curbed using an orifice (β = 0.5) or a loop inclination (> 4°). The small inclination can eliminate multiple steady state solutions and hence stabilizes the loop. Besides, the reduction in natural circulation flow rate with inclination is much smaller compared to the orifice making its heat transport capability higher. Additionally, the steady state results were validated with 1-D analytical model giving a good match.

Stabilization of the flat Poiseuille-type flow for viscoelastic polymeric liquid

This paper presents a numerical study for the problem of the one-dimensional flow of viscoelastic liquid polymers between two parallel plates. The equations of a rheologically modified Vinogradov–Pokrovskii (mVP) model is used for the formulation of the problem. It is shown that the problem could have multiple steady-state solutions. The evaluation of non-steady solutions was performed to see if the time-dependent solutions got eventually attracted by the steady ones. Also for the case of multiple steady solutions, it was checked which one attracts the non-steady solution if any. The evaluation of time-dependent solutions was used to estimate the stability of equilibrium states. It is revealed that stable steady-state regimes of the problem exist under certain conditions, and also there could be no more than one stable regime for any given set of parameters. The calculations were performed to estimate the values of Reynolds and Weissenberg numbers corresponding to either stable or unstable steady regimes. The result indicates that instability of the steady flow could possibly occur for arbitrary low Reynolds numbers under certain balance of viscous and elastic forces.

Gregory-Laflamme encounters Superradiance

We investigate the effect of superradiant scattering of gravitational perturbations on the stability of rotating black strings, focusing on the six dimensional equal-spinning Myers-Perry black string. We find that rapidly rotating black strings are unstable to gravitational superradiant modes within a bounded range of string lengths. The instability occurs because momentum along the string direction creates a potential barrier that allows for the confinement of superradiant modes. Yet, five dimensional Myers-Perry black holes do not have stable particle orbits so, unlike other known superradiant systems, these black strings remain stable to perturbations with sufficiently high azimuthal mode number — this is a ‘finite-m’ superradiant instability. For some parameters, this instability competes with the Gregory-Laflamme instability, but otherwise exists independently. The onset of this instability is degenerate and branches to multiple steady-state solutions. This paper is the first of a trilogy: in the next two, we construct two distinct families of rotating strings emerging from the superradiant onset (the ‘black resonator strings’ and ‘helical black strings’). We argue that similar physics is present in 5-dimensional Kerr black strings, but not in D > 6 equal-spinning Myers-Perry black strings.

Numerical evaluation of static and dynamic stability characteristics of a supercritical CO[formula omitted]-driven natural circulation loop

The stability characteristics of closed supercritical natural circulation loop has hardly been explored owing to the mathematical complexity and absence of explicit boundary condition. Present numerical study investigates the nature of both static and dynamic instability in a rectangular loop with carbon dioxide as the working medium. Sharp deterioration in the steady-state circulation rate can be observed, consistent to existing multidimensional analyses. Loop exhibits Ledinegg instability for an intermediate range of heater power, characterized by multiple steady-state solutions. Appearance of static instability has been substantiated following the pressure drop versus mass flow rate profiles, leading to the development of the stability maps. Dynamic instability has also been identified by introducing perturbation during transient simulations, and the range of both static and dynamic instabilities conforms well with each other. A novel approach of approximating the location of the stability threshold using time-series data has also been proposed.

Structural Features of Microvascular Networks Trigger Blood Flow Oscillations

We analyse mathematical models in order to understand how microstructural features of vascular networks may affect blood flow dynamics, and to identify particular characteristics that promote the onset of self-sustained oscillations. By focusing on a simple three-node motif, we predict that network “redundancy”, in the form of a redundant vessel connecting two main flow-branches, together with differences in haemodynamic resistance in the branches, can promote the emergence of oscillatory dynamics. We use existing mathematical descriptions for blood rheology and haematocrit splitting at vessel branch-points to construct our flow model; we combine numerical simulations and stability analysis to study the dynamics of the three-node network and its relation to the system’s multiple steady-state solutions. While, for the case of equal inlet-pressure conditions, a “trivial” equilibrium solution with no flow in the redundant vessel always exists, we find that it is not stable when other, stable, steady-state attractors exist. In turn, these “nontrivial” steady-state solutions may undergo a Hopf bifurcation into an oscillatory state. We use the branch diameter ratio, together with the inlet haematocrit rate, to construct a two-parameter stability diagram that delineates regimes in which such oscillatory dynamics exist. We show that flow oscillations in this network geometry are only possible when the branch diameters are sufficiently different to allow for a sufficiently large flow in the redundant vessel, which acts as the driving force of the oscillations. These microstructural properties, which were found to promote oscillatory dynamics, could be used to explore sources of flow instability in biological microvascular networks.

Effect of geometrical and operating parameters on performance of a rectangular two-phase natural circulation loop

Natural Circulation Loops (NCLs) find application in a wide range of engineering devices and it is often necessary to predict and ensure the performance of such loops under various operating conditions. In this paper analytical expressions are presented in non-dimensional form for steady-state mass flow rate in typical rectangular, two-phase natural circulation loops having Vertical-Heater Vertical-Cooler (VHVC) and Vertical-Heater Horizontal-Cooler (VHHC) configurations. The normalized loop mass flow rates are expressed in terms of several geometrical and operating variables such as, normalized heat input, density ratio, volume/diameter of heater and cooler sections, aspect ratio of loop, inlet subcooling, fractional length of heater and cooler sections, two-phase friction multiplier and the loss coefficient at heater inlet. The analytical mass flow rate expressions are first validated using published data for various loops and shown to reasonably capture the gravity and friction dominated regimes in two-phase pipe flow. Parametric studies are then conducted to elucidate the effect of key variables on steady-state loop mass flow rates. For the chosen geometrical configuration of the loops, the possibility of existence of multiple steady-state solutions is checked for the entire range of values used in this study for each variable. The discussions in this paper are limited to explaining the influence of individual variables by comparing with a base case analysis.

Flame stabilisation by a highly conductive body: multiple steady-state solutions and time-dependent dynamics

We present an investigation of the stabilisation of premixed laminar flames by means of an isolated highly conductive bluff-body, a circular cylinder, placed in a uniform flow of a combustible mixture. It is shown that the problem has non-unique steady-state solutions for certain values of the parameters. Moreover, we solve the time-dependent equations to check the stability of the solutions and demonstrate the possibility of controlling the convergence to a certain steady-state solution.

Magnetic Field Effect on Multiplicity of Solutions Induced by Thermosolutal Convection in a Bénard Square Porous Cavity Submitted to Horizontal Concentration Gradient

In this work, we present a numerical study of the magnetic field effect on double diffusive natural convection in a square porous cavity saturated with an electrically conducting binary mixture. The cavity is heated from below and cooled from the top, while its vertical walls are adiabatic and maintained at constant but different concentrations. The numerical results are obtained for a Lewis number Le = 10 and the following ranges for the other controlling parameters: 40 ≤ RT ≤ 1000, -0.2 ≤ N ≤ 0.2 and -0.5 ≤ Ha ≤ 0.5, where RT, N and Ha are the thermal Rayleigh number, the buoyancy ratio and the Hartmann parameter, respectively. First, the effect of the Hartmann parameter on the maintenance and disappearance of the multiple steady state solutions obtained in the case of purely thermal convection is examined. Then, the combined effect of N and Ha on the existence of these steady solutions is analyzed. It is found that the critical values of N corresponding to the transitions between the different solutions are modified by the application of a magnetic field. However, the nature of the transitions is unchanged. It is shown that the magnetic field may affect considerably the flow intensity and the heat and mass transfer in the medium.

Bifurcation in a reaction-diffusion model with nonlocal delay effect and nonlinear boundary condition

In this paper, the existence, stability, and multiplicity of steady-state solutions and periodic solutions for a reaction-diffusion model with nonlocal delay effect and nonlinear boundary condition are investigated by using Lyapunov-Schmidt reduction. When the interior reaction term is weaker than the boundary reaction term, it is found that there is no Hopf bifurcation no matter how either of the interior reaction delay and the boundary reaction delay changes. When the interior reaction term is stronger than the boundary reaction term, it is the interior reaction delay instead of the boundary reaction delay that determines the existence of Hopf bifurcation. Moreover, the general results are illustrated by applications to models with either a single delay or bistable boundary condition.

Determination of periodic response limits among multiple solutions for mechanical systems with wedge dampers

Wedge dampers are commonly used to utilize the frictional behavior in many engineering fields such as vehicle dynamics and turbo-machinery. However, the presence of non-unique contact forces in the damper interfaces creates an uncertainty that provides different dynamic response amplitudes even for the same input parameters. The maximum limits of the variability range always take the core attention in most of the damper design processes. In this paper, determination of an upper and a lower boundary among multiple steady-state solutions is presented by using a numerical approach. The method is specifically suitable for the mechanical systems with wedge dampers modeled by macro-slip frictional contact elements in the joint interfaces. In the approach proposed, a criterion that determines the periodic response boundaries according to the limit tangential force values is utilized. The method is demonstrated by illustrating several case studies on a lumped parameter system which represents a turbo-machinery application with a symmetric wedge damper pressed against two vibrating adjacent blades. A point-to-point 1D friction model with varying normal force is used in both contact sides. A parametric investigation on the variability range and response limits is performed for different damper configurations. Harmonic Balance Method with Newton's iteration scheme is used in the numerical solution of the governing equations. The results show that a large variability exists for damper geometries where a strong coupling is present between tangential and normal contact forces. The method proposed successfully captures the limits of the variability range in all cases.

Hysteresis and linear stability analysis on multiple steady-state solutions to the Poisson-Nernst-Planck equations with steric interactions.

In this work, we numerically study linear stability of multiple steady-state solutions to a type of steric Poisson-Nernst-Planck equations with Dirichlet boundary conditions, which are applicable to ion channels. With numerically found multiple steady-state solutions, we obtain S-shaped current-voltage and current-concentration curves, showing hysteretic response of ion conductance to voltages and boundary concentrations with memory effects. Boundary value problems are proposed to locate bifurcation points and predict the local bifurcation diagram near bifurcation points on the S-shaped curves. Numerical approaches for linear stability analysis are developed to understand the stability of the steady-state solutions that are only numerically available. Finite difference schemes are proposed to solve a derived eigenvalue problem involving differential operators. The linear stability analysis reveals that the S-shaped curves have two linearly stable branches of different conductance levels and one linearly unstable intermediate branch, exhibiting classical bistable hysteresis. As predicted in the linear stability analysis transition dynamics, from a steady-state solution on the unstable branch to one on the stable branches, are led by perturbations associated to the mode of the dominant eigenvalue. Further numerical tests demonstrate that the finite difference schemes proposed in the linear stability analysis are second-order accurate. Numerical approaches developed in this work can be applied to study linear stability of a class of time-dependent problems around their steady-state solutions that are computed numerically.

Bistabilities and domain walls in weakly open quantum systems

Weakly pumped systems with approximate conservation laws can be efficiently described by (generalized) Gibbs ensembles if the steady state of the system is unique. However, such a description can fail if there are multiple steady state solutions, for example, a bistability. In this case domains and domain walls may form. In one-dimensional (1D) systems any type of noise (thermal or non-thermal) will in general lead to a proliferation of such domains. We study this physics in a 1D spin chain with two approximate conservation laws, energy and the zz-component of the total magnetization. A bistability in the magnetization is induced by the coupling to suitably chosen Lindblad operators. We analyze the theory for a weak coupling strength \epsilonϵ to the non-equilibrium bath. In this limit, we argue that one can use hydrodynamic approximations which describe the system locally in terms of space- and time-dependent Lagrange parameters. Here noise terms enforce the creation of domains, where the typical width of a domain wall goes as \sim 1/\sqrt{\epsilon}∼1/ϵ while the density of domain walls is exponentially small in 1/\sqrt{\epsilon}1/ϵ. This is shown by numerical simulations of a simplified hydrodynamic equation in the presence of noise.

Multiple steady state solutions for double-diffusive convection in a shallow horizontal rectangular cavity uniformly heated and salted from the side and filled with non-Newtonian power-law fluids

This paper reports on the multiplicity of steady state solutions and the effect of buoyancy ratio due to both heat and mass transfer on natural convection in a horizontal rectangular cavity filled with a power-law non-Newtonian fluid. The short vertical sides of the cavity are submitted to horizontal thermal and solutal flux densities, while its horizontal walls are impermeable and adiabatic. The problem under consideration is governed by the cavity aspect ratio, A, the Lewis number, Le, the buoyancy ratio, N, the power-law behavior index, n, and the generalized Prandtl, Pr, and thermal Rayleigh, RaT, numbers. The equations describing the double-diffusion convection phenomenon are solved numerically using a finite volume method. In the case of a shallow cavity A > > 1, the governing equations are significantly simplified by using the parallel flow approximation, which allows an analytical solution that agrees well with the numerical one. Results expressed in terms of streamlines, isotherms, iso-concentrations, central stream function and average Nusselt and Sherwood numbers are obtained for various values of the governing parameters. The onset and the development of double-diffusive convection, for cooperating and opposing flows, are investigated. The existence of multiple steady state solutions, for a given set of the governing parameters, is demonstrated in the case of opposing double diffusive flow.

Reduced-order modelling of thick inertial flows around rotating cylinders

A new model for the behaviour of a thick, two-dimensional layer of fluid on the surface of a rotating cylinder is presented, incorporating the effects of inertia, rotation, viscosity, gravity and capillarity. Comparisons against direct numerical simulations (DNS) show good accuracy for fluid layers of thickness of the same order as the cylinder radius, even for Reynolds numbers up to . A rich and complex parameter space is revealed, and is elucidated via a variety of analytical and numerical techniques. At moderate rotation rates and fluid masses, the system exhibits either periodic behaviour or converges to a steady state, with the latter generally being favoured by greater masses and lower rotation rates. These behaviours, and the bifurcation structure of the transitions between them, are examined using a combination of both the low-order model and DNS. Specific attention is dedicated to newly accessible regions of parameter space, including the multiple steady state solutions observed for the same parameter values by Lopes et al. (J. Fluid Mech., vol. 835, 2018, pp. 540–574), where the corresponding triple limit point bifurcation structure is recovered by the new low-order model. We also inspect states in which the interface becomes multivalued – and thus outside the reach of the reduced-order model – via DNS. This leads to highly nonlinear multivalued periodic structures appearing at moderate thicknesses and relatively large rotation rates. Even much thicker films may eventually reach steady states (following complex early evolution), provided these are maintained by a combination of forces sufficiently large to counteract gravity.

A static equilibrium condition for the steady-state analysis of variable-stroke swash plate mechanism

Variable-stoke swash plate mechanisms are widely used for compressors in automotive air-conditioning systems, but the dynamics of the system has not been investigated sufficiently. In the present study, we derived a dynamic equation based on the Lagrangian mechanics for the mechanism and solved it numerically. The solutions were compared with the ones obtained from the commercial multibody dynamics software, DAFUL. We proposed a static function that can be conveniently used to predict the steady-state behavior of the system without conducting a full dynamic analysis of the variable-stoke swash mechanism. Using the static function, for example, we showed why the stroke of the variable-stoke swash tends to increase with the driving speed and showed a simple way to reverse the tendency. Especially, we observed that the system can have multiple steady-state solutions depending on initial conditions. Based on this observation, we theoretically showed that a hysteresis phenomenon can occur in the variable-stoke swash mechanism so that the behavior of the system can depend on the operation history of the system.