Nonlinear Instability Analysis Research Articles

Thermal instability and vibration characteristics of laminated composite struts with Graphene reinforcements: An analysis of distribution patterns and geometrical imperfections

This study explores the effect of local buckling on the compressive performance of slender structural elements, particularly those with thin-walled sections. The phenomenon of local buckling significantly reduces the axial compressive stiffness, leading to a notable decrease in the load-bearing capacity of these elements. The main goal of this research is to examine how the post-buckling characteristics of polymeric composite channel section struts can be improved under thermal loading by incorporating multi-layer graphene reinforcements. The solution methodology incorporates the von Karman geometrical nonlinearity and is based on the layerwise third-order shear deformation theory (LW-TSDT). To ascertain the precision and computational performance of the results derived from LW-TSDT, a three-dimensional (3D) finite element model is created in ABAQUS for comparative evaluation. An extensive analysis of nonlinear thermal instability in perfect and geometrically imperfect FG-GRC laminated channel section struts is undertaken to discern the graphene distribution patterns that are most and least effective in elevating the critical buckling temperature and natural frequencies through pre- and post-buckling conditions. The comparative analysis indicates that employing the FG-X graphene distribution pattern across the thickness of the web and flanges in channel section struts leads to a projected increase of 12 % in the critical buckling temperature for clamped channel section struts, in contrast to those that adopt the FGO graphene distribution pattern. For cases with simply-supported boundary conditions, this increase is noted to be approximately 9 %. Moreover, findings confirm that incorporating an asymmetric graphene distribution pattern (FGV) or introducing geometrical imperfections in the flanges and web that generate a bending moment within the structure from the beginning of thermal loading effectively prevents the primary natural frequencies of FG-GRC channel section struts from declining to zero close to the critical buckling temperature. This is significantly different from scenarios involving perfectly structured and symmetrically reinforced graphene distribution patterns such as FGX.

Nonlinear dynamic modeling and global instability analyses of planetary gear trains considering multi-state engagement and tooth-contact temperature effects

Nonlinear dynamic modeling and global instability analyses of planetary gear trains considering multi-state engagement and tooth-contact temperature effects

Weakly nonlinear instability analysis of triple diffusive convection under internal heat generator and modulated boundaries

A nonlinear instability analysis of triple diffusive convection under the time-dependent heat and mass transfer boundary conditions in the presence of internal heat source is evaluated in this study. On various physical parameters, the momentary behavior of both Sherwood and Nusselt number profiles is examined. In the geometry, we have considered two parallel infinite horizontal plates acting gravity vertically downward z-direction. By using the weakly nonlinear analysis, the Ginzburg–Landau equation is generated for the rate of heat and mass transport. Here, we have considered the temperature and concentration of two solutes. The temperature and first concentration of the solute at the lower plate are higher than the upper plate, while the second concentration of the solute at the upper plate is higher compared to that of the lower plate. According to the different modulation, we have considered four cases based on the phase angle of the modulations. The convective heat and mass transports are measured as a function of the Nusselt number (Nu) and Sherwood number (Sh1 and Sh2) for both the concentration. From the results, it is found that the first Lewis number increases all the considered profiles, while Ri increases the Nusselt number profile only. The principal discovery elucidated by this article resides in the observation that the internal heat source, subject to modulated boundaries, maintains the convective instability if different solutes are used from both ends.

Instability of Liquid Film with Odd Viscosity over a Non-Uniformly Heated and Corrugated Substrate.

The effect of odd viscosity on the instability of liquid film along a wavy inclined bottom with linear temperature variation is investigated. By utilizing the long-wave approximation, the non-linear evolution equation of the free surface is derived. By applying the normal mode method, the linear instability of thin film flow is investigated. With the help of multi-scale analysis methods, the weakly non-linear instability of thin film flow is also investigated. The results reveal that the Marangoni effect caused by non-uniform temperature distribution promotes the instability of the liquid film, while the odd viscosity has a stabilizing effect. In addition, for a positive local inclination angle θ, an increase in bottom steepness ζ inhibits the instability of the liquid film flow. In contrast, with a negative local inclination angle θ, increased bottom steepness ζ promotes the instability of the liquid film flow. The results of the temporal linear instability analysis and the weakly non-linear instability analysis have been substantiated through numerical simulations of the non-linear evolution equations.

Instability analysis of double diffusive convection under time dependent solute boundary conditions in the presence of internal heat generator

The present study provides heat and mass transport of double-diffusive convective flow of Newtonian fluid with the presence of internal heat source. Here, we have considered two infinite horizontal plates apart from a constant distance under the solutal modulated boundary condition. To obtain the weakly nonlinear instability analysis, we drive Ginzburg–Landau equation. The effects of physical parameters obtained during the study on the Nusselt number and Sherwood number have been examined graphically. The Prandtl number and Richardson number enhance the rate of heat and mass transport, and Lewis number suppressed it. The major finding of the current study is that the internal heat source has a negligible effect on mass transfer, but a significant effect on the heat transfer.

Weakly nonlinear instability analysis of viscous planar liquid sheet under varicose mode of initial disturbance with co-flowing gas streams and prediction of resulting spray characteristics

A theoretical investigation of the effects of viscosity is presented for breakup of a planar viscous liquid sheet subjected to a fundamental varicose mode of disturbance in the presence of moving gas medium. A classic perturbation technique up to second order has been employed to derive the linear and nonlinear interface disturbances. The breakup model is coupled with Maximum Entropy Formalism to predict the final spray characteristics. Findings reveal that within an initial range of Re, imposition of first harmonic on the fundamental disturbance produces a faster breakup as compared to the fundamental disturbance alone. As a result, first harmonic produces a destabilizing effect in the first regime, whose range is identified as 1 <Re < 20.3, 1<Re < 13.35 and 1<Re < 10.53 for gas velocity ratio 2.25, 2.50 and 2.75, respectively. Above this initial range of Re, first harmonic exhibits a stabilizing effect as it produces a delayed breakup when compared with the fundamental mode alone. However, a larger increase in dominating linear growth rate with Re negates the reduction of first harmonic in the destabilizing regime as well its damping effect in the stabilizing regime, resulting in an overall reduction in breakup time with the increase in Reynolds number. Analysis of spray characteristic suggests formation of finer droplets at higher gas velocity ratios. Moreover, reduction in liquid viscosity decreases the range of droplet diameter for gas velocity ratios 2.25 and 2.50.

Nonlinear instability analysis of internally heated convection in a porous layer with the impact of magnetic field and variable gravity

AbstractThe impact of the magnetic field, heat source, and variable gravity field on the stability of convective phenomena in a porous layer are investigated numerically. Three types of gravity variations, such as linear, parabolic, and cubic functions are considered. For linear and nonlinear theories, the method of normal modes has been employed to solve governing dimensionless equations. It is found that the onset of convection is delayed by increasing the Hartmann number and gravity variation parameter and enhancement of the internal heat source makes the system stable. The subcritical instability region increases as the Hartmann number and gravity variation parameter increase.

Three-dimensional nonlinear thermoacoustic instability analysis based on Green's function approach

In this work, nonlinear thermoacoustic instability analysis is made in a hard-walled box to investigate the limit cycle of thermoacoustic oscillation and the interplay of two modes. We model the flame as an acoustically compact source described by a generic Flame Describing Function, i.e. an amplitude-dependent Flame Transfer Function. The acoustic field in the hard-wall box could be described by an integral equation using a Green's function tailored to a three-dimensional (3-D) rectangular box with hard-wall boundary conditions. The integral equation is solved by two methods. Firstly, an iteration method, stepping forward in time, is used to give the time history of the acoustic velocity. The other method is done in the frequency domain to determine the thermoacoustic eigenfrequency and growth rate of thermoacoustic modes. We could observe the phenomenon of modes "jump" after two different time windows of the time history operated with the Fourier transformation. This result reveals that there are two modes in the system, the interference between modes will occur. In the early stage, the two modes work together. After reaching the limit cycle oscillation, the role of one mode is dominant and the other mode basically disappears.

Effect of confinement on weakly nonlinear temporal instability analysis of a viscous planar liquid sheet sandwiched between two co-flowing gas streams at equal velocities

Extensive number of applications like rocket jet engine and gas turbines carry out atomization within a confined environment as it provides faster atomization and good mixing efficiency. Literature suggests that the effect of confinement on breakup of droplets, liquid threads, and annular jets is not elementary, but it varies with liquid viscosity, surrounding medium viscosity, and the confinement strength. Through the present weakly nonlinear temporal analysis, we unveil if a similar dynamic effect of confinement persists on the breakup of planar two-dimensional liquid sheets of varying viscosities. Moreover, to mimic air assisted atomization, the surrounding gases are considered to be flowing with non-zero velocities. Our study reveals that presence of confinement at higher gas-to-liquid velocity ratios of 2.5 and 3 produces no apparent improvement in atomization at all liquid Reynolds numbers. However, at relatively low gas-to-liquid velocity ratios of 2 and 2.25, confinement produces faster breakup as compared to an unconfined one, more so when Reynolds number is relatively low (Re &amp;lt; 100). The overall influence of confinement on non-dimensional breakup time is, however, weak as compared to how it affects the linear growth rate. A counteracting effect of confinement on the linear maximum growth rate and normalized second order disturbance amplitude is identified as the primary reason. Our study also identifies the minimum confinement height CH-1 which is just enough for the liquid sheet to achieve breakup without touching the solid walls. Interestingly, the maximum linear growth rate and minimum breakup time for a particular flow condition are always registered when confinement height equals to CH-1. The values of CH-1 display a strong dependence upon Reynolds number and gas-to-liquid velocity ratios.

Nonlinear magneto convection in an inclined porous layer with artificial neural network prediction

The onset of magnetoconvection in an inclined porous layer is investigated. The effects of physical parameters, such as the Rayleigh number, the inclination angle, and the Hartmann number, are examined. The system becomes more stable by increasing the inclination angle and Hartmann number. It is noted that the transverse rolls are more stable than the longitudinal rolls. A comparison between the linear and nonlinear instability analysis is discussed in more detail. The threshold values for the longitudinal and transverse rolls provide the subcritical instability region. It is noted that the subcritical instability region increases as the inclination angle also increases, whereas it remains unchanged as the Hartmann number increases. Besides, an artificial neural network (ANN) model using the Levenberg‐Marquardt backpropagation algorithm is employed to predict the distribution of critical Rayleigh numbers for both linear and nonlinear stability analyses. The optimal number of neurons in the hidden layer is selected on the basis of coefficient of determination, root mean square error, and root mean relative error. The simulated critical Rayleigh numbers obtained by the numerical study and the predicted critical Rayleigh numbers by the ANN coincides.

Analysis of nonlinear vibration and instability of electrostatic functionally graded micro-actuator based on nonlocal strain gradient theory considering thickness effect

Analysis of nonlinear vibration and instability of electrostatic functionally graded micro-actuator based on nonlocal strain gradient theory considering thickness effect

A Comparative Assessment on Different Aspects of the Non-Linear Instability Dynamics of Supercritical Fluid in Parallel Channel Systems

The thermal-hydraulic behavior of supercritical water reactors with a parallel channel configuration was examined through a non-linear instability analysis. This analysis was performed under wide-ranging conditions and aspects, including different working supercritical fluids, varied heat-flux and flow-rate conditions, and channel inclinations. The supercritical fluid (SCFs) dynamics were captured using the density, enthalpy, and velocity analytical approximation functions. The major findings show that both SCFs (water and carbon dioxide) experienced density wave oscillations at a low pseudo-subcooling number. Static instability characteristics were observed for supercritical water, at a relatively high subcooling number. Further, it was found that at different heat flux, the hotter channel makes the overall system more unstable, whereas, at equal heat flux, parallel channels perform similar to a single-channel system. However, the effect of the inclination angle was found to be negligible owing to supercritical pressure conditions. Moreover, stable and unstable limit cycles along with out-of-phase oscillation characteristics were observed in dynamic stability regions. The present model was also compared with experimental and numerical data. Moreover, co-dimension and numerical simulations were performed to confirm the observed non-linear characteristics. This study helps to enhance the heat transfer characteristics during safe operation of heated channel systems, such as nuclear reactors and solar thermal systems.

Ginzburg–Landau Equation for Nonlinear Instability Analysis in Cardiac Tissue

Ginzburg–Landau Equation for Nonlinear Instability Analysis in Cardiac Tissue

Nonlinear analysis of flame hydrodynamic instability at large gas expansion ratio

In the context of the large thermal-expansion approximation, we derive an equation describing flame front dynamics under conditions of Darrieus-Landau instability. We show that the second-order theory leads to system of two evolution equations for the flame front perturbations and for the potential of the unburned mixture flow. In the limiting case of long evolution, the system of equations can be reduced to one equation with respect to the additive variable that is the sum of the front perturbations and the flow potential. The equation with respect to the additive variable at large gas expansion coefficients has the form of the Sivashinsky equation obtained for the case of small gas expansion coefficients.

Rigorous proof of the slightly nonlinear Jeans instability in the expanding Newtonian universe

Due to the nonlinearity of the Euler{Poisson equations, it is possible that the nonlinear Jeans instability may lead to a faster density growing rate than the rate in the standard theory of linearized Jeans instability, which motivates us to study the nonlinear Jeans instability. The aim of this article is to develop a method proving the Jeans instability for slightly nonlinear Euler-Poisson equations in the expanding Newtonian universe. The standard proofs of the Jeans instability rely on the Fourier analysis. However, it is difficult to generalize Fourier method to a nonlinear setting, and thus there is no result in the nonlinear analysis of Jeans instability. We firstly develop a non-Fourier-based method to reprove the linearized Jeans instability in the expanding Newtonian universe. Secondly, we generalize this idea to a slightly nonlinear case. This method relies on the Cauchy problem of the Fuchsian system due to the recent developments of this system in mathematics. The fully nonlinear Jeans instability for the Euler-Poisson and Einstein-Euler equations are in progress.

Nonlinear combustion instability analysis of a bluff body burner based on the flame describing function

Lean premixed combustion is a common form of combustion organization in power equipment and propulsion systems. In order to understand the dynamic characteristics of lean premixed flame and predict and control its combustion instability, it is necessary to obtain its flame describing function (FDF). Based on the open source CFD toolbox, OpenFOAM, the dynamic K-equation model, and the finite rate Partially Stirred Reactor (PaSR) model were used to perform large eddy simulations (LES) of lean premixed combustion, and the response of the unsteady heat release rate to single-frequency harmonic disturbances was studied. The response of the unsteady heat release rate was characterized by the FDF, and the response of the unsteady heat release rate to the two-frequency harmonic disturbance was studied. The results show that the quantitative heat release rate response and flame dynamics have very proper accuracy. In the single-frequency harmonic disturbance, as the forcing frequency increases, the curling behavior of the flame surface and the instantaneous vortex structure change; the nonlinear kinematics effect is manifested by the entrainment of the vortex. At lower forcing frequencies, the heat release response changes linearly with the increase of forcing amplitude; at intermediate frequencies, the heat release response exhibits obvious nonlinear behavior; at high frequencies, the heat release response to amplitude changes decreases. The introduction of the second harmonic disturbance will significantly reduce the response range of the total heat release rate and make the combustion more stable.

Nonlinear dynamic analysis of multiple grooved water-lubricated journal bearings using attenuation rate interpolation method

Purpose The stringent requirements for environmental protection have induced the extensive applications of water-lubricated journal bearings in marine propulsion. The nonlinear dynamic analysis of multiple grooved water-lubricated bearings (MGWJBs) has not been fully covered so far in the literature. This study aims to conduct the nonlinear dynamic analysis of the instability for MGWJBs. Design/methodology/approach An attenuation rate interpolation method is proposed for the determination of the critical instability speed. Based on a structured mesh movement algorithm, the transient hydrodynamic force model of MGWJBs is set up. Furthermore, the parameters’ analysis of nonlinear instability for MGWJBs is conducted. The minimum water film thickness, side leakage, friction torque and power loss of friction are fully analyzed. Findings With the increase of speed, the journal orbits come across the steady state equilibrium motion, sub-harmonic motion and limit circle motion successively. At the limit circle motion stage, the orbits are much larger than that of steady state equilibrium and sub-harmonic motion. The critical instability speed increases when the spiral angle decreases or the groove angle increases. The minimum water film thickness peak is at the rotor speed of 4,000 r/min for the MGWJB with Sa = 0°. As rotor speed increases, the side leakage decreases slightly while the friction torque and the power loss of friction increase gradually. Originality/value Present research provides a beneficial reference for the dynamic mechanism analysis and design of MGWJBs.

Nonlinear thermal instability and vibration analysis of pre/post-buckled FG porous nanotubes using nonlocal strain gradient theory

Current research presents the nonlinear static/dynamic response of functionally graded (FG) porous nanotubes in the thermal pre- and post-buckling regimes. The case of temperature-dependent FG nanotubes with even distributed porosities subjected to uniform temperature rise is considered. The stability and vibration characteristics are analyzed for the uniformly heated nanotube surrounded by a nonlinear elastic foundation. The nonlinear equations of motion are derived using the Hamilton principle for the thermally pre/post-buckled nanotube in the framework of nonlocal strain gradient theory. The size-dependent governing equations are established based on the uncoupled thermoelasticity, the von Kármán assumption and the high-order shear deformation theory. The nonlinear governing equations are reduced for the nonlocal strain gradient-based model of nanotubes with immovable simply supported boundary conditions. The system of equations is solved for the nonlinear problems of free vibration and post-buckling of the heated tube using two-step perturbation technique and Galerkin procedure. The novel parametric examples are carried out to investigate the effect of nonlocal and length scale parameters, porosity distribution, functionally graded pattern, elastic foundations and geometrical parameters on the thermoelastic response of tubes. The obtained results highlight the importance of static/dynamic analysis of FG porous nanotubes in the thermal pre-buckling and post-buckling regimes.

Nonlinear instability analysis of a vertical cylindrical magnetic sheet

This paper concerns with the nonlinear instability analysis of double interfaces separated three perfect, incompressible cylindrical magnetic fluids. The cylindrical sheet is acted upon by an axial uniform magnetic field. The current nonlinear approach depends mainly on solving the linear governing equations of motion and subjected to the appropriate nonlinear boundary conditions. This procedure resulted in two nonlinear characteristic equations governed the behavior of the interfaces deflection. By means of the Taylor expansion, together with the multiple time scales, technique, the stability analysis of linear as well as the nonlinear is achieved. The linear stability analysis reveals a quadratic dispersion equation in the square of growth rate frequency of the surface wave. On the other hand, the nonlinear analysis is accomplished by a coupled nonlinear Schrodinger equation of the evolution amplitudes of the surface waves. The stability criteria resulted in a polynomial of the eleventh degree in the square of the magnetic field strength, together with resonance transition curves. Several special cases are reported upon appropriate data choices. The stability criteria are numerically discussed, at which regions of stability and instability are identified. In the stability profile, the magnetic field intensity is plotted versus the wave number. The influences of the parameters on the stability are addressed. The nonlinear stability approach divides the phase plane into several parts of stability/instability. The nonlinear stability shows an in contrast mechanism of the role of the sheet thickness.

On the Instability of Fluid Particle Interface and Shape Oscillations

The dynamic interaction of a fluid particle with a continuous phase has been often a topic of study, for its broad range of applications in various industries. Better understanding of particle-fluid interface instability can assist to enhance the efficiency of different processes and possibly avoid unfavorable consequences. In the present work, a weakly nonlinear instability analysis is performed to study the driven shape oscillations of a fluid particle (droplet or bubble) immersed in another fluid (liquid or gas). The analysis is conducted using a perturbation expansion approach, allowing for shape mode interactions for any arbitrary number of modes. The present work intends to provide an analytical solution to forced shape oscillations of a fluid particle, which accounts for viscosity of both fluids involved, instability nonlinearities, and shape mode interactions. The model proposed is used to study the effect of various parameters on the deformation of fluid particle interface. The cases studied provide an insight into shape mode interactions and resonances due to nonlinearity. Moreover, the results show that the viscosity of the fluid particle can play a role in increasing the chances of greater mode interaction and parametric resonance effects, especially for relatively large initial deformations.