Inertia Terms Research Articles

Kinetic interpretation of the classical Rayleigh-Taylor instability.

Rayleigh-Taylor (RT) instabilities are prevalent in many physical regimes ranging from astrophysical to laboratory plasmas and have primarily been studied using fluid models, the majority of which have been ideal fluid models. This work presents a five-dimensional (two spatial dimensions, three velocity space dimensions) simulation using the continuum-kinetic model to study the effect of the collisional mean free path and transport on the instability growth. The continuum-kinetic model provides noise-free access to the full particle distribution function permitting a detailed investigation of the role of kinetic physics in hydrodynamic phenomena such as the RT instability. For long mean free path, there is no RT instability growth, but as collisionality increases, particles relax towards the Maxwellian velocity distribution, and the kinetic simulations reproduce the fluid simulation results. An important and novel contribution of this work is in the intermediate collisional cases that are not accessible with traditional fluid models and require kinetic modeling. Simulations of intermediate collisional cases show that the RT instability evolution is significantly altered compared to the highly collisional fluidlike cases. Specifically, the growth rate of the intermediate collisionality RT instability is lower than the high collisionality case while also producing a significantly more diffused interface. The higher moments of the distribution function play a more significant role relative to inertial terms for intermediate collisionality during the evolution of the RT instability interface. Particle energy flux is calculated from moments of the distribution and shows that transport is significantly altered in the intermediate collisional case and deviates much more so from the high collisionality limit of the fluid regime.

Convergence Analysis of New Construction Explicit Methods for Solving Equilibrium Programming and Fixed Point Problems

In this paper, we present improved iterative methods for evaluating the numerical solution of an equilibrium problem in a Hilbert space with a pseudomonotone and a Lipschitz-type bifunction. The method is built around two computing phases of a proximal-like mapping with inertial terms. Many such simpler step size rules that do not involve line search are examined, allowing the technique to be enforced more effectively without knowledge of the Lipschitz-type constant of the cost bifunction. When the control parameter conditions are properly defined, the iterative sequences converge weakly on a particular solution to the problem. We provide weak convergence theorems without knowing the Lipschitz-type bifunction constants. A few numerical tests were performed, and the results demonstrated the appropriateness and rapid convergence of the new methods over traditional ones.

Non-uniqueness, stability and bifurcation analyses in elasto-viscoplastic boundary value problems with no inertia

The article focuses on non-uniqueness, bifurcation and stability conditions in elasto-viscoplastic boundary value problems when inertia terms are neglected. Analytical and numerical studies are presented to investigate the capability of an elasto-viscoplastic model to regularize the behavior in the occurrence of strain localization with respect to number of strain bands formed and mesh dependency. It is found that elasto-viscoplasticity in a Cauchy medium neither restores the uniqueness of the solution nor provides mesh independent results. A high value of the viscosity parameter can sometimes provide results that are mesh independent, up to a certain limit strain, as it actually modifies the response of the constitutive law by an ad-hoc increase of its hardening branch. On the contrary, coupling elasto-viscoplasticity with a second gradient model that introduces an internal length parameter reproduces realistically the rate dependent behavior and regularizes the results.

CFD SIMULATION OF A COUGHING DROPLET DIFFUSION IN THE ACCELERATING-MOTION METRO CABIN

Abstract. The metro cabin provides an effective way for the spread of the pneumonia virus when provides convenience to human beings. The cabin undergoes a series of motions of acceleration speed, uniform speed, and deceleration speed during the one-station travelling process. The induced flow of the accelerated moving cabin is the determinant of the air drag force on the cough droplets, and effect on the time-frequency characteristics and motion trajectories of droplets of different sizes. In this study, we established a momentum equation on droplets, affected by the inertial force correction term in the non-inertial frame. Then added the drag force from the velocity difference between the cough droplets and the background airflow, and simulated configurations as: 1) droplet spreading aerodynamic process from different face orientations of infected dummy, 2) six groups of size particles, and 3) variable speed motion phases. The numerical simulation provides a physical analysis idea for studying the relationship between the two-phase flow under the action of inertial force. This design of a ventilation environment in public transportation helps to reach a more profound understanding of the inertial precipitation mechanism of droplet jets.

Account for the Inertial Term in the Navier–Stokes Equation in the Model of Magnetohydrodynamic Domains for Nematics with Negative Anisotropy of Diamagnetic Susceptibility

Account for the Inertial Term in the Navier–Stokes Equation in the Model of Magnetohydrodynamic Domains for Nematics with Negative Anisotropy of Diamagnetic Susceptibility

The notions of inertial balanced viscosity and inertial virtual viscosity solution for rate-independent systems

AbstractThe notion of inertial balanced viscosity (IBV) solution to rate-independent evolutionary processes is introduced. Such solutions are characterized by an energy balance where a suitable, rate-dependent, dissipation cost is optimized at jump times. The cost is reminiscent of the limit effect of small inertial terms. Therefore, this notion proves to be a suitable one to describe the asymptotic behavior of evolutions of mechanical systems with rate-independent dissipation in the limit of vanishing inertia and viscosity. It is indeed proved, in finite dimension, that these evolutions converge to IBV solutions. If the viscosity operator is neglected, or has a nontrivial kernel, the weaker notion of inertial virtual viscosity (IVV) solutions is introduced, and the analogous convergence result holds. Again in a finite-dimensional context, it is also shown that IBV and IVV solutions can be obtained via a natural extension of the minimizing movements algorithm, where the limit effect of inertial terms is taken into account.

The Synchronization Analysis of Cohen-Grossberg Stochastic Neural Networks with Inertial Terms.

The exponential synchronization (ES) of Cohen–Grossberg stochastic neural networks with inertial terms (CGSNNIs) is studied in this paper. It is investigated in two ways. The first way is using variable substitution to transform the system to another one and then based on the properties of integral, differential operator, and the second Lyapunov method to get a sufficient condition of ES. The second way is based on the second-order differential equation, the properties of calculus are used to get a sufficient condition of ES. At last, results of the theoretical derivation are verified by virtue of two numerical simulation examples.

Evaluating resiliency of electric power generators against earthquake to maintain synchronism

Natural disasters, such as earthquakes, can cause significant disruptions in power systems, such as loss of generations and loads. To evaluate the behavior of a generator to remain in synchronism, subjected to the large disturbance of an earthquake, we propose a dynamic generator resiliency model. The proposed approach models the effects of earthquake energy transfer to a generator considering the time-variant behavior of earthquake disturbance. In addition, the proposed model analyzes the transient behavior of generators impacted by an earthquake considering earthquake ground acceleration and generators’ dynamic characteristics. Using this model, we can determine synchronism status of impacted generators and transient stability status of a power system in response to an earthquake. The proposed approach is tested using real-world data including the data of a real earthquake occurred on a real power plant. The obtained results illustrate the effectiveness of the proposed model to correctly predict the impact of an earthquake on a power plant, and to determine the effects of earthquake magnitude and generator robustness (in terms of generator inertia and damping torque) on the response of a generator to an earthquake.

Inertial Modification Using Self-Adaptive Subgradient Extragradient Techniques for Equilibrium Programming Applied to Variational Inequalities and Fixed-Point Problems

Equilibrium problems are articulated in a variety of mathematical computing applications, including minimax and numerical programming, saddle-point problems, fixed-point problems, and variational inequalities. In this paper, we introduce improved iterative techniques for evaluating the numerical solution of an equilibrium problem in a Hilbert space with a pseudomonotone and a Lipschitz-type bifunction. These techniques are based on two computing steps of a proximal-like mapping with inertial terms. We investigated two simplified stepsize rules that do not require a line search, allowing the technique to be carried out more successfully without knowledge of the Lipschitz-type constant of the cost bifunction. Once control parameter constraints are put in place, the iterative sequences converge on a particular solution to the problem. We prove strong convergence theorems without knowing the Lipschitz-type bifunction constants. A sequence of numerical tests was performed, and the results confirmed the correctness and speedy convergence of the new techniques over the traditional ones.

Jeffery–Hamel flow of a shear-thinning fluid that mimics the response of viscoplastic materials

The flow between two divergent plane walls with a source at the vertex (Jeffery–Hamel flow) of a shear-thinning fluid, that mimics the response of a class of seemingly viscoplastic materials, is studied. The semi-inverse approach is used to obtain the governing equations for the velocity profile. The third-order non-linear ordinary differential equation governing the flow is solved numerically, and the effects of the material moduli that characterize the fluid, the angle between plane walls, and the inertial term, on the velocity are reported and discussed. Results show that there is a critical angle beyond which flow reversal occurs, and the number and angular extent of inflow regions varies based on the material moduli and the inertial term.

The inertial iterative extragradient methods for solving pseudomonotone equilibrium programming in Hilbert spaces

In this paper, we present new iterative techniques for approximating the solution of an equilibrium problem involving a pseudomonotone and a Lipschitz-type bifunction in Hilbert spaces. These techniques consist of two computing steps of a proximal-type mapping with an inertial term. Improved simplified stepsize rules that do not involve line search are investigated, allowing the method to be implemented more quickly without knowing the Lipschitz-type constants of a bifunction. The iterative sequences converge weakly on a specific solution to the problem when the control parameter conditions are properly specified. The numerical tests were carried out, and the results demonstrated the applicability and quick convergence of innovative approaches over earlier ones.

Design and validation of a tunable inertial microfluidic system for the efficient enrichment of circulating tumor cells in blood.

The analysis of circulating tumor cells (CTCs) in blood is a powerful noninvasive alternative to conventional tumor biopsy. Inertial‐based separation is a promising high‐throughput, marker‐free sorting strategy for the enrichment and isolation of CTCs. Here, we present and validate a double spiral microfluidic device that efficiently isolates CTCs with a fine‐tunable cut‐off value of 9 μm and a separation range of 2 μm. We designed the device based on computer simulations that introduce a novel, customized inertial force term, and provide practical fabrication guidelines. We validated the device using calibration beads, which allowed us to refine the simulations and redesign the device. Then we validated the redesigned device using blood samples and a murine model of metastatic breast cancer. Finally, as a proof of principle, we tested the device using peripheral blood from a patient with hepatocellular carcinoma, isolating more than 17 CTCs/ml, with purity/removal values of 96.03% and 99.99% of white blood cell and red blood cells, respectively. These results confirm highly efficient CTC isolation with a stringent cut‐off value and better separation results than the state of the art.

Unified theory of tearing mode growth from linear to Rutherford regime in the presence of local equilibrium current gradients

Using the two-field reduced magneto-hydrodynamic (MHD) model, a new theory of tearing mode transition from linear to Rutherford regime in the presence of local equilibrium current gradient (LCG) effects is proposed. By containing the LCG term in the vorticity equation, I reproduce the modified Rutherford equation with saturation terms via a quasi-linear approach. Therefore, the LCG term in the vorticity equation is responsible for the saturation mechanism of the classical tearing mode. The unified formula of the tearing mode evolution from linear exponential growth to quasi-linear saturation is obtained if one contains the LCG and inertial terms in the vorticity equation simultaneously. Since LCG is essential for the linear growth rate of tearing mode in the marginal stable regime, the new theory is also valid when the mode close to marginality.

Chaos over order: mapping 3D rotation of triaxial asteroids and minor planets

ABSTRACT Celestial bodies approximated with rigid triaxial ellipsoids in a two-body system can rotate chaotically due to the time-varying gravitational torque from the central mass. At small orbital eccentricity values, rotation is short-term orderly and predictable within the commensurate spin–orbit resonances, while at eccentricity approaching unity, chaos completely takes over. Here, we present the full three-dimensional rotational equations of motion around all three principal axes for triaxial minor planets and two independent methods of numerical solution based on Euler rotations and quaternion algebra. The domains of chaotic rotation are numerically investigated over the entire range of eccentricity with a combination of trial integrations of Euler’s equations of motion and the GALI(k) (Generalized Alignment Index) method. We quantify the dependence of the order–chaos boundaries on shape by changing a prolateness parameter, and find that the main 1:1 spin–orbit resonance disappears for specific moderately prolate shapes already at eccentricities as low as 0.3. The island of short-term stability around the main 1:1 resonance shrinks with increasing eccentricity at a fixed low degree of prolateness and completely vanishes at approximately 0.8. This island is also encroached by chaos on longer time-scales, indicating longer Lyapunov exponents. Trajectories in the close vicinity of the 3:2 spin–orbit resonance become chaotic at smaller eccentricities, but separated enclaves of orderly rotation emerge at eccentricities as high as 0.8. Initial perturbations of rotational velocity in latitude away from the exact equilibrium result in a spectrum of free libration, nutation, and polar wander, which is not well matched by the linearized analysis omitting the inertial terms.

Distributed [formula omitted]-winners-take-all via multiple neural networks with inertia

This paper is dedicated to solving the k-winners-take-all problem with large-scale input signals in a distributed manner. According to the decomposition of global input signals, a novel dynamical system consisting of multiple coordinated neural networks is proposed for finding the k largest inputs. In the system, each neural network is designed to tackle its available partial inputs only for a local objective ki (ki≤k). Simultaneously, a consensus-based approach is adopted to coordinate multiple neural networks for achieving the global objective k. In addition, an inertial term is introduced in each neural network for regulating its transient behavior, which has the potential of accelerating the convergence. By developing a cocoercive operator, we theoretically prove that the multiple neural networks with inertial terms converge asymptotically/exponentially to the k-winners-take-all solution exactly from arbitrary initial states for whatever decomposition of inputs and objective. Furthermore, some extensions to distributed constrained k-winners-take-all are also investigated. Finally, simulation results are presented to substantiate the effectiveness of the proposed system as well as its superior performance over existing distributed networks.

Fully compressible convection for planetary mantles

SUMMARY The numerical simulations of convection inside the mantle of the Earth or of terrestrial planets have been based on approximate equations of fluid dynamics. A common approximation is the neglect of the inertia term which is certainly reasonable as the Reynolds number of silicate mantles, or their inverse Prandtl number, are infinitesimally small. However various other simplifications are made which we discuss in this paper. The crudest approximation that can be done is the Boussinesq approximation (BA) where the various parameters are constant and the variations of density are only included in the buoyancy term and assumed to be proportional to temperature with a constant thermal expansivity. The variations of density with pressure and the related physical consequences (mostly the presence of an adiabatic temperature gradient and of dissipation) are usually accounted for by using an anelastic approximation (AA) initially developed for astrophysical and atmospheric situations. The BA and AA cases provide simplified but self-consistent systems of differential equations. Intermediate approximations are also common in the geophysical literature although they are invariably associated with theoretical inconsistencies (non-conservation of total energy, non-conservation of statistically steady state heat flow with depth, momentum and entropy equations implying inconsistent dissipations). We show that, in the infinite Prandtl number case, solving the fully compressible (FC) equations of convection with a realistic equation of state (EoS) is however not much more difficult or numerically challenging than solving the approximate cases. We compare various statistical properties of the Boussinesq, AA and FC simulations in 2-D simulations. We point to an inconsistency of the AA approximation when the two heat capacities are assumed constant. We suggest that at high Rayleigh number, the profile of dissipation in a convective mantle can be directly related to the surface heat flux. Our results are mostly discussed in the framework of mantle convection but the EoS we used is flexible enough to be applied for convection in icy planets or in the inner core.

The interplay between fractional damping and nonlinear memory for the plate equation

In this paper, we study the interplay between a fractional damping (− Δ)θut with θ ∈ [0, 1/2), and a nonlinear memory term applied to a plate equation: in space dimension . In different scenarios, we prove the global existence of small data solutions in for supercritical powers, where the critical exponent is determined by the fractional power of the damping term and by the order of the fractional integration in the memory term. In one case, we also feel the influence of the regularity‐loss type decay, which is due to the presence of the rotational inertia term −Δutt in the plate equation.

Existence for a Singular Nonlocal Phase Field System with Inertial Term

This paper deals with a singular nonlocal phase field system with inertial term. The system has the logarithm of the absolute temperature $\theta $ under time derivative. Although the system has a difficult mathematical point caused by the combination of $(\ln \theta )_{t}$ , the inertial term and the nonlocal diffusion term for the order parameter $\varphi $ (see Section 1.1), we can establish existence of solutions by a key estimate (see Remark 1.2).

Double stochastic energetic resonance of underdamped two-scale bistable system

Multistable structures have abundant nonlinear characteristics, a multi-level system with large-scale and small-scale bistable structures is constructed, and its dynamic characteristic is discussed. Based on the complete Langevin equation with an inertial term and a variable damping term, the relationship between the Kramers escape rate and the damping coefficient is deduced and the approximate analytical expression of the output response in the underdamped state is established. From the view of the energy, the mechanism of double resonance and resonance effect enhancement induced by damping is further revealed with the energy dissipation caused by damping as the core. The physical quantity work is introduced to measure the strength of the resonance effect. The results of theoretical analysis and numerical simulation indicate that as the damping coefficient decreases into the underdamping or even the weak damping scope, the double stochastic resonance phenomenon induced by noise still exists. The strongest resonance effect can be obtained with the optimal damping value.

In-plane vibration analysis of horizontally curved beams resting on visco-elastic foundation subjected to a moving mass

This paper deals with the in-plane dynamics of Horizontally Curved Beams (HCBs) supported by a visco-elastic foundation under the excitation induced by a moving mass. What has been included as worthy of detailed exposition is the inevitable contribution of the inertial actions of the mass object into the problem formulation. By taking into account the effect of Coriolis acceleration, centrifugal force and rotary inertia, the governing coupled non-linear differential equations of equilibrium for a simply supported HCB are derived.In the proposed solution, a new system of linear ordinary differential equations is distilled from the governing differential equations of motion, which can be solved utilizing standard numerical techniques. The capability as well as the reliability of the proposed semi-analytical solution is also examined through an extensive parametric study.On that account, non-dimensional parameters are defined to advantage as input data in response calculations and to better interpret the outcomes. To this aim, the Dynamic Amplification Factor (DAF) spectra are introduced with particular attention to the influential parameters such as subtended (central) angle and radius of the HCB, mass and speed of the moving object along with the number of contributing modes.The significance of allowing for the contribution of moving mass inertia is demonstrated in the dynamic response of HCBs, focusing on highly curved elements under the action of a heavy mass at high speeds.It is shown that, in a wide range of design parameters, the simplified moving-force solution would not work to satisfaction if utilized in the analysis of HCB-moving mass dynamics. However, the moving mass framework takes care of all the essential effects induced due to a moving object running on an HCB.In addition, on the basis of the multiple nonlinear regression analysis, simplified expressions have been established which are advantageous in that they supply a means to predict the DAFs in a more practical manner. The findings of the present investigation imply that the influence of additional inertial terms is of fundamental importance in a wide range of design parameters. Therefore, emphasis is placed on implementing the moving mass method, as a more rigorous and reliable representative of the real-world dynamic response of HCBs in comparison with the unsafe results obtained from the moving force approach.Attention is also accorded to the influence of cross-ties in conjunction with the torsional stiffness of rail-tie connection on the in-plane dynamics of HCBs. Also, the results indicate that sharply-curved HCBs, especially at high velocities, are apt to higher DAFs under the action of an in-plane moving inertial load.