Force Balance Equation Research Articles

Силовой баланс в токовых слоях в бесстолкновительной плазме

In this paper, we derive a divergent form of the force balance equation for collisionless plasma in the quasineutrality approximation, in which the electric field and current density are excluded. For a stationary spatially one-dimensional current sheet with a constant normal component of the magnetic field and magnetized electrons, the general form of the force balance equation has been obtained for the first time in the form of a conservation law. An equation in this form is necessary for the correct formulation of boundary conditions when modeling asymmetric current sheets, as well as for the control of the stationarity of the numerical solution obtained in the model. Furthermore, the fulfillment of this equation is considered for two types of stationary configurations of a thin current sheet, which are obtained using a numerical model. The derived equation makes it possible to develop models of asymmetric current sheets, in particular current sheets on the magnetopause flanks in the magnetotail.

Seepage Force and Its Direct Mechanical Effects in Hydrate-Bearing Porous Media

The direct mechanical effects of seepage force on the behavior of hydrate-bearing porous media (HBPM) have often been neglected in previous studies, which may lead to inaccurate predictions of the mechanical behavior of HBPM under seepage conditions. Here, we propose an extended three-phase physical model for unsaturated HBPM, including gas, capillary water, and generalized solid skeleton (GSS). Based on the model, the force balance equations for the three phases are formulated. Performing a force analysis of generalized solid particles under seepage conditions, we find that the tangential force acting on the generalized solid particle is the seepage force in HBPM. By combining this finding with the formulated balance equations, we derive the expression for the seepage force, which can distinguish the mechanical effects between the tangential force and normal force. The stresses, induced by the tangential force (i.e., seepage force), are divided into two types: one acts on the cross sections of generalized solid particles and the other on the contacts between particles. Neither of them is transmitted through the GSS. The former mainly causes the particles themselves to compress, whereas the latter primarily influences the sliding of the particles at contacts. Based on the mechanisms of these two stresses, their effects on the mechanical behavior of HBPM are quantified, which provides a new insight to evaluate the direct mechanical effects of seepage force.

Quasi-static numerical simulation of the dynamic launching process for high-altitude balloon

In this paper, by establishing a set of mathematical models and computing methods, the numerical simulation of the ground dynamic launching process under the quasi-static assumption for a high-altitude balloon system is conducted to reveal the rich mechanical and geometric behaviors involved in it. The mathematical model includes the solution of the bubble shapes and the kinematic analysis, as well as the static model of the “membrane bundle system”. Considering the middle and lower region of the bubble still meet the zero circumferential stress condition under the hydrostatic pressure gradient, the upper fully expanded shape generatrix is the same as the balloon generatrix of the ceiling height. The multiple shooting method combined with Sequential Quadratic Programming (SQP) is used to solve the lower partially expanded bubble generatrix. Based on the force balance equations of the microelement in “membrane bundle system”, and considering the friction force in contact with the horizontal ground and the characteristics that the membrane bundle is a variable mass system along the arc length direction, the “multiple shooting method + SQP” method is still used to solve the shape and internal force of the “membrane bundle system” under two end geometric boundary conditions. In co-simulation, taking the dynamic launching process of the zero-pressure balloon system with the ceiling flight height of 30 km and hanging a load of 300 kg as the object, under the conditions of 1.5 m/s and 4.5 m/s wind speeds, the initial layout and the rising process of dynamic launching is calculated. Through the analyses of the simulation results, four stages of the rising process are concluded, namely, the stage of the “membrane bundle system” relaxation and falling to the ground, the longitudinal rising stage, the main rising stage, and the straightening stage. The geometry and force characteristics of the bubble and the “membrane bundle system” are discussed to provide a reference for the simulation of the actual high-altitude balloon ground dynamic launching testing.

A reformulation of time-dependent Kohn-Sham theory in terms of the second time derivative of the density.

The Kohn-Sham approach to time-dependent density-functional theory (TDDFT) can be formulated, in principle, exactly by invoking the force-balance equation for the density, which leads to an explicit expression for the exchange-correlation potential as an implicit density functional. It is shown that this suggests a reformulation of TDDFT in terms of the second time derivative of the density, rather than the density itself. The result is a time-local Kohn-Sham scheme of second-order in time whose causal structure is more transparent than that of the usual Kohn-Sham formalism. The scheme can be used to construct new approximations at the exchange-only level and beyond, and it offers a straightforward definition of the exact adiabatic approximation.

Mechanical Models of Pattern and Form in Biological Tissues: The Role of Stress\u2013Strain Constitutive Equations

Mechanical and mechanochemical models of pattern formation in biological tissues have been used to study a variety of biomedical systems, particularly in developmental biology, and describe the physical interactions between cells and their local surroundings. These models in their original form consist of a balance equation for the cell density, a balance equation for the density of the extracellular matrix (ECM), and a force-balance equation describing the mechanical equilibrium of the cell-ECM system. Under the assumption that the cell-ECM system can be regarded as an isotropic linear viscoelastic material, the force-balance equation is often defined using the Kelvin–Voigt model of linear viscoelasticity to represent the stress–strain relation of the ECM. However, due to the multifaceted bio-physical nature of the ECM constituents, there are rheological aspects that cannot be effectively captured by this model and, therefore, depending on the pattern formation process and the type of biological tissue considered, other constitutive models of linear viscoelasticity may be better suited. In this paper, we systematically assess the pattern formation potential of different stress–strain constitutive equations for the ECM within a mechanical model of pattern formation in biological tissues. The results obtained through linear stability analysis and the dispersion relations derived therefrom support the idea that fluid-like constitutive models, such as the Maxwell model and the Jeffrey model, have a pattern formation potential much higher than solid-like models, such as the Kelvin–Voigt model and the standard linear solid model. This is confirmed by the results of numerical simulations, which demonstrate that, all else being equal, spatial patterns emerge in the case where the Maxwell model is used to represent the stress–strain relation of the ECM, while no patterns are observed when the Kelvin–Voigt model is employed. Our findings suggest that further empirical work is required to acquire detailed quantitative information on the mechanical properties of components of the ECM in different biological tissues in order to furnish mechanical and mechanochemical models of pattern formation with stress–strain constitutive equations for the ECM that provide a more faithful representation of the underlying tissue rheology.

Force balance model predictions of sliding bubbles velocity in vertical subcooled boiling flow

In flow boiling systems, bubbles tend to slide along the wall as they depart from nucleation sites where they originate. This behavior has a profound impact on the wall heat transfer as evidenced in previous experiments. As a result, the heat transfer modeling associated with sliding bubbles has gained great attention within the CFD boiling heat transfer community. The modeling efforts are primarily aimed at improving the Heat Flux Partitioning (HFP) model by accounting for the heat transfer mechanisms introduced by sliding bubbles. The performance of the HFP model depends inherently on the accuracy of sub-models to predict the fundamental bubble parameters, such as bubble departure and lift-off diameters. Similarly, the accurate prediction of sliding bubble parameters, such as sliding bubble growth and sliding bubble velocity, is critical to improving the HFP model when taking into account the effect of sliding bubbles. Of the many sliding bubble parameters, this paper deals with the sliding bubble velocity. Specifically, a force balance model is presented and assessed in view of the sliding bubble velocity prediction. The experimental results from Maity (2000)[1] and Yoo et al. (2016)[2] are used to demonstrate the force balance model accuracy under various subcooled boiling conditions of both, water and refrigerant (NOVEC-7000) flowing upward in vertical channels. We performed a parametric study of the key sub-models (e.g., drag force, bubble growth) used in the force balance equation, together with a direct comparison with proposed models from other researchers. Then, a new force balance model is proposed which outperforms previous models and consistently well-predicts both sliding bubble velocity and growth.

Experimental Studies and Analysis on Mobilization of the Cohesionless Sediments Through Alluvial Channel: A Review

Entrainment of river bed particles by turbulent flow is a core matter of study in river hydrodynamics. It is of great interest to river engineers to evaluate the shear stress for initiating river bed motion. The main objective is to calculate transport rates for bed load, to predict changes in bed level which are scoured or aggraded and to design a stable channel. Forces acting upon the particle especially fluid forces which give a major role in the incipient motion of the particle on the rough boundary. For calculation generally use shield’s diagram but some other modified methods and approaches are discussed. Modeling criteria are discussed for the hydraulically smooth and rough boundary depending on Reynolds number. In the past, experimental studies on tractive shear stress have been done by many researchers but consideration of lift force to analyze the movement of sediment is very limited. For suspended load transport, a detailed analysis of lift force is required. Based on the study it has been observed that shear stress depends on channel slope not only due to gravitational force but also many other factors like drag force, lift force, friction angle, fluctuations, velocity profile, etc. Complete analysis of these factors provides slope dependency over shear stress. To improve past studies, some factors have been discussed, to give a more correct force balance equation. This is very difficult task to analyze more and more variable’s dependency on the slope. Consideration of the possible number of variable holds complete analysis of experimental study. This paper also reviews the effect of particle Reynolds number and relative submergence over critical shield stress. Doi: 10.28991/cej-2021-03091700 Full Text: PDF

Gyrokinetic and extended-MHD simulations of a flow shear stabilized Z-pinch experiment

Axisymmetric (m = 0) gyrokinetic and extended-MHD simulations of a sheared-flow Z-pinch plasma are performed with the high-order finite volume code COGENT. The present gyrokinetic model solves the long-wavelength limit of the gyrokinetic equation for both ion and electron species coupled to the electrostatic gyro-Poisson equation for the electrostatic potential. The extended-MHD model is electromagnetic and includes the effects of the gyro-viscous pressure tensor, diamagnetic electron and ion heat fluxes, and generalized Ohm's law. A prominent feature of this work is that the radial profiles for the plasma density and temperature are taken from the fusion Z-pinch experiment (FuZE), and the magnetic field profile is obtained as a solution of the MHD force balance equation. Such an approach allows to address realistic plasma parameters and provide insights into the current and planned experiments. In particular, it is demonstrated that the radial profiles play an important role in stabilization, as the embedded guiding center (E×B) drift has a strong radial shear, which can contribute to the Z-pinch stabilization even in the absence of the fluid flow shear. The results of simulations for the FuZE plasma parameters show a decrease in the linear growth rate with an increase in the flow shear; however, full stabilization in the linear regime is not observed even for large (comparable to the Alfvén velocity) radial variations of the axial flow. Nonlinear stability properties of the FuZE plasmas are also studied, and it is found that profile broadening can have a pronounced stabilizing effect in the nonlinear regime.

Aspects of GRMHD in high-energy astrophysics: geometrically thick disks and tori agglomerates around spinning black holes

This work focuses on some key aspects of the general relativistic (GR)—magneto-hydrodynamic (MHD) applications in high-energy astrophysics. We discuss the relevance of the GRHD counterparts formulation exploring the geometrically thick disk models and constraints of the GRMHD shaping the physics of accreting configurations. Models of clusters of tori orbiting a central super-massive black hole (SMBH) are described. These orbiting tori aggregates form sets of geometrically thick, pressure supported, perfect fluid tori, associated to complex instability processes including tori collision emergence and empowering a wide range of activities related expectantly to the embedding matter environment of Active Galaxy Nuclei. Some notes are included on aggregates combined with proto-jets, represented by open cusped solutions associated to the geometrically thick tori. This exploration of some key concepts of the GRMHD formulation in its applications to High-Energy Astrophysics starts with the discussion of the initial data problem for a most general Einstein–Euler–Maxwell system addressing the problem with a relativistic geometric background. The system is then set in quasi linear hyperbolic form, and the reduction procedure is argumented. Then, considerations follow on the analysis of the stability problem for self-gravitating systems with determined symmetries considering the perturbations also of the geometry part on the quasi linear hyperbolic onset. Thus we focus on the ideal GRMHD and self-gravitating plasma ball. We conclude with the models of geometrically thick GRHD disks gravitating around a Kerr SMBH in their GRHD formulation and including in the force balance equation of the disks the influence of a toroidal magnetic field, determining its impact in tori topology and stability.

Walking Characteristics of Dual-Arm Inspection Robot with Flexible-Cable

The overhead transmission line has a catenary shape, which has great influence on the dynamic characteristics of an inspection robot walking along the line and may even cause the walking-wheel to fall from the line. Compared with other similar inspection robots, the unique structure of the dual-arm inspection robot with flexible-cable is introduced. Taking the dual-arm inspection robot with flexible-cable walking along the uphill section of the line as an example, the force states of the robot when it works at acceleration, uniform speed, deceleration, and stopping were studied in detail. The corresponding force balance equations were established, and the walking-wheel torques in each working state were solved. The working states of the robot walking along the catenary shape line were simulated using ADAMS software. Simulation results show that the walking process of the robot is stable, the walking-wheels have good contact with the line, and the forces of two walking-wheels are almost balanced, which enables the robot to have good adaptability and climbing ability for the line. The prototype test that the robot walked along the line was carried out. The results of the simulation and prototype test are consistent with the theoretical analysis, so the rationality of robot structure design is verified. In the future, the navigation control and stability of the robot walking along the line will be researched, so that the robot can complete the patrol task in the real environment.

Research on Deformation of Flexible Side Plates in Intra-Vane Type Pumps and its Effect on Oil Film Thickness of the Flow Distribution Pair

This paper sets up both a force balance equation and a three-dimensional geometric model of a flexible side plate in an intra-vane type pump, after analyzing the structure and mechanical properties of the plate. The formulas for calculating deformation were obtained, and the theoretical curve of change in deformation was determined. Then, the contour of deformation was obtained using software simulation. The calculated and simulated values of deformation of the flexible side plate, in three positions under four working pressures, were found to be close to one another, the relative error between them being less than 2%. Next, the formula for the calculation of oil film thickness was obtained, and the curve of the change in oil film thickness relative to the distance along the radial direction of the side plate was determined, considering the flexible side plate deformation. The oil film thickness for three flexible side plate positions were then tested through experiment. The calculated values of oil film thickness were also close to the tested values. Although there was a small difference, the relative error between them was less than 5% for the oil film thickness measured at its lowest point, which indicates the accuracy of the results obtained by the theoretical calculation.

Modeling and analysis of a magnetoelastic annular membrane placed in an azimuthal magnetic field

This work presents the development of a 2D nonlinear magnetoelastic framework for a thin membrane undergoing large deformations. An asymptotic theory is obtained, starting from the 3D variational magnetostatic and force balance equations for a weakly magnetizable material. The model is subsequently specialized to axisymmetry and applied to a pre-stretched annular membrane deforming under azimuthal magnetic field and transverse pressure loading. Parametric studies are performed by varying the pre-stretch, magnetic field, and transverse pressure inputs.

A Fast Estimation Method of Soil Discharged by an Earth Pressure Balanced Shield Machine

The soil discharged by an earth pressure balanced (EPB) shield machine reflects ground loss and controls surface settlement. In engineering, the discharge was mainly estimated by weighing the tunnelling spoil, however, human disturbance and information lag bring serious hidden security risks to tunnel construction. Through the analysis of conveying flow and force of screw conveyor, an energy model of interaction between screw conveyor and soil discharged was proposed and verified experimentally. Then, the energy model was combined with the discharge equation and the force balance equation, and a semi-analytical method to fast estimate the soil discharged of the EPB shield machine was proposed. Taking an EPB shield construction project as an example, according to the data of earth pressure, the torque and the rotation of the screw conveyor recorded by EPB shield machine, the real-time soil discharged of EPB shield machine can be accurately estimated. The mean absolute error (MAE) of this method is 3.177 × 103 kg compared with the measured weight of tunnelling spoil, which verifies the rationality of the method and provides a method for further effective evaluation and control of ground loss.

Mechanical approach to surface tension and capillary phenomena

Many textbooks dealing with surface tension favor the thermodynamic approach (minimization of some thermodynamic potential such as free energy) over the mechanical approach (balance of forces) to describe capillary phenomena, stating that the latter is flawed and misleading. Yet, a mechanical approach is more intuitive for students than free energy minimization, and does not require any knowledge of thermodynamics. In this paper, we show that capillary phenomena can be correctly described using the mechanical approach, as long as the system on which the forces act is properly defined. After reviewing the microscopic origin of a tangential tensile force at the interface, we derive the Young–Dupré equation, emphasizing that this relation should be interpreted as an interface condition at the contact line, rather than a force balance equation. This correct interpretation avoids misidentification of capillary forces acting on a given system. Moreover, we show that a reliable method to correctly identify the acting forces is to define a control volume that does not embed any contact line on its surface. Finally, as an illustration of this method, we apply the mechanical approach in a variety of ways on a classic example: the derivation of the equilibrium height of capillary rise (Jurin's law).

Electron mobility modulated by optical phonons in Al Ga1-N/In Ga1-N/GaN/AlN heterostructures

Abstract The electron mobility influenced by optical phonons, the size of InGaN/GaN channels and In composition y in wurtzite AlxGa1-xN/InyGa1-yN/GaN/AlN heterostructures as basic structures of high electron mobility transistors (HEMTs) is investigated around room temperature. The electron states affected by the two-dimensional electron gas near the interfaces are obtained by self-consistently solving the Poisson and Schrodinger equations. Based on the dielectric continuum and Loudon's uniaxial crystal models combined with the transfer matrix methods, the dispersion relations and electrostatic potentials of interface (IF), confined (CO), half space, and propagating optical phonons are discussed in detail, as well as the corresponding transformations among different phonon modes. Furthermore, the mobility contributed by optical phonons is given by the Lei and Ting's force balance and energy balance equations. Our results show that the electron mobility in the heterostructures with composite InGaN/GaN channel is mainly dominated by IF and CO phonon modes, and is higher than that in the conventional heterostructures with GaN-channels. Meanwhile, the optimized size and In composition of the composite channel to improve the mobility are also clarified.

Features of the Behavior of a Polymer Solution Jet in Electrospinning

Using the numerical analysis of the force balance equation and the rheological equation of the model of finitely extensible chains, the dynamics of a charged jet during the electrospinning of a polymer solution and the orientation of macromolecules in the jet are studied. In fairly weak electric fields, the jet always remains rectilinear, while in strong fields the straight section of the jet has a finite length, after which the motion of the jet becomes unstable. This behavior is due to the competition between inertial and viscoelastic forces, with viscoelasticity dominating in strong fields. It is found that polymer chains in the jet are strongly stretched along the flow direction.

Continuum model of the simple dielectric fluid: consistency between density based and continuum mechanics methods

ABSTRACT The basic continuum model for polar fluids is deceptively simple. The free energy integral consists of four terms: the coupling of polarisation to an external field, the electrostatic energy of the induced electric field interacting with itself and the stored polarisation energy quadratic in the polarisation. A local function of density accounts for the mechanical state of the fluid. Viewed as a non-equilibrium free energy functional of number density and polarisation, minimisation in these two densities under constraints of the Maxwell field equations should lead to the correct equilibrium state. The alternative is a continuum mechanics approach in which the mechanical degree of freedom is extended to full deformation. We show that the continuum electromechanics method leads to a force balance equation which is consistent with the density functional equilibrium equation. The continuum mechanics procedure is significantly more demanding. The gain is a well-defined pressure tensor derived from deformation of total energy. This resolves the issue of the uncertainty in the pressure tensor obtained from integration of the force density, which is the conventional method in density based thermomechanics. Our derivation is based on the variational electrostatics approach developed by Ericksen (Arch. Rational Mech. Anal. 183 299 (2007)).

Research on the Formation Mechanism of Loess Vertical Joints

The popular occurrence of vertical joints in loess formation aggravates the anisotropism of loess and contributes to various geohazards (e.g., slope failures and soil erosion). To alleviate the geohazards caused by the vertical joints, researches need to be carried out to investigate the formation mechanism of these vertical joints. However, the theoretical analysis of the vertical joints’ formation mechanism is limited up to now. In this study, we conducted a laboratory column experiment to observe joint development. Furthermore, using the unsaturated soil theory, we proposed a theoretical model to investigate the tensile stress that contributes to the formation of the loess vertical joint. In the experiment, the air-dried and crushed soil was sifted into the column, which simulates the free fall deposition process of the natural loess in China and contributes to a uniform state. 2500 ml of water was added at the top of the column. The topsoil experienced a wetting-drying process. During desaturation, 9 similar vertical joints were developed with similar horizontal distance. A theoretical model that calculates the interparticle force or tensile force between two adjacent particles was proposed based on the force balance equations. The theoretical model elucidates the phenomena in a laboratory experiment well and provides an insight into the formation mechanism of vertical joints in a uniform soil. The results highlight the generation of vertical joints in the initial deposition stage of loess with a uniform particle arrangement. Besides, the tensile force that contributes to the joint formation arises from the matric suction and surface tension of the solid-water-air contractile film.

Numerical Analysis of the Nanoparticle Dynamics in a Viscous Liquid: Deterministic Approach

We study the deterministic dynamics of single-domain ferromagnetic nanoparticles in a viscous liquid induced by the joint action of the gradient and uniform magnetic fields. It is assumed that the gradient field depends on time harmonically and the uniform field has two components, perpendicular and parallel to the gradient one. We also assume that the anisotropy magnetic field is so strong that the nanoparticle magnetization lies along the anisotropy axis, i.e., the magnetization vector is "frozen" into the particle body. With these assumptions and neglecting inertial effects we derive the torque and force balance equations that describe the rotational and translational motions of particles. We reduce these equations to a set of two coupled equations for the magnetization angle and particle coordinate, solve them numerically in a wide range of the system parameters and analyze the role of the parallel component of the uniform magnetic field. It is shown, in particular, that nanoparticles perform only periodic rotational and translational motions if the perpendicular component of the uniform magnetic field is absent. In contrast, the nanoparticle dynamics in the presence of this component becomes non-periodic, resulting in the drift motion (directed transport) of nanoparticles. By analyzing the short and long-time dependencies of the magnetization angle and particle coordinate we show that the increase in the parallel component of the uniform magnetic field decreases both the particle displacement for a fixed time and its average drift velocity on each period of the gradient magnetic field.

Justification of vibrating hopper hole parameters during fine grained bulk material unloading

Purpose. The leakage process of the fine grained bulk material (BM) from the unloading hopper is considered. As known the BM behaviour during the unloading process is directly related to the material stress state (active or passive) inside the hopper, which depends on the physical and mechanical bulk material properties, as well as hopper geometric parameters. At the boundary of the transition from the cylindrical part to the conical part of the unloading hopper hole, the horizontal stresses that precede the leakage of the material increase sharply. The use of vibration reduces horizontal stresses in the hopper and thus improves the bulk product fluidity. Also, the use of vibration reduces the probability of the free-fall arch formation over the outlet. Therefore, the present paper is aimed at developing the mathematical model of the fine grained bulk material flow under the vibration action from the conical hopper hole. Methodology. It is used one of the methods of studying the BM behavior, namely the analysis of the stress state of the conditionally stationary (pseudo-stationary) layer of the product and the force balance equation in the free-fall arch is made for further study of the geometric parameters of the unloading hole. Findings. In this paper, the mathematical model of the BM behavior in the conical hopper hole is developed. Since the most determining factor influencing the hopper productivity is the outlet diameter, so based on the developed model, it is established relationships that allow determining all parameters that affect the diameter. Graphical dependences are obtained, which allow to estimate the influence of the angle inclination of the hopper hole walls on the value of the hopper outlet size at active and passive stress states. The dependence of the diameter of the unloading hopper hole on the BM properties is also established. The BM properties directly affect the initial shear resistance of the material. The influence of vibration on the unloading hole geometry is presented. Originality. The pseudo-immobile layer of bulk product is considered in the developed mathematical model, and it is investigated how vibration affects the BM behavior in this layer. Practical value. The practical value of this paper lies in the possibility of further mathematical modeling of the influence of the BM properties, hopper geometric dimensions and vibration on the product behavior in the material leakage process from the hopper. The results of the studies, presented in the form of graphs, can be used during the vibrating hopper design and the selection of rational modes of hopper operation.