Maxwell Formula Research Articles

Insightful inspection of the nonlinear instability of an azimuthal disturbance separating two rotating magnetic liquid columns

The nonlinear stability examination of two revolving magnetized liquid columns connecting two completely submerged fluids in a permeable region is the aim of the existing paper. Two endless vertical cylinders occupied with two magnetic fluids make up the present structure. Significantly, the disturbance at the border displays an azimuthal behavior. The entire structure is activated by an azimuthal unchanging magnetic field (MF). The increasing interest in the atmospheric and oceanic dynamics is the primary motivation in exploring this problem. To relax the complication of the mathematical processes, the viscous potential theory (VPT) is established. The motion is assessed using three basic coexistent field formulations: Maxwell's formula, Brinkman's formula, and the continuity condition, in the construction of the Coriolis force and centrifugal implications. The explanations of the linearized formula of motion produce a nonlinear categorizing diffusion structure because of the implications of the nonlinear boundary conditions (BCs). The non-perturbative approach (NPA) based on the He's frequency formulation (HFF) is employed to transform the nonlinear characteristic ordinary differential equation (ODE) into a linear one. A short description of the NPA is also presented. The nonlinear ODE with real and imaginary coefficients is exposed by the stability analysis. The stability requirements are implemented using only a nonlinear analysis. As demonstrated, as an unusual state, it is exposed that ignoring the Weber number removes all complex items of the nonlinear formulation. Physically, this means the absence of the angular velocities from the physical model. For both the real and complex situations of the original equation, the stability remains unchanged. It is found that the azimuthal MF, rotating parameter, and Darcy’s numeral have a maintenance impact. On the other hand, the azimuthal wave numeral has a destabilizing one. Several polar designs are drawn to agreement the stability situations.

Experimental study on damage of slope under drying and wetting cycle based on DC electric method.

In order to investigate the seepage law and crack development characteristics of dump slopes, as well as the impact on slope stability during drying and wetting cycles, a simulation test slope system was constructed in a rainfall environment, specifically designed to mimic the engineering conditions of dump slope. The apparent resistivity response formula for the seepage and crack development processes was derived based on the three-phase medium theory of rock-soil bodies and Maxwell's conductivity formula. The geoelectric field characteristics pertaining to slope damage and the corresponding patterns of alteration were comprehensively investigated. The results demonstrate a negative correlation between resistivity and slope water content, with resistivity increasing as cracks develop and decreasing with water infiltration. The progression of crack formation in a rainfall environment on a dump slope can be categorized into three stages: The initial phase involves the saturation of the slope as water content increases. Subsequently, the second phase entails the initiation and expansion of capillary zones, along with the formation of dominant waterways. Lastly, the third phase encompasses the formation and expansion of cracks within the dumping site. The occurrence of sudden changes and abnormal fluctuations in apparent resistivity within a saturated slope signifies the presence of cracks and weak surfaces, leading to gradual and irreversible damage. This phenomenon serves as an indicator of slope damage and can be utilized for the early prediction of slope instability.

Bending-torsional coupling vibration characteristics of asynchronous motorized spindle considering electromechanical coupling and multi-excitation effect

For the asynchronous motorized spindle directly driven by the induction motor, the load-dependent slip ratio determines that its transient characteristics are sensitive to load. For the accurate evaluation of transient characteristics, it is necessary to deeply analyze the bending-torsional coupling vibration characteristics of the asynchronous motorized spindle with the non-ideal energy source. In this work, starting from energy conservation and clarifying the multi-physical field coupling mechanism, the bending-torsional coupling dynamic model is developed. Considering the speed fluctuation, the mass eccentricity excitation is deconstructed to explore its contribution to the coupling vibration. From the load-displacement relationship, the unbalanced excitation experienced by the journal from the bearing is modeled, considering the friction. According to the T-type equivalent circuit and mechanical characteristics of the induction motor, the electromagnetic torque considering power conservation is obtained. Furthermore, combined with Maxwell's stress formula, the electromagnetic excitation from electromagnetic torque and magnetic pull is obtained, considering the tangential and normal magnetic stress. Based on the projection and synthesis of forces, the semi-empirical mechanical cutting force model is employed to calculate the milling load. And a numerical algorithm for the fast acquisition of state-dependent delay is proposed. In particular, to reasonably characterize the energy characteristics in the system, the influence of structural vibration as an energy sink on the multi-physical field composed of mechanical, electromagnetic, and thermal fields is explored from the energy conservation. The proposed model was validated experimentally. The effects of electrical and mechanical parameters on the response are numerically discussed.

Prediction of Magnetocaloric Effect Induced by Continuous Modulation of Exchange Interaction: A Monte Carlo Study.

A magnetic-to-thermal energy conversion, derived from the continuous modulation of intrinsic exchange energy, is conceived and studied by performing Monte Carlo simulations. On the basis of thermodynamics and Weiss's molecular field theories, we modified the Maxwell formula, where the magnetic entropy change (∆SM) is calculated by integrating the temperature derivative of magnetization under a continuously increasing exchange interaction, rather than an external magnetic field, from zero to a given value. For the conventional ∆SM induced through increasing magnetic field, the ∆SM maximum value is enhanced with increasing magnetic field, while the ∆SM peak temperature is weakly influenced by the magnetic field. On the contrary, the ∆SM induced by changing the exchange interaction is proportional to the exchange interaction while suppressed by a magnetic field. Another feature is that the relative cooling power calculated from the ∆SM induced by changing the exchange interaction is fully independent of the magnetic field perspective for obtaining the magnetically stabilized self-converted refrigerants. The controlled variation of exchange interaction could be realized by partial substitution or the application of hydrostatic pressure to lower the cost of magnetic energy at no expense of magnetocaloric response, which opens an avenue to develop the practical and energy-saving devices of conversion from magnetic energy to thermal energy, highly extending the material species of the magnetocaloric effect.

Thermal conductivity of nanofluids: influence of particle shape

The paper proposes a modified Maxwell formula applied to the calculation of the thermal conductivity of nanofluids. It is shown that the use of elongated particles or chains of nanoparticles leads to a significant increase in thermal conductivity. Theoretical estimates based on the proposed model are in good agreement with experimental data. Keywords: Thermal conductivity, nanofluids, elongated particles, chains of nanoparticles, Maxwell formula.

Giant Magnetocaloric Response Induced by Tuning the Intrinsic Exchange Field in the Alloys With Random Magnetic Anisotropy Under a Constant Magnetic Field

The use of a limited magnetic field strength to induce a giant magnetocaloric response is desired towards the potential magnetic refrigeration applications. A method to lower the external magnetic field through taking advantage of the exchange field gradient in the alloys with different component ratios and random magnetic anisotropy under a constant magnetic field is conceived, where the conventional thermodynamic Maxwell formula is modified by replacing the external magnetic field by internal exchange field to calculate magnetic entropy change. The Ni46Co3Mn37-x CuxIn14 (x=0, 1, 2, 3, at.%) alloys are fabricated, and with increasing x, the martensitic transformation temperature increases towards room temperature, meanwhile, the ferromagnetism is weakened and the Curie temperature decreases down to room temperature. Under a constant magnetic field of 1.5 T, it is calculated that an exchange field variation of 1 T can induce a giant positive magnetic entropy change of ~14 J kg-1K-1 near the martensitic transformation and a high relative cooling power ~110 J kg-1 near the paramagnetic-to-ferromagnetic phase transition. Other features of this magnetocaloric effect include that it is weakened under stronger magnetic fields and the relative cooling power is independent of magnetic field. The Monte Carlo simulation is also performed to fit the magnetization and magnetic entropy change behaviors near the magnetic phase transition, and the exchange energy in alloys are quantified from 1.438×10-22 to 1.290×10-22 J with the x variation from 0 to 1 through fitting the linear relative cooling power results. This work not only opens a new avenue to develop a household magnetic refrigerant with low external stimuli and high signal-to-noise ratio, but also provides a method to quantify the intrinsic magnetic parameters using magnetocaloric effect.

Теплопроводность наножидкостей: влияние формы частиц

The paper proposes a modified Maxwell formula applied to the calculation of the thermal conductivity of nanofluids. It is shown that the use of elongated particles or chains of nanoparticles leads to a significant increase in thermal conductivity. Theoretical estimates based on the proposed model are in good agreement with experimental data.

Refinement of the Maxwell formula for a composite reinforced by circular cross-section fibres. Part II: using Pad\xe9 approximants

The effective properties of the fiber-reinforced composite materials with fibers of circle cross section are investigated. The novel estimation for the effective coefficient of thermal conductivity refining the classical Maxwell formula is derived. The method of asymptotic homogenization is used. For an analytical solution of the periodically repeated cell problem the Schwarz alternating process (SAP) was employed. Convergence of this method was proved by S. Mikhlin, S. Sobolev, V. Mityushev. Unfortunately, the rate of the convergence is often slow, especially for nondilute high-contrast composite materials. For improving this drawback we used Padé approximations for various forms of SAP solutions with the following additive matching of obtained expressions. As a result, the solutions in our paper are obtained in a fairly simple and convenient form. They can be used even for a volume fraction of inclusion very near the physically possible maximum value as well as for high-contrast composite constituents. The results are confirmed by comparison with known numerical and asymptotic results.

Refinement of the Maxwell formula for composite reinforced by circular cross-section fibers. Part I: using the Schwarz alternating method

The effective properties of the fiber-reinforced composite materials with fibers of circular cross section are investigated. The novel estimation for the effective coefficient of thermal conductivity refining the classical Maxwell formula is derived. The method of asymptotic homogenization is used. For analytical solution of the periodically repeated cell problem, the Schwarz alternating process is employed. The principal term of the refined formula coincides with the classical Maxwell formula. On the other hand, the refined formula can be used far beyond the area of applicability of the Maxwell formula. It can be used for dilute and non-dilute composites. It is confirmed by comparison with known numerical and asymptotic results.

Electrical Performance Design for Circuit Breakers Based on Multi-domain Sequential Coupling

This paper presents an electrical performance design method of mechatronic circuit breakers (CB) based on multi-domain sequential coupling (MSC). Firstly, the attraction force of electromagnetic trip mechanism is obtained by Maxwell formula and Ampere’s circuital law. The multi-body kinematics and dynamics model of CB are built by recursive Euler–Lagrange partial differential equations using generalized coordinates. The electrical performance optimization model of CB is built to obtain the optimal opening angle, clearance and over-travel of the contact. The MSC piecewisely combines one field with another field according to the variation and sensitivity of the unstable field to perform multiphysics simulation such as magnetohydro-dynamics (MHD). The physical experiment of rated short-circuit capacity is carried out to verify the proposed method using infrared thermogram. The peak voltage, Joule integral and on–off time of the CB are optimized and shortened with maximum change of ratio 8.13% which proves that the CB has better reliability with longer mean time between failures.

Structural study and large magnetocaloric entropy change at room temperature of La1-x □ x MnO3 compounds.

In this study, our central focus is to investigate the magnetocaloric characteristics of a La1−x□xMnO3 (x = 0.1, 0.2 and 0.3) series prepared by a sol–gel technique published in Prog. Mater. Sci., 93, 2018, 112–232. The crystallographic study revealed that our compounds crystallize in a rhombohedral structure with R3̄c. Ferromagnetic (FM) and paramagnetic (PM) characters were detected from the variation in magnetization as a function of magnetic fields at different temperatures. The second order transition was verified from the Arrott plots (M2vs. (μ0H/M)), where the slopes have a positive value. In order to verify the second order, we traced the variation of magnetization vs. temperature at different magnetic fields for x = 0.2. This revealed a ferromagnetic (FM)–paramagnetic (PM) transition when temperature increases. Relying on the indirect method while using the Maxwell formula, we determined the variation in the entropy (−ΔSM) as a function of temperature for different magnetic fields for the three samples. We note that all the studied systems stand as good candidates for magnetic refrigeration with relative cooling power (RCP) values of around 131.4, 83.38 and 57.26 J kg−1 with magnetic fields below 2 T, respectively. Subsequently, the magnetocaloric effect was investigated by a phenomenological model for x = 0.2. The extracted data confirm that this phenomenological model is appropriate for the prediction of magnetocaloric properties. The study also demonstrated that this La0.8□0.2MnO3 system exhibits a universal behaviour.

Refinement of the Maxwell Formula for Fiber-Reinforced Composites

The effective properties of the fiber-reinforced composite materials with fibers of square cross-section are investigated. The novel formula for the effective coefficient of thermal conductivity refining the classical Maxwell formula (MF) is derived. The methods of asymptotic homogenization, boundary shape perturbation and Schwarz alternating process are applied. It is shown that the principal term of the asymptotic expansion of the refined formula in powers of small size of inclusions coincides with the classical MF. The corrections to the MF are obtained for different values of geometrical and physical properties of the constituents of the composite material. The analytical and numerical analyses are carried out and illustrated graphically. In particular, the derived refined formula and the MF are compared for the limiting values of the geometric dimensions and physical properties of the composite. It is shown that the refined formula is applicable for the inclusions with any conductivity in the entire range of the geometric sizes of inclusions, including the limiting cases of inclusions with zero thermal conductivity and maximally large inclusions.

Linking Curie constant and phase transition temperature with fundamental ion properties

ABSTRACTThe Curie Law and Curie – Weiss Law, empirically determined relations, with experimentally determined Curie constant(s) and Curie temperature(s), Tc, are related, through correlation, to the permittivity. The Clausius – Mossotti relation, a fundamental equation that links: permittivity; index of refraction through the Lorentz – Lorenz equation; and, conductivity through Maxwell's formula; is here shown to incorporate both the ferroelectric Curie Law and Curie – Weiss Law. In light of recent developments with regard to coordination and temperature dependent ion properties, radii and polarizabilities, of Perovskites, it is now possible to model and design temperature dependent properties: lattice parameters and volumes; permittivity; as well as the temperature of polarization induced structural phase transitions; of at least Perovskites through the Clausius – Mossotti equation using fundamental ion properties which also form the foundational basis upon which the indices of refraction, through the Lorentz – Lorenz equation, and, conductivity, through Maxwell's formula, depend.

The Length of an Extremal Network in a Normed Space: Maxwell Formula

In the present paper we consider local minimal and extremal networks in normed spaces. It is well known that in the case of the Euclidean space these two classes coincide and the length of a local minimal network can be found by using only the coordinates of boundary vertices and the directions of boundary edges (the Maxwell formula). Moreover, as was shown by Ivanov and Tuzhilin [15], the length of a local minimal network in the Euclidean space can be found by using the coordinates of boundary vertices and the structure of the network. In the case of an arbitrary norm there are local minimal networks that are not extremal networks, and an analogue of the formula mentioned above is only true for extremal networks; this is the main result of the paper. Moreover, we generalize the Maxwell formula for the case of extremal networks in normed spaces and give an explicit construction of norming functionals used in the formula for several normed spaces.

Distribution of molecules by kinetic energy revisited.

The distribution of the molecules by their kinetic energy is treated from statistical point of view. Boltzmann's formula for the distribution of the molecules by energy was found as a limit case of digamma function. Maxwell's formula for the ideal monoatomic gas was derived directly from Boltzmann's main result without employing other sketch formulas. The analytical formulas for two speeds - one real and the other virtual - of molecules were obtained for the case when the molecules carry the kinetic energy in more than three independent components, like in polyatomic molecules. Analysis of the generalized distribution function of the molecules by their virtual speeds indicated that certain molecular states the reach a highest mode in the distribution of the molecules by their virtual speed for a deterministic value of the energy components.

Evaporation of a particle into chemically reactive carrier gas

After a general discussion and classification of physicochemical processes contributing to aerosol particle formation, evaporation of a particle into chemically active carrier gas is studied in more detail. The vapor flux from the particle surface is shown to increase if the carrier gas contains components reacting with the evaporated molecules. A method for determining the chemical correction to the Maxwell formula for evaporated flux is proposed. The correction to the heat balance of the evaporation process due to heat production by the chemical reaction is considered. The temperature of the particle is found to depend on the total evaporated flux, specific heat of the reaction, and the latent heat of evaporation. Main results are listed in the concluding part of the paper.

A Critical Assessment of the Maxwell Slip Boundary Condition for Rarified Wall Bounded Flows

The Maxwell slip boundary condition is widely used to compute the slip velocity at the wall under rarefied flow condition. In this paper, we apply the Direct Simulation Monte Carlo (DSMC) method on a Couette flow problem in order to evaluate this boundary condition. The computations are performed at different values of tangential momentum accommodation coefficient (between 0.1 and 1) and Knudsen numbers (Kn between 0.001 and 3), at 500 K for Argon, with the aim of comparing the ensuing slip length to that predicted by Maxwell's formula. It is found that the boundary condition is accurate up to a Kn = 0.1 but not at higher Knudsen numbers. Although it has been known that the slip boundary condition is valid only for small Knudsen numbers, the critical Knudsen number from where the deviation begins is not well documented. The present results indicate that such deviations occur at Kn = 0.1 in flows that are bounded by opposing walls. At higher Knudsen numbers, the Knudsen layers overlap and the collisions of...

Generalized Maxwell formula for the length of a minimal tree with a given topology

The classic Maxwell formula calculates the length of a planar locally minimal binary tree in terms of coordinates of its boundary vertices and directions of incoming edges. However, if an extreme tree with a given topology and a boundary has degenerate edges, then the classic Maxwell formula cannot be applied directly, to calculate the length of the extreme tree in this case it is necessary to know which edges are degenerate. In this paper we generalize the Maxwell formula to arbitrary extreme trees in a Euclidean space of arbitrary dimension. Now to calculate the length of such a tree, there is no need to know either what edges are degenerate, or the directions of nondegenerate boundary edges. The answer is the maximum of some special linear function on the corresponding compact convex subset of the Euclidean space coinciding with the intersection of some cylinders.

Anisotropic thermal conductivity of magnetic fluids

Considering the forces acting on the particles and the motion of the particles, this study uses a numerical simulation to investigate the three-dimensional microstructure of the magnetic fluids in the presence of an external magnetic field. A method is proposed for predicting the anisotropic thermal conductivity of magnetic fluids. By introducing an anisotropic structure parameter which characterizes the non-uniform distribution of particles suspended in the magnetic fluids, the traditional Maxwell formula is modified and extended to calculate anisotropic thermal conductivity of the magnetic fluids. The results show that in the presence of an external magnetic field the magnetic nanoparticles form chainlike clusters along the direction of the external magnetic field, which leads to the fact that the thermal conductivity of the magnetic fluid along the chain direction is bigger than that along other directions. The thermal conductivity of the magnetic fluids presents an anisotropic feature. With the increase of the magnetic field strength the chainlike clusters in the magnetic fluid appear to be more obvious, so that the anisotropic feature of heat conduction in the fluids becomes more evident.

Numerical determination of the effective moments of non-spherical particles

Dielectric characterisation of polarisable particles, and prediction of the forces and torques exerted upon them, relies on the knowledge of the effective, induced dipole moment. In turn, through the mechanism of depolarisation, the induced dipole moment of a particle is strongly dependent upon its shape. Since realistic shapes create modelling difficulties, the ‘spherical particle’ approximation is often invoked. However, in many cases, including biological dielectric spectroscopy and dielectrophoresis, this assumption is a poor one. For example, human erythrocytes are essentially oblate spheroids with indented sides, while viruses and bacteria often have elongated cigar shapes. Since shape-dependent polarisation both strongly influences the accuracy of conventional dielectric characterisation methods using Maxwell's mixture formula and confounds accurate prediction of dielectrophoretic forces and torques, it is important to develop means to treat non-spherical particles. In this paper, we demonstrate a means to extract the dipole moment directly from numerical solutions of the induced electrostatic potential when a particle is placed in a uniform electric field. The accuracy of the method is demonstrated for a range of particle shapes: spherical, ellipsoidal, truncated cylinders and an approximation of an erythrocyte, the red blood cell.