Basic Equations Research Articles

Fracture Analysis of Planar Cracks in 3D Thermal Piezoelectric Semiconductors

The study of piezoelectric semiconductor (PSC) materials and devices is experiencing rapid growth, giving rise to a burgeoning research domain known as piezotronics. Understanding the fracture behavior of PSC materials is essential for designing robust devices and ensuring their long-term functionality. A planar crack of arbitrary shape in the isotropic plane of a three-dimensional (3D) transversely isotropic thermal PSC body subjected to combined thermal-electrical-mechanical loadings is studied via the extended displacement discontinuity (EDD) boundary integral equation method. Under the framework of linearized basic equations and one-way thermal coupling, the hypersingular boundary integral equations with the volume integrals containing the temperature and carrier concentration are obtained and expressed by the EDDs which include displacement, electric potential, carrier concentration and temperature discontinuities across the crack surfaces and are the basic variables in analyzing crack problems. Based on the finite part of hypersingular integrals, the EDDs near the crack border are proved to have the classical behavior of O(r). Furthermore, the asymptotic solutions of extended stresses including the stress, electric displacement, current density and heat flux at the crack front are presented and display classical singular behavior of O(1/r). The extended stress intensity factors are defined and given in terms of EDDs. A 3D numerical model of a penny-shaped crack in the thermal PSC cylinder is established and used to verify the presented formulae.

Nonlinear Vibration of Saturated Porous Functionally Graded Plate in Thermal Environment

In this work we studied investigates the nonlinear vibrations of a rectangular saturated porous functionally graded (FG) plate on elastic foundations in thermal environment. The mechanical properties of the saturated porous material vary smoothly with the thickness in three different distributions of porosity including uniform, symmetrically irregular, and asymmetrically irregular. The basic equations are employed by the Reddy’s higher order shear deformation theory, incorporating the geometrically nonlinear von Kármán strain-displacement relationship, stress-strain relations based on the elastic theory for porous materials by Biot, and an analytical solution obtained through the Galerkin method and Airy’s stress function for the simply supported plate. The influence of the geometrical and material parameters, elastic foundations and temperature increment on the nonlinear vibrations of the saturated porous FG plate were specifically evaluated through numerical investigations. 
 
 

Seismic Isolation Effects on Elevated and Ground-Supported Flexible Concrete Cylindrical Tanks Under Bi-Directional Excitation Using an Advanced Mechanical Model

In this paper, the effectiveness of seismic isolation using lead rubber bearings (LRBs) and friction pendulum systems (FPSs) for slender and broad, grounded and elevated tanks (two seat locations of isolation systems in elevated tanks) is investigated under bi-directional excitation of up to 10 records of earthquakes. An analytical mechanical model for a flexible, concrete cylindrical tank, taking into consideration the effect of wall mass and sloshing, is used. The assumptions underlying the mechanical model are basic equations of the motion according to the theory of fluid dynamics. So, the assumptions are more realistic and the results therefore are more accurate than the previous models. The results show that by selecting the best mechanical properties of base isolation systems, reductions of seismic base shear in the grounded broad tanks are around 35% and 30% and for the grounded slender tanks are around 55% and 58% in the x- and y-directions, respectively. Maximum total hydrodynamic pressure is reduced considerably by about 40% on average in all grounded and elevated models. As a result of isolation, elevated tanks are concluded to be better candidates in terms of more effective application of seismic isolation.

Radon and Its Short-Lived Products in Indoor Air: Present Status and Perspectives

Initially, basic equations are given to express the activity concentrations and concentrations of potential α-energies of radon (222Rn) and thoron (220Tn) and their short-lived products in indoor air. The appearance of short-lived products as a radioactive aerosol is shown, and the fraction of the unattached products is particularly exposed, a key datum in radon dosimetry. This fundamental part is followed by giving the sources of radon and thoron indoors, and thus, their products, and displaying the dependence of their levels on the ground characteristics, building material and practice, and living–working habits of residents. Substantial hourly, daily, and seasonal changes in their activity concentrations are reviewed, as influenced by meteorological parameters (air temperature, pressure, humidity, and wind speed) and human activity (either by ventilation, air conditioning and air filtration, or by generating aerosol particles). The role of the aerosol particle concentration and their size distribution in the dynamics of radon products in indoor air has been elucidated, focusing on the fraction of unattached products. Intensifying combined monitoring of radon short-lived products and background aerosol would improve radon dosimetry approaches in field and laboratory experiments. A profound knowledge of the influence of meteorological parameters and human activities on the dynamics of the behaviour of radon and thoron accompanied by their products in the air is a prerequisite to managing sustainable indoor air quality and human health.

The epoch-making importance of Ervin Bauer's theoretical biology

Ervin Bauer was the only biologist who recognized that the best way to develop theoretical biology on an equal footing with theoretical physics was to follow the method that has ensured the great successes of modern theoretical physics: the general method of science. Following this method, he succeeded to find the universal principle of biology. From this principle he managed to derive all the basic equations of biology, that of metabolism, reproduction, growth, responsiveness and successfully explained all the fundamental phenomena of life. In this paper, I introduce Bauer's theoretical biology and discuss whether he understood it within the framework of the modern physical worldview, or in a broader framework. I point out that the theoretical biology of Ervin Bauer is the first to go beyond the physical worldview, to establish a deeper, biological worldview, and thus to represent a major advance in our understanding of the nature of life, with a significance even greater than that of the Copernican turn. Clarifying the difference between the living and the non-living, it is important to consider the difference between machines and living organisms. It is well known that machines are the manifestations of a dual control; globally, their behavior is controlled by their given structure, while locally, their behavior is governed by the physical laws. Based on Bauer's theoretical biology, it is pointed out that living organisms manifest a three-level causality; the ‘additional’, biological level corresponds to the autonomous, time-dependent control of their structures.

The Post-Quasi-Static Approximation: An Analytical Approach to Gravitational Collapse

A seminumerical approach proposed many years ago for describing gravitational collapse in the post-quasi-static approximation is modified in order to avoid the numerical integration of the basic differential equations the approach is based upon. For doing that we have to impose some restrictions on the fluid distribution. More specifically, we shall assume the vanishing complexity factor condition, which allows for analytical integration of the pertinent differential equations and leads to physically interesting models. Instead, we show that neither the homologous nor the quasi-homologous evolution are acceptable since they lead to geodesic fluids, which are unsuitable for being described in the post-quasi-static approximation. Also, we prove that, within this approximation, adiabatic evolution also leads to geodesic fluids, and therefore, we shall consider exclusively dissipative systems. Besides the vanishing complexity factor condition, additional information is required for a full description of models. We shall propose different strategies for obtaining such an information, which are based on observables quantities (e.g., luminosity and redshift), and/or heuristic mathematical ansatz. To illustrate the method, we present two models. One model is inspired in the well-known Schwarzschild interior solution, and another one is inspired in Tolman VI solution.

DEVELOPMENT OF A VORTEX CHAMBER MODEL TO ESTIMATE THE VALUES OF THE LIQUID AND GASEOUS PHASES

When preparing well products from gas fields at complex gas treatment plants, dry stripped gas and unstable condensate are separated. During the exploitation of a field, development indicators (flow rate, reservoir pressure, composition of extracted products, etc.) change. At the same time, the efficiency of the apparatus for separating the gas and liquid phases decreases, which is accompanied by an increase in the entrainment of the liquid phase from the gaseous phase and can reach about 30–40 % of the total volume. This affects economic efficiency indicators, since the volume of unstable condensate decreases, and, consequently, the profit from its sale.The article is devoted to studying the effectiveness of using vortex chambers to reduce the entrainment of the liquid phase from the gaseous phase during the preparation of well products. The results of the research analysis are presented, methods for assessing phase transitions in vortex chambers and the mathematical models used are described.The work defines the basic equations that describe the process of entrainment of the liquid phase from the gaseous phase. Thus, to describe the phase boundaries, the volume of fluid model is used, and the standard k – ε is used as a flow turbulence model. A model of a vortex chamber is being developed in ANSYS FLUENT, the mesh model is built using elements – tetrahedrons. The size of the mesh elements was determined from the condition of stability of the resulting solution and amounted to 10 mm.The influence of chamber geometry, liquid mass velocity and other parameters on the process of phase entrainment is considered.A comparison of the results of experimental studies and modeling results shows the possibility of reducing the entrainment of the liquid phase as a result of the use of vortex chambers and justify the selected parameters for developing the model.

FEATURES OF DECOMPOSITION OF GAS HYDRATE IN NONEQUILIBRIUM MODE IN A POROUS MEDIUM

The paper presents a mathematical model of the nonequilibrium decomposition of a gas hydrate contained in a porous reservoir of finite extent. The system of basic equations, which includes the law of conservation of mass, the heat inflow equation and Darcy’s law, also contains an equation describing the kinetics of nonequilibrium hydrate decomposition, which is determined by pressure, temperature and surface area of particles. On the basis of the numerical solution of the problem, dependences are obtained that allow us to identify the regularities of the process of nonequilibrium decomposition of gas hydrate in a natural reservoir when warm gas is injected into a porous reservoir. It is shown that at the initial moments of time after the start of injection, the extent of the region saturated with gas, hydrate and water is very small compared to the extent of the formation, which corresponds to the mode of hydrate decomposition on the frontal surface. At long injection times, the nonequilibrium hydrate decomposition process is characterized by the formation of three characteristic zones: the near one, where the pores are saturated with gas and water, the far one, saturated with methane and its gas hydrate, and the intermediate one, where gas, hydrate and water are located. It is established that the pressure and temperature in the extended region are connected by the condition of phase equilibrium. The influence of the main parameters of the system on the distributions of pressure, temperature, water saturation and hydrate saturation, as well as on the dynamics of their changes over time, is analyzed. It is shown that injection pressure, as well as reservoir permeability and porosity have a significant effect on the decomposition rate of gas hydrate. An increase in the values of these parameters contributes to an increase in the rate of nonequilibrium hydrate decomposition. An increase in the injection temperature leads only to an increase in the extent of the area containing methane and water.

Experimental Investigation of a Rapid Calculation and Damage Diagnosis of the Quasistatic Influence Line of a Self-Anchored Suspension Bridge Based On Deflection Theory

Abstract Based on deflection theory (DT), the basic differential equations and deformation compatibility equations of self-anchored suspension bridges (SASBs) are established. The calculation formula for the stiffening girder (SG) deflection of a three-span SASB is derived, and the validity of the formula is confirmed through an example. In addition, a fast calculation method for the theoretical deflection influence line (TDIL) is proposed by using the deflection calculation formula, and a calculation method for the actual deflection influence line (ADIL) is founded on the concept of time and space coordinate transformation. By loading the bridge with a load car and using real-time monitoring of the displacement response of the bridge test section, combined with the proposed intelligent damage evaluation program, the abnormal situation of the damage of the three-span SASB can be quickly and accurately identified. The research results of this paper can provide flexible, economical and effective guidance for detecting the capacity of real bridges.

Distinctive approach for MHD natural bioconvection flow of Reiner–Rivlin nano-liquid due to an isothermal sphere

The analysis of heat excess through non-Newtonian fluid is noticed proper and promising arena for biomedical technological applications than Newtonian liquid. Herein, a type of non-Newtonian (Reiner–Rivlin) nanomaterial has been scrutinized with gyrotactic microorganisms around an isothermal sphere with surface of zero mass flux. Being mindful of that the model of Buongiorno’s nanofluid is utilized. Non-dimensional formulas for the basic dimensional partial differential equation system are obtained using appropriate non-similarity transformations. MATLAB function bvp4c code is adopted to get the non-similar solutions. The computations disclosed that the velocity of Reiner–Rivlin nanofluid and its gradient accelerate away from the surface of sphere to infinity and reduces in proximity to the sphere surface for higher values of Reiner–Rivlin factor K. The rate of surface density transport, rate of surface heat transport, and gradient of velocity diminish as the stream wise coordinate boosts. As the problem is subjected to special circumstances, there is a remarkable agreement between the acquired and prior numerical calculations resulting from the comparison.

A robust Gated-PINN to resolve local minima issues in solving differential algebraic equations

Physics-Informed Neural Network (PINN) has emerged as a promising tool for solving various physical problems with differential equations. However in practice, PINN often suffers from the local minima issue while solving problems with minimal initial conditions. In this paper, we propose a robust Gated-PINN to address this issue, which blends numerical methods and PINN. The Gated-PINN overcomes the local minima issue by controlling the flow of information of the neural network similar to a numerical method. We first investigate the use of conventional PINN to the Cartesian-coordinate simple pendulum problem with minimal initial conditions, as one of basic differential algebraic equation (DAE) problem. Our results show the advantage of PINN over existing numerical methods in that it does not require sophisticated mathematical techniques such as index reduction. But we also observe that conventional PINN can lead to inaccurate solutions; such solutions partially satisfy the differential equation requirement, but does not meet the given initial condition and fail to further improve. We demonstrate the effectiveness of the proposed Gated-PINN by showing that it yields accurate solutions, such that for a pendulum of length 1, the mean Euclidean error between the Gated-PINN model and the traditional numerical method model is less than 0.01 and pointwise maximum Euclidean error is less than 0.04. Moreover, Gated-PINN can operate without any complicated index reduction, and unlike conventional PINN, accurate solution can be obtained consistently without falling into a local minima. Overall, our study presents the potential of the Gated-PINN for solving DAE problems and provides a valuable insight into the challenges and limitations of using PINN for solving physical problems.

DUST COLLECTION EFFICIENCY UNDER IN-PHASE TURBULENCE CONDITIONS IN THE PRESENCE OF PHASE TRANSITIONS

The article presents the results of a study of the dust collection efficiency in a regular packing layer of a rotoclone-type hybrid apparatus on the particle inertia index in the presence of phase transition processes. The effect of phase transitions on the efficiency of gas purification processes under inphase turbulence conditions is considered. Based on the conditions of stability of the in-phase vortex formation regime of the gas phase in a regular packing layer, the dominant influence of turbulent diffusion mechanisms on the intensity of phase transition processes and, as a consequence, on the dust collection efficiency has been established. In this case, the predominant influence is in the area of highly dispersed and finely dispersed dust fractions. The results of the experimental studying the patterns of dust collection efficiency depending on the particle inertia index in the presence of phase transition processes are presented. The studies were carried out for three phase transition modes: condensation of vaporsfrom a vapour-gas mixture, evaporation of a spraying liquid as well as a neutral mode. Using A.N. Kolmogorov’s hypothesis,the basic equations and results of calculating the numerical values of local characteristics of a turbulent gas flow in a regular packing layer are presented under the assumption of isotropic turbulence realized in a layer of a regular packing, whose local characteristics do not depend on coordinates and direction.

Reaction operator in the main equation of spin chemistry

In our previous works, using an exactly solvable model as an example, we demonstrated the validity of the reaction operator in the basic equation of spin chemistry. However, in these works, only recombination from the singlet state was considered. Although the general relations allowed us to consider the general case. In the present paper, such a general case is analyzed. The radical pair comes into contact in a mixed singlet-triplet state, and recombination proceeds from both the singlet and triplet states. The result obtained fully corresponds to the basic equation of spin chemistry.

On the Disproportionate Contribution of Membrane Electron Donor Functionality in Membrane Biofouling.

This study set out to uncover which interfacial properties have the greatest impact on membrane organic fouling, biofouling, and fouling resistance. A relatively simple manipulation of the basic equations used in determining Lifshitz-van der Waals (LW) and Lewis acid-base (AB) surface tensions for solid materials reveals that the high electron accepticity of water makes the electron donicity of membrane and biofouling materials the key component governing their interfacial free energy of adhesion (ΔG132), which defines the favorability (or unfavorability) of one material (1) adhering to another (2) when immersed in a liquid (3). Various biofoulant and membrane LW and AB surface tensions were systematically characterized. Static bacterial adhesion, alginic acid filtration, and wastewater filtration tests were conducted to determine the fouling propensities of three different polymeric membrane materials. Experimental results of microbial adhesion, alginate fouling, and biofouling tests all correlated well with membrane electron density, where higher electron density produced less organic fouling or biofouling. These combined theoretical and experimental results confirm the importance of surface electron donicity in determining the fouling propensity of polymeric membranes.

Influence of the Schottky Junction on the Propagation Characteristics of Shear Horizontal Waves in a Piezoelectric Semiconductor Semi-Infinite Medium

In this paper, a theoretical model of the propagation of a shear horizontal wave in a piezoelectric semiconductor semi-infinite medium is established using the optimized spectral method. First, the basic equations of the piezoelectric semiconductor semi-infinite medium are derived with the consideration of biased electric fields. Then, considering the propagation of a shear horizontal wave in the piezoelectric semiconductor semi-infinite medium, two equivalent mathematical models are established. In the first mathematical model, the Schottky junction is theoretically treated as an electrically imperfect interface, and an interface characteristic length is utilized to describe the interface effect of the Schottky junction. To legitimately confirm the interface characteristic length, a second mathematical model is established, in which the Schottky junction is theoretically treated as an electrical gradient layer. Finally, the dispersion and attenuation curves of shear horizontal waves are numerically calculated using these two mathematical models to discuss the influence of the Schottky junction on the dispersion and attenuation characteristics of shear horizontal waves. Utilizing the equivalence of these two mathematical models and the above numerical results, the numerical value of the interface characteristic length is reliably legitimately confirmed; this value is independent of the thickness of the upper metal layer, the doping concentration of the lower n-type piezoelectric semiconductor substrate, and biasing electric fields. Only the biasing electric field parallel to the Schottky junction can provide an evident influence on the attenuation characteristics of shear horizontal waves and enhance the interface effect of the Schottky junction. Since the second mathematical model is also a validation of our previous mathematical model established through the state transfer equation method, some numerical results calculated using these two mathematical models are compared with those obtained using the previous method to verify the correctness and superiority of the research work presented in this paper. Since these two mathematical models can better calculate the dispersion and attenuation curves of high-frequency waves in micro- and nano-scale piezoelectric semiconductor materials, the establishment of mathematical models and the revelation of physical mechanisms are fundamental to the analysis and optimization of micro-scale resonators, energy harvesters, and amplifications.

Insight into the dynamics of slip and radiative effect on magnetohydrodynamic flow of hybrid ferroparticles over a porous deformable sheet

AbstractIn the present work, we explored the magnetohydrodynamic boundary layer flow in the presence of pressure gradient across a flat plate. The effects of the slip velocity conditions, thermal Radiation the addition of hybrid Ferroparticles (i.e., a mixture of two types of magnetic nanoparticles for example, (magnetite () and cobalt ferrite ()) in base fluid ( ) and Porous wall (suction/ injection) are also considered in this study. Basic partial differential equations are transformed into nonlinear ordinary differential equations using the appropriate similarity transformations. Then, this equation was treated numerically by using the Runge–Kutta–Fehlberg 4th–5th order method with the shooting technique and analytically via an efficient method called the Generalized Decomposition Method. The effects of various physical parameters on the velocity and temperature profiles are shown graphically. It is found that the incorporation of hybrid nanoparticles demonstrates a relatively substantial impact on the behavior of dynamic and thermal field; outperforming both non‐hybrid nanoparticles and regular fluid in terms of speed and temperature enhancement. The slip and permeability parameters significantly alter the flow structure. The positive values of the slip and permeability parameters enhance the flow velocity and reduce the temperature, leading to the desired flow behavior. Conversely, adopting negative values of these parameters reverses the effect. Furthermore, the radiation parameter enhances the temperature distribution. Additionally, the boundary layer separation tends to disappear with the increase in the magnetic parameter. The results obtained clearly show the accuracy of the proposed method and also indicate an excellent agreement between analytical and numerical data.

Activation energy with binary chemical reaction of convective radiative magnetized nanofluid flow with viscous dissipation and pressure gradient

ABSTRACT The present investigation deals with the study of activation energy with the binary chemical reaction on magnetohydrodynamic convective radiative heat and mass transfer of nanofluid flow with internal heat source/sink, viscous dissipation, pressure gradient force, and suction effects. The basic equations of conservation of momentum, energy, and mass diffusion are brought to a set of ordinary differential equations (nonlinear) using suitable similarity transformations, and these are then solved using the Homotopy Perturbation Method (HPM). The validity of the present results is established by comparing them with those obtained by the Runge-Kutta-Fehlberg method (RKFM). The results reveal that the temperature distribution increases with raising the value of the thermal radiation parameter. It is found that the concentration distribution increases by increasing the values of the activation energy parameter. In contrast, the reverse effect is observed by enhancing the chemical reaction parameter, Schmidt number, and pressure gradient parameter. Further, the skin-friction coefficient and Nusselt number decrement by enhancing the value of the Hartmann number, whereas the reverse trend is found by enhancing the pressure gradient and suction parameter values. Also, the Sherwood number retards with an enhancement in the Hartmann number, whereas it rises by increasing the pressure gradient and suction parameters.

Optimising thermal design for cross flow heat exchanger in fan coil units

ABSTRACT Analytical and numerical solutions for determining the dimensions of a cross-flow heat exchanger under different conditions have been developed in this study. The analytical solution using the ε-NTU method is used to find the initial dimensions of the cross-flow heat exchanger. The numerical solution is used to validate the analytical solution and optimise the cooling coil dimensions. The ε-NTU design results show the effect of water mass flow rate, air inlet velocity, and the outer tube diameter of the cooling coil. The distribution of air and water temperatures through the heat exchanger can be predicted by solving the basic Navier-Stokes equation and energy equation using numerical methods.

Semi-Separable Potentials as Solutions to the 3D Inverse Problem of Newtonian Dynamics

We study the motion of a test particle in a conservative force-field. Our aim is to find three-dimensional potentials with symmetrical properties, i.e., V(x,y,z)=P(x,y)+Q(z), or, V(x,y,z)=P(x2+y2)+Q(z) and V(x,y,z)=P(x,y)Q(z), where P and Q are arbitrary C2-functions, which are characterized as semi-separable and they produce a pre-assigned two-parametric family of orbits f(x,y,z) = c1, g(x,y,z) = c2 (c1, c2 = const) in 3D space. There exist two linear PDEs which are the basic equations of the Inverse Problem of Newtonian Dynamics and are satisfied by these potentials. Pertinent examples are presented for all the cases. Two-dimensional potentials are also included into our study. Families of straight lines is a special category of curves in 3D space and are examined separately.

Dynamic evolutions of Bohm sheaths and pre-sheaths

The time evolutions of the Bohm sheath and the related pre-sheath are analyzed as an initial value problem. The standard classical fluid model of a collisionless plasma is used with cold ions and Boltzmann distributed electrons. Numerical solutions of the basic equations show that a stationary plasma sheath itself is established within a few ion plasma periods. It is demonstrated analytically that for infinitely extended homogeneous plasmas in one spatial dimension, the only possible pre-sheath solution is dynamic, giving a steady expansion: no stationary solution exists for this one-dimensional case. The potential and density drops along the dynamic pre-sheath differ notably from the values found for stationary models suggested in the literature. Collisions give only formal changes to the collisionless results, and their substance remains the same in one spatial dimension. Cylindrical and spherical geometries, on the other hand, allow for physically acceptable, fully nonlinear, stationary solutions with analytical results given. These are supported by numerical solutions.