Fundamental Equations Research Articles

Irruption Theory in Phase Transitions: A Proof of Concept With the Haken-Kelso-Bunz Model

Many theoretical studies defend the existence of ongoing phase transitions in the brain dynamics that could explain its enormous plasticity to cope with the environment. However, tackling the ever-changing landscapes of brain dynamics seems a hopeless task with complex models. This paper uses a simple Haken-Kelso-Bunz (HKB) model to illustrate how phase transitions that change the number of attractors in the landscape for the relative phase between two neural assemblies can occur, helping to explain a qualitative agreement with empirical decision-making measures. Additionally, the paper discusses the possibility of interpreting this agreement with the aid of Irruption Theory (IT). Being the effect of symmetry breakings and the emergence of non-linearities in the fundamental equations, the order parameter governing phase transitions may not have a complete microscopic determination. Hence, many requirements of IT, particularly the Participation Criterion, could be fulfilled by the HKB model and its extensions. Briefly stated, triggering phase transitions in the brain activity could thus be conceived of as a consequence of actual motivations or free will participating in decision-making processes.

Thermally oscillatory flow inside an irregular channel through a porous medium with velocity and temperature slips

The purpose of this exploration is to scrutinize the heat transfer characteristics of oscillatory flow on the magnetized second-grade fluid within an asymmetric porous wavy channel with velocity slip and temperature jump conditions. Flow in porous space is characterized by modified Darcy's law. The use of a magnetized second-grade fluid in an asymmetric porous wavy channel could potentially enhance heat transfer within the fluid, leading to better control over drug release rates and spatial distribution. The fundamental flow equations are converted into ordinary differential equations (ODEs) utilizing suitable non-dimensional variables and then resolved analytically. The outcomes are graphically represented to illustrate the physical impacts of governing parameters on the temperature, and velocity profiles. Streamlines are also constructed for specific physical parameters to investigate trapping phenomena. In addition, variations on the coefficient of skin friction, mass flux, and heat transfer coefficients are illustrated with different values of pertinent parameters. The outcomes indicate that fluid velocity is enhanced by increasing the values of Darcy's number and Grashof's number. The fluid temperature decreases by increasing the values of the temperature jump parameter.

Simulation studies on the hemodynamic models for blood flow

Abstract One of the most important fluid flows is blood flow seen in hemodynamics, which is a vital process and carries many ingredients from one place to another in the interior of the body. Blood is a special suspension; it is a non-Newtonian fluid as the blood flow cannot be compressed due to the imbalance in strain force and velocity. The blood flow is modelled by various equations which are based on fundamental equations such as the Korteweg-De Vries (KdV) equation and the nonlinear Schrödinger type equations. In this study, some new solitary solutions of the blood flow models are obtained in explicit form via Bernoulli method which is one of the ansatz-based methods. Moreover, 3D and 2D simulations under the suitable values of the parameters of the solutions obtained are plotted.

Heat and Mass Transfer within Unsteady Nanofluid Movement in the Presence of Sustained Solar Radiation

The unsteady movement of nanofluid on porous inclined media is essential for absorbing and transferring heat from solar radiation. From renewable energy sources, solar is limitless, sustainable and universally accessible without creating conflict. In this study, heat and mass transfer have been explored by unsteadily moving nanofluid with the occurrence of Sun rays and viscous dissipation. Tiwari-Das and Darcy-Forchheimer models are encompassed with convective heat transfer and mass suction/injection. Then, the non-linear higher-order set of ordinary differential equations was obtained from fundamental non-linear partial differential equations by using similarity transformation. Both semi-analytical and numerical strategies have been adopted. Comparisons with published articles have detected and observed similar outcomes. Accordingly, thermal Grashof number elevates nanofluid motion while postponing drag force creation. Permeability and Darcy’s number have publicized a contradictory trend in the nanofluid’s movement and temperature. Nanofluid’s temperature expands by incident solar radiation and Eckert number but not by absorption. There is less heat transfer rate by convective than conductive through magnifying magnetic field and nanoparticles’ concentration. Nanofluid constructed by Cu–H2O produces more drag force and less heat transfer rate than that of Cu–C3H8O2. Heat transfer from solar energy is applicable for cooking, heating water and producing electricity.

Development of an innovative diffraction scattering theory of X-rays and electrons in imperfect crystals.

Fundamental equations describing the X-ray and electron diffraction scattering in imperfect crystals have been derived in the form of the matrix Fredholm-Volterra integral equation of the second kind. A theoretical approach has been developed using the perfect-crystal Green function formalism. In contrast, another approach utilizes the wavefield eigenfunctions related to the diagonalized matrix propagators of the conventional Takagi-Taupin and Howie-Whelan equations. Using the Liouville-Neumann-type series formalism for building up the matrix Fredholm-Volterra integral equation solutions, the general resolvent function solutions of the X-ray and electron diffraction boundary-valued Cauchy problems have been obtained. Based on the resolvent-type solutions, the aim is to reveal the features of the diffraction scattering onto the crystal lattice defects, including the mechanisms of intra- and interbranch wave scattering in the strongly deformed regions in the vicinity of crystal lattice defect cores. Using the two-stage resolvent solution of the second order, this approach has been supported by straightforward calculation of the electron bright- and dark-field contrasts of an edge dislocation in a thick foil. The results obtained for the bright- and dark-field profiles of the edge dislocation are discussed and compared with analogous ones numerically calculated by Howie & Whelan [Proc. R. Soc. A (1962), 267, 206].

Resolution of the Fundamental Equation of the Mathematical Formulation of Nuclear Collision Mechanisms Model

The fundamental equation of the model titled “mathematical formulation of nuclear collision mechanisms” [1] was solved. The solution is compounded of two steps, the first step consists in calculating the values of the parameters of the mechanism-operator underlying a given nuclear collision and the second step, consists in deducing the open channels for this collision. Knowing that the equation is not classic, its resolution has required the development of an innovative mathematical method that sets on the use of physical constraints of the problem.

Clarification of the Nature of Quantum Theory Based on Scientific Philosophy

It is known that there was a disagreement between Bohr and Einstein about the nature of the fundamental equations of quantum mechanics. In 1949, in the work "Remarks on Articles," Einstein clearly expressed his thoughts on this issue. He wrote: "All this discussion was needed only to clarify the following. By attempting to defend the thesis that statistical quantum theory can in principle give a complete description of individual systems, we come to highly improbable theoretical concepts. On the other hand, the aforementioned interpretive difficulties of the theory disappear if quantum-theoretical descriptions are considered as descriptions of ensembles of systems"

The Hidden Clash: Spacetime Outlook and Quantum-State Reductions

It is generally assumed that compatibility with special relativity is guaranteed by the invariance of the fundamental equations of quantum physics under Lorentz transformations and the impossibility of transferring energy or information faster than the speed of light. Despite this, various contradictions persist, which make us suspect the solidity of that compatibility. This paper focuses on collapse theories—although they are not the only way of interpreting quantum theory—in order to examine what seems to be insurmountable difficulties we encounter when trying to construct a space–time picture of such typically quantum processes as state vector reduction or the non-separability of entangled systems. The inescapable nature of such difficulties suggests the need to go further in the search for new formulations that surpass our current conceptions of matter and space–time.

A new method for solving multidimensional heat conduction problems

In this study, we propose an efficient boundary-type method named the half boundary method (HBM) for solving multidimensional heat conduction equations. Unlike conventional numerical methods, the core idea behind the HBM is to concentrate variables along half of the boundary and employ them to represent node variables within the solution domain. This approach streamlines the solving process by dealing with smaller-order matrices, substantially reducing storage requirements. We derived the fundamental equations of the HBM for solving multidimensional heat conduction differential equations. Furthermore, by introducing the Cayley–Hamilton formula, we optimize the program to enhance computational efficiency. Additionally, we explored heat conduction problems in solution domains differing from rectangular regions. Notably, we pioneer the application of the HBM to address three-dimensional (3-D) problems for the first time, ensuring its accuracy through rigorous validation.

Bound-State Relativistic Quantum Electrodynamics: A Perspective for Precision Physics with Atoms and Molecules.

Precision physics aims to use atoms and molecules to test and develop the fundamental theory of matter, possibly beyond the Standard Model. Most of the atomic and molecular phenomena are described by the quantum electrodynamics (QED) sector of the Standard Model. Do we have the computational tools, algorithms, and practical equations for the most possible complete computation of atoms and molecules within the QED sector? What is the fundamental equation to start with? Is it still Schrödinger's wave equation for molecular matter, or is there anything beyond that? This paper provides a concise overview of the relativistic QED framework and recent numerical developments targeting precision physics and spectroscopy applications with common features of the robust and successful relativistic quantum chemistry methodology.

Exploring the role of nanoparticles in enhancing thermal performance of fractional Carreau fluid flow under magnetic field and hall current

Nanofluids garner significant scientific interest due to their exceptional heat-conducting abilities and potential to enhance heat transfer efficiency. This manuscript investigates the behavior of a two-dimensional fractional Carreau fluid model on a stretched sheet over time, incorporating convection, magnetic fields, nanoparticles, diffusion, and thermal radiations. The model elucidates viscoelastic nanofluids’ memory and inheritance properties, employing non-integer Caputo fractional derivatives innovatively. Fundamental equations are transformed into dimensionless form and solved using an explicit finite difference approach. Essential parameters like Skin friction coefficient, Nusselt number, and Sherwood number are accurately determined. Rigorous stability and convergence criteria ensure effective solution convergence. Visual representations demonstrate the significant impact of each parameter on fluid flow. It is noted that temperature gradient increases 49.38% with the increase of fractional exponent α while it decreases 13.39% with the increase of magnetic parameter M. Moreover, concentration gradient increases 66.48% with the increase of thermophoresis parameter Nt and 94.71% with the increase of pedesis parameter Nb.

Cross-diffusion and heat transfer effects on peristaltic flow of Reiner-Philippoff nanofluid with pseudoplastic and dilatant fluids over tapered channel: Multiple solutions and stability analysis

Peristaltic flow is used in many scientific and technological fields, including biology and medical technology. The goal of this research is to investigate the peristaltic behavior in an asymmetric non-uniform (tapering) channel using a Reiner-Philippoff fluid model (RPFM)-based nanofluid under the influence of dilatant and pseudoplastic fluid behavior. The Buongiorno nanofluid model (BNFM) is considered to investigate heat mass transportation. The channel is transformed from static reference to movement frame using the subsequent set of modifications, and the dynamic frame is transformed back to dimensionless form using the dimensionless parameters. The fundamental equations have been streamlined due to the long wavelength and small Reynolds number. Novelty of the current study is to examine the stability of numerical framework for RPFM-based nanofluid with dilatant and pseudoplastic fluid behavior. The boundary value problem fourth order code (bvp4c) technique is used to obtain the numerical results. With the use of graphs and tables, the physical elements that influence fluid dynamics are further explained. Stability evaluation has been performed mathematically and graphically. The most reliable solution is shown by the lowest positive eigenvalues, while the unreliable solution is indicated by the negative eigenvalues. It has been proven that changing the Reiner-Philippoff fluid (RPF) factor results in a fluid's velocity changing from dilatant to Newtonian as well as from Newtonian to pseudoplastic. According to the findings, the temperature curves rise as thermophoretic components and Brownian motion increase while decreasing as the Prandtl number increases. A brief theoretical and graphical analysis of each important parameter's effects on the flow dynamics is also carried out.

Free vibration analysis of laminated anisotropic doubly-curved shell structures reinforced with three-phase polymer/CNT/fiber material

The present work investigates the vibrational response of laminated anisotropic doubly-curved shell structures reinforced with Carbon Nanotubes (CNTs) short fibers. The fundamental equations are derived from a curvilinear reference system of principal coordinates, employing the Equivalent Single Layer (ESL) methodology. Furthermore, a general variation in laminate thickness is considered. The unknown field variable is described using higher order theories through the unified formulation, taking into account a general interpolation of the unknown variables with Lagrange polynomials across the physical domain. The Hamiltonian Principle is adopted for the derivation of the dynamic equilibrium equations, which arediscretized numerically by using the Generalized Differential Quadrature (GDQ) method. In addition, an efficient isogeometric NURBS-based mapping is adopted for the distortion of the physical domain. The boundary conditions of the differential problem are modelled through a general distribution of linear elastic springs along the lateral surfaces of the three-dimensional doubly-curved solid. The theory is then applied to study the vibrational modes of structures with different curvatures and lamination schemes, taking into account a general distribution of CNTs along the thickness direction. The homogenized properties of CNTs layers are obtained from the Mori-Tanaka procedure, which accounts for the agglomeration effects of the dispersed nanofibers within the matrix. Unlike previous studies focusing on doubly-curved shells reinforced with composite materials and CNTs, the present formulation enables to derive the vibration characteristics of structures made of generally anisotropic materials with general orientations. Furthermore, the calibration of the parameters for the linear elastic springs’ distribution leads to general external constraints within a single element, thus reducing the computational cost of the problem.

Accretion of matter by a charged dilaton black hole

Considering accretion onto a charged dilaton black hole, the fundamental equations governing accretion, general analytic expressions for critical points, critical velocity, critical speed of sound, and ultimately the mass accretion rate are obtained. A new constraint on the dilation parameter coming from string theory is found and the case for polytropic gas is delved into a detailed discussion. It is found that the dialtion and the adiabatic index of accreted material have deep effects on the accretion process.

The Dirac equation as a linear tensor equation for one component

The Dirac equation is one of the most fundamental equations of modern physics. It is a spinor equation, but some tensor equivalents of the equation were proposed previously. Those equivalents were either nonlinear or involved several components of the Dirac field. On the other hand, the author showed previously that the Dirac equation in electromagnetic field is equivalent to a fourth-order equation for one component of the Dirac spinor. The equivalency is used in this work to derive a linear tensor equivalent of the Dirac equation for just one component. This surprising result can be used in applications of the Dirac equation, for example, in general relativity or for lattice approximation of the Dirac field, and can improve our understanding of the Dirac equation.

On the vibration responses of orthotropic laminated cylindrical panels with non-uniform porosity distributions using higher-order shear deformation theory

The present study investigates free vibration analysis of cylindrical panels made of porous orthotropic layers. Porous material properties are modeled using uniform and two non-uniform porosity distribution patterns through the thickness. The theoretical formulations are expressed based on the higher-order shear deformation theory. The equations of motion of laminated cylindrical panels are established by performing Hamilton’s principle. These fundamental partial differential equations are solved analytically for the laminated cylindrical panels, which are simply supported on all edges. Galerkin’s solution procedure is applied to obtain the natural frequency equation. The natural frequencies are compared to results in the literature to verify the obtained frequency equation. Afterward, novel parametric studies are developed to discuss the influence of porosity parameters, porosity distribution patterns, geometrical parameters, orthotropy, and lamination sequences on the free vibration response of porous laminated cylindrical panels.

A new methodology in evaluating nonlinear electrohydrodynamic azimuthal stability between two dusty viscous fluids

The objective of the current paper is to scrutinize the azimuthal stability of a circular interface between two viscous fluids across porous medium in the presence of fine dust. Dusty fluid models are employed to comprehend the dynamics of aerosols in the atmosphere, a crucial aspect in the investigation of air quality, climate change, and the dissemination of pollutants. Furthermore, the comprehension and management of azimuthal nonlinear instabilities are essential in diverse engineering applications, where turbulence is a prominent factor. This includes aerospace engineering, where it is necessary for developing more efficient aircraft, as well as fluid dynamics, where it is critical for optimizing industrial processes. The system is subjected to the influence of a uniform electric field (EF) that is directed in the azimuthal direction. Electrospray trustees utilized in spaceship propulsion systems make use of the principles of Electrohydrodynamics (EHD). These representatives employ the process of ionization on a liquid propellant and utilize EFs to accelerate the resulting ions, so producing thrust. Assuming a simplified situation, the prototype is assumed to have a two-dimensional structure. Both fluids are presumed as rotating. An examination of rotating fluids is crucial for comprehending the dynamics of the atmosphere and oceans. Obtaining this knowledge is essential for accurately predicting weather, simulating climate patterns, and investigating oceanic currents. The mathematical approach is simplified by considering the viscous potential theory (VPT). The nonlinear technique relies on the use of linear fundamental partial differential equations (PDEs) and the application of the appropriate nonlinear boundary conditions (BCs). This method produces a nonlinear PDE that controls the displacement of the interface. By disregarding the nonlinear elements, a dispersion equation that is linear in nature is obtained. The stability criteria are graphed using several dimensionless physical quantities. The non-perturbative analysis (NPA) achieves nonlinear stability. As well-known, the primary objective of NPA is to convert a linear ordinary differential equation (ODE) into a linear equation. The NPA distinguishes itself from conventional perturbation methods by its ability to accurately analyze the dynamics of highly nonlinear oscillators. This novel methodology is developed utilizing the He's frequency formula (HFF) for the purpose of designing the displacement of the interface. The computational computations demonstrated that the NPA is efficacious, encouraging, and powerful. The linear and nonlinear techniques demonstrate that the effects of various dimensionless physical parameters remain constant. It is found that the Weber number We and Capillary number Cabecome zero as crucial consequences. The Ohnesorge number, Oh which represents the ratio of viscosity and the mass concentration in dusty particles, has a destabilizing effect.

Minimum Condition Requirements in Laser Driven ICF for Advanced Space Propulsion Applications

Inertial Confinement Fusion (ICF) is one of the candidate methods by which spacecraft can be propelled onto interplanetary and interstellar trajectories. In this paper we discuss the application of laser driven ICF to advanced space propulsion and present fundamental equations which relate the capsule mass and nozzle jet efficiency to propulsion parameters. This is also shown for the scenario of using expellant propellant for the purpose of thrust augmentation. Keywords: Inertial Confinement Fusion, Space Propulsion

Consequence analysis of a small-scale hydrogen leakage from the overhead hydrogen piping based on machine learning and physical modeling

Leakage from the overhead hydrogen piping (OHP) must be contained because of the wide flammability range and low minimum ignition energy of hydrogen; rapid leakage detection and appropriate emergency responses are necessary to ensure safe OHP operation. Consequence analysis after leakage detection is useful for hazardous-area estimation and assists operators in emergency-response planning. However, consequence analysis and leakage-detection methods have not been combined so far, and an integrated method has not yet been developed. This study aimed to develop a method based on machine learning (ML), physical modeling, and consequence analyses for rapid and appropriate decision-making after hydrogen leakage. Initially, an OHP model was constructed using physical modeling; this is a method that models target systems based on fundamental physical equations. Subsequently, an ML model for leakage detection was constructed based on various pressure or flow-rate profiles obtained from the OHP model. Regression models were constructed to estimate the leakage diameter. In addition, consequence analyses were conducted to estimate the effects of hydrogen explosions and fires using the leakage diameters predicted by the regression model. We confirmed that the ML model could distinguish between leakage and non-leakage conditions, and the estimated consequences could be used to visualize the risk level of hazardous areas near the leakage points in the case studies. The results of the case studies demonstrated the effectiveness of the proposed method for rapid decision making when hydrogen leaks from OHP. This method enables improved emergency responses in post-leakage situations.

Numerical study of hybrid binary Al2O3-Cu-H2O nanofluid coating boundary layer flow from an exponentially stretching/shrinking perforated substrate with Cattaneo-Christov heat flux, heat source, suction and multiple slip effects

Modern coating systems are increasingly deploying nanomaterials which offer improved thermal performance. The optimization of these systems can benefit from more elegant fluid dynamic models of the coating process. Inspired by these advancements, the present investigation introduces a novel mathematical model to analyze the boundary layer transport in Al2O3-Cu-H2O hybrid binary nanofluid coating deposition on an exponentially stretching porous substrate (sheet). Lateral mass flux (suction) at the wall is also considered a heat source (generation) for hot spot manufacturing effects. To add further sophistication to the thermal conduction model, a non-Fourier approach is adopted which accurately incorporates thermal relaxation effects i.e. the Cattaneo-Christov heat flux (CCHF) model. The Tiwari-Das volume fraction nanoscale formulation is implemented for different combinations of Alumina (Al2O3) and Copper (Cu) metallic nanoparticles in an aqueous base fluid (H2O). Hydrodynamic wall and thermal slip are also included as they feature in coating processes. The fundamental equations governing mass, momentum, and energy conservation, along with the corresponding conditions at the wall (substrate) and free stream, are made dimensionless through suitable scaling similarity transformations. The resulting nonlinear coupled ordinary differential boundary value problem is subsequently addressed using the efficient MATLAB bvp4c routine. Special cases of the non-Fourier model are validated against published results, and the general model is further confirmed through validation using an Adams-Moulton 2-step predictor-corrector algorithm (AMPC). Furthermore, dual solutions of the generalized model are extracted and visualized. A comprehensive study of the effects of key thermophysical key parameters on momentum and thermal characteristics (velocity, temperature, skin friction and local Nusselt number) is conducted and solutions are visualized graphically. The computations show that higher temperatures are present with intensification in the thermal relaxation parameter (Cattaneo-Christov hyperbolic model parameter), which predicts therefore larger temperatures than the classical Parabolic heat conduction model using Fourier analysis. Effectively, thermal relaxation is responsible for finite heat wave propagation and produces more accurate estimations of heat transfer than the Fourier model. An increase in stretching parameter enhances the skin friction coefficient and elevates the local Nusselt number. As the thermal relaxation parameter increases, the temperature magnitudes corresponding to the first and second solutions are both enhanced as are thermal boundary layer thicknesses. An increase in wall suction induces an escalation in the first solution for velocity whereas it reduces the second solution velocity. Stronger wall suction suppresses temperatures. With increasing thermal slip, the boundary layer is cooled i.e. temperature plummets and thermal boundary layer thickness is decreased. Strong deceleration is induced with greater hydrodynamic (velocity) slip. Both first and second solutions for skin friction are elevated with shrinking whereas they are depressed with stretching. With greater velocity slip both the first and second solutions for skin friction are depleted with greater velocity slip for the shrinking case, whereas they are generally enhanced for the very strong stretching case. The computations offer some new insights into multi-physical thermal fluid dynamics of nanofluid-based coating systems.