Dynamic Evolution Equations for Isolated Smoke Vortexes in Rational Mechanics

Smoke circle vortexes are a typical dynamic phenomenon in nature. The similar circle vortexes phenomenon appears in hurricane, turbulence, and many others. A semi-empirical method is constructed to get some intrinsic understanding about such circle vortex structures. Firstly, the geometrical motion equations for smoke circle is formulated based on empirical observations. Based on them, the mechanic dynamic motion equations are established. Finally, the general dynamic evolution equations for smoke vortex are formulated. They are dynamic evolution equations for exact stress field and dynamic evolution equations for average stress field. For industrial application and experimental data processing, their corresponding approximation equations for viscous fluid are given. Some simple discussions are made.Comment: 25 pages, 2 figure

Quantum Field and Cosmic Field-Finite Geometrical Field Theory of Matter Motion Part Three

This research establishes an operational measurement way to express the quantum field theory in a geometrical form. In four-dimensional spacetime continuum, the orthogonal rotation is defined. It forms two sets of equations: one set is geometrical equations, another set is the motion equations. The Lorentz transformation can be directly derived from the geometrical equations, and the proper time of general relativity is well expressed by time displacement field. By the motion equations, the typical time displacement field of matter motion is discussed. The research shows that the quantum field theory can be established based on the concept of orthogonal rotation. On this sense, the quantum matter motion in physics is viewed as the orthogonal rotation of spacetime continuum. In this paper, it shows that there are three typical quantum solutions. One is particle-like solution, one is generation-type solution, and one is pure wave type solution. For each typical solution, the force fields are different. Many features of quantum field can be well explained by this theoretic form. Finally, the general matter motion is discussed, the main conclusions are: (1). Geometrically, cosmic vacuum field can be described by the curvature spacetime; (2). The spatial deformation of planet is related with a planet electromagnetic field; (3). For electric charge less matter, the volume of matter will be expanding infinitely; (4).For strong electric charge matter, it shows that the volume of matter will be contracting infinitely.Comment: 22 pages, no figure

Gravity Field and Electromagnetic Field-Finite Geometrical Field Theory of Matter Motion Part Two

Gravity field theory and electromagnetic field theory are well established and confirmed by experiments. The Schwarzschild metric and Kerr Metric of Einstein field equation shows that the spatial differential of time gauge is the gravity field. For pure time displacement field, when its spatial differentials are commutative, conservative fields can be established. When its spatial differentials are non-commutative, Maxwell electromagnetic field equations can be established. When the contra-covariant is required for the non-commutative field, both Lorentz gauge and Coulomb gauge are derived in this research. The paper shows that the light is a special matter in that the addition of its Newtonian mass and its Coulomb electric charge is zero. In fact, this conclusion is true for the electromagnetic wave in vacuum. For the conservative field, the research shows that once the mass density and the Coulom charge dendity are given, the macro spacetime feature is completely determined. Both of them are intrisinc features of macro matter in cosimic background. However, for the cosmic ages old events, the spatial curvature may be cannot be ignored. On this sense, the oldest gravity field has the largest curvature of space. This point is very intrinsic for astronomy matters.Comment: 16 pages, no figure

Geometrical Field Formulation of Thermomechanics in Rational Mechanics

In modern science, the thermo mechanics motion can be traced back to quantum motion in micro viewpoint. On the other hand, the thermo mechanics is definitely related with geometrical configuration motion (phase) in macro viewpoint. On this sense, the thermomechanics should be formulated by two kinds of motion: quantum motion and configuration motion. Its principle goal ought to be bridge the gap between atomic physics and engineering practice. In this research, the configuration motion is formulated by deformation geometrical field (motion transformation tensor). The quantum motion is formulated by the wave function of quantum state. Based on these two fields, the thermo stress is formulated as the coupling of quantum motion and configuration motion. Along this line, the entropy is interpreted and formulated according to thermodynamics rules. For scalar entropy, the traditional meaning of entropy is reserved. For infinitesimal configuration variation, the formulation is degenerated to the traditional elasticity deformation. For large random configuration deformation, the formulation is degenerated to the statistical physics methods. This research supplies a possible formulation to bridge the gap between the macro deformation and the micro quantum motion.Comment: 39 pages, no figure

Evolution of Continuum from Elastic Deformation to Flow

Traditionally, the deformation of continuum is divided into elastic, plastic, and flow. For a large deformation with cracking, they are combined together. So, for complicated deformation, a formulation to express the evolution of deformation from elastic to flow will help to understand the intrinsic relation among the related parameters which relate the deformation with a stress field. To this purpose, Eringen polar decomposition and Trusedell polar decomposition are formulated by explicit formulation of displacement field, based on Chen additive decomposition of deformation gradient. Then the strain introduced by the multiplicative decomposition and the strain introduced by the additive decomposition are formulated explicitly with displacement gradient. This formulation clears the intrinsic contents of strains defined by taking the Eringen polar decomposition and Trusedell polar decomposition. After that, it shows that the plastic deformation can be expressed as the irreversible local average rotation. For initial isotropic simple elastic material, the path-dependent feature of classical plasticity theory is naturally expressed in Chen strain definition. It is founded that for initially isotropic material the motion equations require a non-symmetric stress for dynamic deformation and a symmetric stress for static deformation. This controversy between dynamic deformation and static deformation can be used to explain the cracking or buckling of solid continuum. Finally, the research shows that the flow motion of continuum can be expressed by the same formulation system. So, it forms an evolution theory from elastic deformation to flow of continuum.Comment: 25 pages, no figure

Formulation of Deformation Stress Fields and Constitutive Equations in Rational Mechanics

In continuum mechanics, stress concept plays an essential role. For complicated materials, different stress concepts are used with ambiguity or different understanding. Geometrically, a material element is expressed by a closed region with arbitral shape. The internal region is acted by distance dependent force (internal body force), while the surface is acted by surface force. Further more, the element as a whole is in a physical background (exterior region) which is determined by the continuum where the element is embedded (external body force). Physically, the total energy can be additively decomposed as three parts: internal region energy, surface energy, and the background energy. However, as forces, they cannot be added directly. After formulating the general forms of physical fields, the deformation tensor is introduced to formulate the force variations caused by deformation. As the force variation is expressed by the deformation tensor, the deformation stress concept is well formulated. Furthermore, as a natural result, the additive decomposition gives out the definition of static continuum, which determines the material parameters in constitutive equations. Through using the exterior differentials, the constitutive equations are formulated in general form. Throughout the paper, when it is suitable, the related results are simplified to classical results for easier understanding.Comment: 32 pages, 7 figure

Investigating Epithelial-To-Mesenchymal Transition with Integrated Computational and Experimental Approaches

The transition between epithelial and mesenchymal (EMT) is a fundamental cellular process that plays critical roles in development, cancer metastasis, and tissue wound healing. EMT is not a binary process but involves multiple partial EMT states that give rise to a high degree of cell state plasticity. Here, we first reviewed several studies on theoretical predictions and experimental verification of these intermediate states, the role of partial EMT on kidney fibrosis development, and how quantitative signaling information controls cell commitment to partial or full EMT upon transient signals. Next, we summarized existing knowledge and open questions on the coupling between EMT and other biological processes, such as the cell cycle, epigenetic regulation, stemness, and apoptosis. Taken together, EMT is a model system that has attracted increasing interests for quantitative experimental and theoretical studies.Comment: 37 pages, 6 figures, accepted in Physical Biolog

Lyapunov exponents and related concepts for entire functions

Let $f$ be an entire function and denote by $f^\#$ be the spherical derivative of $f$ and by $f^n$ the $n$-th iterate of $f$. For an open set $U$ intersecting the Julia set $J(f)$, we consider how fast $\sup_{z\in U} (f^n)^\#(z)$ and $\int_U (f^n)^\#(z)^2 dx\:dy$ tend to $\infty$. We also study the growth rate of the sequence $(f^n)^\#(z)$ for $z\in J(f)$.Comment: 20 page

Inertial System and Special Relativity Finite Geometrical Field Theory of Matter Motion Part One

Special relativity theory is well established and confirmed by experiments. This research establishes an operational measurement way to express the great theory in a geometrical form. This may be valuable for understanding the underlying concepts of relativity theory. In four-dimensional spacetime continuum, the displacement field of matter motion is measurable quantities. Based on these measurements, a finite geometrical field can be established. On this sense, the matter motion in physics is viewed as the deformation of spacetime continuum. Suppose the spacetime continuum is isotropic, based on the least action principle, the general motion equations can be established. In this part, Newton motion and special relativity are discussed. Based on the finite geometrical field theory of matter motion, the Newton motion equation and the special relativity can be derived simply based on the isotropy of spacetime continuum and the definition of inertial system. This research shows that the Lorentz transformation is required by both of the inertial system definition and the time gauge invariance for inertial systems. Hence, the special relativity is the logic conclusion of time invariance in inertial system. The source independent of light velocity supports the isotropy of inertial system rather than the concept of proper time, which not only causes many paradox, such as the twin-paradox, but also causes many misunderstanding and controversial arguments. The singularity of Lorentz transformation is removed in other parts of finite geometrical field theory, where the gravity field, electromagnetic field, and quantum field will be discussed with the time displacement field.Comment: 19 pages, no gigure

Rational Mechanics Theory of Turbulence

The instant Lagranian coordinator system is used to describe the fluid material motion. By this way, the instant deformation gradient (expressed by spatial velocity gradient) concept is established. Based on this geometrical understanding, the strain rate and stress is expressed by local rotation tensor which is simply based on the fluid material kinetic energy concept (ruling out the concept of fluid-stretching). For fluid filling-in experiments, the characteristic of turbulence is analyzed geometrically and, then, the Navier-Stokes equation is used to get the analytical solution in first-order approximation. For convenient the comprising with experiments, the related typical values are given. As this solution is very basic, it can be expected be valuable for industrial application.Comment: 17 pages, 2 figure