Effects of Topography and Current on Horizontal Irrotational Waves in Shallow Water

Based on the Boussinesq assumption, derived are couple equations of free surface elevation and horizontal velocities for horizontal irrotational flow, and analytical expressions of the corresponding pressure and vertical velocity. After the free surface elevation and horizontal velocity at a certain depth are obtained by numerical method, the pressure and vertical velocity distributions can be obtained by simple calculation. The dispersion at different depths is the same at the O (epsilon) approximation. The wave amplitude will decrease with increasing time due to viscosity, but it will increase due to the matching of viscosity and the bed slope, thus, flow is unstable. Numerical or analytical results show that the wave amplitude, velocity and length will increase as the current increases along the wave direction. but the amplitude will increase, and the wave velocity and length will decrease as the water depth decreases.

The Jaffa Transform for Hessian Matrix Systems and the Laplace Equation

Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.

THE FORMULATION OF THE GENERALIZED CONSTITUTIVE EQUATIONS FROM THEIR IRROTATIONAL FORMS FOR SIMPLE MATERIALS

I. THE GENERALIZED DIENES’ PROBLEMSince Dienes’work, a great number of papers have been published in the last decade on the problems of the rate form constitutive equations, the objective stress rates, and the rotation measures of deforming bodies. Dienes raised the so-called Dienes’problem that given a constitutive equation having the rate

Steady vortex patches near a nontrivial irrotational flow

In this paper, we study the vortex patch problem in an ideal fluid in a planar bounded domain. By solving a certain minimization problem and studying the limiting behavior of the minimizer, we prove that for any harmonic function q corresponding to a nontrivial irrotational flow, there exists a family of steady vortex patches approaching the set of extreme points of q on the boundary of the domain. Furthermore, we show that each finite collection of strict extreme points of q corresponds to a family of steady multiple vortex patches approaching it.

Kelvin-Helmholtz instability with mass transfer through porous media: Effect of irrotational viscous pressure

This paper studies the effect of the irrotational viscous pressure on Kelvin-Helmholtz instability of the plane interface of two viscous and incompressible fluids in a fully saturated porous media with mass and heat transfers across the interface. In the earlier work, the instability of the plane interface of two viscous and streaming miscible fluids through porous media was studied by assuming that the motion and the pressure are irrotational and the viscosity enters the normal stress balance. This theory is called the viscous potential flow theory. Here, we use another irrotational theory in which the discontinuities in the irrotational tangential velocity and shear stress are eliminated in the global energy balance by considering viscous contributions of the irrotational pressure. The Darcy-Brinkman model is used in the investigation and the stability criterion is formulated in terms of a critical value of the relative velocity. It is observed that the heat and mass transfer has a destabilizing effect on the stability of the system while the irrotational shearing stresses stabilize the system.

On the Irrotational Approximation to Adiabatic Chaplygin Gas Dynamic System

In this paper we mainly study the difference of the weak solutions generated by a wave front tracking algorithm for the steady adiabatic Chaplygin gas dynamic system and the steady irrotational system. Under the hypothesis that the initial data are of sufficiently small total variation, we prove that the difference between the solutions to these two systems can be bounded by the cube of the total variation of the initial perturbation.

Mild-slope equation for water waves propagating over non-uniform currents and uneven bottoms

A time-dependent mild-slope equation for the extension of the classic mild-slope equation of Berkhoff is developed for the interactions of large ambient currents and waves propagating over an uneven bottom, using a Hamiltonian formulation for irrotational motions. The bottom topography consists of two components the slowly varying component which satisfies the mild-slope approximation, and the fast varying component with wavelengths on the order of the surface wavelength but amplitudes which scale as a small parameter describing the mild-slope condition. The theory is more widely applicable and contains as special cases the following famous mild-slope type equations: the classical mild-Slope equation, Kirby's extended mild-slope equation with current, and Dingemans's mild-slope equation for rippled bed. Finally, good agreement between the classic experimental data concerning Bragg reflection and the present numerical results is observed.

A New Theory for the Essence and Nature of Electron Charge

Charge is a fundamental physical property of matter that is responsible for its interactions with electromagnetic fields. The real nature and the essence of charge are unknown. In this paper, a new theory is presented to describe the nature and the essence of electric charge is formulated based on the vortex model of the electron which has a finite size and has an irrotational vortex structure. This theory and the vortex model of the electron enables us, for the first time, to describe the origin of bivalency, stability, quantization, equality of the absolute values of the bivalent charges, to derive a simple formulation to calculate the electric charge based on hydrodynamics without the use any constant. The difference between negative and positive charge, is revealed and the charged particles interactions are described. The electric charge is an expression of accelerated spherical mass per area reduced by the stiffness of the vacuum which has the units ε0 ML3/T2. The calculated results based on these equations comply accurately with the experimental results.

ON MULTI-DIMENSIONAL SONIC-SUBSONIC FLOW

In this paper, a compensated compactness framework is established for sonicsubsonic approximate solutions to the n-dimensional （n ≥ 2） Euler equations for steady irrotational flow that may contain stagnation points. This compactness framework holds provided that the approximate solutions are uniformly bounded and satisfy Hloc^-1（Ω） compactness conditions. As illustration, we show the existence of sonic-subsonic weak solution to n-dimensional （n ≥ 2） Euler equations for steady irrotational flow past obstacles or through an infinitely long nozzle. This is the first result concerning the sonic-subsonic limit for n-dimension （n ≥ 3）.

SPLITTING OF THE SPECTRAL DOMAIN ELECTRICAL DYADIC GREEN’S FUNCTION IN CHIRAL MEDIA

A new method of formulating dyadic (Green's) functions in lossless,reciprocal and unbounded chiral medium was presented.Based on Helmholtz theorem and the non-divergence and irrotational splitting of dyadic Dirac delta-function was this method, the electrical vector dyadic (Green's) function equation was first decomposed into the non-divergence electrical vector dyadic (Green's) function equation and irrotational electrical vector dyadic (Green's) function equation,and then (Fourier's) transformation was used to derive the expressions of the non-divergence and irrotational component of the spectral domain electrical dyadic (Green's) function in chiral media.It can avoid having to use the wavefield decomposition method and dyadic (Green's) function eigenfunction expansion technique that this method is used to derive the dyadic (Green's) functions in chiral media.

EXTENDED HELMHOLTZ THEOREM AND ITS APPLICATION IN ELECTROMAGNETIC THEORY

It is proved that in a regular boundary system of rectangular,cylindrical or sphericalcoordinate,an arbitrary vector function can be separated into three orthogonal parameters:TEmodel field,TM model field and irrotational field.Each of these components can be determinedfully by a scalar function.On the basis of this theorem,the completeness of vector wave functionsystem{L,M,and N}is proved also.Thus it is explained that a vector function space can beprojected into three uncrossed subspaces not only in Euclidean space but also in the subspace ofvector wave function space.

A NEW METHOD FOR SOLVING DYADIC GREEN'S FUNCTION OF ELECTROMAGNETIC FIELD

A new method for solving electromagnetic field boundary value problem is given.Byusing this method,the boundary value problem of the vector wave equation can be transformedinto the independent boundary value problem of scalar wave equations and the two additionalvector differential operations.All the dyadic Green’s functions got by eigenfunction expansionof the dyadic Green’s function can be got by this method easily and some of the dyadic Green’sfunctions for complex systems which are very difficult to get by the ordinary method have beengot by this new method.The dyadic Green’s function for a dielectric loaded cavity is one of thegiven examples.

Separability of Stokes Equations in Axisymmetric Geometries

For Stokes flow in non spherical geometries, when separation of variables fails to derive closed form solutions in a simple product form, analytical solutions can still be obtained in an almost separable form, namely in semiseparable form, R-separable form or R-semiseparable form. Assuming a stream function Ψ, the axisymmetric viscous Stokes flow is governed by the fourth order elliptic partial differential equation E4Ψ = 0 where E4 = E2oE2 and E2 is the irrotational Stokes operator. Depending on the geometry of the problem, the general solution is given in one of the above separable forms, as series expansions of particular combinations of eigenfunctions that belong to the kernel of the operator E2. In the present manuscript, we provide a review of the methodology and the general solutions of the Stokes equations, for almost any axisymmetric system of coordinates, which are given in a ready to use form. Furthermore, we present necessary and sufficient conditions that are serving as criterion for identifying the kind of the separation the Stokes equation admits, in each axisymmetric coordinate system. Additionally, as an illustration of the usefulness of the obtained analytical solutions, we demonstrate indicatively their application to particular Boundary Value Problems that model medical problems.

A New Theory on the Origin and Nature of the Fine Structure Constant

The nature and the origin of the fine structure are described. Based on the vortex model and hydrodynamics, a comprehensible interpretation of the fine structure constant is developed. The vacuum considered to have superfluid characteristics and elementary particles such as the electron and Hydrogen molecule are irrotational vortices of this superfluid. In such a vortex, the angular rotation ω is maintained, and the larger the radius, the slower the rotational speed. The fine structure value is derived from the ratio of the rotational speed of the boundaries of the vortex to the speed of the vortex eye in its center. Since the angular rotation is constant, the same value was derived from the ratio between the radius of the constant vortex core and the radius of the hall vortex. Therefore, the constancy of alpha is an expression of the constancy relation in the vortex structure.

A New Theory for the Essence and Origin of Electron Spin

Spin is an intrinsic form of angular momentum carried by elementary particles, composite particles, and atomic nuclei. It is wildly believed that spin is a purely quantum mechanical concept and has no classical analogue. In fact, elementary particles are conceived as point objects which have no axis to “spin” around. Therefore, there is no explaining how spin arises at the fundamental level, why particles have the values they do, and what underpins the Pauli Exclusion principle and Bose-Einstein behavior. However, spin is like a vector quantity;it has a definite magnitude, and it has a “direction”, in order to spin should be composite. In this paper we propose a physical explanation for spin of the electron at the sub-particle level, relying on the vortex model of the electron. The electron is described as a superfluid frictionless vortex which has a mass, angular momentum and spin to provide a complete explanation of all properties of the electron: it composite, spinning around its own axis, produces a tiny magnetic fields independent of those from its orbital motions. The classical hydrodynamic laws are used to describe the quantum properties of the electron, such as spin, angular momentum, magnetic momentum and a magnetic dipole. The circulation in the vortex is constant, and the angular momentum of the vortex is conserved and has the same value of Planck constant. The direction of the angular momentum of a spinning electron vortex is along the axis of rotation and determined by the direction of spin. The spin quantum number 1/2 has a fixed value which represents the gap between the circulation rate of the core of the vortex and the boundaries of the vortex. The changeable values +1/2 “spin-up” or -1/2 “spin-down” indicate the direction of the magnetic dipole of the vortex. The relation between spin and Planck constant is discussed and the origin h/4pi angular momentum units are revealed.

ON SUBSONIC AND SUBSONIC-SONIC FLOWS IN THE INFINITY LONG NOZZLE WITH GENERAL CONSERVATIVES FORCE

In this article, we study irrotational subsonic and subsonic-sonic flows with gen- eral conservative forces in the infinity long nozzle. For the subsonic case, the varified Bernoulli law leads a modified cut-off system. Because of the local average estimate, conservative forces do not need any decay condition. Afterwards, the subsonic-sonic limit solutions are constructed by taking the extract subsonic solutions as the approximate sequences.

RECONSTRUCTED DISCONTINUOUS APPROXIMATION TO STOKES EQUATION IN A SEQUENTIAL LEAST SQUARES FORMULATION

We propose a new least squares finite element method to solve the Stokes problem with two sequential steps.The approximation spaces are constructed by the patch reconstruction with one unknown per element.For the first step,we reconstruct an approximation space consisting of piecewise curl-free polynomials with zero trace.By this space,we minimize a least squares functional to obtain the numerical approximations to the gradient of the velocity and the pressure.In the second step,we minimize another least squares functional to give the solution to the velocity in the reconstructed piecewise divergence-free space.We derive error estimates for all unknowns under both L 2 norms and energy norms.Numerical results in two dimensions and three dimensions verify the convergence rates and demonstrate the great flexibility of our method.

On Subsonic and Subsonic-Sonic Flows with General Conservatives Force in Exterior Domains

In this paper, we study the irrotational subsonic and subsonic-sonic fows with general conservative forces in the exterior domains. The conservative forces indicate the new Bernoulli law naturally. For the subsonic case, we introduce a modified cut-off system depending on the conservative forces which needs the varied Bers skill, and construct the solution by the new variational formula. Moreover, comparing with previous results, our result extends the pressure-density relation to the general case. Afterwards we obtain the subsonic-sonic limit solution by taking the extract subsonic solutions as the approximate sequences.

Global existence of smooth solutions to two-dimensional compressible isentropic Euler equations for Chaplygin gases

In this paper we investigate the two-dimensional compressible isentropic Euler equations for Chaplygin gases. Under the assumption that the initial data is close to a constant state and the vorticity of the initial velocity vanishes, we prove the global existence of the smooth solution to the Cauchy problem for twodimensional flow of Chaplygin gases.

Structural method for solving dyadic Green's functions

It is about fifty years since dyadic Green’s functions (DGF) were used to solveelectromagnetic boundary problems. However. by 1971 the DGF under normal boundaryconditions had been studied systematically with the method of Ohm-Rayleigh by C. T. Tai.