Variables In Cartesian Coordinates Research Articles

Numerical exploration of the features of thermally enhanced chemically reactive radiative Powell–Eyring nanofluid flow via Darcy medium over non-linearly stretching surface affected by a transverse magnetic field and convective boundary conditions

This communication lineup the characteristics of MHD, heat sink/source, and convective boundary conditions in chemically reactive radiative Powell–Eyring nanofluid flow via Darcy channel using a nonlinearly settled stretching sheet/surface. A binary chemical reaction term is considered in the model. Nonlinear radiation is accounted within the flow. The model involves effect of heat sink/source. Convective boundary conditions are employed. Brownian motion and Thermophoresis are considered. An applied transverse magnetic field effect is considered that impacts through an inclined angle for enhancement of electromagnetic conductance of the nanofluid. Furthermore, low Reynolds is assumed to dismiss the induction of magnetic field. The so-formulated boundary layer governing equations consist of two variables in Cartesian coordinates that are converted to ordinary differential equations (ODEs) via suitably moderated transformations. The solutions are obtained numerically and portrayed graphically as well as in the form of data tables. Behavior of the flow profiles is interpreted for various fluid parameters. The outcomes are plotted for both nonlinear and linear stretching rates of surface. The change in Skin-friction, and Nusselt and Sherwood factors is noted for both the linear and nonlinear stretching cases. The outcomes indicate that the enhanced porosity factor is a major source of reduction in fluid velocity and enhancement of the drag force. Furthermore, the involvement of radiation factor sufficiently enhances the temperature distribution. The results obtained here are useful in industrial applications of nanofluids, especially in designing heating equipment, propulsion devices, gas turbines, nuclear plants, space-type vehicles, satellites, and many others.

New infinite families of Nth-order superintegrable systems separating in Cartesian coordinates

A study is presented of superintegrable quantum systems in two-dimensional Euclidean space E 2 allowing the separation of variables in Cartesian coordinates. In addition to the Hamiltonian H and the second order integral of motion X, responsible for the separation of variables, they allow a third integral that is a polynomial of order N (N ⩾ 3) in the components p 1, p 2 of the linear momentum. We focus on doubly exotic potentials, i.e. potentials V(x, y) = V 1(x) + V 2(y) where neither V 1(x) nor V 2(y) satisfy any linear ordinary differential equation (ODE). We present two new infinite families of superintegrable systems in E 2 with integrals of order N for which V 1(x) and V 2(y) are given by the solution of a nonlinear ODE that passes the Painlevé test. This was verified for 3 ⩽ N ⩽ 10. We conjecture that this will hold for any doubly exotic potential and for all N, and that moreover the potentials will always actually have the Painlevé property.

Mathematical Simulation of a 2D Diode System with a Blade-Shaped Field Emitter

The electrostatic potential distribution in a 2D diode system with a blade-shaped field emitter on a flat substrate has been calculated. The anode of the system represents a plane that is parallel to the substrate. In calculating the potential distribution, the influence of the emitter is replaced by the influence of a charged filament, so that the emitting surface serves as a zero equipotential. The boundary-value problem has been solved by the method of separation of variables in Cartesian coordinates. All geometrical dimensions are the parameters of the problem.

Математическое моделирование двумерной диодной системы с полевым эмиттером лезвийной формы

In this paper the calculation of the electrostatic potential distribution for a two-dimensional diode emission system with a blade-shaped field emitter on a plane substrate is presented. The anode is a plane parallel to the substrate. To calculate the potential distribution, the effect of the emitter is simulated as the effect of a charged line, so that the emitter surface is zero equipotential. To solve the boundary problem, the method of separation of variables in Cartesian coordinates is used. All geometrical sizes of the diode system are parameters of the problem.

Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras

Résumé. In recent years, progress toward the classification of superintegrable systems with higher order integrals of motion has been made. In particular, a complete classification of all exotic potentials with a third or a fourth order integrals, and allowing separation of variables in Cartesian coordinates. All doubly exotic potentials with a fifth order integral have also been completely classified. It has been demonstrated how the Chazy class of third order differential equations plays an important role in solving determining equations. Moreover, taking advantage of various operator algebras defined as Abelian, Heisenberg, Conformal and Ladder case of operator algebras, we re-derived these models. These new techniques also provided further examples of superintegrable Hamiltonian with integrals of arbitrary order. It has been conjectured that all quantum superintegrable potentials that do not satisfy any linear equation satisfy nonlinear equations having the Painlevé property. In addition, it has been discovered that their integrals naturally generate finitely generated polynomial algebras and the representations can be exploited to calculate the energy spectrum. For certain very interesting cases associated with exceptional orthogonal polynomials, these algebraic structures do not allow to calculate the full spectrum and degeneracies. It has been demonstrated that alternative sets of integrals which can be build and used to provide a complete solution. This this allow to make another conjecture i.e. that higher order superintegrable systems can be solved algebraically, they require alternative set of integrals than the one provided by a direct approach.

Two-dimensional superintegrable systems from operator algebras in one dimension

We develop new constructions of 2D classical and quantum superintegrable Hamiltonians allowing separation of variables in Cartesian coordinates. In classical mechanics we start from two functions on a one-dimensional phase space, a natural Hamiltonian H and a polynomial of order N in the momentum p . We assume that their Poisson commutator vanishes, is a constant, a constant times H, or a constant times K. In the quantum case H and K are operators and their Lie commutator has one of the above properties. We use two copies of such pairs to generate two-dimensional superintegrable systems in the Euclidean space E2, allowing the separation of variables in Cartesian coordinates. Nearly all known separable superintegrable systems in E2 can be obtained in this manner and we obtain new ones for N = 4.

An infinite family of superintegrable systems from higher order ladder operators and supersymmetry

We discuss how we can obtain new quantum superintegrable Hamiltonians allowing the separation of variables in Cartesian coordinates with higher order integrals of motion from ladder operators. We also discuss how higher order supersymmetric quantum mechanics can be used to obtain systems with higher order ladder operators and their polynomial Heisenberg algebra. We present a new family of superintegrable systems involving the fifth Painlevé transcendent which possess fourth order ladder operators constructed from second order supersymmetric quantum mechanics. We present the polynomial algebra of this family of superintegrable systems.

On the nonlinear development of two-dimensional and three-dimensional perturbations in the presence of Rayleigh-Taylor instability

The nonlinear evolution of two-dimensional and three-dimensional perturbations of finite amplitude in the presence of Rayleigh-Taylor instability is investigated. It is assumed that the problem is one of potential flow. The solution is constructed by the Fourier method [1]. In the two-dimensional case the conformal mapping method [2, 3] is employed, which makes it possible to consider the strongly nonlinear stages of development of the perturbations, including the formation of surfaces with multiply valued dependence of the variables in Cartesian coordinates. The construction of the mappings reduces to the solution of the Hilbert problem, which is given in the form of Schwartz integrals [4]. Explicit expressions for these integrals [5], obtained with the aid of Fourier series, are employed. Effective computational algorithms are developed and a series of numerical investigations is carried out. Inter alia, a destabilizing effect of the short-wave components is detected, the regularizing action of the surface tension is demonstrated, and the characteristic times of nonlinear development of the perturbations and the characteristic spectral distributions are found. The role of three-dimensional effects, characterized by a decrease in the rate of development of perturbations, is investigated.