Numerical solution of MHD free-convection flow in the Stokes problem by the Crank-Nicolson method

A numerical solution of magnetohydrodynamic free-convection flow, in the Stokes' problem, for a porous vertical plate, is obtained, when Prandtl numberP is not equal to one. The magnetic lines of force are assumed to be fixed relative to the plate which is started moving impulsively in its own plane (I.S.P.) or it is uniformly accelerated (U.A.P.). The solution is obtained by the Crank-Nicolson method, which is an implicit second-order method, forP=0.71 (air) andP=7 (water). The obtained results are shown on figures and tables.     

Effects of stepwise variation of single-scattering albedo on reflectivity and transmissivity of a linear-anisotropically scattering slab

Unsteady free-convection flows near an infinite vertical plate in a rotating medium in the presence of a constant transverse magnetic field are investigated under an arbitrary time-dependent heating of the plate. By using the Laplace transform technique, the Green's function of the problem is determined and exact solutions are obtained for special cases of the time-dependent heating effect. The thermal influence on skin friction at the plate and the displacement thickness of the boundary layer are determined, and the structure of the thermal wave trains is discussed.Formerly Kulshrestha.     

The effects of induced magnetic field on unsteady hydromagnetic flows in a rotating medium

Unsteady flows of a viscous incompressible electrically conducting fluid filling the semi-infinite space in contact with an infinite conducting plate in a rotating medium in the presence of a transverse induced magnetic field are investigated under an arbitrary time-dependent forcing effect on the motion of the plate where the plate and the fluid rotate uniformly as a rigid body. By using the Laplace transform technique, the Greens function of the problem is determined and certain asymptotic expansions of the exact solutions are analyzed. the steady-state oscillatory flow problem is solved, and structure of waves and displacement thickness of the boundary layers are discussed for different cases of some natural parameters in the problem. The results are compared with similar flows in the presence of a constant magnetic field.     

Unsteady magnetohydrodynamic flows in a rotating elastico-viscous fluid

This paper points out the errors in the solutions of a research work by N. Nanousis under the same title published in this journal, volume 199, 1993. The correct solutions of the problem for the velocity field and the drag on the plate, by the Laplace transform technique, are presented. The results are discussed for two cases of an arbitrary time-dependent forcing effect. It is shown that the viscoelastic parameterk > 0 influences the velocity and introduces reverse flow. For a suddenly accelerated plate,k > 0 increases the velocity forz < and decreases it forz > . In the case of the ramp-type boundary condition,k > 0 tends to decrease the velocity.     

A numerical solution of MHD free-convection flow in the general nonlinear stokes problem by the finite-difference method

With viscous dissipation and Joule heating taking into account a numerical solution of magnetohydrodynamic free convection flow, in the Stokes's problem, is obtained for different values of Prandtl numberP. The fluid is viscous, incompressible, and electrically conducting and the magnetic lines of force are assumed to be fixed relative to the plate which is started moving impulsively in its own plane (ISP) or it is uniformly accelerated (UAP). The solution is obtained with an implicit second-order method, forP=0.71 (air) andP=7 (water) and the obtained results are shown on figures and tables.     

Unsteady magnetohydrodynamic flows in a rotating elasto-viscous fluid

The unsteady flow of an incompressible electrically-conducting and elasto-viscous fluid (Walter's liquidB), filling the semi-infinite space, in contact with an infinite non-conducting plate, in a rotating medium and in the presence of a transverse magnetic field is investigated. An arbitrary time-dependent forcing effect on the motion of the plate is considered and the plate and fluid rotate uniformly as a rigid body. The solution of the problem is obtained with the help of the Laplace transform technique and the analytical expressions for the velocity field as well as for the skin-friction are given.     

The stability of the triangular points in the elliptic restricted problem

Results are given for the stability regions in the -e plane of the triangular points in the elliptic restricted three-body problem. This has been done by using the two independent second-order sets of equations, as derived by Tschauner (1971a). A new property is being used in the numerical computation, which saves up more than 50% of computer time and is also applicable to the fourth order system. One of Tschauner's (1974) transition curves (derived from analytic approximations) didn't show up here. This seems to confirm Danby's result that thee axis is a transition curve.     

Unsteady hydromagnetic free-convection flow with radiative heat transfer in a rotating fluid of arbitrary optical thickness,III

This paper investigates transient effects on the flow of a thermally-radiating and electrically-conducting compressible gas in a rotating medium bounded by a vertical flat plate. The transience is provoked by a time-dependent perturbation on a constant plate temperature. The problem particularly focusses on an optically thick gas and a gas of arbitrary optical thickness when the difference between the wall and free-stream temperatures is small. Analytical results are possible only for limiting values of time and these results are discussed quatitatively. Indeed the assumption of small temperature difference is more appropriate for plates which are opaque than transparent.     

Unsteady MHD free-convection flows on a porous plate with time-dependent heating in a rotating medium

The three-dimensional unsteady free-convection flows of a viscous fluid near a porous infinite vertical plate in a rotating medium in the presence of a constant transverse magnetic field are investigated under an arbitrary time-dependent heating of the plate. By using the Laplace transform technique, the Green function of the problem is determined and exact solutions are obtained for special cases of the impulsive and the accelerated heating effect for an arbitrary Prandtl number. The thermal influence on skin friction at the plate and the displacement thickness of the boundary layers are discussed.     

Analytical and numerical analysis of the generalized Shkarofsky function

The main analytical properties of the generalized Shkarofsky function and a numerical code for its computation are discussed. The results of a numerical analysis are compared with the results of an asymptotic analysis for parameter values relevant to the problem of whistler-mode propagation in the Earth's magnetosphere. This comparison allows us to specify the range of applicability of different approximations to the generalized Shkarofsky function, which have been used for the analysis of relativistic effects on whistler-mode propagation and instability.     

Similarity solution for free and forced convection hydromagnetic flow over a horizontal semi-infinite plate through a non-homogeneous porous medium

A similarity analysis for the free and forced convection hydromagnetic flow over a horizontal semi-infinite flat plate through a non-homogeneous porous medium is presented, taking into account the hydrostatic pressure variation normal to the flat plate. The similarity solution of the problem under consideration is obtained under certain valid simplifying assumptions when, (i) the plate temperature is inversely proportional to the square root of the distance from the leading edge, (ii) the intensity of the applied magnetic field, normal to the plate, changes with the inverse square root of the distance from the leading edge, and (iii) the permeability of the porous medium, occupying a semi-infinite region of the space bounded by the flat plate, is proportional to the distance measured in the direction of the flow. A numerical solution of the resulting system of ordinary differential equations of motion and energy is obtained, depending on the Prandtl number Pr, the magnetic parameterM _n ,the bouyancy parameter , and the permeability parameterP _m .The variations of the fundamental quantities of the problem are shown graphically followed by a quantitative discussion.     

Radiation Hydrodynamics in Supernovae

We discuss the current status of our hydrodynamical radiation (HYDRA) code for rapidly expanding, low-density envelopes commonly found in core collapse and thermonuclear supernovae. In supernovae, one of the main issues is the coupling between a radiation field and properties of the matter. Due to the low densities, nonthermal excitation by high-energy photons from radioactive decays and the time dependence of the problem, significant departures from local thermodynamical equilibrium (LTE) are common throughout the envelope even at large optical depths. This effect must be taken into account to simulate the evolution of spectra and light curves which are the basic tools to link between explosion physics and observations. The large velocity fields and the non-LTE problem result in a coupling of spatial, frequency space and the level population. This physical system can be described by a large system of coupled integro-differential equations for which the spatial and energy discretization (and its errors) are coupled. For the numerical solution, we use variable separation, analytic solutions and approximations, and iterative schemes. The need for adaptive mesh refinement (AMR) is demonstrated. As example, we show detailed spectra and light curves for the thermonuclear Supernova SN99by.     

MHD thermal-diffusion effects on free-convective and mass-transfer flow over an infinite vertical moving plate

The Soret effect on MHD free-convective and mass-transfer flow of an incompressible, viscous, and electrically-conducting fluid, past a moving vertical infinite plate is studied. The flow is assumed to be at small Reynolds numbers so that the induced magnetic field is neglected. The problem is solved with the help of the Laplace transform method for two different values of the dimensionless functionf(t) signifying two different cases, e.g., (i) when the boundary surface, the flat plate, is impulsively started, moving in its own plane (I.S.P.) and (ii) when it is uniformly accelerated (U.A.P.). The effects on the velocity field as well as on the skin-friction of the various dimensionless parameters occurring into the problem, especially the magnetic parameterM and Soret number So, are discussed with the help of graphs.     

On the vertical families of two-dimensional tori near the triangular points of the Bicircular problem

This paper focuses on some aspects of the motion of a small particle moving near the Lagrangian points of the Earth–Moon system. The model for the motion of the particle is the so-called bicircular problem (BCP), that includes the effect of Earth and Moon as in the spatial restricted three body problem (RTBP), plus the effect of the Sun as a periodic time-dependent perturbation of the RTBP. Due to this periodic forcing coming from the Sun, the Lagrangian points are no longer equilibrium solutions for the BCP. On the other hand, the BCP has three periodic orbits (with the same period as the forcing) that can be seen as the dynamical equivalent of the Lagrangian points. In this work, we first discuss some numerical methods for the accurate computation of quasi-periodic solutions, and then we apply them to the BCP to obtain families of 2-D tori in an extended neighbourhood of the Lagrangian points. These families start on the three periodic orbits mentioned above and they are continued in the vertical (z and ż) direction up to a high distance. These (Cantor) families can be seen as the continuation, into the BCP, of the Lyapunov family of periodic orbits of the Lagrangian points that goes in the (z, ż) direction. These results are used in a forthcoming work [9] to find regions where trajectories remain confined for a very long time. It is remarkable that these regions seem to persist in the real system. This revised version was published online in July 2006 with corrections to the Cover Date.     

Opacity errors in model atmospheres

Different issues involved in the numerical evaluation of continuous absorption by hydrogenlike atoms and ions have been analyzed. Usually, the calculation of the opacity coefficient is explicitly performed for a predetermined number of levels and the higher ones are treated as an integral; it is shown that there are some cases where as many as 17 discrete levels need to be added before switching to the integral approximation if an error smaller than 0.1% is wanted in the results (a condition not always fulfilled with model atmosphere codes being used). Also, approximations to Gaunt factors are analyzed. Though in some cases approximations can be used which are very good, there are other occasions when it should be advisable to switch to interpolating in original tables, or to use more elaborated expressions (such as Hummer's for the Gaunt free-free factor). A numerical procedure for solving the non linear Pg-Pe-T relation is described in an Appendix.     

Free particle modelling of hypervelocity asteroid collisions with the Earth

This paper presents a time-dependent two-dimensional numerical model of the impact phenomena. The model deals with formation and evolution of a crater, formation of an impact jet, and with global deformation and dynamical parameters of the impacted body. The model is applied to study the problem of deformation of the Earth when impacted by an asteroid. A hydrodynamical code of the free particle numerical method (HEFP) is applied to a silicate asteroid (impactor) and to the multilayered spherical Earth (target) described by the PREM model. The asteroids radii are within a range between 5 and 800 km. The velocity range is 20–30 km s⁻¹. Calculations cover the time intervals up to 2000 s.

Each of the material points of the bodies under consideration (the Earth and an asteroid) is described by its time-dependent position, velocity, specific internal energy, pressure and density. The global results, among others, are as follows: (i) deformation of the Earth's surface; (ii) position of the shock wave within the Earth; (iii) deformation of consecutive layers within the Earth's interior, and (iv) morphology of the crater including behavior of the impact jet and deformation of the impactor.     

Symmetric theory of probe-plasma interactions

The theory of interactions between a probe and the surrounding plasma at rest is developed in a spherically and in a cylindrically symmetric model (probe theory). The theory is based on the Vlasov-Poisson system; a general numerical program was developed to solve this system by means of an iterative procedure. Various ambient plasma and charged particle emission properties are described by the complete set of boundary conditions for the distribution functions in the phase space. By use of this numerical method, potential and space charge density in the whole surroundings of the probe as well as the current densities of all plasma constituents are calculated self-consistently.Furthermore, the regions of the phase space with particle trajectories of the same kind can be approximated depending on the plasma properties. Then, the current densities can be estimated analytically. This approach to the problem yields self-consistent approximations and is the only stringent derivation of the thick sheath and of the thin sheath approximation of the classical Langmuir theory. These approximations are generalized with respect to the charged particle emission from the surface.The symmetric probe theory is applied to the following problems of spacecraft environment and spacecraft charging: (i) a spacecraft in the ionosphere with very negative surface potential, (ii) a spacecraft in the solar wind with strong photoelectron emission, and (iii) a spacecraft in the transition region of comet Halley with very strong secondary plasma emission.     

Restricted Three-Body Problem: An Approximation of its General Solution – Part One – the Manifold of Symmetric Periodic Solutions

This paper gives the results of a programme attempting to exploit ‘la seule bréche’ (Poincaré, 1892, p. 82) of non-integrable systems, namely to develop an approximate general solution for the three out of its four component-solutions of the planar restricted three-body problem. This is accomplished by computing a large number of families of ‘solutions précieuses’ (periodic solutions) covering densely the space of initial conditions of this problem. More specifically, we calculated numerically and only for μ = 0.4, all families of symmetric periodic solutions (1st component of the general solution) existing in the domain D:(x ₀ ∊ [−2,2],C ∊ [−2,5]) of the (x ₀, C) space and consisting of symmetric solutions re-entering after 1 up to 50 revolutions (see graph in Fig. 4). Then we tested the parts of the domain D that is void of such families and established that they belong to the category of escape motions (2nd component of the general solution). The approximation of the 3rd component (asymmetric solutions) we shall present in a future publication. The 4th component of the general solution of the problem, namely the one consisting of the bounded non-periodic solutions, is considered as approximated by those of the 1st or the 2nd component on account of the `Last Geometric Theorem of Poincaré' (Birkhoff, 1913). The results obtained provoked interest to repeat the same work inside the larger closed domain D:(x ₀ ∊ [−6,2], C ∊ [−5,5]) and the results are presented in Fig. 15. A test run of the programme developed led to reproduction of the results presented by Hénon (1965) with better accuracy and many additional families not included in the sited paper. Pointer directions construed from the main body of results led to the definition of useful concepts of the basic family of order n, n = 1, 2,… and the completeness criterion of the solution inside a compact sub-domain of the (x ₀, C) space. The same results inspired the ‘partition theorem’, which conjectures the possibility of partitioning an initial conditions domain D into a finite set of sub-domains D i that fulfill the completeness criterion and allow complete approximation of the general solution of this problem by computing a relatively small number of family curves. The numerical results of this project include a large number of families that were computed in detail covering their natural termination, the morphology, and stability of their member solutions. Zooming into sub-domains of D permitted clear presentation of the families of symmetric solutions contained in them. Such zooming was made for various values of the parameter N, which defines the re-entrance revolutions number, which was selected to be from 50 to 500. The areas generating escape solutions have being investigated. In Appendix A we present families of symmetric solutions terminating at asymptotic solutions, and in Appendix B the morphology of large period symmetric solutions though examples of orbits that re-enter after from 8 to 500 revolutions. The paper concludes that approximations of the general solution of the planar restricted problem is possible and presents such approximations, only for some sub-domains that fulfill the completeness criterion, on the basis of sufficiently large number of families.     

Unsteady hydromagnetic free-convection flow with radiative heat transfer in a rotating fluid

We consider the buoyancy-induced flow of an electrically-conducting fluid with radiative heat transfer past a vertical flat plate of infinite length. We assume that the density obeys the simple Boussinesq equation of state while the viscosity and thermal conductivity vary with temperature, that is a compressible fluid. If the temperature of the plate is such that a time-dependent component is superimposed on a constant value, the problem is tackled by asymptotic approximation. The results are compared and contrasted with those of incompressible flow.     

The general analytical expression for computation of generalized relativistic Fermi-Dirac functions

In this paper, general sufficiently analytical formulae are developed for the arbitrary order generalized relativistic Fermi-Dirac (FD) functions. Analytical assessment of relativistic FD function is very important for various fields of physics especially in the theory of relativistic nondegenerate and degenerate electron gas systems. One of the more appropriate and correct approximations is based on a binomial expansion method and incomplete Gamma functions that have been used in the calculations of the generalized relativistic FD functions. Note that, the established expression in special cases of specific values of parameters becomes the evaluation formulae of other type FD functions. Calculation results of the generalized relativistic FD functions are compared with the other approximations methods and available numerical approaches and demonstrated satisfactory agreement.