Analysis of magnetohydrodynamic flow of a fractional viscous fluid through a porous medium

The aim of this paper is to investigate the exact general solutions of the incompressible viscous fluid flow by using the time-fractional Caputo–Fabrizio derivative. The flow of the fluid is subject to the motion of a plane wall, embedded in a porous medium under the influence of magnetic field. The corresponding non-dimensional governing fractional differential equation with appropriate initial and boundary conditions is solved by means of integral transforms namely, Laplace and Fourier transforms. Solutions are expressed as a sum of steady and transient parts, for the sinusoidal oscillations of the plane wall. The influence of involved physical parameters are discussed graphically. Specifically, it has been observed that the effective permeability K_eff reduces the time taken to reach the steady state.     

Flows of a generalized second grade fluid in a cylinder due to a velocity shock

Unsteady axial flows of second grade fluids with generalized fractional constitutive equation in a circular cylinder are studied. Flows are generated by a time-dependent pressure gradient in the axial direction, an external magnetic field perpendicular on the flow direction and by the cylinder motion. Two different problems are analyzed; one in which the cylinder velocity supports a shock at the instant t = 0 and another in which the cylinder motion is a translation with time-dependent velocity along the axis of cylinder. The generalized fractional constitutive equation of second grade fluid is described by the Caputo time-fractional derivative. Analytical solutions for the velocity field are obtained by using the Laplace transform with respect to time variable and the finite Hankel transform of order zero with respect to the radial coordinate. The influence of the fractional parameter of Caputo derivative on the fluid velocity has been studied by numerical simulations and graphical illustrations. It is found that the fractional fluid flows are faster than the ordinary second grade fluid.     

Effects of Hall current on unsteady MHD flows of a second grade fluid

The aim of this present paper is to construct exact solutions corresponding to the motion of magnetohydrodynamic (MHD) fluid in the presence of Hall current, due to cosine and sine oscillations of a rigid plate as well as those induced by an oscillating pressure gradient. A uniform magnetic field is applied transversely to the flow. By using Fourier sine transform steady state and transient solutions are presented. These solutions satisfy the governing equations and all associated initial and boundary conditions. The results for a hydrodynamic second grade fluid can be obtained as a limiting case when B ₀ → 0 and for a Newtonian fluid when α ₁ → 0.     

Dynamics of and radiation from superconducting strings and springs

We present a detailed study of the dynamics of and radiation from superconducting strings. We derive an approximate local action for a current-carrying vortex line and present some exact solutions to its equation of motion. These include stable static “springs” and oscillating “kinky” loops. For one of these “kinky” loops we are able to calculate the radiation exactly, and find (in contrast to previous work) the result is finite. We also argue that the non-local electromagnetic self-interaction of a loop causes the kinks to slowly straighten out. Finally, we discuss the loss of current at “cusp-like” regions and show that the shrinking of loops generally leads to current loss rather than gain.     

Structural physical regularities of the mechanism and kinetics of the formation of piezoelectric and ferroelectric crystals from a liquid phase

The specific features of the kinetics of formation of noncentrosymmetric crystals from electrolyte solutions are investigated for approximately 50 salts of M-N electrolytes. The “solubility product-limiting solution supercooling” set {SP-ΔT _m } for noncentrosymmetric crystals is separated into nine characteristic taxons in which the regularities of formation and growth of crystals are revealed and compared. It is demonstrated that the minimum supercoolings ΔT _m are observed for solutions of crystals with weak “acentric” properties. Higher supercoolings ΔT _m are observed for solutions of noncentrosymmetric crystals that belong to potential ferroelastics and exhibit pronounced acentric properties. The proposed taxonomy of electrolytes makes it possible to predict the conditions for the growth of perfect crystals for fundamental and applied physics.     

Effect of side walls on the motion of a viscous fluid induced by an infinite plate that applies an oscillating shear stress to the fluid

The velocity field corresponding to the unsteady motion of a viscous fluid between two side walls perpendicular to a plate is determined by means of the Fourier transforms. The motion of the fluid is produced by the plate which after the time t = 0, applies an oscillating shear stress to the fluid. The solutions that have been obtained, presented as a sum of the steady-state and transient solutions satisfy the governing equation and all imposed initial and boundary conditions. In the absence of the side walls they are reduced to the similar solutions corresponding to the motion over an infinite plate. Finally, the influence of the side walls on the fluid motion, the required time to reach the steady-state, as well as the distance between the walls for which the velocity of the fluid in the middle of the channel is unaffected by their presence, are established by means of graphical illustrations.     

Selection rules forG 2 vectors in the atomicf shell

In various tabulations of such spectroscopic coefficients as the matrix elements of tensor operators or fractional parentage coefficients, it is found that many entries are unexpectedly zero. A survey is made of all cases that occur in the atomicf shell and that involve the 7-dimensional vector representation of the groupG ₂. Direct explanations are given in terms of the group structure of the electronic configurations that comprise the shell. The techniques used depend on a splitting of the state space into “spin-up” and “spin-down” parts, and, for other cases, the extensive use of the methods of second quantization. TheF terms of the atomicf shell are found to split into three classes. The separation that this classification provides for the twoF terms belonging to the irreducible representation (31) ofG ₂ coincides with Racah's separation. An improved separation of theH states of (31) is described.     

Impossibility of collapse under the law of gravitationR 44=0

Radial motion of a small point mass in the gravitational field of a large point mass is investigated for the law of gravitationR ₄₄ =0. When geodesic equations are expressed in terms of components of acceleration, it is found that the normally “attractive force” of gravitation gradually weakens as the large mass is approached, and becomes “repulsive” inside a critical nonsingular radius close to the origin of coordinates. A particle requires an infinite time to reach the origin, regardless of its initial distance. Gravitational collapse, or at least violent collapse, is thus precluded.     

Natural convection with damped thermal flux in a vertical circular cylinder

Natural convection flows of an incompressible Newtonian fluid inside a circular cylinder are studied. The heat transfer process is described by a generalized fractional constitutive equation for the thermal flux-temperature gradient. Caputo time-fractional derivative operator, which provides the damping of thermal flux, is considered into the studied model.Analytical solutions to the fluid temperature, thermal flux, fluid velocity and volume flow rate are obtained with the integral transforms method (Laplace transform and finite Hankel transform).Temperature behaviors for small and large values of the time t, as well as the post-transient and transient velocity components are determined. The influence of the memory parameter (the order of the time-fractional derivative) on the temperature, thermal flux, velocity and the volume flow rate is numerically and graphically studied.     

The anisorrhopic guiding-center fluid

A study is made of so-called “finite-orbit effects” in a two-dimensional guiding-center plasma. The macroscopic mass motion of the plasma is represented on the basis of a simple incompressible one-fluid model (so-called “representative fluid”), and the guiding-center motions of single particles are then referred to a Lagrangian coordinate network comoving with the representative fluid. The fluid motion defines the network motion. It turns out, however, to have no effect on the guiding-center motion relative to the network (autonomy theorem). It is found, in other words, that the relative trajectories of guiding centers are determinable in advance independently of the network motion (or the fluid motion), and this provides the necessary information to determine all the state parameters of the representative fluid (density of mass, density of gyrational angular momentum, etc.) as functions of the time, t, at any given point of the network. Once this information is available, the fluid motion is then completely determined by the remaining hydrodynamic equations (equation of motion, equation of incompressibility). The so-called “finite-orbit effects” take the form of gyroscopic-quasielastic forces in the equation of motion. No special isorrhopy condition is assumed. (This refers to a special initial condition assumed in an earlier work, for the sake of analytical simplicity. Here, the special initial condition is dropped.) Much attention is devoted to problems of wave propagation and stability. There are two independent sets of wave modes (if a nonvanishing anisorrhopy is allowed): so-called fluid modes, and so-called drift modes, respectively defined as first-order perturbations in the network motion (or the fluid motion) relative to the fixed coordinate frame, and in the guiding-center motion relative to the network. The stability conditions against both sets of modes are found to be quite stringent, much more so than in the earlier isorrhopic case. Nonetheless, a reasonably extensive class of stable solutions is shown to exist.     

Optical solitons in dense resonant media

Exact single-soliton solutions of the modified system of Maxwell-Bloch equations, in which the dipole-dipole interactions of the atoms of a dense resonant medium are taken into account, are obtained. Two propagation regimes are analyzed: “coherent,” where the pulse duration is much shorter than both relaxation times (T _p ? T ₁, T ₂), and “incoherent,” where the pulse duration falls between the relaxation times (T ₂ ? T _p ? T ₁). It is predicted, for the first time, that soliton propagation of an ultrashort pulse is possible in a dense resonant absorbing medium in an incoherent interaction regime. The differences between the amplitude and phase characteristics of the solitons considered and the corresponding characteristics of the solitons for McCall-Hahn self-induced transparency are noted.     

The B=2 sector of the Skyrme model in S3(L)

We consider baryon-number-two solutions to the Skyrme effective lagrangian on the surface of a hypersphere, S₃(L). At high densities we find solutions with “radially” uniform energy and baryon densities analogous to solutions previously found for the baryon-number-one problem on S₃(L) and for the dense matter problem in R₃. The density for this phase transition is essentially the same in the three cases. An interpretation of this phase transition as a simulation of “deconfinement” is offered.     

平行板微管道间Maxwell流体的高Zeta势周期电渗流动

本文研究了两平行板微管道中线性黏弹性流体的周期电渗流动, 其中线性黏弹性流体的本构关系是由广义Maxwell模型描述的. 将电渗力作为体力, 解析求解了非线性的Poisson-Boltzmann (P-B)方程, 柯西动量方程和广义Maxwell本构方程. 通过数值计算, 分析了无量纲壁面Zeta势ψ₀ 、 周期电渗流 (electroosmotic flow, EOF) 振荡雷诺数Re和无量纲弛豫时间λ ₁ω 对速度剖面的影响. 结果表明: 对给定的电动宽度K(表示微管道的特征尺度与双电层厚度的比值)、 弛豫时间λ ₁ω 和振荡雷诺数Re, 高Zeta势ψ₀ 产生较大的EOF速度振幅, 并且速度剖面的变化主要集中在双电层 (electric double-layer, EDL) 的狭窄的区域. 此外, 随着弛豫时间的增长流体的弹性显著增加, 速度的变化可以延伸到整个流动的区域中. 对给定的雷诺数Re, 较长的弛豫时间λ₁ω 导致EOF速度剖面较快的变化, 且速度剖面的振幅逐渐增大.     

Analysis of peristaltic flow of Jeffrey six constant nano fluid in a vertical non-uniform tube

Peristaltic flow of non-Newtonian nano fluid through a non-uniform surface has been investigated in this paper. The fluid motion along the wall of the surface is caused by the sinusoidal wave traveling with constant speed. The governing equations are converted into cylindrical coordinate system and assuming low Reynolds number and long wave length partial differential equations are simplified. Analytically solutions of the problem are obtained by utilizing the homotopy perturbation method (HPM). In order to insight the impact of embedded parameters on temperature, concentration and velocity some graphs are plotted for different peristaltic waves. At the end, some observations were made from the graphical presentation that velocity, pressure rise and nano particle concentration are increasing function of thermophoresis parameter N_t while temperature and frictional forces show opposite trend.     

The 3rd flow component as a QGP signal

Earlier fluid dynamical calculations with QGP show a softening of the directed flow while with hadronic matter this effect is absent. On the other hand, we indicated that a third flow component shows up in the reaction plane as an enhanced emission, which is orthogonal to the directed flow. This is not shadowed by the deflected projectile and target, and shows up at measurable rapidities, y _CM=1?2. To study the formation of this effect initial stages of relativistic heavy ion collisions are studied. An effective string rope model is presented for heavy ion collisions at RHIC energies. Our model takes into account baryon recoil for both target and projectile, arising from the acceleration of partons in an effective field. The typical field strength (string tension) for RHIC energies is about 5–12 GeV/fm, what allows us to talk about “string ropes”. The results show that QGP forms a tilted disk, such that the direction of the largest pressure gradient stays in the reaction plane, but deviates from both the beam and the usual transverse flow directions. The produced initial state can be used as an initial condition for further hydrodynamical calculations. Such initial conditions lead to the creation of third flow component. Recent v ₁ measurements are promising that this effect can be used as a diagnostic tool of the QGP.     

Relaxation in a Duffing potential

The solution of the equation of motion for a particle in a Duffing potential, V(x) = α₁x²/2 + α₂x⁴/4 (α₁, α₂ > 0) for arbitrary anharmonicity strength is characterized by the presence of odd frequencies which implies that velocity and position autocorrelation functions of such an oscillator in a microcanonical ensemble are also characterized by odd frequencies. It is, however, non-trivial to determine whether such “discrete” frequencies also characterize the autocorrelation functions in a canonical ensemble as discussed recently by Fronzoni et al. (J. Stat. Phys. 41 (1985) 553). We recover and extend upon the results of Fronzoni et al. to show analytically, via Mori-Lee theory, that “essentially discrete” (i.e. well-defined peaks with finite but “small” width) temperature-dependent frequencies characterize the autocorrelation functions in a canonical ensemble.     

The Hubble law and the spiral structures of galaxies from equations of motion in general relativity

Fully exploiting the Lie group that characterizes the underlying symmetry of general relativity theory, Einstein's tensor formalism factorizes, yielding a generalized (16-component) quaternion field formalism. The associated generalized geodesic equation, taken as the equation of motion of a star, predicts the Hubble law from one approximation for the generally covariant equations of motion, and the spiral structure of galaxies from another approximation. These results depend on the imposition of appropriate boundary conditions. The Hubble law follows when the boundary conditions derive from the oscillating model cosmology, and not from the other cosmological models. The spiral structures of the galaxies follow from the same boundary conditions, but with a different time scale than for the whole universe. The solutions that imply the spiral motion areFresnel integrals. These predict the star's motion to be along the “Cornu Spiral.” The part of this spiral in the first quadrant is the imploding phase of the galaxy, corresponding to a motion with continually decreasing radii, approaching the galactic center as time increases. The part of the “Cornu Spiral” in the third quadrant is the exploding phase, corresponding to continually increasing radii, as the star moves out from the hub. The spatial origin in the coordinate system of this curve is the inflection point, where the explosion changes to implosion. The two- (or many-) armed spiral galaxies are explained here in terms of two (or many) distinct explosions occurring at displaced times, in the domain of the rotating, planar galaxy.     

Solution of the Equations of Motion for Einstein’s Field in Fractional <Emphasis Type="Italic">D</Emphasis> Dimensional Space-Time

As a continuation of Sadallah et al. work (M. Sadallah, S. Muslih and D. Baleanu, Equations of motion for Einstein’s field in non-integer dimensional space. Czechoslov. J. Phys. 56:323, 2006), the fractional action function S is given as an integration over fractional spatial dimension D _s and fractional time D _t dimension. The variational principle which minimize S leads to Euler-Lagrange equations of motion in D _s +D _t fractional dimensions. As an example we extend our study to obtain the equations of motion for Einstein’s field in fractional D _s +D _t fractional dimensions of N+1 space-time coordinates. It is shown that the time dependent solutions are single valued for only D _s =4 dimensional space. Also the angular solutions are convergent for any value of D _s .