G. Ismail

Hydromagnetic flow of a Cu–water nanofluid past a moving wedge with viscous dissipation

A numerical study is performed to investigate the flow and heat transfer at the surface of a permeable wedge immersed in a copper (Cu)–water-based nanofluid in the presence of magnetic field and viscous dissipation using a nanofluid model proposed by Tiwari and Das (Tiwari I K and Das M K 2007 Int. J. Heat Mass Transfer 50 2002). A similarity solution for the transformed governing equation is obtained, and those equations are solved by employing a numerical shooting technique with a fourth-order Runge–Kutta integration scheme. A comparison with previously published work is carried out and shows that they are in good agreement with each other. The effects of velocity ratio parameter λ, solid volume fraction , magnetic field M, viscous dissipation Ec, and suction parameter Fw on the fluid flow and heat transfer characteristics are discussed. The unique and dual solutions for self-similar equations of the flow and heat transfer are analyzed numerically. Moreover, the range of the velocity ratio parameter for which the solution exists increases in the presence of magnetic field and suction parameter.

Dynamic Susceptibility for Ising‐Spin Clusters in Random Fields

The dynamic susceptibility for a cluster of six coupled random field Ising spins in two different distributions, binary (BD) and Gaussian (GD), are calculated and exact results are obtained. The real and imaginary parts of the dynamic susceptibility display maxima when plotted versus temperature. These maxima can be described by an Arrhenius law. If the logarithm of the susceptibilities is plotted as a function of the logarithm of frequency and if the clusters are frustrated, then the real part displays a sequence of plateau regions and the imaginary part has a sequence of maxima in weak random fields. In the BD case of random field for large amplitudes there is only one plateau and one corresponding maximum as in ferromagnetic (FM) and paramagnetic (PM) cases. Our results confirm that any weak random field will turn out to destroy the ordered state and random field Ising-spin clusters behave like Ising-spin glasses.

Glassy behaviour of random field Ising spins on Bethe lattice in external magnetic field

The thermodynamics and the phase diagram of random field Ising model (RFIM) on Bethe lattice are studied by using a replica trick. This lattice is placed in an external magnetic field (B). A Gaussian distribution of random field (hi) with zero mean and variance 〈h2i〉 = HRF2 is considered. The free-energy (F), the magnetization (M) and the order parameter (q) are investigated for several values of coordination number (z). The phase diagram shows several interesting behaviours and presents tricritical point at critical temperature TC = J/k and when HRF = 0 for finite z. The free-energy (F) values increase as T increases for different intensities of random field (HRF) and finite z. The internal energy (U) has a similar behaviour to that obtained from the Monte Carlo simulations. The ground state of magnetization decreases as the intensity of random field HRF increases. The ferromagnetic (FM)-paramagnetic (PM) phase boundary is clearly observed only when z → ∞. While FM—PM-spin glass (SG) phase boundaries are present for finite z. The magnetic susceptibility (χ) shows a sharp cusp at TC in a small random field for finite z and rounded different peaks on increasing HRF.

A computational method for solving the branched channel flow problem

In the present work, a method is suggested to find the solution of Poissons equation with Dirichlet boundary conditions in polar coordinates. This method is expressed over the unit circle and consequently over the upper-half plane from which geometry we find the solution to Poissons equation over the branched channel domain. The results seems very reasonable and the derived transformation from unit circle to Y-shape channel is accurate. Poissons equation can be solved similarly over much other geometry. The solution looks very encouraging and resembles that is experimentally known.

Dynamics of Random One-Dimensional ±J Systems

Finite chains of a two-state random Potts spin model with periodic boundary conditions are studied within Glauber dynamics. The spin exchange is assumed random with frustration between ferro and antiferromagnetic values (±J). Time-dependent fluctuations are induced by periodic temperature oscillations. Master type differential equations for spin correlation functions are solved within linear response theory. The spectrum of relaxation times are calculated at different temperatures. The ±J Potts glass chains undergo a zero temperature phase transition. The barriers against inversion of the spin chain take only two values; 0 and 2|J|. The temperature behaviour of specific heat is characterized by rounded peaks. The frequency dependence displays two plateaus for the real part of specific heat and two corresponding peaks for the imaginary part. The dynamic specific heat is not affected by the longest relaxing mode like susceptibility. The time separation of the modes is demonstrated by the Cole-Cole plots.

Static Properties of Potts Spin Chains

One-dimensional 2-state Potts spin glasses (SG) disordered ferromagnets (DFM) and ±J systems are studied. The energy minima (EM) and magnetization with their distributions are exactly calculated. The stable local EM of no more than 27-spin chains with periodic boundary conditions are analyzed. All these systems have an exponentially growing numbers of average energy minima versus number of spins. The frustration effect is also discussed for some particular samples. The investigation of metastable states shows that the energy minima distributions for all our systems approach the normal distribution. The SG and DFM have identical distributions of energy minima. The systems studied here differ merely by the properties of the magnetization. The absolute value of magnetization averaged over all the EM decreases logarithmically with the number of spins for the three systems.

Glassy phase and thermodynamics for random field Ising model on spherical lattice in magnetic field

Phase diagram and thermodynamic parameters of the random field Ising model (RFIM) on spherical lattice are studied by using mean field theory. This lattice is placed in an external magnetic field (B). The random field (hi) is assumed to be Gaussian distributed with zero mean and a variance 〈h2i〉 = H2RF. The free energy (F), the magnetization (M) and the order parameter (q) are calculated. The ferromagnetic (FM) spin-glass (SG) phase transition is clearly observed. The critical temperature (TC) is computed under a critical intensity of random field . The phase transition from FM to paramagnetic (PM) occurs at TC = J/k in the absence of magnetic field. The critical temperature decreases as HRF increases in the phase boundary of FM-to-SG. The magnetic susceptibility (X) shows a sharp cusp at TC and the specific heat (C) has a singularity in small random field. The internal energy (U) has a similar behaviour to that obtained from the Monte Carlo simulation.

Metastability of Ising spin chains with nearest-neighbour and next-nearest-neighbour interactions in random fields

One-dimensional Ising systems in random fields (RFs) are studied taking into account the nearest-neighbour and next-nearest-neighbour interactions. We investigate two distributions of RFs: binary and Gaussian distributions. We consider four cases of the exchange couplings: ferro-ferromagnetic (F-F), ferro-antiferromagnetic (F-AF), antiferro-ferromagnetic (AF-F) and antiferro-antiferromagnetic (AF-AF). The energy minima of chains of no more than 30 spins with periodic boundary conditions are analysed exactly. We found that the average number of energy minima grows exponentially with the number of spins in both cases of RFs. The energy distributions across the corresponding energy minima are shown. The effects of RFs on both the average and density of metastable states are explained. For a weak RF, the energy distributions display a multipartitioned structure. We also discuss the frustration effect due to RFs and exchange fields. Finally, the distributions of magnetization are calculated. The absolute value of magnetization averaged over all metastable states decreases logarithmically with the number of spins.

Spectrum of relaxation times for ising spin clusters in random fields

SummaryExact results on the single-spin-flip Glauber dynamics of six-coupled random field Ising spins with the coordination number of four are presented. Two distributions of random fields (RF), binary (BD) and Gaussian (GD) ones, are investigated. The effects of the static magnetic field are discussed. In the zero-magnetic-field case, the number of diverging relaxation times is equal to the number of energy minima minus one. This rule breaks in the presence of a magnetic field. The longest relaxation times in the absence of the field verify the Arrhenius law with the energy barrier determined by the energy needed to invert the ground-state spin configuration. At low temperature, according to the Arrhenius law, the spectrum of relaxation times shows a two-peaked distribution on a logarithmic scale. In the GD case of RF, the energy barrier distribution is continuous, while it is quasi-discrete in the BD case.