Six-coupled Ising spins on a triangular lattice with antiferromagnetic (AF) nearest neighbour and ferromagnetic (F) next-nearest neighbour interactions are investigated by Glauber dynamics. The system is fully frustrated and has seven non-trivial local energy minima in both clusters. The ground state has four degenerate states with energy -6. These states are separated by the energy barrier ∆E = 4.0 to invert spins in the ground state. The dynamics of AF-F and F-AF coupling clusters are solved exactly. Each cluster contributes as many long relaxation times as it has non-trivial local energy minima. The barriers against inversion of the clusters take only two values, 0 and 4|J|. In the paramagnetic case (PM), there is no diverging relaxation time. In the ferromagnetic case (FM), there is only one long-lived mode. The longest relaxation times follow an Arrhenius law. Both AF-F and F-AF clusters undergo a zero-temperature phase transition. The real and imaginary parts of the dynamic susceptibility display maxima if plotted versus temperature. The frequency dependence of the susceptibilities explain the effect of frustration. The real part reveals two plateaus and the imaginary part displays two corresponding maxima if they are plotted as functions of the logarithm of frequency. The Argand plots show two overlapping semicircles for fixed frequency at low temperature, indicating the time separation of the modes.

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 ϕ, magnetic field M, viscous dissipation E c , and suction parameter F w 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.

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.

The advection diffusion equation was solved analytically using separation of variables technique, considering first the wind speed and eddy diffusivity as constants; second as variables dependent on vertical height z. Comparison between predicted two models and observed concentration on Inshas, Cairo (Egypt) is done.

No abstract