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Academic and industrial researches on nanoscale flows and heat transfers are an area of increasing global interest, where fascinating phenomena are always observed, e.g. admirable water or air permeation and remarkable thermal conductivity. The purpose of this paper is to reveal the phenomena by the fractional calculus.

This paper begins with the continuum assumption in conventional theories, and then the fractional Gauss’ divergence theorems are used to derive fractional differential equations in fractal media. Fractional derivatives are introduced heuristically by the variational iteration method, and fractal derivatives are explained geometrically. Some effective analytical approaches to fractional differential equations, e.g. the variational iteration method, the homotopy perturbation method and the fractional complex transform, are outlined and the main solution processes are given.

Heat conduction in silk cocoon and ground water flow are modeled by the local fractional calculus, the solutions can explain well experimental observations.

Particular attention is paid throughout the paper to giving an intuitive grasp for fractional calculus. Most cited references are within last five years, catching the most frontier of the research. Some ideas on this review paper are first appeared.

The purpose of this paper is to study nanoparticles diffusion and coagulation processes in a twin-jet.

Large eddy simulation (LES) and Taylor-series expansion moment method (TEMOM) are employed to deal with a nanoparticle-laden twin-jet flow.

The numerical results show that the interaction of the two jets and turbulence eddy structures rolling-up, paring and shedding in flow sharply affects particles number concentration. Particle diameter grows quickly at the interfaces of jets. Coagulation shows more obvious effect at initial stage than that in the subsequent period. Then diffusion makes the particle diameter distribution much more uniform.

In recent years a great number of attentions have been focussed on the issue of particulate dynamics processes including diffusion, coagulation and deposition, etc. However, up to now few works have been focus on the nanoparticles coagulation and dispersion in turbulent flows. The investigation on the diffusion and coagulation process of nanoparticles using TEMOM in a twin-jet flow has not been found.

The purpose of this paper is to investigate the gas flow characteristics in near wall region and the velocity slip phenomenon on the wall in nano-channels based on the molecular dynamics simulation.

An external gravity force was employed to drive the flow. The density and velocity profiles across the channel, and the velocity slip on the wall were studied, considering different gas temperatures and gas-solid interaction strengths.

The simulation results demonstrate that a single layer of gas molecules is adsorbed on wall surface. The density of adsorption layer increases with the decrease of gas temperature and with increase of interaction strength. The near wall region extents several molecular diameters away from the wall. The density profile is flatter at higher temperature and the velocity profile has the traditional parabolic shape. The velocity slip on the wall increases with the increase of temperature and with decrease of interaction strength linearly. The average velocity decreases with the increase of gas-solid interaction strength.

This research presents gas flow characteristics in near wall region and the velocity slip phenomenon on the wall in nano-channels. Some interesting results in nano-scale channels are obtained.

The purpose of this paper is to theoretically investigate the thermal and hydrodynamic performance of the flow pattern of fluid in the charged jet. The flow pattern includes laminar flow in which all fluid layers move at different accelerated speeds, and shear forces between the fluid layers give rise to friction forces. This is a favorable condition for the parallel arrangement of the branches on polymer molecules.

The dynamic mechanism of the flow pattern is conducted through analyzing the forces acting on the charged jet. The differential equation obtained in the analyzing process has the solution designating the laminar flow pattern of the fluid in the charged jet.

The fluid in the charged jet flows in laminar pattern, which is favorable to the parallel arrangement of the branches on polymer molecules.

Although the flow pattern is conveyed by means of the simple condition of uniformly accelerated motion, it has the similar effect on the arrangement of the polymer molecules in general conditions, such as non-Newtonian fluids and non-uniformly accelerated motions.

The laminar flow introduced by this paper to the charged jet implies anisotropic properties of the electrospun nanofibers.

The purpose of this paper is to present a general framework of Homotopy perturbation method (HPM) for analytic inverse heat source problems.

The proposed numerical technique is based on HPM to determine a heat source in the parabolic heat equation using the usual conditions. Then this shows the pertinent features of the technique in inverse problems.

Using this HPM, a rapid convergent sequence which tends to the exact solution of the problem can be obtained. And the HPM does not require the discretization of the inverse problems. So HPM is a powerful and efficient technique in finding exact and approximate solutions without dispersing the inverse problems.

The essential idea of this method is to introduce a homotopy parameter p which takes values from 0 to 1. When p=0, the system of equations usually reduces to a sufficiently simplified form, which normally admits a rather simple solution. As p is gradually increased to 1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation.

The mathematical model of a two-phase Lamé-Clapeyron-Stefan problem for a semi-infinite material with a density jump is considered. The purpose of this paper is to study the analytical solutions of the models and show the performance of several parameters.

To describe the heat conduction, the Caputo type time fractional heat conduction equation is used and a convective term is included since the changes in density give rise to motion of the liquid phase. The similarity variables are used to simplify the models.

The analytical solutions describing the changes of temperature in both liquid and solid phases are obtained. For the solid phase, the solution is given in the Wright function form. While for the liquid phase, since the appearance of the advection term, an approximate solution in series form is given. Based on the solutions, the performance of the parameters is discussed in detail.

From the point of view of mathematics, the moving boundary problems are nonlinear, so barely any analytical solutions for these problems can be obtained. Furthermore, there are many applications in which a material undergoes phase change, such as in melting, freezing, casting and cryosurgery.

The purpose of this paper is to use comprehensive model to investigate the effects of particle physical properties on in-flight nano-particles behavior for the radio frequency suspension plasma spray.

In this paper, both the effects thermal properties of solvent and solid particle on the evolution of particle size, velocity and temperature are discussed. Besides, the numerical analysis is also conducted to investigate the influences of particle physical properties on the characteristic distributions of particles for poly-disperse cases.

Results show the thermal properties of solvent have critical effects on the discharged point of the solid particles, but little influence on the final particle velocity and size, as well as their distributions. The final state of particle temperature is mainly determined by the solid particle thermal properties, especially depending on the boiling point.

Most of the former studies took the experimental approaches and mainly focussed on the operating conditions effects. While beyond the operating conditions, the variety of particle physical and thermal properties also has strong effect on particle heating performance.

The purpose of this paper is to find an effective numerical method for solving singularly perturbed convection-diffusion problems.

The present method is based on the asymptotic expansion method and the variational iteration method (VIM). First a zeroth order asymptotic expansion for the solution of the given singularly perturbed convection-diffusion problem is constructed. Then the reduced terminal value problem is solved by using the VIM.

Two numerical examples are introduced to show the validity of the present method. Obtained numerical results show that the present method can provide very accurate analytical approximate solutions not only in the boundary layer, but also away from the layer.

The combination of the asymptotic expansion method and the VIM is applied to singularly perturbed convection-diffusion problems.

The purpose of this paper is to investigate tensile properties of TiC particle-reinforced titanium matrix composites (PRTMC) using the elasto-plastic finite element (FE) programs and the homogenization method and the fixed point iteration method.

Two quasi-static and dynamic transient programs of elasto-plastic FE were coded by using FORTRAN. Based on the FE programs, the FE model of the TiC PRTMC with typical microstructures was established by using the fixed point iteration method and the homogenization theory. The hot deformation behavior of TiC PRTMC under different temperatures were analyzed by using the above model and programs.

Calculation results are presented to investigate the influence of different temperatures on the hot deformation behavior of TiC PRTMC. Based on the experimental data, a good agreement was obtained between the numerical predictions and the experimental results, and the feasibility of this method was verified.

The work is original and findings are new, which demonstrates this FE frame combined with the homogenization method and the fixed point iteration method can be used to investigate the tensile behavior of particle-reinforced metal matrix composites.