A Compact Introduction to Fractional Calculus
A Compact Introduction to Fractional Calculus is presented including basic definitions, fractional differential equations and special functions.
math.HO↗
arXiv subjects
Alexander I. Zhmakin
Publications and source records attributed to Alexander I. Zhmakin.
The Zoo of Non-Fourier Heat Conduction Models
The Fourier heat conduction model is valid for most macroscopic problems. However, it fails when the wave nature of the heat propagation or time lags become dominant and the memory or/and spatial non-local effects significant -- in ultrafast heating (pulsed laser heating and melting), rapid solidification of liquid metals, processes in glassy polymers near the glass transition temperature, in heat transfer at nanoscale, in heat transfer in a solid state laser medium at the high pump density or under the ultra-short pulse duration, in granular and porous materials including polysilicon, at extremely high values of the heat flux, in heat transfer in biological tissues. In common materials the relaxation time ranges from $10^{-8}$ to $10^{-14}$ sec, however, it could be as high as 1 sec in the degenerate cores of aged stars and its reported values in granular and biological objects varies up to 30 sec. The paper considers numerous non-Fourier heat conduction models that incorporate time non-locality for materials with memory (hereditary materials, including fractional hereditary materials) or/and spatial non-locality, i.e. materials with non-homogeneous inner structure.
cond-mat.mtrl-sci↗
Modelling of Heat Transfer in Single Crystal Growth
An attempt is made to review the heat transfer and the related problems encountered in the simulation of single crystal growth. The peculiarities of conductive, convective and radiative heat transfer in the different melt, solution, and vapour growth methods are discussed. The importance of the adequate description of the optical crystal properties (semitransparency, specular reflecting surfaces) and their effect on the heat transfer is stresses. Treatment of the unknown phase boundary fluid/crystal as well as problems related to the assessment of the quality of the grown crystals (composition, thermal stresses, point defects, disclocations etc.) and their coupling to the heat transfer/fluid flow problems is considered. Differences between the crystal growth simulation codes intended for the research and for the industrial applications are indicated. The problems of the code verification and validation are discussed; a brief review of the experimental techniques for the study of heat transfer and flow structure in crystal growth is presented. The state of the art of the optimization of the growth facilities and technological processes is discussed. An example of the computations of the heat transfer and crystal growth of bulk SiC crystals by the sublimation method is presented.
physics.flu-dyn↗