Study On Anomalous Diffusion In Fabrics

Air permeability and heat conduction are two kinds of physical properties of materials.Fabric is a special kind of porous media.In our daily life,air permeability and heat transfer of fabrics are closely connected with human comfort.The design of functional fabric products,such as astronaut suit,firefighter uniform,often needs thermostable fiber.Therefore,the diffusion problems of functional fabric products have attracted great attention.It is significant to analyze diffusion theory in fabric and characterize air permeability and thermal conductivity of fabrics by mathematical modeling.Due to the complex geometry structure of porous materials,the diffusion phenomena in fibrous materials are involved in interdisciplinary science including Mathematics,Mechanics,Human Physiology and Textile.Although much effort has been made on this topic with various experimental methods,it's still worth studying the diffusion problem by mathematical modeling and theoretical analysis.Based on the fractal and fractional calculus,the mechanical model of diffusion in fabric is established and non-linearization method for solution of fractional order models is proposed.The dissertation is divided into five chapters accordingly.The first chapter reveals definition,geometry structures and characterization method of porous materials.The conventional research methods of diffusion problem in porous materials are based on Fick's law and Brown motion.Actually,many diffusion phenomena in a system of porous materials cannot be investigated with Euclidean geometrical methods and don't follow to the classical Fick's law.The history of the development and applications of non-Fick's law in porous materials is introduced.Fractals and fractional calculus were often used in studying anomalous diffusion in porous materials.In chapter 2,fractal theory is applied to investigate air permeability of fabrics.At first,theories of fractal geometry and Hausdorff dimension are introduced.Then,the permeability of fabrics is discussed by porous structure and air permeability rate in fabrics.For porous structures,the hierarchical structure of fractals is consistent with the multi-scale structure of pores.Therefore,fibrous material structure can be characterized by fractals.Hausdorff dimension of fractals is an important parameter to describe the porous structure.It depends on the distributions of their wrap and weft yarns.For characterization of permeability of fabrics,traditional methods were all based on Darcy's law and Hagen-Poisenille equation.In fact,the assumption of mechanics in continuous media is not supported in the widely used approaches.In this chapter,a modified Hagen-Poisenille's Law is proposed based on fractal structure of cloth pores.The air permeability versus pore radius curve of plain woven fabric is plotted according to the experimental data,and the fractal dimension of pores is determined by regression.The fractal dimension of pores of plain woven fabric is also calculated based on fractal theory,which is comparative studied with the experimental one.In addition,fractal dimension of airflow tracing line length is determined by air permeability versus thickness curve.Compared with the classical Hagen-Poisenille's law,the results show that the relationship between the air permeability and the cross section as well as the thickness of plain woven fabric follows to generalized power laws.In chapter 3,the basic theory of fractional calculus is introduced.Based on fractal structure of fabric and fractional order integral theory,coordinate porosity and its calculation for woven fabric are presented.A fractal square and fractal round coordinate porosity models are established and illustrated with stair curves.The results suggest that the porosity of woven fabrics depends on the dimensions and the scales of fractals.Coordinate porosity method and former's characterization method of porosity finite-scale fractal are also compared.A new method to calculate the porosity of incomplete fractal media is proposed.In chapter 4,the fractional heat diffusion equation is used to investigate the diffusion problem of fabric,which contains experimental study of heat diffusion in fabric with air layers,determination of thermal diffusion coefficient by dimensional analysis method and finite difference method and numerical simulation of the thermal diffusion equation.Firstly,with the experimental data and theoretical analysis,a double parameter method is utilized to describe the diffusion in multiphase space which contains fibers and air space.Then,the diffusion coefficient of fabric is obtained in the fractional dimensional sense and the value of the coefficient can be determined by the fractional finite difference method.Finally,based on fractional variational approach,this thesis firstly proposes a fractional variational iterational method which could identify a generalized Lagrange multiplier and this method could solve a fractal initial boundary problem.With the algorithm,approximate solutions of fractional diffusion equation with source and variable coefficients are obtained.