Application Of Fractional Calculus In The Field Of Fluid Flow,Heat Transfer And Anomalous Diffusion

Fractional calculus provides mathematical theoretical support for the study of anomalous diffusion in porous media and constitutive relationship of viscoelastic fluid.In this paper,the flow and heat transfer of fractional viscoelastic fluid between coaxial cylinders and the anomalous diffusion of formaldehyde in dry building material are studied,respectively.The contents of the study are as follows:1)In view of the two-dimensional magnetohydrodynamic flow and heat transfer of fractional Oldroyd-B nanofluid between two coaxial cylinders in axial and radial directions,the momentum and energy equations of fractional boundary layer in cylindrical coordinate system are established.The numerical solution of the governing equation is obtained by combining the finite difference method with the L1 algorithm.Furthermore,the validity of the method is verified by constructing an example and comparing with the exact solution.At the same time,the effects of parameters such as time fractional derivative and exponential parameters on the velocity and temperature fields are discussed.2)Based on the fact that wood-based panel is a complex porous medium with fractal structure,the diffusion and release characteristics of volatile organic compounds(VOCs)in indoor dry building materials are studied,and a time-space fractional diffusion model is established.The space fractional derivative is expressed by the weight coefficients of the upward and downward transfer probabilities,which describes the spatial nonlocality;the time fractional derivative describes the historical channel dependence of formaldehyde diffusion,and the influence of the surface porosity of materials is considered.The finite difference method and parameter estimation method are used to simulate the equations.The simulation results are in good agreement with the experimental data,and the error value(67.44?g/m~3)is much less than the model of Deng and Kim(190?g/m~3).It is proved that the fractional mass transfer model can better describe the early breakthrough and heavy-tailed phenomena,and the influence of key physical parameters on the emission of formaldehyde is revealed.3)Based on the time-space fractional diffusion model,the release characteristics of formaldehyde from wood-based panel to the environmental chamber are studied.The diffusion equation of formaldehyde in the wood-based panel and the mass balance equation in the environmental chamber are modified by considering the fractional chemical reaction with characteristic of delay,time-varying effective diffusion coefficient and environmental factor.The finite difference method combined with L1,L2 and R1 algorithm are used for numerical simulation,and the improved hybrid Nelder Mead simplex search and particle swarm optimization(MH-NMSS-PSO)method is used for parameter estimation.The simulation results of the model are in good agreement with the experimental data.In addition,comparing the emission rate of formaldehyde from different building materials into the environmental chamber,the effect of fractal structure of wood-based panel on the anomalous diffusion of pollutants is revealed.In this paper,the flow and heat transfer model of fractional viscoelastic fluid,the fractional mass transfer model of anomalous diffusion and the method of numerical analysis are established,which lay the foundation for the subsequent study.Through the error analysis of numerical examples,the effectiveness of the numerical method is verified,which provides a way for the modeling and solution of the nonlinear fractional partial differential equation to describe the actual complex problems.