Study On Some Problems Of Anomalous Transport Based On Fractional Calculus Theory

In recent years,due to the research on the theory and application of fractional calculus by scholars from different fields,fractional calculus has attracted extensive attention and has been successfully applied to the constitutive relations of viscoelastic materials,anomalous diffusion,control,biomedicine and other fields.This thesis mainly studies the application of fractional calculus in laser heating,bioheat transfer and fluid mechanics.Compared with classical integer differential operators,fractional differential operators have time memory and spatial global correlation so that they can better describe these abnormal transport processes.However,owing to the nonlocality of fractional differential operators,the solutions of fractional differential equations are still very challenging.Therefore,using fractional differential equations to model practical problems and developing efficient numerical algorithms is a very meaningful and challenging research.In this thesis,we introduce fractional calculus into laser heating,bioheat transfer and hydrodynamics,and develop efficient numerical algorithms to solve the corresponding fractional partial differential equations.The main innovations and contributions of the thesis:Firstly,the efficient and accurate numerical difference schemes are developed for coupled and nonlinear fractional partial differential equations with multi-term fractional derivatives.Also,the unconditional stability and convergence of the difference scheme are proved by theoretical analysis and numerical examples.Secondly,a time-space fractional bioheat transfer model is proposed to characterize the spatially heterogeneous biological tissues.The rationality of the model is verified based on the thermophysical properties of actual biological tissue and laser parameters,and the research results have great significance for medical doctors in the clinical therapeutic for hyperthermia treatment of tumors or cancerous cells.Thirdly,in view of the fractional derivative model,the complex flows of viscoelastic fluids are studied.For the flow and transport problem at the high potential,the nonlinear Poisson-Boltzmann potential equation in cylindrical coordinates is solved for the first time.Fourth,the fractional viscoelastic fluid model is used to simulate the unsteady flow and heat transfer of blood in small vessels under the action of magnetic field and heat flux.This study has broad application prospects in the treatment of cardiovascular diseases and tumors.The research results of this thesis are helpful to promote the application of fractional calculus in engineering mechanics and other fields,and can provide a good reference for solving fractional partial differential equations.Specifically:In chapter 1,we briefly introduce the development history of fractional calculus,and give the definitions and numerical approximations of several commonly used fractional calculus operators.Then,we present the definitions of several integral transformation methods used in this thesis.Finally,the main contents of this thesis are summarized.In chapter 2,we numerically discuss the non-Fourier heat conduction behavior in a finite medium based on time fractional dual-phase-lag model.Firstly,the time fractional dual-phase-lag heat conduction equation for short pulse laser heating is established by considering two different types of pulse laser distributions,namely Gaussian and non-Gaussian.Based on the L1 approximation for the Caputo derivative,the CN difference scheme is developed for the short pulse laser heating problem.The solvability,stability and convergence of the scheme are proved.Meanwhile,the accuracy of the method has been verified by using three numerical examples.Finally,we graphically describe the effect of the ratio between two phase lags on the non-Fourier heat conduction behavior in finite media.In chapter 3,we develop a time-space fractional heat conduction model to study the non-Fourier bioheat transfer process within the skin tissues during laser irradiation.Based on the L1 approximation for the Caputo time fractional derivative and the central difference scheme for the Riesz space fractional derivative,the implicit difference scheme for laser heating biological tissue is presented.The efficiency and accuracy of this scheme are verified by using three numerical examples.According to the thermophysical properties of the biological tissues,the effects of related parameters on temperature distribution within living biological tissues are analyzed and shown graphically.The peak values of the temperature with the hyperthermia position and time are listed in the tables.In chapter 4,we study the electroosmotic flow and rotating electroosmotic flow of viscoelastic fluids in microchannels under high potential.Firstly,the electroosmotic flow of fractional Oldroyd-B fluids in a narrow circular tube with high potential is presented.We solve the nonlinear Poisson-Boltzmann equation in cylindrical coordinates for the first time.The difference schemes for the velocity distribution,flow rate and shear stress are derived by using the finite difference method.The accuracy of the difference scheme is verified by comparing our results with that in the existing literature.Then,the rotating electroosmotic flow of fractional Maxwell fluid in parallel plate microchannels at high potential is studied.Based on the L1 approximation of Caputo derivative,a CN numerical scheme is developed.We compare the results with that given in the previous work,a good agreement can be found.Finally,the effects of related parameters on velocity distribution are discussed in detail.In chapter 5,we study the electromagnetohydrodynamic(EMHD)slip flow and heat transfer of viscoelastic fluid in microchannels.Firstly,we consider the unsteady EMHD slip flow of fractional Maxwell fluid.The analytical solution of velocity distribution is derived by means of integral transformation method,and the second-order difference schemes of the velocity equations are developed by using the second-order Lubich approximation for the time fractional derivative.A comparison between numerical and analytical solutions under different parameter values is also given.Then,we analyze the EMHD flow and heat transfer of fractional Oldroyd-B fluids under pressure gradient,electric field,magnetic field and heat flux.The analytic and numerical solutions of velocity distribution are given by using integral transformation method and finite difference algorithm,respectively.Additionally,based on the velocity distribution of fully developed flow,the energy equation including volume Joule heating,electromagnetic coupling effect and energy dissipation is also solved by using the finite difference method,and then the important heat transfer parameter,Nusselt number,is also obtained.In chapter 6,we study the unsteady flow and heat transfer of blood in small blood vessels under the action of magnetic field and heat flux.Research shows that the blood exhibit the characteristics of non-Newtonian fluid at relatively low shear rate.Since the shear rate in small blood vessels is relatively low,we use a non-Newtonian fluid model,namely fractional Maxwell fluid,to simulate the rheological behavior of blood.Using the finite difference method,the difference schemes of the nonlinear and coupled governing equations of velocity and temperature under pulsatile pressure gradient,magnetic field and thermal radiation are given,and the convergence as well as stability of the difference schemes are verified.Finally,the effects of relevant parameters on velocity,flow rate and temperature are analyzed.In chapter 7,We summarize the dissertation and look forward to the future research direction.