The Applications Of Fractional Differential Equations And Lie Group Methods In Studies Of Complex Fluids

It is an important research topic to explore the flow of complex fluids and further the phenomena of heat and mass transfer.Viscoelastic fluids,with many kinds and dif-ferences in rheological properties,are complex fluids.Most of the classical constitutive equations for viscoelastic fluids are complicated in mathematics.Therefore,some new constitutive models of viscoelastic fluids are needed to be explored.Nanofluids are al-so complex fluids that it has great potential in industrial applications.Although a large number of experiments have shown that nanoparticles enhance the heat transfer of the base liquid,the physical mechanism is still under discussion.Moreover,the data are quite different in the experiments.Therefore,it is of great significance to further study the mechanism of heat transfer of nanofluid.In this paper,the boundary layer flow of viscoelastic fluid,anomalous heat and mass transfer of nanofluids are studied.For the boundary layer problem of viscoelastic fluid,the governing equation with the spatial fractional derivative is derived in this paper.Using the Lie analysis method,the similarity transformation formula of the equation has been established for the first time,and the problem of the flow around the plate and the wall jet is further studied.The conservation law of the flow problem is studied by the nonlinear self-adjoint method,because the solution of the wall jet problem needs to be combined with the physical conservation law.Two methods of Runge-Kutta-Grunwald scheme and Runge-Kutta-predictor-correction scheme are proposed in this paper to solve the mixed order of or-dinary differential equations.Finally,the velocity of viscoelastic fluid in the boundary layer is analyzed with the change of the fractional order.The anomalous heat and mass transfer of nanofluids in a complex medium are al-so studied.The effects of the anomalous migration of nanoparticles on the convection heat transfer of nanofluids are considered.By using the random walk model,the gov-erning equation of the anomalous motion of the nanoparticles is derived.Based on the single-phase model of nanofluids,this paper studies the anomalous heat transfer in the plate boundary layer in porous media,the problem of mixed convection and the non-Newtonian base liquid.The numerical results show that the local Nusselt number be-comes larger with the decrease of the fractional order.The thickness of the thermal boundary layer becomes thinner and the heat transfer ability of nanofluids is enhanced with the increase of the anomalous movement strength of nanoparticles.Based on the two phase flow model,the governing equations of heat and mass transfer of nanofluids are established.The concentration distribution of nanoparticles and the changes of the temperature distribution of nanofluids with the fractional order are discussed.The effects of complex medium on the diffusion of nanoparticles are further studied by using the two phase model.The new temperature equation and the diffusion equation of nanoparticles are derived.The numerical results show that with the increase of the fractional order in the model,the diffusion speed of the nanoparticles becomes larger.That is,the retarda-tion effect of the medium on the particle diffusion is reduced,and the particle diffusion tends to the classical Cattaneo diffusion process.