Study On Several Types Of Viscoelastic Fluid Models Based On Fractional Calculus Theory

Recently,the application of fractional calculus theory has attracted much attention in the flow and heat transfer field of viscoelastic fluid.The fractional model of viscoelastic fluids is derived from the forced balance equation,and it is obtained by substituting integer derivatives of time in classical equations into fractional calculus operators.The flow and heat transfer of viscoelastic fluid are more accurately characterized through the nonlinear relation between shear stress and velocity gradient.Based on the investigation of the flow and heat transfer for two types of viscoelastic fluid under different physical conditions,the governing equations with mixed time-space fractional derivative are established and solved numerically.Due to the physical properties of the viscoelastic fluid,the flow resistance can be reduced in the flow process.Moerover,the effect of heat transfer can be enhanced by adding nanoparticals into the fluids.Therefore,this paper deeply studies the flow and heat transfer of the fractional viscoelastic nanofluid in the electromagnetic field and the porous medium.IIn addition to using the modified Darcy's law and the fractional Cattaneo model to describe describe porous media and the thermal transfer process of fluids,respectively,this paper introduces the Hamilton-Crosse model to study the heat transfer of viscoelastic fluids firstly.The Hamilton-Crosse model is used to approximate value of thermal conductivity by considering the parameter of nanoparticle shape.Moreover,the flow and heat transfer of the fractional order Maxwell viscoelastic fluid in porous media is considered under the conditions of slip boundary and variable viscosity.Because of the application and research of fractional calculus theory in the coupling problem of flow and heat transfer of viscoelastic fluid,the problem of obtaining analytical and numerical solutions of the corresponding fractional differential equations becomes an important research topic.In this paper,the discrete scheme is established by the finite difference method combined with the L1-algorithm,and the numerical solution is obtained by using the Thomas algorithm.The flow and heat transfer characteristics of the fluid are discussed in three different physical models.The effects of fractional derivative parameters,porosity,slip parameter and other involved physical parameters on the velocity,temperature,average skin friction coefficient and average Nusselt number are discussed and graphically illustrated in detail.