Numerical Study On Unsteady Convection,Heat And Mass Transfer Of Fractional Viscoelastic Fluid

Recently,the fractional derivatives were found to be quite flexible in the description of complex dynamics(such as constitutive relation and heat conduction law)in viscoelastic fluids due to its nonlocal property or global dependency of fractional operators.In general,the constitutive relationships for generalized viscoelastic fluids are modified from well-known fluid models by replacing the time derivative of an integer order by the so-called fractional calculus operator.However,in classical study on viscoelastic fluid complex flow and heat transfer,the authors commonly ignored the effects of nonlinear convection and dealt only with the cases when the governing equations are linear.Their methods,i.e.,integral transforms,are difficult to solve the problems when the equations are nonlinear.Very little efforts have so far been made to discuss nonlinear convection terms with fractional derivatives.This paper presents a research on unsteady boundary layer convection heat and mass transfer of viscoelastic fluid.Fractional shear stress and heat flux models are introduced and the fractional boundary layer governing equations are firstly formulated and derived.From such derivation,the model constitutes nonlinear coupled equations with mixed time-space derivatives in the convection and diffusion terms,which are solved by a newly developed finite difference method combined with an L1-algorithm.Moreover,the fractional Marangoni boundary condition is firstly established via the balance between the surface tension and shear stress.Modified Darcy's law is employed to investigate flow and heat transfer characteristics of fractional viscoelastic fluid through a porous medium.Some novel phenomena are found that both the velocity and temperature boundary layers manifest short-term memory and basic relaxation time characteristics,shear stress shows slow response to external body force.The Marangoni surface tension and local Nusselt number show different variation tendencies dependent on temperature.The effects of fractional derivative parameters and other involved parameters on velocity,temperature and concentration fields are presented graphically and analyzed in detail to characterize the complexity of heat and mass transfer of viscoelastic fluid.The Marangoni number plays a connecting role between the velocity and temperature gradients on the boundary surface and only has slight influence on the thickness of the boundary layers.The temperature distributions decline with the increase of porosity,which demonstrate a loss of the thickness of the thermal boundary layer.Better permeability in porous medium not only promotes momentum transmission of the fluid,but also reduces the heat conduction loss in the convection flow.Brownian motion number accelerates the convection flow and improves the efficiency of mass transfer,while thermophoresis number has strong effect on nanoparticle diffusion during the convection flow.