Stokes’ second problem of viscoelastic fluids with constitutive equation of distributed-order derivative

Abstract

The steady-state periodic flow of Stokes’ second problem for viscoelastic fluids with constitutive equation in terms of the distributed-order derivative was considered. The distributed-order derivative involves an integration with respect to the order of fractional derivative, and the order is associated with a weight function p(α). With a general weight function p(α), the flow velocity was obtained. The amplitude, the penetration depth and the wavelength were given analytically. Results of Newtonian fluid and single fractional constitutive equation were derived as special cases of weight function p(α). Also we considered other three cases of weight function p(α): linear combination of Dirac delta functions, constant and parameterized exponential function.