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In this paper, the mixed convective heat transfer mechanism of nanofluids is investigated. Based on the Buongiorno model, we develop a novel Cattaneo–Buongiorno model that reflects the non-local properties as well as Brownian motion and thermophoresis diffusion. Due to the highly non-linear character of the equations, the finite difference method is employed to numerically solve the governing equations. The effectiveness of the numerical method and the convergence order are presented. The results show that the rise in the fractional parameter δ enhances the energy transfer process of nanofluids, while the fractional parameter γ has the opposite effect. In addition, the effects of Brownian motion and thermophoresis diffusion parameters are also discussed. We infer that the flow and heat transfer mechanism of the viscoelastic nanofluids can be more clearly revealed by controlling the parameters in the Cattaneo–Buongiorno model. [ABSTRACT FROM AUTHOR]

• Double fractional Maxwell model and generalized Fourier's law are introduced to the constitutive relationships. • Reliability of numerical solutions is proved by constructing the exact solutions with the source items. • Convergence order of numerical schemes can achieve the expected first order. • Double fractional Maxwell model presents solid-like or fluid-like behavior. This paper presents research on the fractional boundary layer flow and heat transfer over a stretching sheet with variable thickness. Based on the Caputo operators, the double fractional Maxwell model and generalized Fourier's law are introduced to the constitutive relationships. The governing equations are solved numerically by utilizing the finite difference method. The effects of fractional parameters on the velocity and temperature field are analyzed. The results indicate that the larger is the fractional stress parameter, the stronger is the elastic characteristic. However, fluids show viscous fluid-like behavior for a larger value of fractional strain parameter. Moreover, the numerical solutions are in good agreement with the exact solution and the convergence order can achieve the expected first order. The numerical method in this study is reliable and can be extended to other fractional boundary layer problems over a variable thickness sheet. [ABSTRACT FROM AUTHOR]

A double fractional Maxwell model is developed to study the boundary layer flow of viscoelastic MHD fluid over a stretching sheet. Based on the Caputo operators, fractional boundary layer equations are solved numerically by the finite difference method. The effects of stress fractional parameter α and strain fractional parameter β on the flow characteristics are analyzed. It is worth mentioning that the viscoelastic MHD fluid described by the double fractional Maxwell model presents solid-like behavior or fluid-like behavior for different α and β , which agree well with the classical work of Friedrich. The results indicate that the greater the α is, the stronger the elastic characteristic and the oscillating phenomenon is. However, as the β decreasing, the fluid represents viscous fluid-like behavior. [ABSTRACT FROM AUTHOR]

Lie-group is introduced for studying boundary layer flow and heat transfer of fractional viscoelastic MHD fluid over a stretching sheet. Fractional boundary layer equations, based on Riemann–Liouville operators, are reduced and solved numerically by Grünwald scheme approximation. Results show that the skin friction and thermal conductivity are strongly affected by magnetic field parameter, fractional derivative and wall stretching exponent. The bigger of the fractional order derivative leads to the faster velocity of viscoelastic fluids near the plate but not to hold near the outer flow. Skin friction increases with increase of magnetic field parameter M , while the heat transfer decreases. For wall stretching exponent parameter β = 1 . 0 , the velocity profile decreases with the increase of similarity variable η . However, for β = − 1 . 5 , the velocity profile increases initially and then decreases afterwards with the biggest velocity at the interior of boundary layer. [ABSTRACT FROM AUTHOR]