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• 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]

The distributed-order model is equivalent to weighting the fractional-order model on a given interval. In this paper, a double time-distributed order Maxwell model is developed and formulated to study the characteristics of a start-up pipe flow. The numerical solution of the distributed-order governing equation is obtained by the finite difference method. In order to test the convergence of numerical solutions, the truncation error of the proposed numerical method be calculated. The characteristics of the flow and the effects of fractional parameters and weight functions on the velocity distribution are discussed in detail. By comparing the velocity distribution diagrams, it can be inferred that the distributed-order model controlled by the weight function can describe the memory and elastic properties of fluid more flexibly. [ABSTRACT FROM AUTHOR]