This study investigates viscoelastic fluid, which possesses both viscous and elastics properties, by employing a fractional derivative model to reveal the stress relaxation phenomenon with distance. The spatial-fractional derivative used in the momentum conservation equation is the Riemann–Liouville type derivative. Further, we use non-uniform boundary conditions subject to the boundary layer equations of the fluid flowing through a semi-infinite permeable flat surface. Owing to the fractional derivative model and non-uniform boundary conditions, this problem is complex. Thus, a finite difference scheme is applied after the coupled continuity equation and momentum equation are decoupled and linearized. The accuracy, convergence, and stability of the numerical method are presented. It is shown that non-uniform mass transfer through a permeable surface considerably affects the velocity boundary layer. By illustrating the physical interactions between the velocity fields in the boundary layer and the permeation mode in the surface, this paper predicts the velocity distributions with varying permeable surface, and also provides the possibility of changing the velocity fields by altering the permeable sheet. The results and numerical technique used in this study will help in the understanding of fractional calculus investigation in engineering.

本研究通过采用分数导数模型揭示了随距离变化的应力松弛现象，研究了既具有粘性又具有弹性的粘弹性流体。动量守恒方程中使用的空间分数导数是黎曼-利维尔类型的导数。此外，我们使用非均匀边界条件，服从于流经半无限可渗透平面的流体的边界层方程式。由于分数导数模型和非均匀边界条件，这个问题很复杂。因此，在将耦合的连续性方程和动量方程解耦并线性化之后，将应用有限差分方案。给出了数值方法的准确性，收敛性和稳定性。结果表明，通过可渗透表面的不均匀传质大大影响了速度边界层。通过说明边界层速度场与表面渗透模式之间的物理相互作用，预测了渗透面变化时的速度分布，并提供了通过改变渗透层来改变速度场的可能性。本研究中使用的结果和数值技术将有助于理解工程中的分数演算研究。并且还提供了通过改变可渗透片来改变速度场的可能性。本研究中使用的结果和数值技术将有助于理解工程中的分数微积分研究。并且还提供了通过改变可渗透片来改变速度场的可能性。本研究中使用的结果和数值技术将有助于理解工程中的分数演算研究。