Artificial boundary method for the fractional second-grade fluid flow on a semi-infinite plate with the effects of magnetic field and a power-law viscosity

As a typical representative of viscoelastic fluids, second-grade fluids have many applications, such as paints, food products, and cosmetics. In this paper, the equation for describing the fractional second-grade fluid with the power-law viscosity on a semi-infinite plate under the influence of a magnetic field is studied. The numerical solution is obtained using the finite difference method. To handle the semi-unbounded region, the (inverse) $z$-transform is applied to establish the absorbing boundary condition (ABC) for the solution at the cut-off point. In addition, the numerical example analyzes the superiority of the ABC over the directly truncated boundary condition and the effects of different parameters on the velocity distribution. The conclusion is that the slip parameter, power-law exponent parameter, and power-law index parameter promote the fluid flow, while the magnetic field and fractional parameter hinder the fluid flow.