High order numerical method for the simulation of Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative on regular and irregular domains

The Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives, plays a significant role in describing the dynamic behaviour of some non-Newtonian fluids.The presence of the time-fractional derivative in the equation is to capture the viscoelastic behaviour of the flow. In the current paper, we propose a high order numerical method to solve the two dimensional case of mentioned equation on regular and irregular regions. To this end, we use an unconditionally stable scheme of order \(\mathcal {O}(\tau ^{2})\) to discretize this problem in temporal direction. Afterwards, using a high order method to discretize this problem in space directions, fully discrete scheme is achieved. Error estimate of fully discrete scheme is given. Using numerical simulation, we investigate the accuracy of the proposed scheme on regular and irregular domains, including convex and non-convex domains. Also, the numerical results are compared with the results of other methods in literature to show the accuracy and efficiency of the proposed method.