This paper examines two nonlinear time-fractional Navier-Stokes (NS) systems, in the context of the ϕ-Caputo fractional derivative that offers a generalized and versatile form of the memory and hereditary influence in fluid motion. In order to achieve the semi-analytical approximate solutions, two semi-analytical methods are designed and constructed, including the q-Homotopy ZZ Transform Method (q-HZZTM) and the ZZ Transform Variational Iteration Method (ZZ-VIM). The proposed strategies take the benefits of the ZZ transform, homotopy and variational iteration structures to deal with strong nonlinearities. Numerical tables are used to check and validate the accuracy and convergence of the obtained solutions against the exact solutions for various fractional orders. Moreover, the graphical illustrations are provided to examine the response behavior and the effects of the fractional-order parameter. It is found that both q-HZZTM and ZZ-VIM converge fast and are quite consistent with the exact solutions in the classical case and are stable in the case of fractional-order. The suggested framework is highly efficient, accurate, and flexible, and it is a potent analysis of nonlinear fractional fluid models emerging in applied science and engineering.
Al-sawalha et al. (Mon,) studied this question.