In this study, we aimed to derive analytical solutions for a system of nonlinear time-fractional Navier–Stokes equations in Cartesian coordinates by employing the residual power series method. Moreover, we showed that the $\varphi$-Caputo fractional derivative describes these equations in time, enabling the Riemann-Liouville, Hadamard, and Katugampola fractional derivatives to be generalized into a unified form. Additionally, we provide results for certain cases that are given in the literature. Therefore, the solutions obtained for the time-fractional Navier–Stokes equations are presented graphically in the Caputo–Hadamard sense.