We propose a hybrid numerical framework for solving time-fractional Navier–Stokes equations with nonlinear damping. The method combines the finite difference L1 scheme for time discretization of the Caputo derivative ($0<\alpha <1$) with mixed finite element methods (P1b–P1 and $P_2$–$P_1$) for spatial discretization of velocity and pressure. This approach addresses the key challenges of fractional models, including nonlocality and memory effects, while maintaining stability in the presence of the nonlinear damping term ${\gamma \left|\mathbf{u}\right|}^{r-2}\mathbf{u}$, for $r\ge 2$. We prove unconditional stability for both semi-discrete and fully discrete schemes and derive optimal error estimates for the velocity and pressure components. Numerical experiments validate the theoretical results. Convergence tests using exact solutions, along with benchmark problems such as backward-facing channel flow and lid-driven cavity flow, confirm the accuracy and reliability of the method. The computed velocity contours and streamlines show close agreement with analytical expectations. This scheme is particularly effective for capturing anomalous diffusion in Newtonian and turbulent flows, and it offers a strong foundation for future extensions to viscoelastic and biological fluid models.

This evaluation inspects an analytical computation of magneto carboxymethylcellulose (CMC)–water-based hybrid nanofluid containing molybdenum disulfide (MoS ₂ ) and zinc oxide (ZnO) nanoparticles through an immersed porous surface. The flow is considered an exponential surface along a hybrid nanofluid. The energy and mass expression has been computed through thermal radiation, heat generation/absorption. The plate temperature obtains higher to T w {T}_{w} , and the concentration of species C w {C}_{w} surrounding the plate climbs linearly over time. Employing nondimensional factors allows a model’s leading equation to transform into dimensionless. The Laplace transformation methods are employed to address the nondimensional differential equations of hybrid nanofluid in the MATHEMATICA 12.0 software. The impact of developing factors on temperature, concentration, velocity, drag force, Nusselt, and Sherwood number is explored which are examined graphically and tabular form for the MoS ₂ + ZnO/CMC–water and MoS ₂ /CMC–water. The flow variables such as porosity flow, radiative variable, heat source variable, Prandtl number, chemical reaction, and Schmidt number. It is concluded that the greater estimation of porosity and magnetic variable declined the velocity distribution for the MoS ₂ + ZnO/CMC–water and MoS ₂ /CMC–water. Thermal radiation causes the temperature to rise distribution but declining phenomena is observed for Prandtl number. The porosity of flow and magnetic field enhances the drag force by 32.13–55.76%.