This work presents a comprehensive framework for the sensitivity analysis of the Navier–Stokes equations, with an emphasis on the stability estimate of the discretized first‐order sensitivity of the Navier–Stokes equations. The first‐order sensitivity of the Navier–Stokes equations is defined using the polynomial chaos method, and a finite element‐volume numerical scheme for the Navier–Stokes equations is suggested. This numerical method is integrated into the open‐source industrial code TrioCFD developed by the CEA. The finite element‐volume discretization is extended to the first‐order sensitivity Navier–Stokes equations, and the most significant and original point is the discretization of the nonlinear term. A stability estimate for continuous and discrete Navier–Stokes equations is established. Finally, numerical tests are presented to evaluate the polynomial chaos method and to compare it to the Monte Carlo and Taylor expansion methods.