The cold ion limit of the local gyrokinetic model is rigorously taken to produce a nonlinear system of fluid equations that includes background flow shear. No fluid closure is required. By considering a simple slab geometry with magnetic drifts, but no magnetic shear, these fluid equations reduce to the Charney–Hasegawa–Mima model in the presence of flow shear. Analytic solutions to this model are found to study the impact of E × B flow shear on the stability of a single parallel velocity gradient driven mode. Additionally, the model is used to investigate the effect of background E × B flow shear on the basic three-mode nonlinear coupling, which reveals differences between zonal and non-zonal modes. These analytic results agree with gyrokinetic simulations and can serve to benchmark the numerical implementation of flow shear and nonlinear coupling.