The paper discusses the existence of solutions for fluid flow equations over a spatial domain under prescribed body forces and initial velocity. For a nonlinear Reiner-Rivlin type of deviatoric stress, the paper proves the global existence of strong solutions under suitable restrictions on the body forces and initial velocity and establishes exponential decay estimates for the solutions. In particular, the rest state with zero body force and zero initial velocity is stable, and the perturbed solution decays exponentially to zero if it is true also for the perturbed body force.