Obstacle problems, mathematical models of some nonlinear phenomena accompanying a free boundary, have been well studied. In this paper, the existence and uniqueness of a system between the obstacle problem and the Navier-Stokes equations is considered. The abstract theory for evolution equations governed by a subdifferential of the indicator functional on a time-dependent, closed, and convex set is applied to show the main theorem. $L^\infty $-estimate is an important lemma to prove the existence theorem.