We consider an Einstein-aether type Lorentz-violating theory of gravity in which the aether vector field Vμ is represented as the gradient of a scalar field ϕ, Vμ=∇μϕ. A self interacting potential for the scalar aether field is considered, as well as the possibility of a coupling between the hydrodynamic matter flux and the aether field, with the imposition of the timelike nature of the aether vector. The gravitational field equations and the equation of motion of the scalar field are derived by varying the action with respect to the metric and ϕ. In the absence of matter flux and scalar field coupling the effective energy-momentum tensor of the scalar aether is conserved. The matter flux-aether coupling generates an extra force acting on massive test particles and consequently the motion becomes non-geodesic. The Newtonian limit of the theory is investigated and the generalized Poisson equation for weak gravitational fields is obtained. The cosmological implications of the theory is also considered and it is shown that in the framework of the Scalar Einstein-aether theory both decelerating and accelerating cosmological models can be constructed.