Force and moment in incompressible flows

General formulas for the force and moment acting on a rigid body immersed in an incompressible flow are obtained. The pressure boundary value relationship is handled with a decomposition theorem of the Hilbert space into three orthogonal subspaces which allows elimination of the pressure at the boundary in favor of the velocity field. Closed formulas for the force and moment exerted on a fixed rigid body are obtained in terms of the global solenoidal velocity field, without reference to the pressure variable. The expressions of force and moment are found to be quite similar, the only difference being in the harmonic function to be used for their evaluation. As an example, steady creeping flow and unsteady inviscid flow are considered in order to reobtain the classical expressions for the drag exerted on a sphere by a uniform far field flow.