  The incompressible Navier-Stokes equations are re-formulated to involve an
arbitrary time dilation; and in this manner, the modified Navier-Stokes
equations are obtained which have some penalization terms in the right hand
side. Then, the solid rigid bodies are modeled as the regions where time is
dilated infinitely. The physical and mathematical properties of the modified
equations and the penalization terms are investigated, and it is shown that the
modified equations satisfy the no-slip, no-diffusion, no-advection, and
no-pressure coupling conditions. The modified equations can be used in exact
imposition of the solid rigid bodies on the incompressible Navier-Stokes
equations. To show the capability of the modified equations, three classical
exact solutions of the Navier-Stokes equations, that is, the Stokes first
problem, the plane stagnation point flow, and the stokes flow over a sphere are
re-solved exactly, this time in the presence of a solid region.
