Consider the case of a solid sphere moving in a stationary liquid with a constant velocity.
The liquid is modeled as an incompressible fluid (i.e. with constant density), and being stationary means that its velocity tends towards zero as the distance from the sphere approaches infinity.
For a real body there will be a transient effect due to its acceleration as it begins its motion; however after enough time it will tend towards zero, so that the fluid velocity everywhere will approach the one obtained in the hypothetical case in which the body is already moving for infinite time.
Thus we assume a sphere of radius a moving at a constant velocity , in an incompressible fluid that is at rest at infinity. We will work in coordinates  that move along with the sphere with the coordinate center located at the sphere's center. We have:
Since these boundary conditions, as well as the equation of motions, are time invariant (i.e. they are unchanged by shifting the time ) when expressed in the  coordinates, the solution depends upon the time only through these coordinates.
The equations of motion are the Navier-Stokes equations defined in the resting frame coordinates . While spatial derivatives are equal in both coordinate systems, the time derivative that appears in the equations satisfies:
where the derivative  is with respect to the moving coordinates . We henceforth omit the m subscript.
Oseen's approximation sums up to neglecting the term non-linear in . Thus the incompressible Navier-Stokes equations become:
for a fluid having density &rho; and kinematic viscosity &nu; = &mu;/&rho; (&mu; being the dynamic viscosity). p is the pressure.
Due to the continuity equation for incompressible fluid , the solution can be expressed using a vector potential . This turns out to be directed at the  direction and its magnitude is equivalent to the stream function used in two-dimensional problems. It turns out to be:
where  is Reynolds number for the flow close to the sphere.
Note that in some notations  is replaced by  so that the derivation of  from  is more similar to its derivation from the stream function in the two-dimensional case (in polar coordinates).