A fundamental property of Oseens equation is that the general solution can be split into longitudinal and transversal' waves.
A solution  is a longitudinal wave if the velocity is irrotational and hence the viscous term drops out. The equations become
In consequence
Velocity is derived from potential theory and pressure is from linearized Bernoulli's equations.
A solution  is a transversal wave if the pressure  is identically zero and the velocity field is solenoidal. The equations are
Then the complete Oseen solution is given by
a splitting theorem due to Horace Lamb. The splitting is unique if conditions at infinity (say ) are specified.
For certain Oseen flows, further splitting of transversal wave into irrotational and rotational component is possible  Let  be the scalar function which satisfies  and vanishes at infinity and conversely let  be given such that , then the transversal wave is
where  is determined from  and  is the unit vector. Neither  or  are transversal by itself, but  is transversal. Therefore,
The only rotational component is being .