Consider uniform, steady, incompressible fluid flow in an unbounded domain past a fixed, closed body. In the far field the Oseen equations approximately hold. We give the Oseen velocity and pressure expansions in the far field for two– and three–dimensional flow by using two approaches: the first decomposes the velocity into a potential velocity and a wake velocity, and was introduced by Lamb and Goldstein; the second approach uses the Oseen representation of the velocity and pressure by a Green's integral distribution of singularities called ‘Oseenlets’. These are the force (drag and lift) singular solutions.
We show that, in general, the Lamb–Goldstein approach will not model the Oseen velocity in the far field wake. In contrast, by using Oseen's representation to expand each Oseenlet in a Taylor series, we obtain velocity and pressure expansions everywhere in the far field.
Nevertheless, there are exceptions for which the Lamb–Goldstein approach can be used, notably: (i) the singular drag and lift solutions; (ii) axisymmetric flow, such that the axis of rotational symmetry is parallel to the uniform stream velocity direction; (iii) low Reynolds number flow, and (iv) two–dimensional flow. For these special cases, the two approaches are shown to be equivalent.