A slender body theory in Oseen flow obtained by expanding the Oseenlets in the Green's integral representation

Consider steady, uniform flow at a small angle of incidence to a slender body. Let us assume that Oseen flow, the far field approximation obtained by linearizing the velocity about the uniform flow velocity, also holds in a near-field region that continues right up to the body boundary layer where the impermeability (slip) boundary condition is applied. Let us assume that in this near field the standard slender body approximation also holds, that changes along the body length are much smaller than in the transverse plane. A Green's integral representation is then given over a near-field surface that encloses the body. The Taylor series expansion is then further applied to give a representation along a line axis within the slender body. An interesting consequence of the analysis, a jump in lift towards the rear of the slender body, is presented and discussed.