Aysmptotic equations for thin dipole layers

We develop asymptotic, thin-layer equations that describe the evolution of two adjacent, semi-infinite material fluid strips, each with spatially uniform but generally differing vorticity. This diplole layer moves in an otherwise irrotational, unbounded inviscid incompressible fluid. The configuration is viewed as a simple model for the wake formed by two boundary layers separating from a splitter plate. At leading order, the closed initial-value, nonlinear, long-wavelength equations describe the dipole layer center line motion together with functions representing both the local thickness-weighted mean velocity and velocity difference. For unequal far-field velocities, stability analysis of the linearized equations reveals Kelvin-Helmholz instability. Equal velocities gives a pure dipole layer where solution of the initial-value problem shows a double pole in Laplace transform space leading to linear algebraic growth. This agrees with the long wavelength limit of the full linearized, three-curve stability equations for sinuous modes.