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Help to understand a limit $\varepsilon\rightarrow 0$ computation on a fluid mechanic paper

In Córdoba and Gancedo - Contour dynamics of incompressible 3-D fluids in a porous medium with different densities (page 4) I read that if $$ v (x_1,x_2,x_3,t)=-\frac{\rho_2-\rho_1}{4\pi} \...
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