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The incompressibility equation turns the advective term into a divergence:because $v_y=-u_x$. For a plume in an otherwise stationary environment, the boundary-layer pressure equals the ambient pressure and has no imposed $x$-gradient. Integrating the vertical momentum equation across $y$ therefore givesBoth boundary terms vanish as the velocity and shear decay away from the plume. Hence the integral momentum-flux balance for a two-dimensional plume is