Remarks to 12.6 corrections, hand-in 3, FYTA14 2018
The hint v_x ~ v, or v_y << v_x, cannot be used to simplify Leonardo's law. v_y << v_x does not necessarily mean d(v_y)/dy << d(v_x)/dx. Consider a rectangle of height g, where g <= h, starting at x_1 and ending at x_2. No net flow through the surfaces means
v_x1*d*g - v_x2*d*g - v_y*d*(x_2-x_1) = 0.
Even if v_y << v_x, we may also have g << (x_2-x_1), so the final term cannot be neglected. If we instead let the height be h(x), so that no flow is passing through the "roof", we find
v_x1*d*h_1 - v_x2*d*h_2 = 0.
where h_1 and h_2 may differ.
The approximation v_x ~ v is instead needed to simplify the comoving derivative.