Asymptotics of liquid velocity to the problem of fluid outflow from a rectangular duct

We have constructed and analyzed the first-order asymptotics of liquid velocity for the two-dimensional steady problem of energy-conserving fluid outflow from a rectangular duct. These asymptotics are constructed using the first-order asymptotics for the liquid depth, which was investigated in [V. V. Ostapenko, “Asymptotics of solutions to the problem of fluid outflow from a rectangular duct,” Phys. Fluids 33, 047106 (2021)] and found to be in good agreement with the Wilkinson laboratory experiment [D. L. Wilkinson, “Motion of air cavities in long horizontal ducts,” J. Fluid Mech. 118, 109 (1982)] on modeling the energy-conserving steady flow predicted by the classical piecewise constant Benjamin solution [T. B. Benjamin, “Gravity currents and related phenomena,” J. Fluid Mech. 31, 209 (1968)]. Two cases are considered: in the first case, it is assumed that in the entire asymptotics constructing region, the Green–Naghdi condition is satisfied; in the second case, this a priori assumption is not used. It is shown that the first-order asymptotics for the liquid depth and the horizontal fluid velocity are the same in both of these cases, wherein the asymptotics of the horizontal velocity is a quadratic function of the vertical coordinate z. At the same time, the first-order asymptotic of the vertical fluid velocity is a cubic function of the vertical coordinate z in the first case, and it is a linear function of the coordinate z in the second general case.