Consider a tank open to atmospheric pressure, with a large cross-section area A1 and height H, that contains an incompressible fluid. Two vertical pipes with length H and cross-section area A2 are connected to the bottom of the tank. These pipes are connected to a third, horizontal pipe with a cross-section area A3, which begins at the end of the left vertical pipe, merges with the right vertical pipe, and continues right (so in general the pressure there is unknown and not atmospheric).
Assuming steady, irrotational flow, is it possible to infer anything regarding the ratio of flow in the left and right vertical pipes?
Bernoulli's principle applies for a streamline connecting the top of the tank with each of the left/right vertical pipes, as well as with the horizontal pipe, so
$2\rho gH+\frac{1}{2}\rho(\frac{dH}{dt})^2+P_{atm}=\rho gH+\frac{1}{2}\rho v_{left}^2+P_{left}=\rho gH+\frac{1}{2}\rho v_{right}^2+P_{right}=\frac{1}{2}\rho v_{bottom}^2+P_{bottom}$
where $v_{bottom}, P_{bottom}$ refer to the velocity and pressure in a position right to the rightmost pipe.
From continuity we get
$A_1\frac{dH}{dt}=A_2(v_{left}+v_{right})=A_3v_{bottom}$
So we have 5 equations, but we have 7 properties (velocities and pressures), and since the system should in general be predictable, it seems to me that there is another constraint missing. Additional equations can be made for any point in the vertical pipes (since potential energy is exchanged for kinetic energy), as well as for the bottom pipe, left of the rightmost intersection, but that doesn't seem to provide any helpful information.
