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Tank and pipes

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.

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Pressure drop is the driving force for flow. Knowing that, you can see that the flow from the left vertical pipe has a longer distance to travel than flow from the right vertical pipe. You also know that the vertical pipes meets at a junction, and the pressure there is only one (unknown) value, which gives you a differential pressure equation for the individual flows to that junction. Finally, from the junction to the outlet on the right hand side of the picture, there has to be a differential pressure to support the flow in that common line. From this information, there should be enough equations to solve for the unknowns.

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  • $\begingroup$ Thanks for your answer. I'm a bit puzzled though, because I thought that in each of the horizontal pipes (left and right of the junction), the pressure (and velocity) would remain constant (from continuity), and that there would be a pressure change only in junctions. So I'm not quite sure on which variable the pressure should depend on. $\endgroup$ – occd2000 Aug 7 '19 at 20:36
  • $\begingroup$ The pressure in all of the piping is continuously changing due to friction of fluid against the walls of the pipe, and due to viscous dissipation withing the flowing fluid. $\endgroup$ – David White Aug 7 '19 at 22:50

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