A fluid jet in a gas-filled pipe

A jet of liquid in a pipe filled with gas is investigated. A one-dimensional approximation is used. The liquid is fed into the pipe through the inlet at a given velocity. Two situations are considered: when the liquid is fed as a continuous jet and when the liquid jet at the inlet consists of individual droplets. For the jet in the form of drops, the effect of gravity is taken into account, while for the solid jet, gravity is neglected. The gas velocity near the inlet is assumed to be zero. The liquid and gas are considered in an incompressible approximation. The pressure at the outlet of the pipe is set. Laminar and turbulent flow regimes are investigated. The pipe is inclined at an arbitrary angle to the horizontal. Analytical solutions for stationary flow regimes have been found. For the case of the solid jet, the solution has been obtained in implicit form. For the case of the jet in the form of drops, the solution is implicit in the laminar flow regime and explicit in the turbulent flow regime. The conditions for the applicability of the solutions are considered.