Steady-state two-phase flow of compressible and incompressible fluids in a capillary tube of varying radius

We study immiscible two-phase flow of a compressible and an incompressible fluid inside a capillary tube of varying radius under steady-state conditions. The incompressible fluid is Newtonian and the compressible fluid is an inviscid ideal gas. The surface tension associated with the interfaces between the two fluids introduces capillary forces that vary along the tube due to the variation in the tube radius. The interplay between effects due to the capillary forces and the compressibility results in a set of properties that are different from incompressible two-phase flow. As the fluids move towards the outlet, the bubbles of the compressible fluid grow in volume due to the decrease in pressure. The volumetric growth of the compressible bubbles makes the volumetric flow rate at the outlet higher than at the inlet. The growth is not only a function of the pressure drop across the tube, but also of the ambient pressure. Furthermore, the capillary forces create an effective threshold below which there is no flow. Above the threshold, the system shows a weak non-linearity between the flow rates and the effective pressure drop, where the non-linearity also depends on the absolute pressures across the tube.