If one of the two phases in an immiscible liquids simulation of lattice gas has an attractive interaction with the walls, (e.g. walls are blue-philic), then there will be a pressure difference between the phases that depends on the angle that the meniscus makes and the surface tension, as well as the curvature of the interface. Depending on the sign of this pressure difference, the blue phase can displace the red, or vice versa. The rate of flow is proportional to this capillary pressure.

Below are snapshots of the progression of a blue front upon the red phase. At t=0 the two phases occupy equal areas in the two dimensional world, and shortly afterwards a meniscus forms and the blue phase starts to move to the right.

The system size is 512x64, particle density is 70%, with the system confined to a periodic tube with blue-philic walls. The peridicity of the tube implies that red particles going over from the right side to the left will turn into blue particles, and vice versa. Throught the simulation there is no external driving force, but as the simulation progresses, a flow rate develops to the right.

At 30,000 steps the average X momentum is 0.025 (per site), which coincides with
(independent) measurements of the particle flux.

t=0 |
t=10,000 |

t=200 |
t=20,000 |

t=500 |
t=30,000 |

t=1,000 |
t=40,000 |

t=5,000 |