Table 1. Boundary conditions Patch...

Table 1.

Boundary conditions

Patch	Velocity	Pressure	Phase fraction	Charge density	Electric potential
Liquid inlet	$u = u_{0}$	$\partial P / \partial n = 0$ Pa/m	$α = 1$	$\partial ρ_{E} / \partial n = 0 C {/m}^{4}$	$\partial ϕ / \partial n = 0 V$ /m
Capillary	$u = 0$	$\partial P / \partial n = 0$ Pa/m	$\partial α / \partial n = 0$	$\partial ρ_{E} / \partial n = 0 C {/m}^{4}$	$ϕ = ϕ_{0}$
Gas inlet	pressureInlet^a	$P = P_{0}$	$α = 0$	$\partial ρ_{E} / \partial n = 0 C {/m}^{4}$	$\partial ϕ / \partial n = 0 V$ /m
Electrode	$u = 0$	$\partial P / \partial n = 0$ Pa/m	$\partial α / \partial n = 0$	$\partial ρ_{E} / \partial n = 0 C {/m}^{4}$	$ϕ = 0 V$
Atmosphere	pressureOutlet^b	$P = 0 Pa$	inletOutlet^c	$\partial ρ_{E} / \partial n = 0 C {/m}^{4}$	$\partial ϕ / \partial n = 0 V$ /m
Outlet	pressureOutlet	$P = 0 Pa$	inletOutlet	$\partial ρ_{E} / \partial n = 0 C / m^{4}$	$ϕ = 0 V$
Front	wedge^d
Back	wedge

Patch	Velocity	Pressure	Phase fraction	Charge density	Electric potential
Liquid inlet	$u = u_{0}$	$\partial P / \partial n = 0$ Pa/m	$α = 1$	$\partial ρ_{E} / \partial n = 0 C {/m}^{4}$	$\partial ϕ / \partial n = 0 V$ /m
Capillary	$u = 0$	$\partial P / \partial n = 0$ Pa/m	$\partial α / \partial n = 0$	$\partial ρ_{E} / \partial n = 0 C {/m}^{4}$	$ϕ = ϕ_{0}$
Gas inlet	pressureInlet^a	$P = P_{0}$	$α = 0$	$\partial ρ_{E} / \partial n = 0 C {/m}^{4}$	$\partial ϕ / \partial n = 0 V$ /m
Electrode	$u = 0$	$\partial P / \partial n = 0$ Pa/m	$\partial α / \partial n = 0$	$\partial ρ_{E} / \partial n = 0 C {/m}^{4}$	$ϕ = 0 V$
Atmosphere	pressureOutlet^b	$P = 0 Pa$	inletOutlet^c	$\partial ρ_{E} / \partial n = 0 C {/m}^{4}$	$\partial ϕ / \partial n = 0 V$ /m
Outlet	pressureOutlet	$P = 0 Pa$	inletOutlet	$\partial ρ_{E} / \partial n = 0 C / m^{4}$	$ϕ = 0 V$
Front	wedge^d
Back	wedge

Note(s): Parameters $u_{0}$ ⁠, $P_{0}$ and $ϕ_{0}$ vary between cases. ^apressureInletVelocity: a boundary condition where the inflow velocity is obtained from the flux with a direction normal to the patch faces (Greenshields, 2015). ^bpressureInletOutletVelocity: a boundary condition that applies a zero-gradient condition for outflow and obtains velocity from the flux with a specified inlet direction (Greenshields, 2015). ^cA boundary condition that provides an outflow condition, with specified inflow for the case of return flow (Greenshields, 2015). ^dA boundary condition enforcing the cycling conditions between the two patches

Source(s): Authors’ own work

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