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Numerical Investigation on the Effects of a Precursor Wetting Film on the Displacement of Two Immiscible Phases Along a Channel

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A set of numerical experiments has been conducted to study the effect of a precursor fluid layer on the motion of two phase system in a channel. This system is characterized by coupled Cahn-Hillard and Navier-Stokes system together with slip boundary conditions. The solution of the governing equation involves first the solution of Cahn-Hillard equation with semi-implicit and Mixed finite element discritization with a convex splitting scheme. The Navier-Stokes equations are then solved with a P2-P0 mixed finite element method. Three cases have been investigated; in the first the effect of different wettability scenarios with no precursor layer has been investigated. In the second scenario, the effect of the precursor layer for different wettability conditions is investigated. In the third case, the effect of the thickness of the precursor layer is investigated. It is found that, wettability conditions have considerable effect on the flow of the considered two-phase system. Furthermore the existence of the precursor layer has additional influence on the breakthrough of the phases.

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Latin letters :

K, r, u :

parameters (Eq. 4), [ML/T 2], [M/LT 2], [M/LT2]

L o :

reference length [L]

M h :

The space of polynomials of zero degree

p :

pressure [M/LT]

P 0,P 1,P 2 :

Polynomials of zero, first and second degree, respectively

R :

Reynolds number []

u :

velocity vector

u n ,u τ :

velocity components in the normal and tangential directions

U h :

The space of polynomials of second degree

V :

reference velocity [L/T]

W h :

The space of polynomials of first degree

Greek letters :

γ :

interfacial tension [M/T2]

ϕ :

phase field function []

μ :

chemical potential [M/LT 2]

η :

dynamic viscosity [M/LT]

ρ :

Density [M/L3]


physical domain


domain boundary

ξ :

interface thickness, [L]

𝜃 :

contact angle

β :

slip coefficient []

Subscript :


normal direction

τ :

tangential direction


slip boundary



Superscript :

n :

Current time step


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Correspondence to Amgad Salama.

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Bao, K., Salama, A. & Sun, S. Numerical Investigation on the Effects of a Precursor Wetting Film on the Displacement of Two Immiscible Phases Along a Channel. Flow Turbulence Combust 96, 757–771 (2016).

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