Cahn–Hilliard–Navier–Stokes systems with moving contact lines

Abstract

We consider a well-known diffuse interface model for the study of the evolution of an incompressible binary fluid flow in a two or three-dimensional bounded domain. This model consists of a system of two evolution equations, namely, the incompressible Navier-Stokes equations for the average fluid velocity u coupled with a convective Cahn–Hilliard equation for an order parameter $\phi$. The novelty is that the system is endowed with boundary conditions which account for a moving contact line slip velocity. The existence of a suitable global energy solution is proven and the convergence of any such solution to a single equilibrium is also established.

Appendix: Derivation of the model

In this section, we briefly recall the physical derivation of the model (1.1)–(1.10) introduced in [58, 59]. To this end, we first consider the energy functional associated with the fluid motion

$$\begin{aligned} \mathcal {F}_{NS}\left( u\right) =\int _{\Omega }\nu \left| D\left( u\right) \right| ^{2}dx+\int _{\Gamma }\frac{\beta }{2}\left| u_{\tau }\right| ^{2}dS. \end{aligned}$$

(8.1)

Here dS denotes the surface measure, while the integral over $\Gamma$ accounts for fluid wall slip. Concerning the free energy functional stabilizing the interface separating the two fluids, we postulate

$$\begin{aligned} \mathcal {F}_{tot}\left( \phi \right) =\int _{\Omega }\left( \frac{\varepsilon }{2}\left| \nabla \phi \right| ^{2}+\varepsilon ^{-1}F\left( \phi \right) \right) dx+\int _{\Gamma }\left( \frac{\gamma }{2}\left| \nabla _{\tau }\phi \right| ^{2}+\frac{\zeta }{2}\left| \phi \right| ^{2}+G\left( \phi \right) \right) dS. \end{aligned}$$

(8.2)

Note that only the case $\gamma =0$ was considered in [59] (and any of its references). Here a boundary contribution has been added to $\mathcal {F} \left( \phi \right)$ [cf. (1.4)] to account for possible fluid-solid interactions taking place at the boundary. Based on the definitions of the functionals above, a hydrodynamic model for the contact line motion can be derived as follows. Let us observe now that the incompressible Navier-Stokes Eqs. (1.1)–(1.2) is nothing but the momentum equation for the (average) velocity field u. The last term $\mu \nabla \phi$ in (1.1) accounts for stress forces at the fluid-fluid interface [61], while Eq. (1.3) is a reformulation of mass conservation (see, for instance, [59]). In order to derive the boundary conditions (1.6 )–(1.8), we also need to introduce a further dissipative functional

$$\begin{aligned} \mathcal {F}_{CH}\left( \phi \right) =\int _{\Omega }\frac{\varrho _{0}}{2} \left| \nabla \mu \right| ^{2}dx+\int _{\Gamma }\frac{1}{2l_{0}} \left| \partial _{t}^{\tau }\phi \right| ^{2}dS, \end{aligned}$$

(8.3)

in order to construct the total rate of energy dissipation, namely,

$$\begin{aligned} \mathcal {F}_{\text {dis}}\left( u,\phi \right) =\mathcal {F}_{NS}\left( u\right) +\mathcal {F}_{CH}\left( \phi \right) . \end{aligned}$$

(8.4)

Observe that $\partial _{t}^{\tau }\phi$ [see (8.3)] denotes the material time derivative of $\phi$ at the solid surface $\Gamma$, i.e.,

$$\begin{aligned} \partial _{t}^{\tau }\phi =\partial _{t}\phi +u_{\tau }\cdot \nabla _{\tau }\phi . \end{aligned}$$

Note that the right-hand side of Eq. (8.4) consists of four quadratic terms corresponding to the four physically distinct sources of dissipation: the shear viscosity in the bulk, the fluid wall slipping, the bulk diffusion and the wall relaxation.

The total free energy $\mathcal {F}_{tot}\left( \phi \right)$ rate change can be written as

$$\begin{aligned} \frac{\partial }{\partial t}\mathcal {F}_{tot}\left( \phi \left( t\right) \right) =\int _{\Omega }\mu \left( t\right) \partial _{t}\phi \left( t\right) dx+\int _{\Gamma }\mathcal {L}\left( \phi \left( t\right) \right) \partial _{t}\phi \left( t\right) dS, \end{aligned}$$

(8.5)

where $\mu$ is the chemical potential (1.6) and $\mathcal {L}$ is defined in (1.9). Using the Cahn–Hilliard Eq. (1.3) and the fact that $\partial _{n}\mu =0$ on $\Gamma \times (0,\infty ),$ we obtain from (8.5)

$$\begin{aligned} \frac{\partial }{\partial t}\mathcal {F}_{tot}\left( \phi \right) =\int _{\Omega }\left( \varrho _{0}\left| \nabla \mu \right| ^{2}-\mu u\cdot \nabla \phi \right) dx+\int _{\Gamma }\mathcal {L}\left( \phi \right) \left( \partial _{t}^{\tau }\phi -u_{\tau }\cdot \nabla _{\tau }\phi \right) dS. \end{aligned}$$

(8.6)

On the other hand, the first law of thermodynamics requires

$$\begin{aligned} \frac{\partial }{\partial t}\mathcal {F}_{tot}\left( \phi \right) =-T\frac{ \partial E}{\partial t}+\frac{\partial W}{\partial t}, \end{aligned}$$

where E and W denote the entropy and work, respectively. Here T denotes temperature which is assumed to be uniform in the fluid. Note that the entropy part $-T\partial S/\partial t$ must arise from the bulk diffusion and the wall relaxation, while the work rate $\partial W/\partial t$ is due to the work done by the flow at the fluid–fluid interface. More precisely, we have

$$\begin{aligned} -T\frac{\partial E}{\partial t}=\int _{\Omega }\varrho _{0}\left| \nabla \mu \right| ^{2}dx+\int _{\Gamma }\mathcal {L}\left( \phi \right) \partial _{t}^{\tau }\phi dS \end{aligned}$$

and

$$\begin{aligned} \frac{\partial W}{\partial t}=-\int _{\Omega }u\cdot \left( \mu \nabla \phi \right) dx-\int _{\Gamma }u_{\tau }\cdot \left( \mathcal {L}\left( \phi \right) \nabla _{\tau }\phi \right) dS. \end{aligned}$$

(8.7)

We can see from (8.7) that $\mu \nabla \phi$ and $\mathcal {L}\left( \phi \right) \nabla _{\tau }\phi$ are forces (stresses) exerted by the interface on the flow. In order to derive the correct boundary conditions for two-phase flows, one may employ Onsager’s variational principle of energy dissipation for incompressible single-phase flows (cf. [56, 57] ; see also [59, (3.1)–(3.5)]). Therefore, a hydrodynamic model for the contact-line motion for two-phase flows can be derived by minimizing

$$\begin{aligned} \mathcal {F}_{\text {dis}}\left( u,\phi \right) +\partial _{t}\mathcal {F} _{tot}\left( \phi \right) \end{aligned}$$

with respect to the rates $\{u,\nabla \mu ,\partial _{t}^{\tau }\phi \}$ subject to the incompressibility condition (1.2) and (1.6)–(1.9) (see, for instance, [59, (3.28)–(3.32)]). It is worth mentioning that an important point of this derivation is that the uncompensated Young stress at the boundary $\mathcal {L}\left( \phi \right) \nabla _{\tau }\phi$ must accompany, according to (8.7), the capillary force density in the bulk (i.e., the term $\mu \nabla \phi$), both being interfacial forces. Therefore, the term $\mathcal {L}\left( \phi \right) \nabla _{\tau }\phi$ is simply an indicator of the fluid-fluid interfacial tension at the solid boundary $\Gamma$.