Consider an incompressible two-phase flow problem consisting of fluids 1 and 2 in a fixed two-dimensional square domain Ω = [a, b]×[c, d] = Ω1∪Ω2 where an interface Σ separates Ω1 from Ω2, see Fig. 2.2. Actually, almost all ma-terials in reality are compressible to some extent, and the incompressibility refers to flow, not the material property. This means that under certain cir-cumstances, a compressible material can nearly behave as an incompressible flow. Strictly speaking, an incompressible fluid is a fluid which is not reduced in volume by an increase in pressure. With this ideal assumption, the density of the fluid element is a constant as it moves from one point to another. This implies that the material derivative of density is zero, Dρ/Dt = 0. Then Eq.
(2.3) can be reduced in the following:
0 = ∂ρ
∂t + ∇ · (ρu) = Dρ
Dt + ρ∇ · u ⇒ ∇ · u = 0.
Take advantage of the divergence free property of the velocity, λδij∂uk
∂xk is zero, then Eq. (2.12) is simply reduced as
σij = −pδij + µ µ∂ui
∂xj +∂uj
∂xi
¶ .
Since these equation are satisfied in each bulk fluid, we can write the corre-sponding Navier-Stokes equations of the two-phase flow as
ρi µ∂ui
∂t + (ui· ∇)ui
¶
= ∇ · Ti+ ρifi in Ωi (2.13)
∇ · ui = 0, in Ωi, (2.14) where for i = 1, 2 in each fluid domain, Ti = −piI + µi(∇ui+ ∇uTi) is the stress tensor, pi is the pressure, ui is the fluid velocity, ρi is the density, µi is the viscosity, and fi is the external force such as the gravitational force.
Jump conditions cross the interface
When two fluids contact with each other, they form a thin layer (a few nanometers for most fluids) due to the influence from the bulk phases. Al-though this layer is very thin, the intermolecular forces acting on it from the bulk phases are so strong that asymmetry in these forces is very important
a b c
d
Ω
fluid1
fluid2 µ2
µ1
ρ1
ρ2
[T]=−F
Figure 2.2: The diagram of a bubble in a two-phase interfacial flow.
for the overall dynamics of the system. For simplicity, the interface is consid-ered as a geometric surface without thickness, and the boundary conditions for the bulk parameters to be formulated on this surface have to incorporate both the universal conservation laws as well as the specific physics of the processes in the interfacial layer for a particular system. Let φ(r, t) be a level function such that φ(r, t) = 0 represents the interface to separate fluid 1 (φ ≤ 0) and fluid 2 (φ ≥ 0) with the superscript + and −, respectively;
n = ∇φ/|∇φ| is a unit normal pointing from fluid 1 to fluid 2.
Suppose that δr is a distance which an element of the interface at position r traveled in the normal direction n in a very short period δt. Then both φ(r, t) and φ(r + δtn, t + δt) are zeros, and we have
0 = φ(r + δtn, t + δt) = φ(r, t) + ∂φ
∂t (r, t) δt + δrn · ∇φ(r, t) + o(δr, δt), Divide the above equality by δt and take δt → 0, we obtain an approximation of the shape of the interface
∂φ
∂t + vs· ∇φ = 0, (2.15)
where vs is the normal projection of the interface velocity.
The conservation laws can be translated into the corresponding boundary conditions in several equivalent ways. Here, an approach based on considering
fluxes across the interface will be used.
Since the interface is very thin and massless (or the density there is of the same order as in the bulk), a sink or source of mass can be neglected compared to the mass fluxes across the boundary. As a result, in the general case of a permeable interface we have the continuity of mass flux across it
ρ+¡
u+− vs¢
· n = ρ−¡
u−− vs¢
· n at φ(r, t) = 0. (2.16) Note that Eq. (2.16) holds for both permeable and impermeable interface, it is sufficient to prescribe a specific mass flux for the concerned physics,
ρ+¡
u+− vs¢
· n = χ, (2.17)
where χ has to be specified in terms of parameters determining a particular physical mechanism responsible for mass transfer across the interface, such as, chemical reactions, evaporation-condensation, mutual dissolution of fluids, etc. In particular, a case of an impermeable interface has χ = 0, and this implies that
u+· n = u−· n, ∂φ
∂t + u+· ∇φ = 0, at φ(r, t) = 0 (2.18) Alternatively, we consider another formulation for the movement of the in-terface and also consider a continuous tangential velocity, then the jump condition of the velocity is
[u]Σ = u|Σ,2− u|Σ,1 = 0. (2.19) Similarly, the conservation of momentum can be translated into the bound-ary conditions by using the momentum flux tensor Π defined by Π = ρuu−T.
The momentum fluxes in fluid 1 and 2 across a moving interface are n · Π+ and n · Π+, respectively, where
Π± = ρ±¡
u±− vs¢ ¡
u±− vs¢
− T±. (2.20)
Note that if the source or sink effect of momentum from the interface is neglected compared with nΠ±, then the momentum flux across an interface is continuous:
n · Π+= n · Π−, (2.21)
for instance, flows with very large length scales such as tidal waves, lava flows, large-scale free-surface flows industry, etc. However, in many situations the dynamics of the interfaces contributes significantly to the overall dynamics
of the system, and we have to consider this contribution here.
The source/sink of momentum due to the presence of an interface simply composes of two components, external forces Fs and surface tension σ.
For the external forces, the interface may possess properties which in a dynamic sense would ”compensate” its negligible thickness. For example, an electrically charged interface with electric current in a electrically neutral and nonconducting fluid can significantly influence the dynamics of the system by an external electromagnetic field. If the interface has no properties con-cerning external forces, then one can neglect the effect from external forces due to the negligible thickness of the interface. For instance, the effect of gravity is proportional to the mass of liquid contained in the interfacial layer and hence practically never pays any role in the interfacial dynamics.
The second way in which an interface can contribute to the overall dy-namics of the system is through its intrinsic dynamic properties the most important being the surface tension. Physically, the surface tension appears as a result of an asymmetric action on the interfacial layer of intermolecular forces from the bulk phases. These forces are singularly strong compared to those considered in fluid mechanics so that their strength compensates the negligible thickness of the interfacial layer making the resultant dynamic effect finite.
Mathematically, σ is a function defined along the interface and, in
Figure 2.3: (a) The surface tension with which an element of the interface acts on its boundary is normal to the boundary and tangential to the interface;
it tries to minimize the area of this element. (b) The resultant action of the surface tension on a surface element from a surrounding surface has a normal component if the interface in not flat, and a tangential component if the surface tension varies along the interface.
the framework of fluid-mechanical modeling, it has to be included in a two-dimensional surface stress tensor Ts. In order to imbed Ts into the
three-dimensional space, it is convenient to use a tensor (I − nn), where as before I is the metric tensor in space, and n is a unit normal to the interface. This tensor generates a metric on the surface and singles out the tangential com-ponents of vectors: if a = ann + ak, where ak is tangential to the interface, then (I − nn) · a = ak.
Taking
Ts= σ (I − nn) , (2.22)
we have n · Ts = 0, while for any two unit vectors t1 and t2 lying in the interface and normal to each other, we have t1· Ts· t2 = 0 and t1· Ts· t1 = σ.
Hence for a line lying in the interface and normal to t1 the stress tensor defined by (2.22) describes a force directed along t1 with the magnitude σ per unit length of the line. This is exactly how we defined the surface tension. In what follows, it is important to remember also that since σ is defined only on the interface, its derivative in the direction normal to it is zero by definition, that is n · ∇σ = 0. This constraint allows one to use σ formally as a function of all three space coordinates. The same applies to other surface characteristics if the interface possesses other mechanical and/or thermal properties, and in particular one has n · ∇n.
Consider the momentum flux across an interface. Given the requirement of momentum conservation and taking into account the contribution from the interface, we have
n ·¡
Π+− Π−¢
= ∇ · Ts+ Fs. (2.23)
Here Fsis the density of external forces per unit area acting on the interface;
the contribution from the surface stress in the form of ∇ · Ts is analogous to the corresponding contribution of bulk stresses.
Substituting expressions (2.20) and (2.22) for Π± and Ts and take the advantage of the mass flux continuity condition (2.16), we have
ρ+u+¡
u+− vs¢
·n−ρ−u−¡
u−− vs¢
·n−n·¡
T+− T−¢
= ∇σ−σn∇·n+Fs. (2.24) The first two terms on the left-hand side written down as
ρ+u+¡
u+− vs¢
· n − ρ−u−¡
u−− vs¢
· n = χ¡
u+− u−¢
, (2.25) show that mass transfer across an interface has an impact on the momentum balance when (a) it is significant in itself, and (b) it is associated with a considerable hump in the bulk velocity. This is the case, for example, in shock waves. On the contrary, for liquid-fluid interfaces the effect of mass
transfer on the momentum balance is practically always negligible compared to capillary effects and especially to the bulk stress. The first to terms on the right-hand side give the tangential and normal components of the force acting on a surface element due to surface tension and its gradient, see Fig.
2.3 (b). In writing them down we used that (a) (I − nn) · ∇σ = ∇σ since the surface-tension gradient is directed along the interface and (b) n · ∇n = 0.
In the simplest case of an impermeable interface (χ = 0) and negligible external surface forces (Fs= 0), the normal projection of Eq. (2.23),
−¡
p+− p−¢
+ n ·¡
σ+− σ−¢
· n = σ∇ · n, (2.26) is known as the capillarity equation, and the normal stress jump is balanced by the interfacial force F (defined only on Σ) as
[Tn]Σ+ F = 0, (2.27)
where n is the unit normal vector on Σ directed towards fluid 2.
Since it is not easy to solve the Navier-Stokes equations (2.13)-(2.14) in Ω with jump conditions (2.19) and (2.27) on Σ, especially when the interface is moving. In order to formulate the problem using the immersed boundary approach, we simply treat the interface as an immersed boundary that exerts force F to the fluids and moves with local fluid velocity. We will discuss the detail later.