Local and instantaneous Navier-Stokes equations

Two-fluid formulation without phase-change

That section is a summary of [1] (section 3.2 page 205). We consider a control volume Fig. 39 with two fluids of index 1 and 2 seprated by one interface. The volume of each fluid is \(V_k\) with \(k=1,2\), the surrounding surface is noted \(A_k\) and the surface of separation is \(A_i\) where the index \(i\) means interface.

../../../_images/Two-Phase_Volume-Control.png

Fig. 39 Control volume of two-phase flows

Mass balance

The mass balance for each fluid writes

\[\begin{split}\begin{aligned} \frac{\partial\rho_{1}}{\partial t}+\boldsymbol{\nabla}\cdot(\rho_{1}\boldsymbol{u}_{1}) & =0\\ \frac{\partial\rho_{2}}{\partial t}+\boldsymbol{\nabla}\cdot(\rho_{2}\boldsymbol{u}_{2}) & =0 \end{aligned}\end{split}\]

In more compact form with index \(k=1,2\) for fluid 1 and fluid 2:

\[\boxed{\frac{\partial\rho_{k}}{\partial t}+\boldsymbol{\nabla}\cdot(\rho_{k}\boldsymbol{u}_{k})=0}\]

The local and instantaneous equation at interface writes:

(314)\[\rho_{1}(\boldsymbol{u}_{1}-\boldsymbol{u}_{i})\cdot\boldsymbol{n}_{1}=\rho_{2}(\boldsymbol{u}_{2}-\boldsymbol{u}_{i})\cdot\boldsymbol{n}_{2}=0\]

It expresses the mass balance through the interface. It is common to set:

\[\dot{m}_{k}:=\rho_{k}(\boldsymbol{u}_{k}-\boldsymbol{u}_{i})\cdot\boldsymbol{n}_{k}\]

where \(\dot{m}_{k}\) is the local mass flux of phase \(k\) which goes out the domain occupied by phase \(k\) at one point of the interface. With that quantity Eq. (314) writes:

\[\dot{m}_{1}+\dot{m}_{2}=0\]
\[\begin{split}\begin{aligned} (\boldsymbol{u}_{1}-\boldsymbol{u}_{i})\cdot\boldsymbol{n}_{1} & =0\\ (\boldsymbol{u}_{2}-\boldsymbol{u}_{i})\cdot\boldsymbol{n}_{2} & =0 \end{aligned}\end{split}\]
\[\boldsymbol{u}_{1}\cdot\boldsymbol{n}_{1}+\boldsymbol{u}_{2}\cdot\boldsymbol{n}_{2}=0\]

If we assume that both fluids do not slip, then the tangential velocities are identical \(\boldsymbol{u}_{1}^{t}=\boldsymbol{u}_{2}^{t}\) and

\[\boxed{\boldsymbol{u}_{1}=\boldsymbol{u}_{2}}\]

Without phase change and no-slip of both fluids, the velocity of each phase is identical.

Impulsion balance with surface tension

The impulsion balance for each bulk phase writes:

\[\boxed{\frac{\partial(\rho_{k}\boldsymbol{u}_{k})}{\partial t}+\boldsymbol{\nabla}\cdot(\rho_{k}\boldsymbol{u}_{k}\boldsymbol{u}_{k})=\boldsymbol{\nabla}\cdot\overline{\overline{\boldsymbol{T}}}_{k}+\rho_{k}\boldsymbol{g}}\]

where \(\overline{\overline{\boldsymbol{T}}}_{k}\) is the stress tensor which is defined by Eq. (267) for each phase \(k\). At interface, the impulsion balance through an element of interface writes

\[\begin{split}\begin{aligned} \frac{d}{dt}\int_{V_{1}}\rho_{1}\boldsymbol{u}_{1}dV+\frac{d}{dt}\int_{V_{2}}\rho_{2}\boldsymbol{u}_{2}dV & =-\int_{A_{1}}\rho_{1}\boldsymbol{u}_{1}(\boldsymbol{u}_{1}\cdot\boldsymbol{n}_{1})dA-\int_{A_{2}}\rho_{2}\boldsymbol{u}_{2}(\boldsymbol{u}_{2}\cdot\boldsymbol{n}_{2})dA+\int_{V_{1}}\rho_{1}\boldsymbol{F}dV+\int_{V_{2}}\rho_{2}\boldsymbol{F}dV\\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\int_{A_{1}}\boldsymbol{n}_{1}\cdot\overline{\overline{\boldsymbol{T}}}_{1}dA+\int_{A_{1}}\boldsymbol{n}_{1}\cdot\overline{\overline{\boldsymbol{T}}}_{1}dA+\oint_{C}\sigma\boldsymbol{n}_{i}dC \end{aligned}\end{split}\]

where \(\sigma\) is the surface tension and \(\boldsymbol{n}_{i}\) is the unit normal vector of \(C\) and directed toward outside the volume \(V\). The impulsion balance the local instantaneous relation at interface:

\[\dot{m}_{1}\boldsymbol{u}_{1}+\dot{m}_{2}\boldsymbol{u}_{2}-\boldsymbol{n}_{1}\cdot\overline{\overline{\boldsymbol{T}}}_{1}-\boldsymbol{n}_{2}\cdot\overline{\overline{\boldsymbol{T}}}_{2}+\underbrace{\frac{d\sigma}{ds}\boldsymbol{t}}_{\text{Marangoni term}}-\underbrace{\frac{\sigma}{R}\boldsymbol{n}_{1}}_{\text{Young-Laplace term}}=0\]

where \(s\) is the curvilinear coordinate at interface, \(\boldsymbol{t}\) is the tangential unit vector at interface and \(R\) is the curvature radius. The term \(d\sigma/ds\) is related to Marangoni effects due to variation of surface tension with temperature or concentration at interface. The term \(\sigma\boldsymbol{n}_{1}/R\) is the Young-Laplace term. In static case, we obtain:

(315)\[\boxed{P_{1}-P_{2}=\frac{\sigma}{R}}\]

One-fluid formulation for incompressible fluids

In one-fluid formulation, there are only one mean velocity \(\boldsymbol{u}\) and one mean pressure for two fluids of bulk densities \(\rho_1\) and \(\rho_2\) and dynamic viscosity \(\eta_1\) and \(\eta_2\). We must introduce a new numerical quantity to track the interface and respect the jump conditions.

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