# Conservation of momentum at interface

For applications involving thin film evaporation and condensation, the effects of surface tension and disjoining pressure will create additional forces on the interface. The momentum balance at the interface is

$\begin{array}{l} \left( {{{{\mathbf{\tau '}}}_\ell } - {{{\mathbf{\tau '}}}_v}} \right) \cdot {\mathbf{n}} + \sigma (T)\left( {\frac{1}{{{R_I}}} + \frac{1}{{{R_{II}}}}} \right){\mathbf{n}} - {p_d}{\mathbf{n}} \\ - \left( {\frac{{d\sigma }}{{dT}}} \right)(\nabla {T_\delta } \cdot {\mathbf{t}}){\mathbf{t}} = {{\dot m''}_\delta }({{\mathbf{V}}_\ell } - {{\mathbf{V}}_v}) \\ \end{array} \qquad \qquad(1)$

On the left-hand side, the first term is the stress tensor, the second term is the capillary pressure, the third term is the disjoining pressure, and the fourth term is the Marangoni stress (Faghri and Zhang 2006). The right-hand side is the momentum transfer due to inertia. In this equation, the tangential direction, t, can either be t1 or t2. The stress tensor is:

$\tau ' = - p{\mathbf{I}} + 2\mu {\mathbf{D}} - \frac{2}{3}\mu \left( {\nabla \cdot {\mathbf{V}}} \right){\mathbf{I}} \qquad \qquad(2)$

The deformation tensor can be written for a reference frame that is adjusted to the interface:

${\mathbf{D}} = \frac{1}{2}\left[ {\nabla {\mathbf{V}} + {{\left( {\nabla {\mathbf{V}}} \right)}^T}} \right] = \left[ {\begin{array}{*{20}{c}} {\frac{{\partial {V_{\mathbf{n}}}}}{{\partial {x_{\mathbf{n}}}}}} & {\frac{1}{2}\left( {\frac{{\partial {V_{\mathbf{n}}}}}{{\partial {x_{{{\mathbf{t}}_1}}}}} + \frac{{\partial {V_{{{\mathbf{t}}_1}}}}}{{\partial {x_{\mathbf{n}}}}}} \right)} & {\frac{1}{2}\left( {\frac{{\partial {V_{\mathbf{n}}}}}{{\partial {x_{{{\mathbf{t}}_2}}}}} + \frac{{\partial {V_{{{\mathbf{t}}_2}}}}}{{\partial {x_{\mathbf{n}}}}}} \right)} \\ {\frac{1}{2}\left( {\frac{{\partial {V_{\mathbf{n}}}}}{{\partial {x_{{{\mathbf{t}}_1}}}}} + \frac{{\partial {V_{{{\mathbf{t}}_1}}}}}{{\partial {x_{\mathbf{n}}}}}} \right)} & {\frac{{\partial {V_{{{\mathbf{t}}_1}}}}}{{\partial {x_{{{\mathbf{t}}_1}}}}}} & {\frac{1}{2}\left( {\frac{{\partial {V_{{{\mathbf{t}}_1}}}}}{{\partial {x_{{{\mathbf{t}}_2}}}}} + \frac{{\partial {V_{{{\mathbf{t}}_2}}}}}{{\partial {x_{{{\mathbf{t}}_1}}}}}} \right)} \\ {\frac{1}{2}\left( {\frac{{\partial {V_{\mathbf{n}}}}}{{\partial {x_{{{\mathbf{t}}_2}}}}} + \frac{{\partial {V_{{{\mathbf{t}}_2}}}}}{{\partial {x_{\mathbf{n}}}}}} \right)} & {\frac{1}{2}\left( {\frac{{\partial {V_{{{\mathbf{t}}_1}}}}}{{\partial {x_{{{\mathbf{t}}_2}}}}} + \frac{{\partial {V_{{{\mathbf{t}}_2}}}}}{{\partial {x_{{{\mathbf{t}}_1}}}}}} \right)} & {\frac{{\partial {V_{{{\mathbf{t}}_2}}}}}{{\partial {x_{{{\mathbf{t}}_2}}}}}} \\ \end{array}} \right] \qquad \qquad(3)$

The normal direction of the interface is [1 0 0], the first tangential direction is [0 1 0] and the second tangential direction is [0 0 1]. Therefore,

$\begin{array}{l} \tau ' \cdot {\mathbf{n}} = \left[ { - \begin{array}{*{20}{c}} p & 0 & 0 \\ \end{array}} \right] + \\ 2\mu \left[ {\begin{array}{*{20}{c}} {\frac{{\partial {V_{\mathbf{n}}}}}{{\partial {x_{\mathbf{n}}}}}} & {\frac{1}{2}\left( {\frac{{\partial {V_{\mathbf{n}}}}}{{\partial {x_{{{\mathbf{t}}_1}}}}} + \frac{{\partial {V_{{{\mathbf{t}}_1}}}}}{{\partial {x_{\mathbf{n}}}}}} \right)} & {\frac{1}{2}\left( {\frac{{\partial {V_{\mathbf{n}}}}}{{\partial {x_{{{\mathbf{t}}_2}}}}} + \frac{{\partial {V_{{{\mathbf{t}}_2}}}}}{{\partial {x_{\mathbf{n}}}}}} \right)} \\ \end{array}} \right] - \frac{2}{3}\mu \left[ {\begin{array}{*{20}{c}} {\nabla \cdot {\mathbf{V}}} & 0 & 0 \\ \end{array}} \right] \\ \end{array} \qquad \qquad(4)$

This can be reduced to the three components to obtain:

$\tau ' \cdot {\mathbf{n}} \cdot {\mathbf{n}} = - p + 2\mu \frac{{\partial {V_{\mathbf{n}}}}}{{\partial {x_{\mathbf{n}}}}} - \frac{2}{3}\mu \nabla \cdot {\mathbf{V}} = - p + \frac{4}{3}\mu \frac{{\partial {V_{\mathbf{n}}}}}{{\partial {x_{\mathbf{n}}}}} - \frac{2}{3}\mu \left( {\frac{{\partial {V_{{{\mathbf{t}}_1}}}}}{{\partial {x_{{{\mathbf{t}}_1}}}}} + \frac{{\partial {V_{{{\mathbf{t}}_2}}}}}{{\partial {x_{{{\mathbf{t}}_2}}}}}} \right) \qquad \qquad(5)$
$\tau ' \cdot {\mathbf{n}} \cdot {{\mathbf{t}}_1} = \mu \left( {\frac{{\partial {V_{\mathbf{n}}}}}{{\partial {x_{{{\mathbf{t}}_1}}}}} + \frac{{\partial {V_{{{\mathbf{t}}_1}}}}}{{\partial {x_{\mathbf{n}}}}}} \right) \qquad \qquad(6)$
$\tau ' \cdot {\mathbf{n}} \cdot {{\mathbf{t}}_2} = \mu \left( {\frac{{\partial {V_{\mathbf{n}}}}}{{\partial {x_{{{\mathbf{t}}_2}}}}} + \frac{{\partial {V_{{{\mathbf{t}}_2}}}}}{{\partial {x_{\mathbf{n}}}}}} \right) \qquad \qquad(7)$

The momentum equation balance at the interface is then broken into its three components, as follows:

Normal Direction

$\begin{array}{l} - {p_\ell } + {p_v} + \frac{4}{3}\left( {{\mu _\ell }\frac{{\partial {V_{\ell ,{\mathbf{n}}}}}}{{\partial {x_{\ell ,{\mathbf{n}}}}}} - {\mu _v}\frac{{\partial {V_{v,{\mathbf{n}}}}}}{{\partial {x_{v,{\mathbf{n}}}}}}} \right) \\ - \frac{2}{3}\left[ {{\mu _\ell }\left( {\frac{{\partial {V_{\ell ,{{\mathbf{t}}_1}}}}}{{\partial {x_{{{\mathbf{t}}_1}}}}} + \frac{{\partial {V_{\ell ,{{\mathbf{t}}_2}}}}}{{\partial {x_{{{\mathbf{t}}_2}}}}}} \right) - {\mu _v}\left( {\frac{{\partial {V_{v,{{\mathbf{t}}_1}}}}}{{\partial {x_{{{\mathbf{t}}_1}}}}} + \frac{{\partial {V_{v,{{\mathbf{t}}_2}}}}}{{\partial {x_{{{\mathbf{t}}_2}}}}}} \right)} \right] \\ + \sigma \left( {\frac{1}{{{R_I}}} + \frac{1}{{{R_{II}}}}} \right) - {p_d} = {{\dot m''}_\delta }\left( {{V_{\ell ,{\mathbf{n}}}} - {V_{v,{\mathbf{n}}}}} \right) \\ \end{array} \qquad \qquad(8)$

Tangential 1

${\mu _\ell }\left( {\frac{{\partial {V_{\ell ,{\mathbf{n}}}}}}{{\partial {x_{{{\mathbf{t}}_1}}}}} + \frac{{\partial {V_{\ell ,{{\mathbf{t}}_1}}}}}{{\partial {x_{\mathbf{n}}}}}} \right) - {\mu _v}\left( {\frac{{\partial {V_{v,{\mathbf{n}}}}}}{{\partial {x_{{{\mathbf{t}}_1}}}}} + \frac{{\partial {V_{v,{{\mathbf{t}}_1}}}}}{{\partial {x_{\mathbf{n}}}}}} \right) - \left( {\frac{{d\sigma }}{{dT}}} \right)\left( {\frac{{\partial {T_\delta }}}{{\partial {x_{{{\mathbf{t}}_1}}}}}} \right) = {\dot m''_\delta }\left( {{V_{\ell ,{{\mathbf{t}}_1}}} - {V_{v,{{\mathbf{t}}_1}}}} \right) \qquad \qquad(9)$

Tangential 2

${\mu _\ell }\left( {\frac{{\partial {V_{\ell ,{\mathbf{n}}}}}}{{\partial {x_{{{\mathbf{t}}_2}}}}} + \frac{{\partial {V_{\ell ,{{\mathbf{t}}_2}}}}}{{\partial {x_{\mathbf{n}}}}}} \right) - {\mu _v}\left( {\frac{{\partial {V_{v,{\mathbf{n}}}}}}{{\partial {x_{{{\mathbf{t}}_2}}}}} + \frac{{\partial {V_{v,{{\mathbf{t}}_2}}}}}{{\partial {x_{\mathbf{n}}}}}} \right) - \left( {\frac{{d\sigma }}{{dT}}} \right)\left( {\frac{{\partial {T_\delta }}}{{\partial {x_{{{\mathbf{t}}_2}}}}}} \right) = {\dot m''_\delta }\left( {{V_{\ell ,{{\mathbf{t}}_2}}} - {V_{v,{{\mathbf{t}}_2}}}} \right) \qquad \qquad(10)$

The non-slip condition at the liquid-vapor interface requires that ${V_{\ell ,{{\mathbf{t}}_1}}} = {V_{v,{{\mathbf{t}}_1}}}$ and ${V_{\ell ,{{\mathbf{t}}_2}}} = {V_{v,{{\mathbf{t}}_2}}}.$ The momentum balance at the tangential directions becomes

${\mu _\ell }\left( {\frac{{\partial {V_{\ell ,{\mathbf{n}}}}}}{{\partial {x_{{{\mathbf{t}}_1}}}}} + \frac{{\partial {V_{\ell ,{{\mathbf{t}}_1}}}}}{{\partial {x_{\mathbf{n}}}}}} \right) = {\mu _v}\left( {\frac{{\partial {V_{v,{\mathbf{n}}}}}}{{\partial {x_{{{\mathbf{t}}_1}}}}} + \frac{{\partial {V_{v,{{\mathbf{t}}_1}}}}}{{\partial {x_{\mathbf{n}}}}}} \right) + \left( {\frac{{d\sigma }}{{dT}}} \right)\left( {\frac{{\partial {T_\delta }}}{{\partial {x_{{{\mathbf{t}}_1}}}}}} \right) \qquad \qquad(11)$
${\mu _\ell }\left( {\frac{{\partial {V_{\ell ,{\mathbf{n}}}}}}{{\partial {x_{{{\mathbf{t}}_2}}}}} + \frac{{\partial {V_{\ell ,{{\mathbf{t}}_2}}}}}{{\partial {x_{\mathbf{n}}}}}} \right) = {\mu _v}\left( {\frac{{\partial {V_{v,{\mathbf{n}}}}}}{{\partial {x_{{{\mathbf{t}}_2}}}}} + \frac{{\partial {V_{v,{{\mathbf{t}}_2}}}}}{{\partial {x_{\mathbf{n}}}}}} \right) + \left( {\frac{{d\sigma }}{{dT}}} \right)\left( {\frac{{\partial {T_\delta }}}{{\partial {x_{{{\mathbf{t}}_2}}}}}} \right) \qquad \qquad(12)$

For most applications, the evaporation or condensation rate –${\dot m''_\delta }$– is not very high; therefore, it can be assumed that ${\dot m''_\delta }({V_{\ell}} - {V_v}) = 0$. If the liquid and vapor phases are further assumed to be inviscid (${{\tau}_{\ell}} = {{\tau}_v} = 0$), the momentum equation at the interface can be reduced to

${p_v} - {p_\ell } = \sigma (T)\left( {\frac{1}{{{R_I}}} + \frac{1}{{{R_{II}}}}} \right) - {p_d} \qquad \qquad(13)$

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA
Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.