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Basic Equations

We discuss free-surface flow motion of viscous liquid falling down a solid wall, referred to as falling film flows. If we take the $x$-axis downward along the wall and the $z$-axis perpendicular to it, denoting the velocity field by $\mathbf {u}= {}^t\left[\, u(x,z,t),\, w(x,z,t) \, \right]$ and the surface location by $z=h(x,t)$, then the basic equations are written as follows:

\begin{displaymath}
\left\{%%
\begin{array}{ll}
\mbox{Incompressibility}: &
...
...thrm{sym}\,\mathrm{grad}\,\mathbf {u}
\end{array}%%
\right.
\end{displaymath} (1)


\begin{displaymath}
\left\{%%
\begin{array}{ll}
\mbox{Dynamical B.C.}: &
\le...
...ial_x {h} - \left.w\right\vert _{z=h} = 0
\end{array} \right.
\end{displaymath} (2)


\begin{displaymath}
\phantom{\{}\ \
\mbox{No-slip condition}:\ \left.\mathbf {u}\right\vert _{z=0} = \mathbf {0}
\end{displaymath} (3)

It is the convention to introduce the representative film thickness $h_0$ and thereby define two non-dimensional parameters (besides the inclination angle $\alpha$), namely

\begin{eqnarray*}
R &=& \frac{U_{\mathrm{N}}h_0}{\nu}
= \frac{(g\sin\alpha )h_...
...au_{\mathrm{N}}}
= \frac{T}{(\rho g\sin\alpha )\, h_0^2} \quad
\end{eqnarray*}



as indicators of the inertia and the surface tension, respectively. For water films at room temperature on a vertical wall we have $W R^{2/3} \approx 3000$.




Author: OOSHIDA Takeshi
ooshida@damp.tottori-u.ac.jp 2000-08-24