Falling film flow

A steady film down a wall has a parabolic velocity profile set by gravity, ambient pressure gradient, and surface shear.

An inverted falling film flows on the underside of an inclined solid surface. Taking the normal coordinate from the solid into the liquid reverses the normal component of gravity relative to an upper-surface film. In the Benney equation, the hydrostatic contribution to the volume flux per unit width therefore changes from $-2\epsilon h^3{h}_x\cot \alpha /3$ to its negative. Normal gravity then amplifies disturbances of long wavelengths, while surface tension resists short wavelengths.

For downslope length scale $L$, film thickness scale $h_0$ and surface velocity scale $U_0=\rho g{h}_0^2\sin \alpha /(2\mu )$, this normalization makes the capillary part of the Benney equation flux $\epsilon S{h}^3{h}_{xxx}$. The dimensionless capillary pressure is $-3S{h}_{xx}/2$. Keeping $S=O(1)$ specifies a distinguished long-wave approximation in which surface tension remains relevant.

The Benney equation is a long-wave approximation for a gravity-driven thin film that expresses its volume flux per unit width locally in terms of height and its derivatives. For surface-velocity scaling on a plane inclined by $\alpha$, one normalization isHere $\epsilon$ is the film aspect ratio, $\mathrm{Re}$ the thickness-based Reynolds number, and $S$ the capillary parameter for a falling film. The leading velocity is parabolic. Its first inertial correction uses the leading mass conservation equation to eliminate the time derivative of the height. This local closure requires the correction to remain small; it does not describe arbitrary large-amplitude short-wave dynamics.

A uniform steady film of thickness $d$ on a plane inclined by $\theta$ above horizontal has downslope coordinate $s$ and outward normal coordinate $x$. Neglecting air shear, the pressure is $p_{\mathrm{atm}}+\rho g\cos \theta (d-x)$ and its downslope gradient vanishes. The Newtonian fluid momentum equation is $\eta v^{\prime \prime }+\rho g\sin \theta =0$. The no-slip boundary condition $v(0)=0$ and shear-free free surface $v^{\prime }(d)=0$ give the displayed parabolic profile. Integrating over thickness gives volume flux per unit width $q$. The wall shear $\rho gd\sin \theta$ balances the film's downslope weight, and the vertical-wall case is $\theta =\pi /2$.

Let the lower fluid have thickness $h$ and viscosity $\mu$, and the upper fluid thickness $\alpha h$ and viscosity $\beta \mu$, with common density $\rho$. For steady flow down a plane inclined at angle $\theta$, set $H=(1+\alpha )h$, $A=\rho g\sin \theta /\mu$. The common hydrostatic pressure is $p_{\mathrm{atm}}+\rho g\cos \theta (H-y)$. No slip at the wall, continuous velocity and shear stress at $y=h$, and zero upper surface shear give $u_1=A(Hy-y^2/2)$ and $u_2(h+z)=A{h}^2(\alpha +1/2)+(A/\beta )(\alpha hz-z^2/2)$. These follow by integrating ${\mu }_i{u}_i^{\prime \prime }=-\rho g\sin \theta$ in each layer. The stress applied to the lower fluid equals the downslope weight of the upper layer, independent of its viscosity; the upper viscosity affects its shear rate but not that transmitted stress.

A falling free surface does not instantaneously dry previously wetted rigid walls. Thin films remain above the bulk liquid and drain under the component of gravitational acceleration parallel to the walls. For a film of thickness $b$ over a wetted distance $\ell$, falling film flow gives a speed of order $g{b}^2/\nu$, hence $b\sim (\nu \ell /(gt){)}^{1/2}$ at late times for an order-one wall angle. These films can be visible in a cross-section while holding asymptotically negligible volume. In a V-shaped channel gravity current, local bulk depth falls only as $t^{-1/7}$; at a fixed nonzero station with a finite previous maximum wetted height, film volume decays as $t^{-1/2}$ while bulk-sectional volume decays as $t^{-2/7}$. Thus the film does not change the leading spreading law.

For a film of thickness $h$ flowing down a slope of angle $\alpha$, an upslope surface stress of magnitude $\tau$ givesThe surface velocity, total flux, and basal shear reverse atrespectively.