# Flow of thin films

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## Flow on inclined plane

Krotov, p. 205f

For laminar conditions, the average flow (m/s) across the layer is: $\text{v}_{ave}=\frac{h^{2}g\sin \phi }{3\mu }$

Where μ is the kinematic viscosity (m2/s)

This flow is stable as long as the viscosity is high enough, that is, $\mu >\frac{\text{v}_{ave}e}{20}$

At flow rates above this, the flow develop waves which depends on σ.

The transition to turbulent flow occurs when: $\mu <\frac{\text{v}_{ave}e}{1500}$

The liquid could be above or below the substrate!

## Illustration of instability in the flow of thin film

An illustration of the unstable fluid fronts in the flow of a thin liquid film.

The pictures on both the top and bottom illustrate the flow of a thin liquid film. All in all, thin film flows can be driven by gravity, or various other forces of mechanical, thermal, or electromagnetic origins. In many situations, the fluid fronts become unstable, leading to the formation of fingerlike or triangular sawtooth patterns, and resulting in uneven or partial surface coverage. Very often, these instabilities are undesirable in technological applications since they may lead to the formation of dry regions or other defects. In experiments, it is found that the degree of instabilities of the fluid fronts is directly dependent on the angle of inclination. The picture below shows the difference in instabilities of the fluid fronts with different inclination angles, α.

An illustration showing the difference in instabilities of the fluid fronts of the thin liquid film with different angles of inclination.

## Flow on a rotating disc

The simplest case is when the average fluid flow (m/s) is small compared to the angular velocity (s-1m): The simplest case is when the average fluid flow (m/s) is small compared to the angular velocity (s-1m):

The dependence of thickness, h, on radial position, r, is: $h=\left( \frac{3\mu \dot{V}}{2\pi \omega ^{2}r^{2}} \right)^{1/3}$

The volumetric flow is $\dot{V}=2\pi r\text{v}_{ave}h$

The requirement that the fluid velocity be small compared to the angular velocity (above) can be re-expressed as: $h\ll \sqrt{{3\mu }/{\omega }\;}$

(Krotov, pp 207-208)

## Flow in a capillary

The driving force is a combination of the Laplace pushing liquid into the capillary and that component of the hydrostatic head in the direction of flow: $F_{z}=\frac{2\sigma \cos \theta }{R}+\rho gl\cos \phi$

Krotov, pp 218-223.

The Poiseuille equation for the rate of laminar flow in a tube is: $\frac{dl}{dt}=\frac{R^{2}}{8\mu l}F_{z}$

For a vertical capillary,

$\phi =\pi ,\text{ }l\equiv h,\text{ }\theta =0$
$t=\frac{8\eta }{\rho gR^{2}}\left( h_{0}\ln \frac{h_{0}}{h_{0}-h}-h \right)$
$h_{0}=\frac{2\sigma }{\rho gR}$

For a horizontal capillary,

$\phi =\frac{\pi }{2},\text{ }l\equiv h,\text{ }\theta =0$
$l=\sqrt{\frac{\sigma Rt}{2\eta }}$

## Thin Film Flow Movies

A drop of isopropyl alcohol (IPA) is deposited on a Si substrate, and during the spreading process, the drop ejects fluid in front, which then nucleates into smaller drops still. These small drops form periodic structures that remind the authors of octopi. Newtonian interference fringes form at the edges due to the differences in film thickness.

Evaporation and drying: http://m.njit.edu/~kondic/thin_films/octopi/movie2.mpg