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Pattern formation and mixing in three-dimensional film flow
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Image of FIG. 1.
FIG. 1.

Viscous film flowing down a corrugated incline with mean inclination angle α. In the (x, y, z) coordinate system the topography is given by y = b(x, z), the free surface position by y = h(x, z), and the film thickness by y = f(x, z), respectively. The main flow is in the x direction.

Image of FIG. 2.
FIG. 2.

Schematic overview of the experimental setup. The highspeed camera is responsible for recording the pathlines and velocities of the carbon powder tracer particles at the free surface. The second camera, mounted to the movable x-y-z-traverse, records the tip of the needle. When the needle contacts the liquid this camera detects the capillary elevation, see subfigure at the left top, which yields the free surface position.

Image of FIG. 3.
FIG. 3.

(a) Analytical and experimental free surface shape for dimensional Nusselt film thickness . The lower surface indicates the topography and the upper one the analytical free surface shape, respectively. The points show the experimental data. (b) Comparison of numerical, analytical, and experimental data for the cross sections of the free surface at z = 0 and z = π.

Image of FIG. 4.
FIG. 4.

(a) Analytical and experimental velocity data for over one period. Crosses indicate the experiment and the gray surface indicates the analytical solution. (b) Cross sections of the analytical, numerical and experimental free surface velocity at z = 0 and z = π.

Image of FIG. 5.
FIG. 5.

Comparison of the numerical and experimental free surface data. The figures at the left-hand side show the three-dimensional free surface shape, the points indicate the experimental data and the isolines the numerical results. The figures at the right-hand side depict cross sections at z = 0 and z = π. (a) and (b), and (c), and (d) .

Image of FIG. 6.
FIG. 6.

Each figure shows the experimental (overall) and numerical (right) streamline patterns at the free surface of the film. The flow direction is from top to bottom. The global maximum of the topography is indicated by the circle at x = z = 0. (a) Re = 7.69, (b) Re = 5.24 (c) Re = 4.81, (d) Re = 2.38.

Image of FIG. 7.
FIG. 7.

Eddy size as a function of the Reynolds number for the experiment B.

Image of FIG. 8.
FIG. 8.

Streamlines of colored silicone oil (red online) injected close to the bottom and close to the symmetry plane z = π. The lower lines (red online) shows the experimental image, the numerical streamlines are the upper (blue online) lines. Re = 8.2. The filled circles indicate the global maxima while the open circles the global minima of the topography. The flow direction is from the left to the right.

Image of FIG. 9.
FIG. 9.

Numerical streamlines in the (y, z) view. The colored silicone oil is injected at x = 0.


Generic image for table
Table I.

Data of the two experimental systems. The fluid parameters are given at 24 °C.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Pattern formation and mixing in three-dimensional film flow