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(316w) Coating Flows on a Rotating Vertical Disc

Authors 

Parmar, N. H. - Presenter, Indian Institute of Technology, Bombay


Introduction We report the experimental and theoretical investigation of viscous liquids coated on a rotating vertical disc (figure 1). The Reynold's number for the flow was sufficiently small to neglect the inertial effects. Here, the balance between viscous, gravitational and surface tension forces supports the liquid on the vertical surface. This work was motivated by experiments on a shear thinning fluid (commercially available shampoo) coating a rotating vertical disc where we observed some interesting flow behavior. At steady state, though the fluid coated the entire surface of the disc, most of the liquid collected into a ring like structure that was displaced horizontally with respect to the axis of rotation. Figure 2 shows the photograph for 6 ml of shampoo coated on a 9 cm diameter disc at rotation rate of 2 rpm at steady state. The ring formation is reproducible. The flow achieved steady state within 10 minutes after complete injection of the liquid. To understand the phenomenon better, we carried out detailed experiments with a high viscosity Newtonian fluids (5 and 35 $Pa\cdot s$) on the 7 cm diameter disc where the fluid completely coated the disc. Although the ring like structure observed in the case of the shear thinning fluid was absent here, the film thickness exhibited spatial variation. The film thickness profile for the Newtonian fluid was measured with an XYZ traverse having a least count of 0.01 mm. Experiments also showed that though the maximum fluid supported by the rotating disc varied with rotation rate and fluid viscosity, the numerical value of the dimensionless number signifying the ratio of gravity to viscous force was same in all the cases.

Figure 1: The schematic of the experimental setup, (a) Front view of the rotating disc (b) Side view of the set up.
\includegraphics{schematic}

Figure 2: Ring formation in the case of shear thinning liquid at rotation rate of 2 rpm on 9 cm diameter disc for 6 ml of volume.
\includegraphics{2rpm_6ml}

Model: Lubrication analysis

A lubrication analysis was performed for the flow of a thin film of Newtonian liquid on the aforementioned geometry. On neglecting the surface tension terms, the Navier-Stokes equation reduced to a single non-linear, first order equation for the time evolution of free surface $h(r, \theta ,t)$, where $r$ and $\theta$ are the radial and tangential coordinates, respectively, and $t$ is the time. At steady state, for low fluid volumes the equation predicts height contours that compare well with the experimental measurements.

On including the surface tension terms, the analysis results in a nonlinear, fourth order PDE (Oron et al., 1997), which was solved using time marching finite difference scheme. Here, as the contact line was pinned at the disc edge, no-flux boundary condition was used to ensure the volume conservation of the liquid. The solution of the equation depends on two parameters $\alpha$, the ratio of gravitational force to viscous force, and the average film thickness $h_0$. Here, $\mu$ is the viscosity, $\rho$ is the density, $R$ radius of the disc and $\Omega$ is the rotation speed.


Figure 3: Comparison of numerical results (solid line) including surface tension terms with experiments along $y=0$. Here, $h_0$ is the average film thickness.
\includegraphics{Y_0}

Results

The numerical results are in close quantitative agreement with the experiments. Figure 3 shows the comparison for 5 ml of silicone oil on 7 cm diameter disc at 1 rpm. The dimensionless thickness $(\frac{h}{h_0})$ profile is plotted along $y=0$, where $h_0$ is the average film thickness. Experiments were performed to determine the maximum volume supported by the disc for varying rotation rates and for fluids of viscosity 5 $Pa\cdot s$ and 35 $Pa\cdot s$. It was found that the value of $\alpha$ evaluated at the maximum volume was constant and approximately equal to 0.3 for all the experiments. Interestingly, the predicted value using lubrication approximation of $\alpha_{max}$

( $\alpha \approx 0.285$

) is close to the observed result. In other words, the maximum supported volume is $V_{max} \approx 1.72 \sqrt{\frac{\mu \Omega R^{5}}{\rho g}}$. This is reminiscent of a similar result obtained for the flow of viscous liquid coating the outside surface of a rotating cylinder where the maximum supported volume is again characterized by a balance between gravity and viscous forces (Moffat, 1977). Bibliography

Moffat, H. K., ``Behavior of a viscous film on the outer surface of a rotating cylinder,'' J. Mech., 16, 651 (1977).
Oron, A., S. Davis, and S. Bankoff, ``Long-scale evolution of thin liquid film,'' Review of Modern Physics, 69, 931-980 (1997).

  

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