(316w) Coating Flows on a Rotating Vertical Disc
AIChE Annual Meeting
2006
2006 Annual Meeting
Engineering Sciences and Fundamentals
Poster Session in Fluid Mechanics
Tuesday, November 14, 2006 - 6:30pm to 9:00pm
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
![$Pa\cdot s$](https://www.aiche.org/sites/default/files/aiche-proceedings/conferences/119773/papers/59661/59661-0.jpg)
Model: Lubrication analysis
![$h(r, \theta ,t)$](https://www.aiche.org/sites/default/files/aiche-proceedings/conferences/119773/papers/59661/59661-3.jpg)
![$r$](https://www.aiche.org/sites/default/files/aiche-proceedings/conferences/119773/papers/59661/59661-4.jpg)
![$\theta$](https://www.aiche.org/sites/default/files/aiche-proceedings/conferences/119773/papers/59661/59661-5.jpg)
![$t$](https://www.aiche.org/sites/default/files/aiche-proceedings/conferences/119773/papers/59661/59661-6.jpg)
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 , the ratio of gravitational force to viscous force, and the average film thickness
. Here,
is the viscosity,
is the density,
radius of the disc and
is the rotation speed.
Results
![$(\frac{h}{h_0})$](https://www.aiche.org/sites/default/files/aiche-proceedings/conferences/119773/papers/59661/59661-16.jpg)
![$y=0$](https://www.aiche.org/sites/default/files/aiche-proceedings/conferences/119773/papers/59661/59661-13.jpg)
![$h_0$](https://www.aiche.org/sites/default/files/aiche-proceedings/conferences/119773/papers/59661/59661-18.jpg)
![$Pa\cdot s$](https://www.aiche.org/sites/default/files/aiche-proceedings/conferences/119773/papers/59661/59661-19.jpg)
![$Pa\cdot s$](https://www.aiche.org/sites/default/files/aiche-proceedings/conferences/119773/papers/59661/59661-20.jpg)
![$\alpha$](https://www.aiche.org/sites/default/files/aiche-proceedings/conferences/119773/papers/59661/59661-21.jpg)
![$\alpha_{max}$](https://www.aiche.org/sites/default/files/aiche-proceedings/conferences/119773/papers/59661/59661-22.jpg)
) is close to the observed result. In other words, the maximum supported volume is . 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