The effect of the Coriolis force on axisymmetric rotating thin film flows (original) (raw)
Related papers
Investigation of the effect of the Coriolis force on a thin fluid film on a rotating disk
International Journal of Non-Linear Mechanics, 1998
The effect of the Coriolis force on the evolution of a thin film of Newtonian fluid on a rotating disk is investigated. The thin-film approximation is made in which inertia terms in the Navier-Stokes equation are neglected. This requires that the thickness of the thin film be less than the thickness of the Ekman boundary layer in a rotating fluid of the same kinematic viscosity. A new first-order quasi-linear partial differential equation for the thickness of the thin film, which describes viscous, centrifugal and Coriolis-force effects, is derived. It extends an equation due to Emslie et al. [J. Appl. Phys. 29, 858 (1958)] which was obtained neglecting the Coriolis force. The problem is formulated as a Cauchy initial-value problem. As time increases the surface profile flattens and, if the initial profile is sufficiently negative, it develops a breaking wave. Numerical solutions of the new equation, obtained by integrating along its characteristic curves, are compared with analytical solutions of the equation of Emslie et al. to determine the effect of the Coriolis force on the surface flattening, the wave breaking and the streamlines when inertia terms are neglected.
A Numerical Study of Thin Liquid Film Flow over a Topographically Patterned Rotating Cylinder
2017
The discoveries in the field of thin liquid film flows over flat and non-flat objects have been thriving over the years. The designing and optimization of such process is quite difficult, owing to it being a complex phenomena. In the present work we are be focusing on understanding the fundamentals. We consider a thin-liquid film is flowing over a rotating cylinder with a sinusoidal pattern over the surface. To account for the intermolecular forces a body force term describing the van der Waals attraction is added to the Navier-Stokes equation. The film behaviour is studied by the application of lubrication approximation. The evolution equation for film thickness as a function of angular position $ \theta $ and time is derived. This evolution equation studies the film behaviour under the influence of gravity, surface tension, intermolecular forces, viscous forces, surface topography and rotating rate is derived. A semi-implicit technique is used to numerically solve the evolution eq...
Thin film dynamics on a vertically rotating disk partially immersed in a liquid bath
Applied Mathematical Modelling, 2008
The axisymmetric flow of a thin liquid film is considered for the problem of a vertically rotating disk that is partially immersed in a liquid bath. A model for the fully three-dimensional free-boundary problem of the rotating disk, that drags a thin film out of the bath is set up. From this, a dimension-reduced extended lubrication approximation that includes the meniscus region is derived. This problem constitutes a generalization of the classic drag-out and drag-in problem to the case of axisymmetric flow. The resulting nonlinear fourth-order partial differential equation for the film profile is solved numerically using a finite element scheme. For a range of parameters steady states are found and compared to asymptotic solutions. Patterns of the film profile, as a function of immersion depth and angular velocity are discussed.
2011
In this paper we consider the flows of a thin films for which the Reynolds number is small. This is true in many practical situations. We deduced the lubrication model for slider bearings (a slider bearing consists of a thin layer of viscous fluid confined between nearly parallel walls that are in relative tangential motion). Another application of the lubrication model is the flow of a thin film with a free boundary, with zero surface tension and respectively flow driven by surface tension gradient (Marangoni flow). 2000 Mathematics Subject Classification: 76D27, 76D05, 76D08, 76D45.
Coating flow of viscous Newtonian liquids on a rotating vertical disk
Physics of Fluids, 2009
We study a Newtonian viscous liquid coating a vertical rotating disk in the creeping flow regime. Experiments were performed at varying disk rotation speeds and liquid volumes, and the thickness profile at steady state was measured. While the maximum liquid supported by the rotating disk varied with rotation rate and liquid viscosity, the numerical value of a dimensionless number signifying the ratio of gravity to viscous forces was the same in all the cases, ␥ = 0.30. A lubrication analysis for the time evolution of the film thickness that accounted for gravity, surface tension, and viscous forces was solved numerically to steady state. The predicted thickness profiles are in quantitative agreement with those obtained experimentally. The lubrication equation at steady state was solved analytically in the absence of surface tension to obtain constant height contours that were circular and symmetric about the horizontal axis. However to obtain a complete solution, knowledge of the height variation across the contours is required, and this is controlled by the surface tension. On including this effect, we derived an asymptotic solution to predict thickness profiles that agree well with measurements for large values of viscosity or rotation rates.
Application of non-Newtonian models to thin film flow
Physical Review E, 2005
The paper describes an investigation into the use of lubrication models on thin film flow. Power law, Ellis, and Carreau models are compared for free surface flow and flow within a channel. It is shown that the Ellis law ͑or a slight modification͒ can give very similar viscosity curves to Carreau. The three models are then compared for thin film flow with a constant height free surface. For low shear rates the power law model can give very inaccurate predictions. Having shown Carreau and Ellis may produce similar results we then study flow in a channel for Ellis and power law fluids. Again the power law can give inaccurate results due to the high viscosity around the turning point for the velocity. Finally, we briefly describe the modification to include surface tension in the free surface flow model.
Two-layer Film Flow over a Rotating Disk
… in Nonlinear Science and Numerical Simulation, 2011
Unsteady two-layer liquid film flow on a horizontal rotating disk is analyzed using asymptotic method for small values of Reynolds number. This analysis of non-linear evolution equation elucidates how a two-layer film of uniform thickness thins when the disk is set in uniform rotation. It is observed that the final film thickness attains an asymptotic value at large time. It is also established that viscous force dominates over centrifugal force and upper layer thins faster than lower layer at large time.
Analytical and numerical studies of the stability of thin-film rimming flow subject to surface shear
Journal of Fluid Mechanics, 2005
Motivated by applications in rapidly rotating machinery, we have previously extended the lubrication model of the thin-film flow on the inside of a rotating circular cylinder to incorporate the effect of a constant shear applied to the free surface of the film and discovered a system rich in film profiles featuring shock structures. In this paper, we extend our model to include the effects of surface tension at leading order and take into account higher-order effects produced by gravity in order to resolve issues regarding existence, uniqueness and stability of such weak solutions to our lubrication model. We find, by analytical and numerical means, a set of feasible steady two-dimensional solutions that fit within a rational asymptotic framework. Having identified mathematically feasible solutions, we study their stability to infinitesimal two-dimensional disturbances. Based on our findings, we conjecture which of the possible weak solutions are physically meaningful.
Thin film flow over spinning discs: The effect of surface topography and flow rate modulation
2008
We examine the effect of disc topography and time modulation of the liquid flow rate at the inlet on the dynamics of a thin film flowing over a spinning disc. We use a combination of boundary-layer theory and the Kármán-Polhausen approximation to derive coupled equations for the film thickness, and radial and azimuthal flow rates. Substrate patterning is taken into account in the limit of small-amplitude topography. Our numerical results indicate that the combined effects of flow rate modulation at the inlet and disc patterning can lead to a significant increase in interfacial waviness, which greatly exceeds that associated with the constant flow rate, smooth disc case. ᭧