Thin-film and curtain flows on the outside of a rotating horizontal cylinder

1999 ◽  
Vol 394 ◽  
pp. 29-49 ◽  
Author(s):  
B. R. DUFFY ◽  
S. K. WILSON
Keyword(s):  
Thin Film ◽  
Finite Thickness ◽  
Surface Profile ◽  
Rotating Cylinder ◽  
Horizontal Cylinder ◽  
Lubrication Approximation ◽  
Coating Flows ◽  
Stagnation Points ◽  
Free Surface Profile ◽  
Two Dimensional Flow

We use the lubrication approximation to investigate the steady two-dimensional flow of a thin film of viscous fluid on the outside of a rigid circular cylinder that is rotating about its (horizontal) axis. Primarily we are concerned with the flow that ensues when fluid is supplied continuously as a ‘curtain’ from above the cylinder, so that it flows round the cylinder and eventually falls off near the bottom. This problem may be thought of as a ‘hybrid’ of the two classical problems studied by Nusselt (1916a, b) and Moffatt (1977), concerning, respectively, flow on a stationary cylinder with a prescribed supply flux, and flow on a rotating cylinder when the supply flux is zero. For all these problems there are indeterminacies in the steady lubrication solution; we present a variety of possible solutions, including both ‘full-film’ and ‘partial-film’ solutions, and solutions that involve smooth ‘jumps’ in the free-surface profile. We show, for example, that stagnation points can occur in the flow, that solutions exist that do not have top-to-bottom symmetry, that in curtain flows the curtain generally takes a characteristic ‘buckled’ shape, and that in full-film curtain flows there is always some fluid that is ‘trapped’ near the rotating cylinder, never escaping as part of the curtain that detaches at the bottom of the cylinder. Also we show that finite-thickness films involving jumps cannot occur in these coating flows (though they are known to occur in rimming flows).

Inertial instability of flows on the inside or outside of a rotating horizontal cylinder

2013 ◽  
Vol 736 ◽  
pp. 107-129 ◽  
Author(s):  
E. S. Benilov ◽  
V. N. Lapin
Keyword(s):  
Stokes Equations ◽  
Liquid Films ◽  
Horizontal Cylinder ◽  
Thin Liquid Films ◽  
Navier Stokes ◽  
Lubrication Approximation ◽  
Navier Stokes Equations ◽  
Coating Flows ◽  
Inertial Instability ◽  
Rimming Flows

AbstractWe consider thin liquid films on the inside (rimming flows) or outside (coating flows) of a cylinder with horizontal axis, rotating about this axis. If the liquid’s net volume is small, such films are known to evolve towards a steady state with a smooth surface, whereas, for larger amounts, the flow develops a ‘shock’ similar to a tidal bore. In this work, smooth films are shown to be unstable. Since the strongest instability occurs at wavelengths comparable to the film’s thickness, our analysis is based on the full Navier–Stokes equations, not on the lubrication approximation (which has been traditionally used in this problem). It is also shown that, for cylinders of sufficiently small radii, the instability can be suppressed by surface tension.

scholarly journals Numerical Investigation of Thin Film Spreading Driven by Surfactant Using Upwind Schemes

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
E. Momoniat ◽  
M. M. Rashidi ◽  
R. S. Herbst
Keyword(s):  
Thin Film ◽  
Free Surface ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Time Derivative ◽  
Surface Profile ◽  
Coupled System ◽  
Upwind Scheme ◽  
Upwind Schemes ◽  
Free Surface Profile

Numerical solutions of a coupled system of nonlinear partial differential equations modelling the effects of surfactant on the spreading of a thin film on a horizontal substrate are investigated. A CFL condition is obtained from a von Neumann stability analysis of a linearised system of equations. Numerical solutions obtained from a Roe upwind scheme with a third-order TVD Runge-Kutta approximation to the time derivative are compared to solutions obtained with a Roe-Sweby scheme coupled to a minmod limiter and a TVD approximation to the time derivative. Results from both of these schemes are compared to a Roe upwind scheme and a BDF approximation to the time derivative. In all three cases high-order approximations to the spatial derivatives are employed on the interior points of the spatial domain. The Roe-BDF scheme is shown to be an efficient numerical scheme for capturing sharp changes in gradient in the free surface profile and surfactant concentration. Numerical simulations of an initial exponential free surface profile coupled with initial surfactant concentrations for both exogenous and endogenous surfactants are considered.

The motion of rotating cylinders sliding on pebbled ice

2000 ◽  
Vol 77 (11) ◽  
pp. 847-862 ◽  
Author(s):  
MRA Shegelski ◽  
M Reid ◽  
R Niebergall
Keyword(s):  
Thin Film ◽  
Liquid Film ◽  
Contact Area ◽  
Relative Velocity ◽  
Thin Liquid Film ◽  
Center Of Mass ◽  
Translational Motion ◽  
Rotating Cylinder ◽  
Outer Edge ◽  
Slow Rotation

We consider the motion of a cylinder with the same mass and sizeas a curling rock, but with a very different contact geometry.Whereas the contact area of a curling rock is a thin annulus havinga radius of 6.25 cm and width of about 4 mm, the contact area of the cylinderinvestigated takes the form of several linear segments regularly spacedaround the outer edge of the cylinder, directed radially outward from the center,with length 2 cm and width 4 mm. We consider the motion of this cylinderas it rotates and slides over ice having the nature of the ice surfaceused in the sport of curling. We have previously presented a physicalmodel that accounts for the motion of curling rocks; we extend this modelto explain the motion of the cylinder under investigation. In particular,we focus on slow rotation, i.e., the rotational speed of the contact areasof the cylinder about the center of mass is small compared to thetranslational speed of the center of mass.The principal features of the model are (i) that the kineticfriction induces melting of the ice, with the consequence that thereexists a thin film of liquid water lying between the contact areasof the cylinder and the ice; (ii) that the radial segmentsdrag some of the thin liquid film around the cylinder as it rotates,with the consequence that the relative velocity between the cylinderand the thin liquid film is significantly different than the relativevelocity between the cylinder and the underlying solid ice surface.Since it is the former relative velocity that dictates the nature of themotion of the cylinder, our model predicts, and observations confirm, thatsuch a slowly rotating cylinder stops rotating well before translationalmotion ceases. This is in sharp contrast to the usual case of most slowlyrotating cylinders, where both rotational and translational motion ceaseat the same instant. We have verified this prediction of our model bycareful comparison to the actual motion of a cylinder having a contactarea as described.PACS Nos.: 46.00, 01.80+b

Intelligent Profile Measurement for Wide-Area Resist Surface Using Multi-Sensor AFM System

2008 ◽  
Vol 381-382 ◽  
pp. 407-410
Author(s):  
Shu Jie Liu ◽  
K. Watanabe ◽  
Satoru Takahashi ◽  
Kiyoshi Takamasu
Keyword(s):  
Thin Film ◽  
Semiconductor Industry ◽  
Surface Profile ◽  
Vertical Direction ◽  
Profile Measurement ◽  
Measuring Device ◽  
Scanning Method ◽  
Angle Measuring ◽  
Experimental Apparatus ◽  
Sensor Errors

In the semiconductor industry, a device that can measure the surface-profile of photoresist is needed. Since the photoresist surface is very smooth and deformable, the device is required to measure vertical direction with nanometer resolution and not to damage it at the measurement. We developed the apparatus using multi-cantilever and white light interferometer to measure the surface-profile of thin film. But, this system with scanning method suffers from the presence of moving stage and systematic sensor errors. So, in this paper, an error separation approach used coupled distance sensors, together with an autocollimator as an additional angle measuring device, was consulted the potentiality for self-calibration of multi-cantilever. Then, according to this method, we constructed the experimental apparatus and do the measurement on the resist film. The results demonstrated the feasibility that the constructed multi-ball-cantilever AFM system combined with an autocollimator could measure the thin film with high accuracy.

scholarly journals Steady rimming flows with surface tension

2008 ◽  
Vol 597 ◽  
pp. 91-118 ◽  
Author(s):  
E. S. BENILOV ◽  
M. S. BENILOV ◽  
N. KOPTEVA
Keyword(s):  
Thin Film ◽  
Surface Tension ◽  
Viscous Fluid ◽  
Small Parameter ◽  
Outer Region ◽  
Lubrication Approximation ◽  
Steady Flows ◽  
Matched Asymptotics ◽  
Weak Surface ◽  
Rimming Flows

We examine steady flows of a thin film of viscous fluid on the inside of a cylinder with horizontal axis, rotating about this axis. If the amount of fluid in the cylinder is sufficiently small, all of it is entrained by rotation and the film is distributed more or less evenly. For medium amounts, the fluid accumulates on the ‘rising’ side of the cylinder and, for large ones, pools at the cylinder's bottom. The paper examines rimming flows with a pool affected by weak surface tension. Using the lubrication approximation and the method of matched asymptotics, we find a solution describing the pool, the ‘outer’ region, and two transitional regions, one of which includes a variable (depending on the small parameter) number of asymptotic zones.

scholarly journals Pulsating fields of a thin film in a static magnetic field under vertical vibrations

2021 ◽  
Vol 2103 (1) ◽  
pp. 012233
Author(s):  
I V Volodin ◽  
A A Alabuzhev
Keyword(s):  
Magnetic Field ◽  
Thin Film ◽  
Surface Wave ◽  
Film Thickness ◽  
Static Magnetic Field ◽  
Surface Profile ◽  
Vertical Vibration ◽  
Vibrational Amplitude ◽  
Gravitational Forces ◽  
Vertical Vibrations

Abstract In the present paper a dynamics of a thin ferrofluid film under the vertical vibration in a static magnetic field is examined. The vibrational amplitude is assumed to be greater than film thickness so that vibrational force is greater than magnetic and gravitational forces. The pulsating part and the averaged part of the hydrodynamics fields are obtained. The solution of pulsating part for the traveling surface wave is found. The equation for the averaged surface profile is found.

scholarly journals The Effects of Surfactant on the Evolution of a Thin Film under a Moving Liquid Drop

2020 ◽  
Vol 5 (1) ◽  
pp. 75-85
Author(s):  
Kartika Yulianti ◽  
Agus Yodi Gunawan ◽  
Edy Soewono
Keyword(s):  
Thin Film ◽  
Film Thickness ◽  
Solid Surface ◽  
Surfactant Concentration ◽  
Liquid Drop ◽  
Nonlinear Partial Differential Equations ◽  
Normal Pressure ◽  
Lubrication Approximation ◽  
Moving Liquid

The effect of surfactant on the thickness of a thin film bounded by a solid surface and a moving liquid drop was investigated. We proposed a model so that parameters from the liquid drop can be stated in a parameter that acts as normal pressure to the thin film. Using the lubrication approximation, the model was reduced to a set of nonlinear partial differential equations in terms of the film thickness and surfactant concentration. Since we were interested in the role of the surfactant in lifting up the drop, we assumed that the density of the drop is higher than the density of the thin film. Numerically, the results show that the presence of the surfactant tends to delay the decrease of the film thickness insignificantly. However, when the surfactant was added into the system, it tends to significantly increase the film thickness for a certain range value of the normal pressure.

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