Dynamics and stability of a thin liquid film on a heated rotating disk film with variable viscosity
TL;DR: In this article, a nonlinear evolution equation describing the shape of the film interface has been derived as a function of space and time and its stability characteristics have been examined using linear theory.
Abstract: A theoretical analysis of the thermal effects on the dynamics of a thin nonuniform film of a nonvolatile incompressible viscous fluid on a heated rotating disk has been considered and the effects of temperature-dependent viscosity and surface tension have been analyzed. A nonlinear evolution equation describing the shape of the film interface has been derived as a function of space and time and its stability characteristics have been examined using linear theory. It has been observed that the infinitesimal disturbances decay for small wave numbers and are transiently stable for large wave numbers, for both zero and nonzero values of Biot number.
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TL;DR: The dynamics and stability of thin liquid films have fascinated scientists over many decades: the observations of regular wave patterns in film flows along a windowpane or along guttering, the patterning of dewetting droplets, and the fingering of viscous flows down a slope are all examples that are familiar in daily life.
Abstract: The dynamics and stability of thin liquid films have fascinated scientists over many decades: the observations of regular wave patterns in film flows down a windowpane or along guttering, the patterning of dewetting droplets, and the fingering of viscous flows down a slope are all examples that are familiar in daily life. Thin film flows occur over a wide range of length scales and are central to numerous areas of engineering, geophysics, and biophysics; these include nanofluidics and microfluidics, coating flows, intensive processing, lava flows, dynamics of continental ice sheets, tear-film rupture, and surfactant replacement therapy. These flows have attracted considerable attention in the literature, which have resulted in many significant developments in experimental, analytical, and numerical research in this area. These include advances in understanding dewetting, thermocapillary- and surfactant-driven films, falling films and films flowing over structured, compliant, and rapidly rotating substrates, and evaporating films as well as those manipulated via use of electric fields to produce nanoscale patterns. These developments are reviewed in this paper and open problems and exciting research avenues in this thriving area of fluid mechanics are also highlighted.
1,226 citations
TL;DR: In this paper, a comprehensive description of the two-dimensional steady gravity-driven flow with prescribed volume flux of a thin film of Newtonian fluid with temperature-dependent viscosity on a stationary horizontal cylinder is obtained.
Abstract: A comprehensive description is obtained of the two-dimensional steady gravity-driven flow with prescribed volume flux of a thin film of Newtonian fluid with temperature-dependent viscosity on a stationary horizontal cylinder. When the cylinder is uniformly hotter than the surrounding atmosphere (positive thermoviscosity), the effect of increasing the heat transfer to the surrounding atmosphere at the free surface is to increase the average viscosity and hence reduce the average velocity within the film, with the net effect that the film thickness (and hence the total fluid load on the cylinder) is increased to maintain the fixed volume flux of fluid. When the cylinder is uniformly colder than the surrounding atmosphere (negative thermoviscosity), the opposite occurs. Increasing the heat transfer at the free surface from weak to strong changes the film thickness everywhere (and hence the load, but not the temperature or the velocity) by a constant factor which depends only on the specific viscosity model considered. The effect of increasing the thermoviscosity is always to increase the film thickness and hence the load. In the limit of strong positive thermoviscosity, the velocity is small and uniform outside a narrow boundary layer near the cylinder leading to a large film thickness, while in the limit of strong negative thermoviscosity, the velocity increases from zero at the cylinder to a large value at the free surface leading to a small film thickness.
19 citations
TL;DR: In this article, a comprehensive description of steady thermoviscous coating and rimming flow on a uniformly rotating horizontal cylinder that is uniformly hotter or colder than the surrounding atmosphere is given.
Abstract: A comprehensive description is obtained of steady thermoviscous (i.e. with temperature-dependent viscosity) coating and rimming flow on a uniformly rotating horizontal cylinder that is uniformly hotter or colder than the surrounding atmosphere. It is found that, as in the corresponding isothermal problem, there is a critical solution with a corresponding critical load (which depends, in general, on both the Biot number and the thermoviscosity number) above which no ``full-film'' solutions corresponding to a continuous film of fluid covering the entire outside or inside of the cylinder exist. The effect of thermoviscosity on both the critical solution and the full-film solution with a prescribed load is described. In particular, there are no full-film solutions with a prescribed load M for any value of the Biot number when M is greater than or equal to M_{c0} divided by the square root of f for positive thermoviscosity number and when M is greater than M_{c0} for negative thermoviscosity number, where f is a monotonically decreasing function of the thermoviscosity number and M_{c0} = 4.44272 is the critical load in the constant-viscosity case. It is also found that when the prescribed load M is less than 1.50315 there is a narrow region of the Biot number - thermoviscosity number parameter plane in which backflow occurs.
16 citations
TL;DR: In this paper, the authors apply the lubrication approximation to the field equations for a thin viscous drop to yield an evolution equation that captures the dependence of viscosity, surface tension, gravity, centrifugal forces and thermocapillarity.
Abstract: An axisymmetric drop spreads on a radially heated, partially wetting solid substrate in a rotating geometry. The lubrication approximation is applied to the field equations for this thin viscous drop to yield an evolution equation that captures the dependence of viscosity, surface tension, gravity, centrifugal forces and thermocapillarity. We study the quasi-static spreading regime, whereby droplet motion is controlled by a constitutive law that relates the contact angle to the contact-line speed. Non-uniform heating of the substrate can generate both vertical and radial temperature gradients along the drop interface, which produce distinct thermocapillary forces and equivalently flows that affect the spreading process. For the non-rotating system, competition between surface chemistry (wetting) and thermocapillary flows induced by the thermal gradients gives rise to bistability in certain regions of parameter space in which the droplets converge to an equilibrium shape. The centrifugal forces that develop in a rotating geometry enlarge the bistability regions. Parameter regimes in which the droplet spreads indefinitely are identified and spreading laws are computed to compare with experimental results from the literature.
9 citations
TL;DR: At the onset of the fingering instability of small drops placed at the center of the rotating substrate in the absence of a temperature gradient, the height profile for small drops is more complex than previously assumed.
Abstract: This paper presents an experimental study on thin liquid drops and films under the combined action of centrifugal forces due to rotation and radial Marangoni forces due to a corresponding temperature gradient. We shall examine thinning of a given liquid layer both with and without rotation and also consider the onset of the fingering instability in a completely wetting liquid drop. In many of the experiments described here, we use an interferometric technique which provides key information on height profiles. For thick rotating films in the absence of a temperature gradient, when an initially thick layer of fluid is spun to angular velocities where the classical Newtonian solution is negative, the fluid never dewets for the case of a completely wetting fluid, but leaves a microscopic uniform wet layer in the center. Similar experiments with a radially inward temperature gradient reveal the evolution of a radial height profile given by h(r) = A(t)rα, where A(t) decays logarithmically with time, and . In the case where there is no rotation, small centrally placed drops show novel retraction behavior under a sufficiently strong temperature gradient. Using the same interferometric arrangement, we observed the onset of the fingering instability of small drops placed at the center of the rotating substrate in the absence of a temperature gradient. At the onset of the instability, the height profile for small drops is more complex than previously assumed.
8 citations
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2,029 citations
TL;DR: In this paper, it was shown that initially irregular fluid distributions tend toward uniformity under centrifugation, and means of computing times required to produce uniform layers of given thickness at given angular velocity and fluid viscosity are demonstrated.
Abstract: Equations describing the flow of a Newtonian liquid on a rotating disk have been solved so that characteristic curves and surface contours at successive times for any assumed initial fluid distribution may be constructed. It is shown that centrifugation of a fluid layer that is initially uniform does not disturb the uniformity as the height of the layer is reduced. It is also shown that initially irregular fluid distributions tend toward uniformity under centrifugation, and means of computing times required to produce uniform layers of given thickness at given angular velocity and fluid viscosity are demonstrated. Contour surfaces for a number of exemplary initial distributions (Gaussian, slowly falling, Gaussian plus uniform, sinusoidal) have been constructed. Edge effects on rotating planes with rising rims, and fluid flow on rotating nonplanar surfaces, are considered.
696 citations
TL;DR: This paper is a review of work on thin fluid films where surface tension is a driving mechanism and discusses asymptotic results, travelling waves, stability, and similarity solutions, as well as analytical work on the resultant equations.
Abstract: This paper is a review of work on thin fluid films where surface tension is a driving mechanism. Its aim is to highlight the substantial amount of literature dealing with relevant physical models and also analytic work on the resultant equations.
In general the introduction of surface tension into standard lubrication theory leads to a fourth-order nonlinear parabolic equation
$$
\pad{h}{t}+\pad{}{x}\left(C \frac{h^3}{3}\frac{\partial^3 h}{\partial x^3} +f(h,h_x,h_{xx})\right) = 0 ,\label{abeq1}
$$
where $h=h(x,t)$ is the fluid film height. For steady situations this equation may be integrated once and a third-order ordinary differential equation is obtained. Appropriate forms of this equation have been used to model fluid flows in physical situations such as coating, draining of foams, and the movement of contact lenses.
In the introduction a form of the above equation is derived for flow driven by surface tension, surface tension gradients, gravity, and long range molecular forces. Modifications to the equation due to slip, the effect of two free surfaces, two phase fluids, and higher dimensional forms are also discussed. The second section of this paper describes physical situations where surface tension driven lubrication models apply and the governing equations are given. The third section reviews analytical work on the model equations as well as the "generalized lubrication equation"
$$
\pad{h}{t}+\pad{}{x}\left(h^n h_{xxx}\right) = 0.
$$
In particular the discussion focusses on asymptotic results, travelling waves, stability, and similarity solutions. Numerical work is also discussed, while for analytical results the reader is directed to existing literature.
433 citations
TL;DR: In this article, the authors give a review of three prototypal problems having sinusoidal time variation: parallel shear flows, convective instabilities, and centrifugal instabilities.
Abstract: The stability of periodic states of mechanical systems has long been an object of study. Dynamic stabilization and destabilization can lead to dramatic modifications of behavior depending on the proper tuning of the amplitude and frequency of the modulation. It has only been in the recent past that attention has been focused on such possibilities in hydrodynamics. The interest lies not only with the mechanics of this new class of problems but with the possibilities for applications. If an imposed modulation can destabilize an otherwise stable state, then there can be a major enhancement of heat/mass/momentum transport. If an imposed modulation can stabilize an otherwise unstable state, then higher efficiencies can be attained in various processing techniques. The aim here is to give reviews of three prototypal problems having sinusoidal time variation: parallel shear flows, convective instabilities, and centrifugal insta bilities.1 These will be used as vehicles for a discussion of scale analysis, a procedure which is crucial to the understanding of these as well as more general flows. Before proceeding with the examination of periodic basic states, a word must be said in reference to the definition of stability. Since the basic state is unsteady, it seems natural to compare the disturbance growth rate with the rate of change of the basic state (Shen 1961). However, in periodic states the repeating sequence of basic-state acceleration followed by basic-state deceleration leads to ambiguities in interpretation. As a result, there is fairly common agreement to follow Rosenblat (1968) and term a basic periodic state unstable if there exists a disturbance that experiences net growth over each modulation cycle. A state on which every disturbance decays at every instant is called stable. namely, monotonically stable. It may happen that a state is neither unstable nor stable, i.e. the basic state is subject
285 citations
TL;DR: In this article, the authors derived an evolution equation for two-dimensional disturbances of a uniform viscous liquid in a uniformly heated inclined plate and derived a linear theory to describe the competition among the instabilities, and derived the finite-amplitude behaviour that determines the propensity for dryout.
Abstract: A layer of volatile viscous liquid drains down a uniformly heated inclined plate. Long-wave instabilities of the uniform film are studied by deriving an evolution equation for two-dimensional disturbances. This equation incorporates viscosity, gravity, surface tension, thermocapillarity, and evaporation eifects. The linear theory derived from this describes the competition among the instabilities. Numerical solution of the evolution equation describes the finite-amplitude behaviour that determines the propensity for dryout of the film. Among the phenomena that appear are the tendency to wave breaking, the creation of secondary structures, and the preemption of dryout by mean flow.
258 citations