Showing papers on "Lubrication theory published in 2014"

Thermocapillary-Driven Motion of a Sessile Drop: Effect of Non-Monotonic Dependence of Surface Tension on Temperature

Abstract: We study the thermocapillary-driven spreading of a droplet on a nonuniformly heated substrate for fluids associated with a non-monotonic dependence of the surface tension on temperature. We use lubrication theory to derive an evolution equation for the interface that accounts for capillarity and thermocapillarity. The contact line singularity is relieved by using a slip model and a Cox-Voinov relation; the latter features equilibrium contact angles that vary depending on the substrate wettability, which, in turn, is linked to the local temperature. We simulate the spreading of droplets of fluids whose surface tension-temperature curves exhibit a turning point. For cases wherein these turning points correspond to minima, and when these minima are located within the droplet, then thermocapillary stresses drive rapid spreading away from the minima. This gives rise to a significant acceleration of the spreading whose characteristics resemble those associated with the "superspreading" of droplets on hydrophobic substrates. No such behavior is observed for cases in which the turning point corresponds to a surface tension maximum.

Modelling the dynamics of a sphere approaching and bouncing on a wall in a viscous fluid

Abstract: The canonical configuration of a solid particle bouncing on a wall in a viscous fluid is considered here, focusing on rough particles as encountered in most of the laboratory experiments or applications. In that case, the particle deformation is not expected to be significant prior to solid contact. An immersed boundary method (IBM) allowing the fluid flow around the solid particle to be numerically described is combined with a discrete element method (DEM) in order to numerically investigate the dynamics of the system. Particular attention is paid to modelling the lubrication force added in the discrete element method, which is not captured by the fluid solver at very small scale. Specifically, the proposed numerical model accounts for the surface roughness of real particles through an effective roughness length in the contact model, and considers that the time scale of the contact is small compared to that of the fluid. The present coupled method is shown to quantitatively reproduce available experimental data and in particular is in very good agreement with recent measurement of the dynamics of a particle approaching very close to a wall in the viscous regime St

Electrostatic Suppression of the "Coffee Stain Effect"

Abstract: The dynamics of a slender, evaporating, particle-laden droplet under the effect of electric fields are examined. Lubrication theory is used to reduce the governing equations to a coupled system of evolution equations for the interfacial position and the local, depth-averaged particle concentration. The model incorporates the effects of capillarity, viscous stress, Marangoni stress, elecrostatically induced Maxwell stress, van der Waals forces, concentration-dependent rheology, and evaporation. Via a parametric numerical study, the one-dimensional model is shown to recover the expected inhomogeneous ring-like structures in appropriate parameter ranges due to a combination of enhanced evaporation close to the contact line, and resultant capillarity-induced flow. It is then demonstrated that this effect can be significantly suppressed via the use of carefully chosen electric fields. Finally, the three-dimensional behavior of the film and the particle concentration field is briefly examined.

Sagging of evaporating droplets of colloidal suspensions on inclined substrates.

Abstract: A droplet of a colloidal suspension placed on an inclined substrate may sag under the action of gravity. Solvent evaporation raises the concentration of the colloidal particles, and the resulting viscosity changes may influence the sag of the droplet. To investigate this phenomenon, we have developed a mathematical model for perfectly wetting droplets based on lubrication theory and the rapid-vertical-diffusion approximation. Precursor films are assumed to be present, the colloidal particles are taken to be hard spheres, and particle and liquid dynamics are coupled through a concentration-dependent viscosity and diffusivity. Evaporation is assumed to be limited by how rapidly solvent molecules can transfer from the liquid to the vapor phase. The resulting one-dimensional system of nonlinear partial differential equations describing the evolution of the droplet height and particle concentration is solved numerically for a range of initial particle concentrations and substrate temperatures. The solutions re...

Moving towards the cold region or the hot region? Thermocapillary migration of a droplet attached on a horizontal substrate

Abstract: We study computationally thermocapillary migration of a two-dimensional droplet attached on a horizontal substrate with a constant temperature gradient A level-set approach is employed to track the droplet interface and a Navier slip boundary condition is imposed to alleviate a stress singularity at the moving contact lines The present numerical model allows us to consider droplets with large contact angles and to take into account effects of the fluid outside the droplet, both have not been well studied so far In the limits of a zero contact angle hysteresis and a small viscosity ratio of the fluids outside and inside the droplet (μ out /μ in ⩽ 01), we find the droplet finally migrates towards the cold region, and both the steady migration speed and the velocity field inside the droplet obtained from numerical simulation agree very well with the lubrication theory of Ford and Nadim [“Thermocapillary migration of an attached drop on a solid surface,” Phys Fluids6, 3183–3185 (1994)] when the contact angles are small (⩽45°) Beyond this regime, increasing the contact angles leads to increased deviations between numerical simulation and the lubrication theory, and the steady migration speed of the droplet towards the cold side decreases with the contact angles The simulation results show that the droplet could fall in a motionless regime when its contact angles are around 100° even without any contact angle hysteresis It is very interesting to find that a droplet with even larger contact angles migrates towards the hot region in a steady speed We also find the transition of the migration direction of a droplet could strongly depend on the viscosity ratio With increasing the viscosity of the external fluid, the transition could happen at much smaller values of contact angles We summarize the results in a phase diagram and discuss the effects of other system parameters, including the contact angle hysteresis, the effective Marangoni number, the Prandtl number, and the slip length, on thermocapillary migration of the droplet

Forced spreading of films and droplets of colloidal suspensions

Abstract: When a thin film of a colloidal suspension flows over a substrate, uneven distribution of the suspended particles can lead to an uneven coating. Motivated by this phenomenon, we analyse the flow of perfectly wetting films and droplets of colloidal suspensions down an inclined plane. Lubrication theory and the rapid-vertical-diffusion approximation are used to derive a coupled pair of one-dimensional partial differential equations describing the evolution of the interface height and particle concentration. Precursor films are assumed to be present, the colloidal particles are taken to be hard spheres, and particle and liquid dynamics are coupled through a concentration- dependent viscosity and diffusivity. We find that for sufficiently high Peclet numbers, even small initial concentration inhomogeneities produce viscosity gradients that cause the film or droplet front to evolve continuously in time instead of travelling without changing shape as happens in the absence of colloidal particles. At high enough particle concentrations, particle diffusion can lead to the formation of long-lived secondary flow fronts in films. Our results suggest that particle concentration gradients can have a dramatic influence on interface evolution in flowing films and droplets, a finding which may be relevant for understanding the onset of patterns that are observed experimentally.

A fluid-mechanical model of elastocapillary coalescence

Abstract: We present a fluid-mechanical model of the coalescence of a number of elastic objects due to surface tension. We consider an array of spring–block elements separated by thin liquid films, whose dynamics are modelled using lubrication theory. With this simplified model of elastocapillary coalescence, we present the results of numerical simulations for a large number of elements, . A good quantitative comparison between the cluster-size statistics from noisy perturbations and this ‘frozen-in’ cluster size suggests that propagating fronts may play a crucial role in the dynamics of coalescence.

The Biomimetic Shark Skin Optimization Design Method for Improving Lubrication Effect of Engineering Surface

Abstract: Nature has long been an important source of inspiration for mankind to develop artificial ways to mimic the remarkable properties of biological systems. In this work, a new method was explored to fabricate a biomimetic engineering surface comprising both the shark-skin, the shark body denticle, and rib morphology. It can help reduce water resistance and the friction contact area as well as accommodate lubricant. The lubrication theory model was established to predict the effect of geometric parameters of a biomimetic surface on tribological performance. The model has been proved to be feasible to predict tribological performance by the experimental results. The model was then used to investigate the effect of the grid textured surface on frictional performance of different geometries. The investigation was aimed at providing a rule for deriving the design parameters of a biomimetic surface with good lubrication characteristics. Results suggest that: (i) the increase in depression width ratio [Formula: see text] decreases its corresponding coefficient of friction, and (ii) the small coefficient of friction is achievable when [Formula: see text] is beyond 0.45. Superposition of depth ratio Γ and angle's couple under the condition of [Formula: see text] < 0.45 affects the value of friction coefficient. It shows the decrease in angle decreases with the increase in dimension depth [Formula: see text].

Asymptotic behaviour of Stokes flow in a thin domain with a moving rough boundary

Abstract: We consider a problem that models fluid flow in a thin domain bounded by two surfaces. One of the surfaces is rough and moving, whereas the other is flat and stationary. The problem involves two small parameters ϵ and μ that describe film thickness and roughness wavelength, respectively. Depending on the ratio λ=ϵ/μ, three different flow regimes are obtained in the limit as both of them tend to zero. Time-dependent equations of Reynolds type are obtained in all three cases (Stokes roughness, Reynolds roughness and high-frequency roughness regime). The derivations of the limiting equations are based on formal expansions in the parameters ϵ and μ.

Analysis of a circular journal bearing lubricated with micropolar fluids including EHD effects

Abstract: Purpose – The performance of finite circular journal bearing lubricated with micropolar fluids taking into account the elastic deformation of the bearing liner is presented. The paper aims to discuss these issues. Design/methodology/approach – The modified Reynolds equation is obtained using the micropolar lubrication theory. The solution of the modified Reynolds equation is determined using finite difference technique. The static characteristics in terms of load-carrying capacity, attitude angle, side leakage and friction coefficient for micropolar and Newtonian fluids are determined for various values of eccentricity ratio and different values of elastic coefficient. Findings – Compared with Newtonian fluids, the micropolar fluids produce an increase in the load-carrying capacity and a reduction in the attitude angle, the friction factor and side leakage for both the rigid and deformable bearings. Originality/value – It is concluded that the influence of elastic deformation on the bearing characteristic...

Physics of the granite sphere fountain

Abstract: A striking example of levitation is encountered in the “kugel fountain” where a granite sphere, sometimes weighing over a ton, is kept aloft by a thin film of flowing water. In this paper, we explain the working principle behind this levitation. We show that the fountain can be viewed as a giant ball bearing and thus forms a prime example of lubrication theory. It is demonstrated how the viscosity and flow rate of the fluid determine (i) the remarkably small thickness of the film supporting the sphere and (ii) the surprisingly long time it takes for rotations to damp out. The theoretical results compare well with measurements on a fountain holding a granite sphere of one meter in diameter. We close by discussing several related cases of levitation by lubrication.

Speeding up thermocapillary migration of a confined bubble by wall slip

Abstract: It is usually believed that wall slip contributes small effects to macroscopic flow characteristics. Here we demonstrate that this is not the case for the thermocapillary migration of a long bubble in a slippery tube. We show that a fraction of the wall slip, with the slip length much smaller than the tube radius , can make the bubble migrate much faster than without wall slip. This speedup effect occurs in the strong-slip regime where the film thickness is smaller than when the Marangoni number is below the critical value , where is the driving thermal stress and is the surface tension. The resulting bubble migration speed is found to be , which can be more than a hundred times faster than the no-slip result (Wilson, J. Eng. Math., vol. 29, 1995, pp. 205–217; Mazouchi & Homsy, Phys. Fluids, vol. 12, 2000, pp. 542–549), with being the fluid viscosity. The change from the fifth power law to the cubic one also indicates a transition from the no-slip state to the strong-slip state, albeit the film thickness always scales as . The formal lubrication analysis and numerical results confirm the above findings. Our results in different slip regimes are shown to be equivalent to those for the Bretherton problem (Liao, Li & Wei, Phys. Rev. Lett., vol. 111, 2013, 136001). Extension to polygonal tubes and connection to experiments are also made. It is found that the slight discrepancy between experiment (Lajeunesse & Homsy, Phys. Fluids, vol. 15, 2003, pp. 308–314) and theory (Mazouchi & Homsy, Phys. Fluids, vol. 13, 2001, pp. 1594–1600) can be interpreted by including wall slip effects.

Electrohydrodynamic deformation of thin liquid films near surfaces with topography

Abstract: Motivated by the use of electrostatic assist to improve liquid transfer in gravure printing, we use theory and experiment to understand how electric fields deform thin liquid films near surfaces with cavity-like topographical features. Lubrication theory is used to describe the film dynamics, and both perfect and leaky dielectric materials are considered. For sinusoidal cavities, we apply asymptotic methods to obtain analytical results that relate the film deformation to the other problem parameters. For trapezoidal-like cavities, we numerically solve evolution equations to study the influence of steep topographical features and the spacing between cavities. Results from flow visualization experiments are in qualitative agreement with the theoretical predictions. In addition to being relevant to printing processes, the model problems we consider are also of fundamental interest in and represent novel contributions to the areas of electrohydrodynamics and thin-liquid-film flows.

Viscoelastic effects and anomalous transient levelling exponents in thin films

Abstract: We study theoretically the profile evolution of a thin viscoelastic film supported onto a no-slip flat substrate. Due to the nonconstant initial curvature at the free surface, there is a flow driven by Laplace pressure and mediated by viscoelasticity. In the framework of lubrication theory, we derive a thin-film equation that contains local viscoelastic stress through the Maxwell model. Then, considering a sufficiently regular small perturbation of the free surface, we linearise the equation and derive its general solution. We analyse and discuss in details the behaviour of this function. We then use it to study the viscoelastic evolution of a Gaussian initial perturbation through its transient levelling exponent. For initial widths of the profile that are smaller than a characteristic length scale involving both the film thickness and the elastocapillary length, this exponent is shown to reach anomalously high values at the elastic-to-viscous transition. This prediction should in particular be observed at sufficiently short times in experiments on thin polymer films.

Boundary lubrication with a liquid crystal monolayer.

Abstract: We study boundary lubrication characteristics of a liquid crystal (LC) monolayer sheared between two crystalline surfaces by nonequilibrium molecular dynamics simulations, using a simplified rigid bead-necklace model of the LC molecules. We consider LC monolayers confined by surfaces with three different atomic structures, subject to different shearing velocities, thus approximating a wide variety of materials and driving conditions. The time dependence of the friction force is studied and correlated with that of the orientational order exhibited by the LC molecules, arising from the competition between the effect of the structure of the confining surfaces and that of the imposed sliding direction. We show that the observed stick-slip events for low shear rates involve order-disorder transitions, and that the LC monolayer no longer has enough time to reorder at high shear rates, resulting in a smooth sliding regime. An irregular stick-slip phase between the regular stick-slip and smooth sliding is observed for intermediate shear rates regardless of the surface structure.

Influence of wear on the performance of hole-entry hybrid misaligned journal bearing in turbulent regime

Abstract: Purpose – The present work aimed to study analytically the influence of wear on the performance of a capillary-compensated hole-entry hybrid misaligned journal bearing system operating in a turbulent regime. The numerically simulated results are presented for the chosen values of restrictor design parameter, Reynolds numbers, wear depth and misalignment parameters. Design/methodology/approach – The wear caused on the bearing surface due to start/stop operations is modeled using the Dufrane’s abrasive wear model. The modified Reynolds equation based on Constantinescu’s lubrication theory is solved using finite element method together with capillary restrictor flow equation. Findings – It is found that the value of minimum fluid-film thickness increases significantly for a constant value of restrictor design parameter when unworn aligned bearing operates in turbulent regime vis-a-vis laminar regime. Further, it has also been observed that when a worn bearing operates in laminar/turbulent regimes, the reduct...

Thin-liquid-film flow on a topographically patterned rotating cylinder

Abstract: The flow of thin liquid films on rotating surfaces is directly relevant to the coating of discrete objects. To begin understanding how surface topography influences such flows, we consider a model problem in which a thin liquid film flows over a rotating cylinder patterned with a sinusoidal surface topography. Lubrication theory is applied to develop a partial differential equation that governs the film thickness as a function of time and the angular coordinate. Static situations are considered first in order to determine the parameter regime in which the lubrication approximation is expected to be valid. When gravitational forces are relatively weak, cylinder rotation leads to the formation of droplets connected by very thin films. The number of droplets is equal to the pattern frequency at low and high rotation rates, with the droplets located at the pattern troughs at low rotation rates and the pattern crests at high rotation rates. When gravitational forces become significant, the film thickness never reaches a steady state, in contrast to the case of an unpatterned cylinder. The results of this work clearly establish that the flow of thin liquid films on rotating surfaces can be very sensitive to the presence of surface topography.

Slender axisymmetric Stokesian swimmers

Abstract: Slender-body theory is used to study axisymmetric swimmers propelled by motions of their surfaces. To leading order, the locomotion speed is given by an integral involving the fluid velocity at the surface of the slender body. Locomotion speeds are calculated for fixed-shape swimmers with prescribed fluid surface velocities and for impermeable swimmers driven by propagating surface waves. Next, the internal mechanics is considered, modelling the swimmer as a viscous fluid bounded by an elastic shell. Prescribed forces are exerted on the shell to drive both the internal and external fluid flow and the surface waves. The internal fluid mechanics is determined using lubrication theory. Locomotion speeds are calculated for transverse and longitudinal waves of surface deformation, and the efficiency of the motions is determined. Transverse surface waves are both weaker and less efficient at driving locomotion than longitudinal waves. The results indicate how estimates of swimming speed based on nearly spherical swimmers with low-amplitude surface waves can be adapted for slender swimmers with nonlinear surface deformations.

A pinned or free-floating rigid plate on a thin viscous film

Abstract: A pinned or free-floating rigid plate lying on the free surface of a thin film of viscous fluid, which itself lies on top of a horizontal substrate that is moving to the right at a constant speed is considered. The focus of the present work is to describe how the competing effects of the speed of the substrate, surface tension, viscosity, and, in the case of a pinned plate, the prescribed pressure in the reservoir of fluid at its upstream end, determine the possible equilibrium positions of the plate, the free surface, and the flow within the film. The present problems are of interest both in their own right as paradigms for a range of fluid–structure interaction problems in which viscosity and surface tension both play an important role, and as a first step towards the study of elastic effects.

On the mathematical paradoxes for the flow of a viscoplastic film down an inclined surface

Abstract: In this paper we consider the motion of thin visco-plastic Bingham layer over an inclined surface whose profile is not flat. We assume that the ratio between the thickness and the length of the layer is small, so that the lubrication approach is suitable. Under specific hypotheses (e.g. creeping flow) we analyze two cases: finite tilt angle and small tilt angle. In both cases we prove that the physical model generates two mathematical problems which do not admit non-trivial solutions. We show that, though the relevant physical quantities (e.g. stress, velocity, shear rate, etc.) are well defined and bounded, the mathematical problem is inherently ill posed. In particular, exploiting a limit procedure in which the Bingham model is retrieved from a linear bi-viscous model we eventually prove that the underlying reason of the inconsistency has to be sought in the hypothesis of perfect stiffness of the unyielded part. We therefore conclude that: either the Bingham model is inappropriate to describe the lubrication motion over a non-flat surface, or the lubrication technique fails in approximating thin Bingham films.

Simulation of radial journal bearings using the FSI approach and a multi-phase model with integrated cavitation

Abstract: Journal bearings are an essential component in mechanical engineering. While the fundamental functional principle is well known, the internal processes in a bearing and the complex interactions between lubrication film and bearing structure have hitherto not been fully researched. Traditionally journal bearing analysis is carried out by special simulation codes based on lubrication theory or use of the Reynolds equation. To take phenomena such as turbulence, cavitation or heat transfer into account, empirical and numerical models are integrated into the calculations. These codes have proven to be efficient and sufficiently accurate for fundamental bearing analysis but are also subject to certain limitations in that they do not allow visualisation of physical phenomena such as cavitation, recirculation or elastohydrodynamic effects in complex geometries. This paper presents an approach based on state of the art numerical fluid structure interaction (FSI) methods. Application of three dimensional computational fluid dynamics (CFD) and finite element methods (FEM) allows the analysis of arbitrary bearing geometries. Furthermore, this approach also permits a detailed analysis of flow phenomena inside the bearing.

강한 측력이 작용하는 피스톤 펌프의 왕복동 피스톤 기구 부에서의 윤활모형에 관한 연구

Abstract: The objective of this study is to model and simulate the nonlinear lubrication performance of the sliding part between the piston and cylinder wall in a hydrostatic swash-plate-type axial piston pump. A numerical algorithm is developed that facilitates simultaneous calculation of the rotating body motion and fluid film pressure to observe the fluid film geometry and power loss. It is assumed that solid asperity contact, so-called mixed lubrication in this study, invariably occurs in the swash-plate-type axial piston pump, which produces a higher lateral moment on the pistons than other types of hydrostatic machines. Two comparative mixed lubrication models, rigid and elastic, are used to determine the reaction force and sliding friction. The rigid model does not allow any elastic deformation in the partial lubrication area. The patch shapes, reactive forces, and virtual local elastic deformation in the partial lubrication area are obtained in the elastic contact model using a simple Hertz contact theory. The calculation results show that a higher reaction force and friction loss are obtained in the rigid model, indicating that solid deformation is a significant factor on the lubrication characteristics of the reciprocating piston part.

Forced spreading of films and droplets of colloidal suspensions

Abstract: When a thin film of a colloidal suspension flows over a substrate, uneven distribution of the suspended particles can lead to an uneven coating. Motivated by this phenomenon, we analyse the flow of perfectly wetting films and droplets of colloidal suspensions down an inclined plane. Lubrication theory and the rapid-vertical-diffusion approximation are used to derive a coupled pair of one-dimensional partial differential equations describing the evolution of the interface height and particle concentration. Precursor films are assumed to be present, the colloidal particles are taken to be hard spheres, and particle and liquid dynamics are coupled through a concentration- dependent viscosity and diffusivity. We find that for sufficiently high Peclet numbers, even small initial concentration inhomogeneities produce viscosity gradients that cause the film or droplet front to evolve continuously in time instead of travelling without changing shape as happens in the absence of colloidal particles. At high enough particle concentrations, particle diffusion can lead to the formation of long-lived secondary flow fronts in films. Our results suggest that particle concentration gradients can have a dramatic influence on interface evolution in flowing films and droplets, a finding which may be relevant for understanding the onset of patterns that are observed experimentally.