Showing papers on "Lubrication theory published in 2012"

Modeling of Hydraulic Fracturing in a Poroelastic Cohesive Formation

Abstract: This paper investigates the main parameters that affect the propagation of a fluid driven-fracture in a poroelastic medium. The fracture results from the pumping of an incompressible Newtonian viscous fluid at the fracture inlet, and the flow in the fracture is modelled by the lubrication theory. Rock deformation is assumed as porous-elastic. Leak-off in the host rock is considered to account for the diffusion effects in the surrounding formation. The propagation criterion is of the cohesive type. Finite element analysis was performed to compute the fracturing pressure and fracture dimensions as a function of the time and length. It was found that higher pressures are needed to extend a fracture in a poroelastic medium than in an elastic medium, and the created profiles of poroelastic fracture are wider. It was found that grain compressibility plays a minor role and does not result any significant difference in the fluid pressures and fracture dimensions. Wider fracture profiles are obtained with ...

Coefficient of restitution for wet particles.

Abstract: The influence of a liquid film on the coefficient of restitution (COR) is investigated experimentally by tracing freely falling particles bouncing on a wet surface. The dependence of the COR on the impact velocity and various properties of the particle and liquid is presented and discussed in terms of dimensionless numbers that characterize the interplay between inertial, viscous, and surface forces. In the Reynolds number regime where lubrication theory does not apply, the ratio of the film thickness to the particle size is found to be a crucial parameter determining the COR.

Spatiotemporal evolution of thin liquid films during impact of water bubbles on glass on a micrometer to nanometer scale

Abstract: Collisions between millimeter-size bubbles in water against a glass plate are studied using high-speed video. Bubble trajectory and shape are tracked simultaneously with laser interferometry between the glass and bubble surfaces that monitors spatial-temporal evolution of the trapped water film. Initial bubble bounces and the final attachment of the bubble to the surface have been quantified. While the global Reynolds number is large (� 10 2 ), the film Reynolds number remains small and permits analysis with lubrication theory with tangentially immobile boundary condition at the air-water interface. Accurate predictions of dimple formation and subsequent film drainage are obtained.

On the λ ratio range of mixed lubrication

Abstract: Mixed lubrication is a mode of fluid lubrication in which both hydrodynamic lubricant film and rough surface asperity contact coexist. Mixed lubrication problems are usually associated with signifi...

Delaying the onset of dynamic wetting failure through meniscus confinement

Abstract: Dynamic wetting is crucial to processes where liquid displaces another fluid along a solid surface, such as the deposition of a coating liquid onto a moving substrate. Numerous studies report the failure of dynamic wetting past some critical process speed. However, the hydrodynamic factors that influence the transition to wetting failure remain poorly understood from an empirical and theoretical perspective. The objective of this investigation is to determine the effect of meniscus confinement on the onset of dynamic wetting failure. A novel experimental system is designed to simultaneously view confined and unconfined wetting systems as they approach wetting failure. The experimental apparatus consists of a scraped steel roll that rotates into a bath of glycerol. Confinement is imposed via a gap formed between a coating die and the roll surface. Flow visualization is used to record the critical roll speed at which wetting failure occurs. Comparison of the confined and unconfined data shows a clear increase in the relative critical speed as the meniscus becomes more confined. A hydrodynamic model for wetting failure is developed and analysed with (i) lubrication theory and (ii) a two-dimensional finite-element method (FEM). Both approaches do a remarkable job of matching the observed confinement trend, but only the two-dimensional model yields accurate estimates of the absolute values of the critical speeds due to the highly two-dimensional nature of the stress field in the displacing liquid. The overall success of the hydrodynamic model suggests a wetting failure mechanism primarily related to viscous bending of the meniscus.

Thin film dynamics on a prolate spheroid with application to the cornea

Abstract: The tear film on the front of the eye is critical to proper eyesight; in many mathematical models of the tear film, the tear film is assumed to be on a flat substrate. We re-examine this assumption by studying the effect of a substrate which is representative of the human cornea. We study the flow of a thin fluid film on a prolate spheroid which is a good approximation to the shape of the human cornea. Two lubrication models for the dynamics of the film are studied in prolate spheroidal coordinates which are appropriate for this situation. One is a self-consistent leading-order hyperbolic partial differential equation (PDE) valid for relatively large substrate curvature; the other retains the next higher-order terms resulting in a fourth-order parabolic PDE for the film dynamics. The former is studied for both Newtonian and Ellis (shear thinning) fluids; for typical tear film parameter values, the shear thinning is too small to be significant in this model. For larger shear thinning, we find a significant effect on finite-time singularities. The second model is studied for a Newtonian fluid and allows for a meniscus at one end of the domain. We do not find a strong effect on the thinning rate at the center of the cornea. We conclude that the corneal shape does not have a significant effect on the thinning rate of the tear film for typical conditions.

Flow of fluids with pressure- and shear-dependent viscosity down an inclined plane

Abstract: In this paper we consider a fluid whose viscosity depends on both the mean normal stress and the shear rate flowing down an inclined plane. Such flows have relevance to geophysical flows. In order to make the problem amenable to analysis, we consider a generalization of the lubrication approximation for the flows of such fluids based on the development of the generalization of the Reynolds equation for such flows. This allows us to obtain analytical solutions to the problem of propagation of waves in a fluid flowing down an inclined plane. We find that the dependence of the viscosity on the pressure can increase the breaking time by an order of magnitude or more than that for the classical Newtonian fluid. In the viscous regime, we find both upslope and downslope travelling wave solutions, and these solutions are quantitatively and qualitatively different from the classical Newtonian solutions.

Wavy regime of a power-law film flow

Abstract: We consider a power-law fluid flowing down an inclined plane under the action of gravity. The divergence of the viscosity at zero strain rate is taken care of by introducing a Newtonian plateau at small strain rate. Two-equation models are formulated within the framework of lubrication theory in terms of the exact mass balance and an averaged momentum equation, which form a set of evolution equations for the film thickness , a local velocity amplitude or the flow rate . The models account for the streamwise diffusion of momentum. Comparisons with Orr–Sommerfeld stability analysis and with direct numerical simulation (DNS) show convincing agreement in both linear and nonlinear regimes. The influence of shear-thinning or shear-thickening on the primary instability is shown to be non-trivial. A destabilization of the base flow close to threshold is promoted by the shear-thinning effect, whereas, further from threshold, it tends to stabilize the base flow when the viscous damping of short waves becomes dominant. A reverse situation is observed in the case of shear-thickening fluids. Shear-thinning accelerates solitary waves and promotes a subcritical onset of travelling waves at larger wavenumber than the linear cut-off wavenumber. A conditional stability of the base flow is thus observed. This phenomenon results from a reduction of the effective viscosity at the free surface. When compared with DNS, simulations of the temporal response of the film based on weighted residual models satisfactorily capture the conditional stability of the film.

Peristaltic transport of rheological fluid: model for movement of food bolus through esophagus

Abstract: Fluid mechanical peristaltic transport through esophagus is studied in the paper. A mathematical model has been developed to study the peristaltic transport of a rheological fluid for arbitrary wave shapes and tube lengths. The Ostwald-de Waele power law of a viscous fluid is considered here to depict the non-Newtonian behaviour of the fluid. The model is formulated and analyzed specifically to explore some important information concerning the movement of food bolus through esophagus. The analysis is carried out by using the lubrication theory. The study is particularly suitable for the cases where the Reynolds number is small. The esophagus is treated as a circular tube through which the transport of food bolus takes place by periodic contraction of the esophageal wall. Variation of different variables concerned with the transport phenomena such as pressure, flow velocities, particle trajectory, and reflux is investigated for a single wave as well as a train of periodic peristaltic waves. The locally variable pressure is seen to be highly sensitive to the flow index “n”. The study clearly shows that continuous fluid transport for Newtonian/rheological fluids by wave train propagation is more effective than widely spaced single wave propagation in the case of peristaltic movement of food bolus in the esophagus.

Internal dynamics of Newtonian and viscoplastic fluid avalanches down a sloping bed

Abstract: We experimentally investigated the spreading of fluid avalanches (i.e., fixed volumes of fluid) down an inclined flume. Emphasis was given to the velocity field within the head. Using specific imaging techniques, we were able to measure velocity profiles within the flowing fluid far from the sidewalls. We studied the behavior of Newtonian and viscoplastic fluids for various flume inclinations and initial masses. For the Newtonian fluids tested (glycerol and Triton X100), we compared the measured velocity field with that predicted by lubrication theory. Provided that the flow Reynolds number Re was sufficiently low (typically Re < 1), there was excellent agreement between theory and experiment except for the very thin region just behind the contact line. For higher Reynolds numbers (typically Re ∼ 10), the discrepancy between theory and experiment was more marked (relative errors up to 17% for the body). As viscoplastic materials, we used Carbopol ultrez 10. For the body, agreement between theoretical and ...

Inertial Lubrication Theory

Abstract: Thin fluid films can have surprising behavior depending on the boundary conditions enforced, the energy input and the specific Reynolds number of the fluid motion. Here we study the equations of motion for a thin fluid film with a free boundary and its other interface in contact with a solid wall. Although shear dissipation increases for thinner layers and the motion can generally be described in the limit as viscous, inertial modes can always be excited for a sufficiently high input of energy. We derive the minimal set of equations containing inertial effects in this strongly dissipative regime.

Capillary-driven flow induced by a stepped perturbation atop a viscous film

Abstract: Thin viscous liquid films driven by capillarity are well described in the lubrication theory through the thin film equation. In this article, we present an analytical solution of this equation for a particular initial profile: a stepped perturbation. This initial condition allows a linearization of the problem making it amenable to Fourier analysis. The solution is obtained and characterized. As for a temperature step in the heat equation, self-similarity of the first kind of the full evolution is demonstrated and a long-term expression for the excess free energy is derived. In addition, hydrodynamical fields are described. The solution is then compared to experimental profiles from a model system: a polystyrene nanostep above the glass transition temperature which flows due to capillarity. The excellent agreement enables a precise measurement of the capillary velocity for this polymeric liquid, without involving any numerical simulation. More generally, as these results hold for any viscous system driven by capillarity, the present solution may provide a useful tool in hydrodynamics of thin viscous films.

Flow transport in a microchannel induced by moving wall contractions: a novel micropumping mechanism

Abstract: A novel micropumping mechanism based on a theoretical model that describes flow transport in a microchannel induced by moving wall contractions in the low Reynolds number flow regime is presented. The channel is assumed to have a length that is much greater than its width (\({\delta = W/L \ll 1}\)) and the upper wall is subjected to prescribed, non-peristaltic, localized moving contractions. Lubrication theory for incompressible viscous flow at low Reynolds number (Re ~ δ) is used to model the problem mathematically and to derive expressions for the velocity components, pressure gradient, wall shear stress, and net flow produced by the wall contractions. The effect of contraction parameters such as amplitude and phase lag on the time-averaged net flow over a single cycle of wall motions is studied. The results presented here are supported by passive particle tracking simulations to investigate the possibility of using this system as a pumping mechanism. The present study is motivated by collapse mechanisms observed in entomological physiological systems that use multiple contractions to transport fluid, and the emerging novel microfluidic devices that mimic these systems.

Capillary-driven flow induced by a stepped perturbation atop a viscous film

Abstract: Thin viscous liquid films driven by capillarity are well described in the lubrication theory through the thin film equation. In this article, we present an analytical solution of this equation for a particular initial profile: a stepped perturbation. This initial condition allows a linearization of the problem making it amenable to Fourier analysis. The solution is obtained and characterized. As for a temperature step in the heat equation, self-similarity of the first kind of the full evolution is demonstrated and a long-term expression for the excess free energy is derived. In addition, hydrodynamical fields are described. The solution is then compared to experimental profiles from a model system: a polystyrene nanostep above the glass transition temperature which flows due to capillarity. The excellent agreement enables a precise measurement of the capillary velocity for this polymeric liquid, without involving any numerical simulation. More generally, as these results hold for any viscous system driven by capillarity, the present solution may provide a useful tool in hydrodynamics of thin viscous films.

A model for the human tear film with heating from within the eye

Abstract: A model for tear film dynamics and cooling during the interblink period is formulated that includes heat transfer from the interior of the eye. Lubrication theory is used to derive an equation for the thickness of the film; the nonlinear partial differential equation for the thickness is solved subject to either a fixed temperature at the substrate or with heat diffusion from within two different model rectangular domains. The model domains are simplified geometries that represent the anterior eye and that may include the cornea and some aqueous humor; one model domain is asymptotically thin (thin substrate) and the other has finite thickness (thick substrate). The thick substrate case captures temperature decreases that are observed in vivo, while the thin substrate and fixed temperature models do not. Parameters to reproduce observed temperature decreases are found.

Modelling of elastohydrodynamic lubrication and fatigue of rough surfaces: The effect of lambda ratio:

Abstract: The article describes a numerical model for the analysis of elastohydrodynamic lubrication under conditions in which the nominal lubricant film thickness is small compared to the roughness present on the surfaces (lambda ratio less than unity). These conditions occur in most types of gear tooth contacts and in many other heavily loaded machine elements. In these situations, lubrication occurs at the roughness asperity level (micro-elastohydrodynamic lubrication), and in extreme cases ‘mixed’ lubrication behaviour occurs in which momentary solid contacts between the surfaces take place. In both micro-elastohydrodynamic lubrication and mixed lubrication regimes high, localised pressures are applied cyclically to asperity features as they move through the nominal contact leading to fatigue at the asperity level, which culminates in micropitting wear. Results of the modelling show the effect of lambda ratio in lubricated gear tooth contacts, and demonstrate the transition from full-film to micro-elastohydrodynamic lubrication and mixed lubrication, and the consequences in terms of predicted fatigue damage.

Film flow for power-law fluids: Modeling and linear stability

Abstract: The paper deals with modeling of a power-law fluid film flowing down an inclined plane for small to moderate Reynolds numbers. A model, accurate up to second order [first order] for dilatant [pseudoplastic] fluids is proposed to describe the nonlinear behavior of the flow. The modeling procedure consists of a combination of the lubrication theory and the weighted residual approach using an appropriate projection basis. A suitable choice of weighting functions allows a significant reduction of the dimension of the problem. The resulting model is naturally unique, i.e., independent of the particular form of the trial functions. Reduced models are proposed for the evolution of the local film thickness and flow rate; their linear spectra are compared to that obtained from the full Orr–Sommerfeld numerical solution. To obtain the latter, a new formulation of the eigenvalue problem is proposed to overcome the classical divergence of the apparent viscosity at the free surface. The full model and its reduced forms all have the advantage of the Benney like model close to criticality. Far from the instability threshold the full model continues to follow the Orr–Sommerfeld solution up to sufficiently large Reynolds numbers and gives better predictions than the depth averaging model. An incomplete regularization procedure is performed to cure the rapid divergence of the reduced two-equation model. Due to its relative simplicity the latter might be preferred in practice to the full model, at least at the initial stage of the nonlinear regime. It is also shown that the convective nature of the instability is not affected by the variation of the power law index.

Rigorous derivation of the thin film approximation with roughness-induced correctors ∗

Abstract: We derive the thin film approximation including roughness-induced correctors. This corresponds to the description of a confined Stokes flow whose thickness is of order~$\eps$ (designed to be small)~; but we also take into account the roughness patterns of the boundary that are described at order~$\eps^2$, leading to a perturbation of the classical Reynolds approximation. The asymptotic expansion leading to the description of the scale effects is rigorously derived, through a sequence of Reynolds-type problems and Stokes-type (boundary layer) problems. Well-posedness of the related problems and estimates in suitable functional spaces are proved, at any order of the expansion. In particular, we show that the micro-/macro-scale coupling effects may be analysed as the consequence of two features: the interaction between the macroscopic scale (order~1) of the flow and the microscopic scale (order~$\eps$ of the thin film) is perturbed by the interaction with a microscopic scale of order~$\eps^2$ related to the roughness patterns (as expected through the classical Reynolds approximation)~; moreover, the converging-diverging profile of the confined flow, which is typical in lubrication theory (note that the case of a constant cross-section channel has no interest) provides additional micro-macro-scales coupling effects.

Agglomeration and de-agglomeration of rotating wet doublets

Abstract: In this work, experiments using a pendulum apparatus were conducted for two particles engaged in oblique, wetted collisions over a range of impact angles, impact velocities, coating thicknesses, liquid viscosities, particle materials, and particle radii. From previous studies on normal or head-on collisions, the two particles bounce apart if the Stokes number (a ratio of particle inertia to viscous forces) exceeds a critical value, whereas they stick together if the Stokes number is below this critical value. However, for oblique collisions, an additional outcome is observed at moderate Stokes numbers and impact angles, in which the spheres initially stick together, rotate as a doublet, and then separate due to centrifugal forces. We refer to this outcome as ‘stick–rotate–separate’. For subcritical Stokes numbers exhibiting this new outcome, the experimental results for the apparent coefficient of normal restitution and angle of rotation from impact to separation show only weak dependence on the fluid viscosity and thickness and the dry restitution coefficient, whereas they both decrease with increasing particle radius. These results are in contrast with those for supercritical Stokes numbers in which the spheres bounce upon impact. An accompanying theory based on lubrication forces, the glass transition of the liquid layer, and solid deformation and rebound agrees well with experimental results and gives insight into the observed trends.

Theoretical analysis of the calendered exiting thickness of viscoelastic sheets

Abstract: A theoretical model was developed to describe the calendering process in viscoelastic sheets of finite initial thickness. The rheological constitutive equation of the fluid under consideration follows a common form of the Simplified–Phan–Thien–Tanner (SPTT) fluid model. We predict the influence of the viscoelastic effects on the leave-off distance that is related to the exiting sheet thickness in the calendering process. The mass and momentum balance equations, which are based on lubrication theory, were nondimensionalized and solved for the velocity and pressure fields by using perturbation and numerical techniques, where the leave-off distance represents an eigenvalue of the mathematical problem. When the above variables were obtained, the dimensionless leave-off distance in the calendering process was determined, considering the influence of the viscoelastic effects in the process. Moreover, quantities of engineering interest were calculated, including the maximum pressure, the roll-separating force and the power transmitted to the fluid by the rolls. The results show that the inclusion of the viscoelastic effect substantially modifies all dimensionless variables in comparison with those obtained for the Newtonian case.

Viscoplastic lubrication theory with application to bearings and the washboard instability of a planing plate

Abstract: We consider lubrication theory for a two-dimensional viscoplastic fluid confined between rigid moving boundaries. A general formulation is presented which allows the flow field and pressure to be calculated given an arbitrary rheological model; the Herschel–Bulkley law is used for illustration. The theory is first applied to a (full) viscoplastic journal bearing with arbitrary motions allowed for the inner cylinder (either prescribed, or arising from an imposed load and torque). Conditions are derived determining when motion is arrested by the yield stress. We next apply the theory to a slider bearing filled with Bingham fluid, computing the lift force on the bearing and the fluid flux through it. The results are then extended to model an inclined plate that is towed at constant horizontal speed over a shallow viscoplastic layer but is able to move vertically. Steady planing solutions are stable at low towing speeds, but give way to unstable vertical oscillations of the plate at higher speed; the yield stress has a relatively weak effect on this instability. The pattern imprinted on the fluid layer by the oscillations provides an analogue of the washboard phenomenon on gravel roads.

The screen printing of a power-law fluid

Abstract: We present a two-dimensional large-aspect-ratio model for the off-contact screen printing of a power-law fluid. We extend the work of White et al. (J Eng Math 54:49–70, 2005) by explicitly including the fluid/air free surface that is present beneath the screen ahead of the squeegee. In the distinguished parameter limit of greatest interest to industry, the process is quasi-steady on the time-scale of a print and can be analysed in three separate regions using the method of matched asymptotic expansions. This allows us to predict where the fluid transfers through the screen, the point at which it first makes contact with the substrate, and the amount of fluid deposited on the substrate during a print stroke. Finally, we show that using a shear-thinning fluid will decrease the amount of fluid transferred ahead of the squeegee, but increase the amount of fluid deposited on the substrate.

Stratified flows with vertical layering of density: experimental and theoretical study of flow configurations and their stability

Abstract: A vertically moving boundary in a stratified fluid can create and maintain a horizontal density gradient, or vertical layering of density, through the mechanism of viscous entrainment. Experiments to study the evolution and stability of axisymmetric flows with vertically layered density are performed by towing a narrow fibre upwards through a stably stratified viscous fluid. The fibre forms a closed loop and thus its effective length is infinite. A layer of denser fluid is entrained and its thickness is measured by implementing tracking analysis of dyed fluid images. Thickness values of up to 70 times that of the fibre are routinely obtained. A lubrication model is developed for both a two-dimensional geometry and the axisymmetric geometry of the experiment, and shown to be in excellent agreement with dynamic experimental measurements once subtleties of the optical tracking are addressed. Linear stability analysis is performed on a family of exact shear solutions, using both asymptotic and numerical methods in both two dimensions and the axisymmetric geometry of the experiment. It is found analytically that the stability properties of the flow depend strongly on the size of the layer of heavy fluid surrounding the moving boundary, and that the flow is neutrally stable to perturbations in the large-wavelength limit. At the first correction of this limit, a critical layer size is identified that separates stable from unstable flow configurations. Surprisingly, in all of the experiments the size of the entrained layer exceeds the threshold for instability, yet no unstable behaviour is observed. This is a reflection of the small amplification rate of the instability, which leads to growth times much longer than the duration of the experiment. This observation illustrates that for finite times the hydrodynamic stability of a flow does not necessarily correspond to whether or not that flow can be realised from an initial-value problem. Similar instabilities that are neutral to leading order with respect to long waves can arise under the different physical mechanism of viscous stratification, as studied by Yih (J. Fluid Mech., vol. 27, 1967, pp. 337–352), and we draw a comparison to that scenario.