Showing papers on "Lubrication theory published in 2015"
TL;DR: This study quantified the hydrophobic interaction in asymmetric system between air bubble and Hydrophobic surfaces, and provided a feasible method for synchronous measurements of the interaction forces with sub-nN resolution and the drainage dynamics of thin films down to nm thickness.
Abstract: A combination of atomic force microscopy (AFM) and reflection interference contrast microscopy (RICM) was used to measure simultaneously the interaction force and the spatiotemporal evolution of the thin water film between a bubble in water and mica surfaces with varying degrees of hydrophobicity. Stable films, supported by the repulsive van der Waals-Casimir-Lifshitz force were always observed between air bubble and hydrophilic mica surfaces (water contact angle, θ(w) < 5°) whereas bubble attachment occurred on hydrophobized mica surfaces. A theoretical model, based on the Reynolds lubrication theory and the augmented Young-Laplace equation including the effects of disjoining pressure, provided excellent agreement with experiment results, indicating the essential physics involved in the interaction between air bubble and solid surfaces can be elucidated. A hydrophobic interaction free energy per unit area of the form: WH(h) = -γ(1 - cos θ(w))exp(-h/D(H)) can be used to quantify the attraction between bubble and hydrophobized solid substrate at separation, h, with γ being the surface tension of water. For surfaces with water contact angle in the range 45° < θ(w) < 90°, the decay length DH varied between 0.8 and 1.0 nm. This study quantified the hydrophobic interaction in asymmetric system between air bubble and hydrophobic surfaces, and provided a feasible method for synchronous measurements of the interaction forces with sub-nN resolution and the drainage dynamics of thin films down to nm thickness.
164 citations
TL;DR: In this paper, high-speed interferometry was used to directly observe the thickness evolution of the air layer during the entire bubble entrapment process, and the initial disc radius and thickness showed excellent agreement with available theoretical models, based on adiabatic compression.
Abstract: When a drop impacts on a solid surface, its rapid deceleration is cushioned by a thin layer of air, which leads to the entrapment of a bubble under its centre. For large impact velocities the lubrication pressure in this air layer becomes large enough to compress the air. Herein we use high-speed interferometry, with 200 ns time-resolution, to directly observe the thickness evolution of the air layer during the entire bubble entrapment process. The initial disc radius and thickness shows excellent agreement with available theoretical models, based on adiabatic compression. For the largest impact velocities the air is compressed by as much as a factor of 14. Immediately following the contact, the air disc shows rapid vertical expansion. The radial speed of the surface minima just before contact, can reach 50 times the impact velocity of the drop.
85 citations
TL;DR: An atomic force microscope (AFM) bubble probe technique was employed, for the first time, to directly measure the interaction forces between an air bubble and sphalerite mineral surfaces under various hydrodynamic conditions, providing insights into the basic understanding of the interaction mechanism between bubbles and minerals at nanoscale in froth flotation processes.
Abstract: The interaction between air bubbles and solid surfaces plays important roles in many engineering processes, such as mineral froth flotation. In this work, an atomic force microscope (AFM) bubble probe technique was employed, for the first time, to directly measure the interaction forces between an air bubble and sphalerite mineral surfaces of different hydrophobicity (i.e., sphalerite before/after conditioning treatment) under various hydrodynamic conditions. The direct force measurements demonstrate the critical role of the hydrodynamic force and surface forces in bubble–mineral interaction and attachment, which agree well with the theoretical calculations based on Reynolds lubrication theory and augmented Young–Laplace equation by including the effect of disjoining pressure. The hydrophobic disjoining pressure was found to be stronger for the bubble–water–conditioned sphalerite interaction with a larger hydrophobic decay length, which enables the bubble attachment on conditioned sphalerite at relatively...
83 citations
TL;DR: In this paper, the authors consider the motion of a negatively buoyant particle in the vicinity of a thin compressible elastic wall, and use scaling arguments to establish different regimes of sliding, and complement these estimates using thin-film lubrication dynamics to determine an asymptotic theory for the sedimentation, sliding and spinning motions of a cylinder.
Abstract: We consider the motion of a fluid-immersed negatively buoyant particle in the vicinity of a thin compressible elastic wall, a situation that arises in a variety of technological and natural settings. We use scaling arguments to establish different regimes of sliding, and complement these estimates using thin-film lubrication dynamics to determine an asymptotic theory for the sedimentation, sliding and spinning motions of a cylinder. The resulting theory takes the form of three coupled nonlinear singular-differential equations. Numerical integration of the resulting equations confirms our scaling relations and further yields a range of unexpected behaviours. Despite the low-Reynolds-number feature of the flow, we demonstrate that the particle can spontaneously oscillate when sliding, can generate lift via a Magnus-like effect, can undergo a spin-induced reversal effect and also shows an unusual sedimentation singularity. Our description also allows us to address a sedimentation–sliding transition that can lead to the particle coasting over very long distances, similar to certain geophysical phenomena. Finally, we show that a small modification of our theory allows us to generalize the results to account for additional effects such as wall poroelasticity.
58 citations
TL;DR: In this paper, the authors investigate single and two-phase displacement flows in which the localised injection of fluid at a constant flow rate is accommodated by the inflation of the sheet and the outward propagation of an axisymmetric front beyond which the cell remains approximately undeformed.
Abstract: The injection of fluid into the narrow liquid-filled gap between a rigid plate and an elastic membrane drives a displacement flow that is controlled by the competition between elastic and viscous forces. We study such flows using the canonical set-up of an elastic-walled Hele-Shaw cell whose upper boundary is formed by an elastic sheet. We investigate both single- and two-phase displacement flows in which the localised injection of fluid at a constant flow rate is accommodated by the inflation of the sheet and the outward propagation of an axisymmetric front beyond which the cell remains approximately undeformed. We perform a direct comparison between quantitative experiments and numerical simulations of two theoretical models. The models couple the Foppl–von Karman equations, which describe the deformation of the thin elastic membrane, to the equations describing the flow, which we model by (i) the Navier–Stokes equations or (ii) lubrication theory. We identify the dominant physical effects that control the behaviour of the system and critically assess modelling assumptions that were made in previous studies. The insight gained from these studies is then used in Part 2 of this work, where we formulate an improved lubrication model and develop an asymptotic description of the key phenomena.
44 citations
TL;DR: In this paper, the authors investigate the injection of inviscid gas into the narrow liquid-filled gap between a rigid base plate and an overlying elastic sheet, and derive expressions for the speed of spreading of the bubble, which reveal that the effect of the capillary pressure drop at the bubble tip is to suck down the sheet over the liquid wedge and thereby reduce the speed.
Abstract: We investigate the injection of inviscid gas into the narrow liquid-filled gap between a rigid base plate and an overlying elastic sheet. After an early-time transient in which the gas deflects the sheet into a large blister, the viscous liquid displaced by the expanding bubble starts to accumulate in a wedge which advances as the elastic sheet peels away from the base. We analyse theoretically the subsequent interaction between viscous forces, elastic (bending or tension) forces and capillary forces. Asymptotic expressions are derived for the speed of spreading of the bubble, which reveal that the effect of the capillary pressure drop at the bubble tip is to suck down the sheet over the liquid wedge and thereby reduce the speed. We show that the system passes through three different asymptotic regimes in sequence. At early times, capillary effects are weak and hence the spreading of the bubble is controlled dominantly by the viscous-peeling process at the wedge tip. The capillary forces grow in importance with time, and at late times they dominate viscous effects and balance with elastic forces, leading to quasi-static spreading. Finally, at very late times, the capillary suction generates a narrow bottleneck at the wedge tip, which pushes a large ridge of liquid ahead of it. These results hold in the framework of standard lubrication theory as well as with an improved lubrication model, which takes into account films of wetting liquid deposited behind the advancing bubble tip. The predictions of the model are shown to be in excellent agreement with the Navier–Stokes simulations and experimental results from Part 1 of this work.
43 citations
TL;DR: In this article, the authors derive the rigid plug equation using an integral approach based on Newton's second law, where the unyielded part is treated as an evolving non material volume.
Abstract: In this paper we present a novel approach for modelling the lubrication flow of a Bingham fluid in a channel whose amplitude is non uniform. The novelty consists in deriving the rigid plug equation using an integral approach based on Newton’s second law, where the unyielded part is treated as an evolving non material volume. Such an approach leads to an integro-differential equation for the pressure that can be solved with an iterative procedure. We prove that a true unyielded plug exists even when the maximum width variation is not “small” and we find constraints on the amplitude of the channel that prevent the plug from “breaking”. We also extend our model to the case of a pressure-dependent viscosity.
35 citations
TL;DR: It is shown that chemical patterning of the wall is not required to generate and control a net flux within the channel, rather channel geometry alone is sufficient, and the results analytically in the asymptotic limit of lubrication theory.
Abstract: Many microfluidic devices use macroscopic pressure differentials to overcome viscous friction and generate flows in microchannels. In this work, we investigate how the chemical and geometric properties of the channel walls can drive a net flow by exploiting the autophoretic slip flows induced along active walls by local concentration gradients of a solute species. We show that chemical patterning of the wall is not required to generate and control a net flux within the channel, rather channel geometry alone is sufficient. Using numerical simulations, we determine how geometric characteristics of the wall influence channel flow rate, and confirm our results analytically in the asymptotic limit of lubrication theory.
34 citations
TL;DR: In this paper, the authors study the energy conversion during a bounce series by analyzing the droplet motion and its shape (decomposed into eigenmodes), showing that viscous dissipation associated with the in-flight oscillations accounts for less than 20 % of the total energy loss.
Abstract: Millimetre-sized droplets are able to bounce multiple times on flat solid substrates irrespective of their wettability, provided that a micrometre-thick air layer is sustained below the droplet, limiting $\mathit{We}$We to ${\lesssim}4$≲4. We study the energy conversion during a bounce series by analysing the droplet motion and its shape (decomposed into eigenmodes). Internal modes are excited during the bounce, yet the viscous dissipation associated with the in-flight oscillations accounts for less than 20 % of the total energy loss. This suggests a significant contribution from the bouncing process itself, despite the continuous presence of a lubricating air film below the droplet. To study the role of this air film we visualize it using reflection interference microscopy. We quantify its thickness (typically a few micrometres) with sub-millisecond time resolution and ${\sim}30~\text{nm}$∼30 nm height resolution. Our measurements reveal strong asymmetry in the air film shape between the spreading and receding phases of the bouncing process. This asymmetry is crucial for effective momentum reversal of the droplet: lubrication theory shows that the dissipative force is repulsive throughout each bounce, even near lift-off, which leads to a high restitution coefficient. After multiple bounces the droplet eventually hovers on the air film, while continuously experiencing a lift force to sustain its weight. Only after a long time does the droplet finally wet the substrate. The observed bounce mechanism can be described with a single oscillation mode model that successfully captures the asymmetry of the air film evolution.
29 citations
TL;DR: In this article, a model based on a combination of lubrication theory and capillary effects is developed to predict the shape and pressure drop of long bubbles flowing in circular tubes in pressure-driven flows.
Abstract: A model based on a combination of lubrication theory and capillary effects (ignoring inertial effects and gravity) is developed to predict the shape and pressure drop of long bubbles flowing in circular tubes in pressure-driven flows. An analytical solution for the thickness of the wetting film left on the tube wall as a function of the Capillary number (the ratio between viscous effects and surface tension) was derived by Klaseboer et al. (Phys Fluids 26:032107, 2014), which considerably extends the original result of Bretherton (J Fluid Mech 10:166–188, 1961) and confirms the empirical law of Aussillous and Quere (Phys Fluids 12:2367–2371, 2000). It is based on a crucial condition that requires that the bubble must fit inside the tube. An extension of this formulation allows for an analytical expression of the pressure drop across the bubble by applying the tube fit condition for the front and the back of the bubble and a force balance. The complete shape of the bubble can then be obtained numerically by applying boundary conditions at the tube centre. The interesting physics occurring at the back of the bubble is also investigated. A theoretical condition for the minimal length of such a bubble is given. Comparisons with experimental and numerical data for the shape of the bubble, pressure drop and curvature at the front and rear of the bubble for small to intermediate Capillary numbers give excellent agreement.
27 citations
TL;DR: In this article, a theoretical model is presented to analyze the calendering process wherein, the material to be calendered, is represented by the constitutive equation of a FENE-P fluid.
Abstract: A theoretical model is presented to analyze the calendering process wherein, the material to be calendered, is represented by the constitutive equation of a FENE-P fluid. The continuity and momentum equations are used in conjunction with lubrication theory to derive the governing equation of the flow under consideration. Exact expressions for the velocity and pressure gradients are obtained. Numerical integration is performed to compute the pressure for a given dimensionless leave-off distance. The quantities of interest in the mechanical design of calendering system such as, the force separating the two rolls and total power input into both rolls are calculated and shown graphically over a wide range of Deborah number. It is found that rheological features of the material modify the pressure, flow characteristics and all other operating variables significantly. In fact, the present analysis highlights some interesting features of the pressure and other operating variables of the calendering problem which are not reported in the available literature.
TL;DR: In this article, the influence of Joule heating on the slip velocity in an electro-osmotic flow (EOF) of viscoelastic fluids is taken into account by employing the simplified Phan-Thien and Tanner constitutive model (sPTT).
TL;DR: The problem of computing the hydrodynamic forces and torques among $N$ solid spherical particles moving with given rotational and translational velocities in Stokes flow is addressed and the original fluid–particle model is considered without introducing new hypotheses or models.
Abstract: We address the problem of computing the hydrodynamic forces and torques among solid spherical particles moving with given rotational and translational velocities in Stokes flow. We consider the original fluid–particle model without introducing new hypotheses or models. Our method includes the singular lubrication interactions which may occur when some particles come close to one another. The main new feature is that short-range interactions are propagated to the whole flow, including accurately the many-body lubrication interactions. The method builds on a pre-existing fluid solver and is flexible with respect to the choice of this solver. The error is the error generated by the fluid solver when computing non-singular flows (i.e. with negligible short-range interactions). Therefore, only a small number of degrees of freedom are required and we obtain very accurate simulations within a reasonable computational cost. Our method is closely related to a method proposed by Sangani & Mo (Phys. Fluids, vol. 6, 1994, pp. 1653–1662) but, in contrast with the latter, it does not require parameter tuning. We compare our method with the Stokesian dynamics of Durlofsky et al. (J. Fluid Mech., vol. 180, 1987, pp. 21–49) and show the higher accuracy of the former (both by analysis and by numerical experiments).
TL;DR: The contribution of micron-order oil droplets on film forming is discussed in this paper, where a parameter η describing the oil supply effects is 30 times higher and the film thickness reduces in starved regime much slower under oil-air lubrication compared with that under oil−jet lubrication.
TL;DR: In this paper, the authors investigate the main pumping parameters that influence a fluid-driven fracture in cohesive poroelastic and poro-elastoplastic weak formations and demonstrate that pumping parameters influence the fracture geometry and fluid pressures in weak formations through the viscous fluid flow and the diffusion process that create back stresses and large plastic zones as the fracture propagates.
Abstract: Summary
In this work, we investigate the main pumping parameters that influence a fluid-driven fracture in cohesive poroelastic and poroelastoplastic weak formations. These parameters include the fluid viscosity and the injection rate. The first parameter dominates in the mapping of the propagation regimes from toughness to viscosity, whereas the second parameter controls the storage to leak-off dominated regime through diffusion.
The fracture is driven in weak permeable porous formation by injecting an incompressible viscous fluid at the fracture inlet assuming that the fracture propagates under plane strain conditions. Fluid flow in the fracture is modeled by lubrication theory. Pore fluid movement in the porous formation is based on the Darcy law. The coupling follows the Biot theory, whereas the irreversible rock deformation is modeled with the Mohr–Coulomb yield criterion with associative flow rule. Fracture propagation criterion is based on the cohesive zone approach. Leak-off is also considered. The investigation is performed numerically with the FEM to obtain the fracture opening, length, and propagation pressure versus time.
We demonstrate that pumping parameters influence the fracture geometry and fluid pressures in weak formations through the viscous fluid flow and the diffusion process that create back stresses and large plastic zones as the fracture propagates. It is also shown that the product of the propagation velocity and fluid viscosity, µv that appears in the scaling controls the magnitude of the plastic zones and influences the net pressure and fracture geometry. These findings may explain partially the discrepancies in net pressures between field measurements and conventional model predictions for the case of weak porous formation. Copyright © 2014 John Wiley & Sons, Ltd.
TL;DR: In this article, the authors investigate how the chemical and geometric properties of the channel walls can drive a net flow by exploiting the autophoretic slip flows induced along active walls by local concentration gradients of a solute species.
Abstract: Many microfluidic devices use macroscopic pressure differentials to overcome viscous friction and generate flows in microchannels. In this work, we investigate how the chemical and geometric properties of the channel walls can drive a net flow by exploiting the autophoretic slip flows induced along active walls by local concentration gradients of a solute species. We show that chemical patterning of the wall is not required to generate and control a net flux within the channel, rather channel geometry alone is sufficient. Using numerical simulations, we determine how geometric characteristics of the wall influence channel flow rate, and confirm our results analytically in the asymptotic limit of lubrication theory.
TL;DR: In this article, the physical mechanisms of thermal droplet actuation on a wall through direct numerical simulation are investigated and clarified. But the authors focus on the physical mechanism and acting forces in the environment of a non-uniform temperature field and offer some explanations.
TL;DR: Based on the augmented Young-Laplace equation and lubrication theory, a detailed analytical model predicting the heat and mass transport characteristics of the evaporating meniscus in a micro-channel is developed as discussed by the authors.
TL;DR: In this article, an experimental and numerical study of the different lubrication regimes occurring in the sealing gap of a mechanical seal with water as sealed fluid is presented. But the results were obtained on a multi-scale mixed lubrication model considering heat transfer in the solids and seal rings deflections.
TL;DR: In this paper, the dispersion relation including phase change is derived based on the lubrication approximation, and the most dangerous wavelength should be λ d = 2 π [ 2 σ / ( Δ ρ g ) ] 1 / 2 instead of λ D = 2 ε [ 3 σ/ ( Δ ) g ] 1/2 for thin viscous gas film.
TL;DR: In this article, the effects of a fluid's thixotropic behavior on the viscous fingering phenomenon in a rectangular Hele-Shaw cell assuming that the displacing fluid is Newtonian while the displaced fluid obeys the Moore model for viscous fluids.
Abstract: The effects of a fluid’s thixotropic behavior is investigated on the viscous fingering phenomenon in a rectangular Hele-Shaw cell assuming that the displacing fluid is Newtonian while the displaced fluid obeys the Moore model for thixotropic fluids. Lubrication theory is used to simplify the gap-averaged governing equations in which the interfacial tension is treated as a body force. It is shown that the shapes of the fingers are dramatically affected by the displaced fluid’s thixotropic behavior. For highly thixotropic fluids, a chaotic behavior, accompanied by a blowup at prolonged times, is predicted to occur for certain set of parameter values. The viscosity ratio of the Moore fluid is also predicted to influence the shapes of the fingers provided that the zero-shear viscosity of the displaced fluid is higher than the viscosity of the displacing fluid. It is shown that the amplitude and wavenumber of the initial perturbation plays a crucial role on its time evolution. Also, a partial slip of the displaced and/or the displacing fluid is predicted to have a stabilizing effect on the viscous fingering phenomenon.
TL;DR: In this article, a nano-to-elastohydrodynamic lubrication (EHL) multiscale approach, developed to integrate molecular-scale phenomena into macroscopic lubrication models based on the continuum hypothesis, is applied to a lubricated contact problem with a ceramic-steel interface and a nanometric film thickness.
Abstract: A novel nano-to-elastohydrodynamic lubrication (EHL) multiscale approach, developed to integrate molecular-scale phenomena into macroscopic lubrication models based on the continuum hypothesis, is applied to a lubricated contact problem with a ceramic–steel interface and a nanometric film thickness. Molecular dynamics (MD) simulations are used to quantify wall slip occurring under severe confinement. Its dependence on the sliding velocity, film thickness, pressure, and different wall materials is described through representative analytical laws. These are then coupled to a modified Reynolds equation, where a no-slip condition applies to the ceramic surface and slip occurring on the steel wall is described through a Navier-type boundary condition. The results of this nano-to-EHL approach can contradict the well-established lubrication theory for thin films. In fact, slip can occur over the whole contact length, leading to a significant modification of the lubricant flow and consequently of the film thickness. If both walls move at the same velocity, the flow is reduced at the contact inlet and the film thickness decreases. If the nonslipping wall entrains the fluid, this one is accelerated resulting in a larger mass flow; nevertheless, the surface separation is reduced as the lubricant flows even faster in the contact center. The opposite effect occurs if the slipping surface entrains the fluid, causing a lower mass flow but higher film thickness. Finally, friction is generally smaller compared to the classical no-slip case and becomes independent of the sliding velocity as total slip is approached.
TL;DR: In this article, the authors present a theoretical and experimental study of viscous gravity currents lubricated by another viscous fluid from below, and they use lubrication theory to model both layers as Newtonian fluids spreading under their own weight in two-dimensional and axisymmetric settings over a smooth rigid horizontal surface.
Abstract: We present a theoretical and experimental study of viscous gravity currents lubricated by another viscous fluid from below. We use lubrication theory to model both layers as Newtonian fluids spreading under their own weight in two-dimensional and axisymmetric settings over a smooth rigid horizontal surface and consider the limit in which vertical shear provides the dominant resistance to the flow in both layers. There are contributions from Poiseuille-like flow driven by buoyancy and Couette-like flow driven by viscous coupling between the layers. The flow is self-similar if both fluids are released simultaneously, and exhibits initial transient behaviour when there is a delay between the initiation of flow in the two layers. We solve for both situations and show that the latter converges towards self-similarity at late times. The flow depends on three key dimensionless parameters relating the relative dynamic viscosities, input fluxes and density differences between the two layers. Provided the density difference between the two layers is bounded away from zero, we find an asymptotic solution in which the front of the lubricant is driven by its own gravitational spreading. There is a singular limit of equal densities in which the lubricant no longer spreads under its own weight in the vicinity of its nose and ends abruptly with a non-zero thickness there. We explore various regimes, from thin lubricating layers underneath a more viscous current to thin surface films coating an underlying more viscous current and find that although a thin film does not greatly influence the more viscous current if it forms a surface coating, it begins to cause interesting dynamics if it lubricates the more viscous current from below. We find experimentally that a lubricated gravity current is prone to a fingering instability.
TL;DR: In this article, a film flow model based on the lubrication theory is proposed to analyze the physics underlying gravity-driven runoff of thin wavy films, and is solved with computational fluid dynamics.
TL;DR: Adopting the lubrication theory, highly nonlinear coupled governing equations involving power law index as an exponent have been linearized and perturbation solutions are obtained about the Sisko fluid parameter.
Abstract: This paper looks at the effects of radiative heat transfer on the peristaltic transport of a Sisko fluid in an asymmetric channel with nonuniform wall temperatures. Adopting the lubrication theory, highly nonlinear coupled governing equations involving power law index as an exponent have been linearized and perturbation solutions are obtained about the Sisko fluid parameter. Analytical solutions for the stream function, axial pressure gradient, axial velocity, skin friction, and Nusselt number are derived for three different cases (i.e., shear thinning fluid, viscous fluid, and shear thickening fluid). The effects of Grashof number, radiation parameter, and other configuration parameters on pumping, trapping, temperature, Nusselt number, and skin friction have been examined in detail. A good agreement has been found for the case of viscous fluid with existing results.
TL;DR: In this paper, the authors quantify the amount of entrained air at the bottom of the drop during the impact and compare their results to various experimental data and find excellent agreement for the air that is entrapped during impact onto a pool.
Abstract: When a mm-sized liquid drop approaches a deep liquid pool, both the interface of the drop and the pool deform before the drop touches the pool. The build up of air pressure prior to coalescence is responsible for this deformation. Due to this deformation, air can be entrained at the bottom of the drop during the impact. We quantify the amount of entrained air numerically, using the Boundary Integral Method (BIM) for potential flow for the drop and the pool, coupled to viscous lubrication theory for the air film that has to be squeezed out during impact. We compare our results to various experimental data and find excellent agreement for the amount of air that is entrapped during impact onto a pool. Next, the impact of a rigid sphere onto a pool is numerically investigated and the air that is entrapped in this case also matches with available experimental data. In both cases of drop and sphere impact onto a pool the numerical air bubble volume V_b is found to be in agreement with the theoretical scaling V_b/V_{drop/sphere} ~ St^{-4/3}, where St is the Stokes number. This is the same scaling that has been found for drop impact onto a solid surface in previous research. This implies a universal mechanism for air entrainment for these different impact scenarios, which has been suggested in recent experimental work, but is now further elucidated with numerical results.
TL;DR: In this article, the authors considered the fracture as an opened part of a preexisting closed fracture of larger length and proposed a finite element method to solve the problem of fracture propagation in linear elastic porous media driven by injection of non-Newtonian power-law fluid.
TL;DR: In this article, numerical and theoretical results on the circular and polygonal hydraulic jumps in the framework of inertial lubrication theory are presented. But they do not consider the nonlinear term in the radial coordinate.
Abstract: This article contains numerical and theoretical results on the circular and polygonal hydraulic jumps in the framework of inertial lubrication theory. The free surface and velocity fields are computed along with cross-sections of the vorticity and pressure, in agreement with experimental data. The forces that drive and resist the instability are identified with the radial shear force, the azimuthal surface tension and the hydrostatic azimuthal force, in addition to a nonlinear term in the radial coordinate. Periodic solutions are obtained from the first orders of a perturbation theory by considering azimuthal symmetries. The thresholds of the instability are defined at closed jumps for discontinuous solutions and at one-sided hydraulic jumps for continuous curves that conserve fluid mass density.
TL;DR: In this article, the authors explored how the simultaneous application of an electric field and temperature gradient can be used to further influence thin-liquid-film instabilities and found that the results of the linear stability analysis of the lubrication equations hold even when the interfacial perturbations are no longer small.
Abstract: Both electric fields and temperature gradients can destabilize the surface of a thin liquid film and lead to the self-assembly of patterns composed of pillar-like structures. Such instabilities offer a relatively simple way to tailor the surface topography of coatings, which in turn can be used to influence coating appearance, texture, and functionality. The present work explores how the simultaneous application of an electric field and temperature gradient can be used to further influence thin-liquid-film instabilities. Both perfect and leaky dielectric materials are considered, and lubrication theory is applied to develop a system of nonlinear partial differential equations for the interfacial height and charge. Linear stability analysis of the lubrication equations shows that in perfect dielectric films, thermal effects tend to dominate the process, often rendering the electric field unimportant. However, in leaky dielectric films, both the thermal and electric fields play important roles and together can produce an increase in the growth rate and a reduction in the dominant wavelength of the instability. Nonlinear simulations of the lubrication equations show that the predictions of the linear theory hold even when the interfacial perturbations are no longer small. The effects of viscoelasticity are considered within the linear theory, and it is found that the growth rate of the instability, but not the length scale, depends on the rheological parameters. The findings of this work suggest that the simultaneous use of an electric field and temperature gradient will allow thin films to be patterned at length scales not accessible when only one of these destabilizing forces is used.
TL;DR: In this article, a modified three-dimensional lubrication model for the dewetting corner structure at the rear of the moving droplets was developed by taking into account the internal flow pattern and scaling arguments.
Abstract: We study a partial dewetting corner flow with a moving contact line at a finite Reynolds number, . When the speed of the moving contact line increases, the receding contact line appears with a corner shape that is also observed in a gravity-driven liquid droplet on an incline and on a plate withdrawn from a bath. In the current problem, is larger than unity, where is the aspect ratio of the flow structure. Therefore, classical lubrication theory is no longer appropriate. We develop a modified three-dimensional lubrication model for the dewetting corner structure at by taking into account the internal flow pattern and by scaling arguments. The key requirement is that the streamlines in the corner are straight and (nearly) parallel. In this case, we can obtain a modified pressure consisting of the capillary pressure and the dynamic pressure. This model describes the three-dimensional dewetting corner structure at the rear of the moving droplets at and furthermore shows that the dynamic pressure effects become dominant at a small half-opening angle. Additionally, this model provides analytical results for the internal flow, which is a self-similar flow pattern. To validate the analytical results, we perform high-speed shadowgraphy and tomographic particle image velocimetry (PIV). We find a good agreement between the theoretical and the experimental results.