Breakdown of the Bretherton law due to wall slippage

2014 ◽  
Vol 741 ◽  
pp. 200-227 ◽  
Author(s):  
Yen-Ching Li ◽  
Ying-Chih Liao ◽  
Ten-Chin Wen ◽  
Hsien-Hung Wei
Keyword(s):  
Slip Length ◽  
Dynamic Contact Angle ◽  
Flow Characteristics ◽  
Wall Slip ◽  
Dynamic Contact ◽  
Precursor Film ◽  
Minor Role ◽  
Textured Surfaces ◽  
Slip Regime ◽  
A Minor

AbstractAgainst the common wisdom that wall slip plays only a minor role in global flow characteristics, here we demonstrate theoretically for the displacement of a long bubble in a slippery channel that the well-known Bretherton $2/3$ law can break down due to a fraction of wall slip with the slip length $\lambda $ much smaller than the channel depth $R$. This breakdown occurs when the film thickness $h_{\infty } $ is smaller than $\lambda $, corresponding to the capillary number $Ca$ below the critical value $Ca^{\ast } \sim (\lambda /R)^{3 / 2}$. In this strong slip regime, a new quadratic law $h_{\infty } /R \sim Ca^{2} (R/\lambda )^{2}$ is derived for a film much thinner than that predicted by the Bretherton law. Moreover, both the $2/3$ and the quadratic laws can be unified into the effective $2/3$ law, with the viscosity $\mu $ replaced by an apparent viscosity $\mu _{app}= \mu h_{\infty } /({\lambda } + h_{\infty })$. A similar extension can also be made for coating over textured surfaces where apparent slip lengths are large. Further insights can be gained by making a connection with drop spreading. We find that the new quadratic law can lead to $\theta _{d} \propto Ca^{1 / 2} $ for the apparent dynamic contact angle of a spreading droplet, subsequently making the spreading radius grow with time as $r \propto t^{1 / 8}$. In addition, the precursor film is found to possess $\ell _{f} \propto Ca^{ - 1 / 2}$ in length and therefore spreads as $\ell _{f} \propto t^{1 / 3}$ in an anomalous diffusion manner. All these features are accompanied by no-slip-to-slip transitions sensitive to the amount of slip, markedly different from those on no-slip surfaces. Our findings not only provide plausible accounts for some apparent departures from no-slip predictions seen in experiments, but also offer feasible alternatives for assessing wall slip effects experimentally.

Speeding up thermocapillary migration of a confined bubble by wall slip

2014 ◽  
Vol 746 ◽  
pp. 31-52 ◽  
Author(s):  
Ying-Chih Liao ◽  
Yen-Ching Li ◽  
Yu-Chih Chang ◽  
Chih-Yung Huang ◽  
Hsien-Hung Wei
Keyword(s):  
Film Thickness ◽  
Slip Length ◽  
Flow Characteristics ◽  
Wall Slip ◽  
Fluid Viscosity ◽  
Tube Radius ◽  
Thermocapillary Migration ◽  
Slip Regime ◽  
Slip Effects ◽  
Slight Discrepancy

AbstractIt 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 $\lambda $ much smaller than the tube radius $R$, can make the bubble migrate much faster than without wall slip. This speedup effect occurs in the strong-slip regime where the film thickness $b$ is smaller than $\lambda $ when the Marangoni number $S= \tau _{T} R/\sigma _{0}~ (\ll 1)$ is below the critical value $S^* \sim (\lambda /R)^{1/2}$, where $\tau _{T}$ is the driving thermal stress and $\sigma _{0}$ is the surface tension. The resulting bubble migration speed is found to be $U_{b} \sim (\sigma _{0}/\mu )S^{3}(\lambda /R)$, which can be more than a hundred times faster than the no-slip result $U_{b} \sim (\sigma _{0}/\mu )S^{5}$ (Wilson, J. Eng. Math., vol. 29, 1995, pp. 205–217; Mazouchi & Homsy, Phys. Fluids, vol. 12, 2000, pp. 542–549), with $\mu $ 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 $b\sim RS^{2}$. 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.

scholarly journals Numerical Bifurcation Analysis of a Film Flowing over a Patterned Surface through Enhanced Lubrication Theory

Fluids ◽  
2021 ◽  
Vol 6 (11) ◽  
pp. 405
Author(s):  
Nicola Suzzi ◽  
Giulio Croce
Keyword(s):  
Contact Angle ◽  
Bifurcation Analysis ◽  
Dynamic Contact Angle ◽  
Flow Characteristics ◽  
Dynamic Contact ◽  
Contact Angles ◽  
Inclined Plate ◽  
Lubrication Equation ◽  
Static Contact ◽  
Numerical Bifurcation

The bifurcation analysis of a film falling down an hybrid surface is conducted via the numerical solution of the governing lubrication equation. Instability phenomena, that lead to film breakage and growth of fingers, are induced by multiple contamination spots. Contact angles up to 75∘ are investigated due to the full implementation of the free surface curvature, which replaces the small slope approximation, accurate for film slope lower than 30∘. The dynamic contact angle is first verified with the Hoffman–Voinov–Tanner law in case of a stable film down an inclined plate with uniform surface wettability. Then, contamination spots, characterized by an increased value of the static contact angle, are considered in order to induce film instability and several parametric computations are run, with different film patterns observed. The effects of the flow characteristics and of the hybrid pattern geometry are investigated and the corresponding bifurcation diagram with the number of observed rivulets is built. The long term evolution of induced film instabilities shows a complex behavior: different flow regimes can be observed at the same flow characteristics under slightly different hybrid configurations. This suggest the possibility of controlling the rivulet/film transition via a proper design of the surfaces, thus opening the way for relevant practical application.

Slipping moving contact lines: critical roles of de Gennes’s ‘foot’ in dynamic wetting

2019 ◽  
Vol 873 ◽  
pp. 110-150
Author(s):  
Hsien-Hung Wei ◽  
Heng-Kwong Tsao ◽  
Kang-Ching Chu
Keyword(s):  
Contact Angle ◽  
Contact Line ◽  
Capillary Number ◽  
Dynamic Contact Angle ◽  
Wall Slip ◽  
Dynamic Contact ◽  
Colloid Interface ◽  
Dynamic Wetting ◽  
Moving Contact Line ◽  
Moving Contact

In the context of dynamic wetting, wall slip is often treated as a microscopic effect for removing viscous stress singularity at a moving contact line. In most drop spreading experiments, however, a considerable amount of slip may occur due to the use of polymer liquids such as silicone oils, which may cause significant deviations from the classical Tanner–de Gennes theory. Here we show that many classical results for complete wetting fluids may no longer hold due to wall slip, depending crucially on the extent of de Gennes’s slipping ‘foot’ to the relevant length scales at both the macroscopic and microscopic levels. At the macroscopic level, we find that for given liquid height $h$ and slip length $\unicode[STIX]{x1D706}$, the apparent dynamic contact angle $\unicode[STIX]{x1D703}_{d}$ can change from Tanner’s law $\unicode[STIX]{x1D703}_{d}\sim Ca^{1/3}$ for $h\gg \unicode[STIX]{x1D706}$ to the strong-slip law $\unicode[STIX]{x1D703}_{d}\sim Ca^{1/2}\,(L/\unicode[STIX]{x1D706})^{1/2}$ for $h\ll \unicode[STIX]{x1D706}$, where $Ca$ is the capillary number and $L$ is the macroscopic length scale. Such a no-slip-to-slip transition occurs at the critical capillary number $Ca^{\ast }\sim (\unicode[STIX]{x1D706}/L)^{3}$, accompanied by the switch of the ‘foot’ of size $\ell _{F}\sim \unicode[STIX]{x1D706}Ca^{-1/3}$ from the inner scale to the outer scale with respect to $L$. A more generalized dynamic contact angle relationship is also derived, capable of unifying Tanner’s law and the strong-slip law under $\unicode[STIX]{x1D706}\ll L/\unicode[STIX]{x1D703}_{d}$. We not only confirm the two distinct wetting laws using many-body dissipative particle dynamics simulations, but also provide a rational account for anomalous departures from Tanner’s law seen in experiments (Chen, J. Colloid Interface Sci., vol. 122, 1988, pp. 60–72; Albrecht et al., Phys. Rev. Lett., vol. 68, 1992, pp. 3192–3195). We also show that even for a common spreading drop with small macroscopic slip, slip effects can still be microscopically strong enough to change the microstructure of the contact line. The structure is identified to consist of a strongly slipping precursor film of length $\ell \sim (a\unicode[STIX]{x1D706})^{1/2}Ca^{-1/2}$ followed by a mesoscopic ‘foot’ of width $\ell _{F}\sim \unicode[STIX]{x1D706}Ca^{-1/3}$ ahead of the macroscopic wedge, where $a$ is the molecular length. It thus turns out that it is the ‘foot’, rather than the film, contributing to the microscopic length in Tanner’s law, in accordance with the experimental data reported by Kavehpour et al. (Phys. Rev. Lett., vol. 91, 2003, 196104) and Ueno et al. (Trans. ASME J. Heat Transfer, vol. 134, 2012, 051008). The advancement of the microscopic contact line is still led by the film whose length can grow as the $1/3$ power of time due to $\ell$, as supported by the experiments of Ueno et al. and Mate (Langmuir, vol. 28, 2012, pp. 16821–16827). The present work demonstrates that the behaviour of a moving contact line can be strongly influenced by wall slip. Such slip-mediated dynamic wetting might also provide an alternative means for probing slippery surfaces.

scholarly journals Dynamic contact angle of a liquid spreading on an unsaturated wettable porous substrate

2013 ◽  
Vol 715 ◽  
pp. 273-282 ◽  
Author(s):  
Yulii D. Shikhmurzaev ◽  
James E. Sprittles
Keyword(s):  
Contact Angle ◽  
Dynamic Contact Angle ◽  
Flow Characteristics ◽  
Dynamic Contact ◽  
Solid Substrate ◽  
Porous Substrate ◽  
Wetting Front ◽  
Existence Of A Solution ◽  
Effective Contact ◽  
Effective Dynamic

AbstractThe spreading of an incompressible viscous liquid over an isotropic homogeneous unsaturated porous substrate is considered. It is shown that, unlike the dynamic wetting of an impermeable solid substrate, where the dynamic contact angle has to be specified as a boundary condition in terms of the wetting velocity and other flow characteristics, the ‘effective’ dynamic contact angle on an unsaturated porous substrate is completely determined by the requirement of existence of a solution, i.e. the absence of a non-integrable singularity in the spreading fluid’s pressure at the ‘effective’ contact line. The obtained velocity dependence of the ‘effective’ contact angle determines the critical point at which a transition to a different flow regime takes place, where the fluid above the substrate stops spreading whereas the wetting front inside it continues to propagate.

Motion of drops on inclined surfaces in the inertial regime

2013 ◽  
Vol 726 ◽  
pp. 26-61 ◽  
Author(s):  
Baburaj A. Puthenveettil ◽  
Vijaya K. Senthilkumar ◽  
E. J. Hopfinger
Keyword(s):  
Contact Angle ◽  
Slip Length ◽  
Water Drop ◽  
Dynamic Contact Angle ◽  
Dynamic Contact ◽  
Drop Diameter ◽  
Angle Variation ◽  
Hydrodynamic Description ◽  
Receding Contact ◽  
Receding Contact Angle

AbstractWe present experimental results on high-Reynolds-number motion of partially non-wetting liquid drops on inclined plane surfaces using: (i) water on fluoro-alkyl silane (FAS)-coated glass; and (ii) mercury on glass. The former is a high-hysteresis ($3{5}^{\circ } $) surface while the latter is a low-hysteresis one (${6}^{\circ } $). The water drop experiments have been conducted for capillary numbers $0. 0003\lt Ca\lt 0. 0075$ and for Reynolds numbers based on drop diameter $137\lt Re\lt 3142$. The ranges for mercury on glass experiments are $0. 0002\lt Ca\lt 0. 0023$ and $3037\lt Re\lt 20\hspace{0.167em} 069$. It is shown that when $Re\gg 1{0}^{3} $ for water and $Re\gg 10$ for mercury, a boundary layer flow model accounts for the observed velocities. A general expression for the dimensionless velocity of the drop, covering the whole $Re$ range, is derived, which scales with the modified Bond number ($B{o}_{m} $). This expression shows that at low $Re$, $Ca\sim B{o}_{m} $ and at large $Re$, $Ca \sqrt{Re} \sim B{o}_{m} $. The dynamic contact angle (${\theta }_{d} $) variation scales, at least to first-order, with $Ca$; the contact angle variation in water, corrected for the hysteresis, collapses onto the low-$Re$ data of LeGrand, Daerr & Limat (J. Fluid Mech., vol. 541, 2005, pp. 293–315). The receding contact angle variation of mercury has a slope very different from that in water, but the variation is practically linear with $Ca$. We compare our dynamic contact angle data to several models available in the literature. Most models can describe the data of LeGrand et al. (2005) for high-viscosity silicon oil, but often need unexpected values of parameters to describe our water and mercury data. In particular, a purely hydrodynamic description requires unphysically small values of slip length, while the molecular-kinetic model shows asymmetry between the wetting and dewetting, which is quite strong for mercury. The model by Shikhmurzaev (Intl J. Multiphase Flow, vol. 19, 1993, pp. 589–610) is able to group the data for the three fluids around a single curve, thereby restoring a certain symmetry, by using two adjustable parameters that have reasonable values. At larger velocities, the mercury drops undergo a change at the rear from an oval to a corner shape when viewed from above; the corner transition occurs at a finite receding contact angle. Water drops do not show such a clear transition from oval to corner shape. Instead, a direct transition from an oval shape to a rivulet appears to occur.

Inertial and viscous effects on dynamic contact angles

1998 ◽  
Vol 357 ◽  
pp. 249-278 ◽  
Author(s):  
R. G. COX
Keyword(s):  
Reynolds Number ◽  
Solid Surface ◽  
Contact Line ◽  
Capillary Number ◽  
Slip Length ◽  
Stress Singularity ◽  
Dynamic Contact Angle ◽  
Dynamic Contact ◽  
Contact Angles ◽  
Viscous Effects

An investigation is made into the dynamics involved in the movement of the contact line when a single liquid with an interface moves into a vacuum over a smooth solid surface. In order to remove the stress singularity at the contact line, it is postulated that slip between the liquid and the solid or some other mechanism occurs very close to the contact line. It is assumed that the flow produced is inertia dominated with the Reynolds number based on the slip length being very large. Following a procedure similar to that used by Cox (1986) for the viscous-dominated situation (in which the Reynolds number based on the macroscopic length scale was assumed very small) using matched asymptotic expansions, we obtain the dependence of the macroscopic dynamic contact angle on the contact line velocity over the solid surface for small capillary number and small slip length to macroscopic lengthscale ratio. These results for the inertia-dominated situation are then extended (at the lowest order in capillary number) to an intermediate Reynolds number situation with the Reynolds number based on the slip length being very small and that based on the macroscopic lengthscale being very large.

Description of Dynamic Contact Angle on a Rough Solid Surface

2003 ◽  
Author(s):  
X. F. Peng ◽  
X. D. Wang ◽  
D. J. Lee
Keyword(s):  
Contact Angle ◽  
Solid Surface ◽  
Contact Line ◽  
Mechanical Performance ◽  
Solid Wall ◽  
Dynamic Contact Angle ◽  
Intrinsic Property ◽  
Dynamic Contact ◽  
Characteristic Parameter ◽  
Precursor Film

An investigation was conducted to understand the contact line movement and associated contact angle phenomena. Contact line was supposed to move on a thin precursor film caused by molecular interaction between solid and liquid and asperity of solid surface. It is expected that contact line has a velocity and is subject to viscous stress on the film or geometrically on the solid surface. With the introduction of a characteristic parameter, λ′, the movement of contact line and contact angle phenomena were very well described in both physics and mathematics. The viscous shearing stress exerted by liquid on solid surface was derived, and the behavior of dynamic contact angle was recognized on rough solid surfaces. The analyses indicate that characteristic parameter, λ′, is dependent upon solid wall intrinsic property and mechanical performance, not liquid property. The comparison of theoretical predictions with available experimental data in open literature showed a quite good agreement with each other.

The Effect of Path Length and Display Size on Memory for Spatial Information

2012 ◽  
Vol 59 (3) ◽  
pp. 147-152 ◽  
Author(s):  
Katherine Guérard ◽  
Sébastien Tremblay
Keyword(s):  
Path Length ◽  
Potential Role ◽  
Spatial Information ◽  
Recall Performance ◽  
Minor Role ◽  
Length Effect ◽  
Serial Memory ◽  
Fixed Reference ◽  
A Minor

In serial memory for spatial information, some studies showed that recall performance suffers when the distance between successive locations increases relatively to the size of the display in which they are presented (the path length effect; e.g., Parmentier et al., 2005) but not when distance is increased by enlarging the size of the display (e.g., Smyth & Scholey, 1994). In the present study, we examined the effect of varying the absolute and relative distance between to-be-remembered items on memory for spatial information. We manipulated path length using small (15″) and large (64″) screens within the same design. In two experiments, we showed that distance was disruptive mainly when it is varied relatively to a fixed reference frame, though increasing the size of the display also had a small deleterious effect on recall. The insertion of a retention interval did not influence these effects, suggesting that rehearsal plays a minor role in mediating the effects of distance on serial spatial memory. We discuss the potential role of perceptual organization in light of the pattern of results.

Use of Different Tissue Thromboplastins in the Control of Anticoagulant-Therapy

1958 ◽  
Vol 02 (05/06) ◽  
pp. 462-480 ◽  
Author(s):  
Marc Verstraete ◽  
Patricia A. Clark ◽  
Irving S. Wright
Keyword(s):  
Prothrombin Time ◽  
Anticoagulant Therapy ◽  
Factor Vii ◽  
Rabbit Brain ◽  
Minor Role ◽  
Rabbit Lung ◽  
Coagulation Mechanism ◽  
Time Test ◽  
The One ◽  
A Minor

SummaryAn analysis of the results of prothrombin time tests with different types of thromboplastins sheds some light on the problem why the administration of coumarin is difficult to standardize in different centers. Our present ideas on the subject, based on experimental data may be summarized as follows.Several factors of the clotting mechanism are influenced by coumarin derivatives. The action of some of these factors is by-passed in the 1-stage prothrombin time test. The decrease of the prothrombin and factor VII levels may be evaluated in the 1-stage prothrombin time determination (Quick-test). The prolongation of the prothrombin times are, however, predominantly due to the decrease of factor VII activity, the prothrombin content remaining around 50 per cent of normal during an adequate anticoagulant therapy. It is unlikely that this degree of depression of prothrombin is of major significance in interfering with the coagulation mechanism in the protection against thromboembolism. It may, however, play a minor role, which has yet to be evaluated quantitatively. An exact evaluation of factor VII is, therefore, important for the guidance of anticoagulant therapy and the method of choice is the one which is most sensitive to changes in factor VII concentration. The 1-stage prothrombin time test with a rabbit lung thromboplastin seems the most suitable method because rabbit brain preparations exhibit a factor VII-like activity that is not present in rabbit lung preparations.

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