Sharp-interface limit of the Cahn–Hilliard model for moving contact lines

2010 ◽  
Vol 645 ◽  
pp. 279-294 ◽  
Author(s):  
PENGTAO YUE ◽  
CHUNFENG ZHOU ◽  
JAMES J. FENG
Keyword(s):  
Contact Line ◽  
Diffusion Length ◽  
Slip Length ◽  
Sharp Interface ◽  
Diffuse Interface ◽  
Contact Lines ◽  
Sharp Interface Limit ◽  
Interface Models ◽  
Moving Contact ◽  
Moving Contact Lines

Diffuse-interface models may be used to compute moving contact lines because the Cahn–Hilliard diffusion regularizes the singularity at the contact line. This paper investigates the basic questions underlying this approach. Through scaling arguments and numerical computations, we demonstrate that the Cahn–Hilliard model approaches a sharp-interface limit when the interfacial thickness is reduced below a threshold while other parameters are fixed. In this limit, the contact line has a diffusion length that is related to the slip length in sharp-interface models. Based on the numerical results, we propose a criterion for attaining the sharp-interface limit in computing moving contact lines.

scholarly journals Moving contact line dynamics: from diffuse to sharp interfaces

2015 ◽  
Vol 788 ◽  
pp. 209-227 ◽  
Author(s):  
H. Kusumaatmaja ◽  
E. J. Hemingway ◽  
S. M. Fielding
Keyword(s):  
Excellent Agreement ◽  
Contact Line ◽  
Scaling Laws ◽  
Slip Length ◽  
Dynamic Contact Angle ◽  
Sharp Interface ◽  
Diffuse Interface ◽  
Fluid Viscosity ◽  
Moving Contact Line ◽  
Moving Contact

We reconcile two scaling laws that have been proposed in the literature for the slip length associated with a moving contact line in diffuse interface models, by demonstrating each to apply in a different regime of the ratio of the microscopic interfacial width $l$ and the macroscopic diffusive length $l_{D}=(M{\it\eta})^{1/2}$, where ${\it\eta}$ is the fluid viscosity and $M$ the mobility governing intermolecular diffusion. For small $l_{D}/l$ we find a diffuse interface regime in which the slip length scales as ${\it\xi}\sim (l_{D}l)^{1/2}$. For larger $l_{D}/l>1$ we find a sharp interface regime in which the slip length depends only on the diffusive length, ${\it\xi}\sim l_{D}\sim (M{\it\eta})^{1/2}$, and therefore only on the macroscopic variables ${\it\eta}$ and $M$, independent of the microscopic interfacial width $l$. We also give evidence that modifying the microscopic interfacial terms in the model’s free energy functional appears to affect the value of the slip length only in the diffuse interface regime, consistent with the slip length depending only on macroscopic variables in the sharp interface regime. Finally, we demonstrate the dependence of the dynamic contact angle on the capillary number to be in excellent agreement with the theoretical prediction of Cox (J. Fluid Mech., vol. 168, 1986, p. 169), provided we allow the slip length to be rescaled by a dimensionless prefactor. This prefactor appears to converge to unity in the sharp interface limit, but is smaller in the diffuse interface limit. The excellent agreement of results obtained using three independent numerical methods, across several decades of the relevant dimensionless variables, demonstrates our findings to be free of numerical artefacts.

scholarly journals Reflections on the article “Moving contact lines and dynamic contact angles: a ‘litmus test’ for mathematical models and some new challenges” by Yulii D. Shikhmurzaev

2020 ◽  
Vol 229 (10) ◽  
pp. 1979-1987 ◽  
Author(s):  
Dieter Bothe
Keyword(s):  
Mathematical Models ◽  
Contact Line ◽  
Dynamic Contact ◽  
Contact Angles ◽  
Sharp Interface ◽  
Contact Lines ◽  
Moving Contact ◽  
Moving Contact Lines ◽  
New Challenges ◽  
Litmus Test

Abstract We carefully consider the ‘litmus test’ proposed by Yulii D. Shikhmurzaev [Y.D. Shikhmurzaev, Eur. Phys. J. Special Topics 229, 1945 (2020)] in the context of the sharp-interface/sharp-contact line model.

The dynamics of three-dimensional liquid bridges with pinned and moving contact lines

2012 ◽  
Vol 707 ◽  
pp. 521-540 ◽  
Author(s):  
Shawn Dodds ◽  
Marcio S. Carvalho ◽  
Satish Kumar
Keyword(s):  
Contact Line ◽  
Liquid Bridge ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Liquid Bridges ◽  
Contact Lines ◽  
Flat Surfaces ◽  
Moving Contact ◽  
Moving Contact Lines ◽  
Contact Line Friction

AbstractLiquid bridges with moving contact lines are relevant in a variety of natural and industrial settings, ranging from printing processes to the feeding of birds. While it is often assumed that the liquid bridge is two-dimensional in nature, there are many applications where either the stretching motion or the presence of a feature on a bounding surface lead to three-dimensional effects. To investigate this we solve Stokes equations using the finite-element method for the stretching of a three-dimensional liquid bridge between two flat surfaces, one stationary and one moving. We first consider an initially cylindrical liquid bridge that is stretched using either a combination of extension and shear or extension and rotation, while keeping the contact lines pinned in place. We find that whereas a shearing motion does not alter the distribution of liquid between the two plates, rotation leads to an increase in the amount of liquid resting on the stationary plate as breakup is approached. This suggests that a relative rotation of one surface can be used to improve liquid transfer to the other surface. We then consider the extension of non-cylindrical bridges with moving contact lines. We find that dynamic wetting, characterized through a contact line friction parameter, plays a key role in preventing the contact line from deviating significantly from its original shape as breakup is approached. By adjusting the friction on both plates it is possible to drastically improve the amount of liquid transferred to one surface while maintaining the fidelity of the liquid pattern.

scholarly journals Wetting failure and contact line dynamics in a Couette flow

2008 ◽  
Vol 614 ◽  
pp. 471-493 ◽  
Author(s):  
M. SBRAGAGLIA ◽  
K. SUGIYAMA ◽  
L. BIFERALE
Keyword(s):  
Capillary Number ◽  
Slip Length ◽  
Immiscible Fluids ◽  
Sharp Interface ◽  
Diffuse Interface ◽  
Control Parameters ◽  
Interface Models ◽  
Critical Capillary Number ◽  
Stationary Interface ◽  
Interface Treatment

Liquid–liquid wetting failure is investigated in a two-dimensional Couette system with two immiscible fluids of arbitrary viscosity. The problem is solved exactly using a sharp interface treatment of hydrodynamics (lubrication theory) as a function of the control parameters – capillary number, viscosity ratio and separation of scale – i.e. the slip length versus the macroscopic size of the system. The transition at a critical capillary number, from a stationary to a non-stationary interface, is studied while changing the control parameters. Comparisons with similar existing analyses for other geometries, such as the Landau–Levich problem, are also carried out. A numerical method of analysis is also presented, based on diffuse interface models obtained from multiphase extensions of the lattice Boltzmann equation. Sharp interface and diffuse interface models are quantitatively compared, indicating the correct limit of applicability of the diffuse interface models.

A theoretical study of two-phase flow through a narrow gap with a moving contact line: viscous fingering in a Hele-Shaw cell

1990 ◽  
Vol 221 ◽  
pp. 53-76 ◽  
Author(s):  
Steven J. Weinstein ◽  
E. B. Dussan ◽  
Lyle H. Ungar
Keyword(s):  
Thin Films ◽  
Contact Angle ◽  
Contact Line ◽  
Viscous Fingering ◽  
Contact Lines ◽  
Fluid Interface ◽  
Moving Contact Line ◽  
Moving Contact ◽  
Moving Contact Lines ◽  
Hele Shaw Cell

The problem of viscous fingering in a Hele-Shaw cell with moving contact lines is considered. In contrast to the usual situation where the displaced fluid coats the solid surface in the form of thin films, here, both the displacing and the displaced fluids make direct contact with the solid. The principal differences between these two situations are in the ranges of attainable values of the gapwise component of the interfacial curvature (the component due to the bending of the fluid interface across the small gap of the Hele-Shaw cell), and in the introduction of two additional parameters for the case with moving contact lines. These parameters are the receding contact angle, and the sensivity of the dynamic angle to the speed of the contact line. Our objective is the prediction of the shape and widths of the fingers in the limit of small capillary number, Uμ/σ. Here, U denotes the finger speed, μ denotes the dynamic viscosity of the more viscous displaced fluid, and σ denotes the surface tension of the fluid interface. As might be expected, there are similarities and differences between the two problems. Despite the fact that different equations arise, we find that they can be analysed using the techniques introduced by McLean & Saffman and Vanden-Broeck for the thin-film case. The nature of the multiplicity of solutions also appears to be similar for the two problems. Our results indicate that when contact lines are present, the finger shapes are sensitive to the value of the contact angle only in the vicinity of its nose, reminiscent of experiments where bubbles or wires are placed at the nose of viscous fingers when thin films are present. On the other hand, in the present problem at least two distinct velocity scales emerge with well-defined asymptotic limits, each of these two cases being distinguished by the relative importance played by the two components of the curvature of the fluid interface. It is found that the widths of fingers can be significantly smaller than half the width of the cell.

Asymptotic theory of fluid entrainment in dip coating

2018 ◽  
Vol 844 ◽  
pp. 1026-1037 ◽  
Author(s):  
Jian Qin ◽  
Peng Gao
Keyword(s):  
Contact Line ◽  
Analytical Formula ◽  
Dip Coating ◽  
Wetting Transition ◽  
Contact Lines ◽  
Moving Contact ◽  
Moving Contact Lines ◽  
Two Fluid ◽  
Large Speed ◽  
Line Movement

When a contact line moves with a sufficiently large speed, liquid or gas films can be entrained on a solid depending on the direction of contact-line movement. In this work, the contact-line dynamics in the situation of a generic two-fluid system is investigated. We demonstrate that the hydrodynamics of a contact line, no matter whether advancing or receding, can formally reduce to that of a receding one with small interfacial slopes. Since the latter can be well treated under the classical lubrication approximation, this analogy allows us to derive an asymptotic solution of the interfacial profiles for arbitrary values of contact angle and viscosity ratio. For the dip-coating geometry, we obtain, with no adjustable parameters, an analytical formula for the critical speed of wetting transition, which in particular predicts the onset of both liquid and gas entrainment. Moreover, the present analysis also builds a novel connection between the Cox–Voinov law and classical lubrication theory for moving contact lines.

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