scholarly journals Calculation of a key function in the asymptotic description of moving contact lines

Author(s):  
Julian F Scott

Summary An important element of the asymptotic description of flows having a moving liquid/gas interface which intersects a solid boundary is a function denoted $Q_i \left( \alpha \right)$ by Hocking and Rivers (The spreading of a drop by capillary action, J. Fluid Mech. 121 (1982) 425–442), where $0 < \alpha < \pi$ is the contact angle of the interface with the wall. $Q_i \left( \alpha \right)$ arises from matching of the inner and intermediate asymptotic regions introduced by those authors and is required in applications of the asymptotic theory. This article describes a new numerical method for the calculation of $Q_i \left( \alpha \right)$, which, because it explicitly allows for the logarithmic singularity in the kernel of the governing integral equation and uses quadratic interpolation of the non-singular factor in the integrand, is more accurate than that employed by Hocking and Rivers. Nonetheless, our results show good agreement with theirs, with, however, noticeable departures near $\alpha = \pi $. We also discuss the limiting cases $\alpha \to 0$ and $\alpha \to \pi $. The leading-order terms of $Q_i \left( \alpha \right)$ in both limits are in accord with the analysis of Hocking (A moving fluid interface. Part 2. The removal of the force singularity by a slip flow, J. Fluid Mech. 79 (1977) 209–229). The next-order terms are also considered. Hocking did not go beyond leading order for $\alpha \to 0$, and we believe his results for the next order as $\alpha \to \pi $ to be incorrect. Numerically, we find that the next-order terms are $O\left( {\alpha ^2} \right)$ for $\alpha \to 0$ and $O\left( 1 \right)$ as $\alpha \to \pi $. The latter result agrees with Hocking, but the value of the $O\left( 1 \right)$ constant does not. It is hoped that giving details of the numerical method and more precise information, both numerical and in terms of its limiting behaviour, concerning $Q_i \left( \alpha \right)$ will help those wanting to use the asymptotic theory of contact-line dynamics in future theoretical and numerical work.

1993 ◽  
Vol 137 ◽  
pp. 535-537 ◽  
Author(s):  
I.W. Roxburgh ◽  
S.V. Vorontsov

AbstractWe extend the second-order asymptotic description developed by Tassoul (1980, 1990) to the forth order, taking into account both gravity perturbations and realistic (non-polytropic) structure of the stellar envelope. We examine the accuracy of the asymptotic description by the direct computations for a solar model.


1977 ◽  
Vol 79 (2) ◽  
pp. 209-229 ◽  
Author(s):  
L. M. Hocking

If the no-slip condition is used to determine the flow produced when a fluid interface moves along a solid boundary, a non-integrable stress is obtained. In part 1 of this study (Hocking 1976), it was argued that, when allowance was made for the presence of irregularities on the solid boundary, an effective slip coefficient could be found, which might remove the difficulty.This paper examines the effect of a slip coefficient on the flow in the neighbourhood of the contact line. Particular cases which are solved in detail are liquid–gas interfaces at an arbitrary angle, and normal contact of fluids of arbitrary viscosity. The contribution of the vicinity of the contact line to the force on the boundary is obtained.The inner region, near the contact line, must be matched with an outer flow, in which the no-slip condition can be applied, in order to obtain the total value of the force on the boundary. This force is determined for the flow of two fluids between parallel plates and in a pipe, with a plane interface. The enhanced resistance produced by the presence of the interface is calculated, and it is shown to be equivalent to an increase in the length of the column of fluid by a small multiple of the pipe radius.


1997 ◽  
Vol 337 ◽  
pp. 49-66 ◽  
Author(s):  
Q. CHEN ◽  
E. RAMÉ ◽  
S. GAROFF

The dynamics of a spreading liquid body are dictated by the interface shape and flow field very near the moving contact line. The interface shape and flow field have been described by asymptotic models in the limit of small capillary number, Ca. Previous work established the validity and limitations of these models of the interface shape (Chen et al. 1995). Here, we study the flow field near the moving contact line. Using videomicroscopy, particle image velocimetry, and digital image analysis, we simultaneously make quantitative measurements of both the interface shape and flow field from 30 μm to a few hundred microns from the contact line. We compare our data to the modulated-wedge solution for the velocity field near a moving contact line (Cox 1986). The measured flow fields demonstrate quantitative agreement with predictions for Ca[les ]0.1, but deviations of ∼5% of the spreading velocity at Ca≈0.4. We observe that the interface shapes and flow fields become geometry independent near the contact line. Our experimental technique provides a way of measuring the interface shape and velocity field to be used as boundary conditions for numerical calculations of the macroscopic spreading dynamics.


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