Yulii D. Shikhmurzaev

Moving contact lines in liquid/liquid/solid systems

A general mathematical model which describes the motion of an interface between immiscible viscous fluids along a smooth homogeneous solid surface is examined in the case of small capillary and Reynolds numbers. The model stems from a conclusion that the Young equation, σ 1 cos θ = σ 2 − σ 3 , which expresses the balance of tangential projection of the forces acting on the three-phase contact line in terms of the surface tensions σ i and the contact angle θ, together with the well-established experimental fact that the dynamic contact angle deviates from the static one, imply that the surface tensions of contacting interfaces in the immediate vicinity of the contact line deviate from their equilibrium values when the contact line is moving. The same conclusion also follows from the experimentally observed kinematics of the flow, which indicates that liquid particles belonging to interfaces traverse the three-phase interaction zone (i.e. the ‘contact line’) in a finite time and become elements of another interface – hence their surface properties have to relax to new equilibrium values giving rise to the surface tension gradients in the neighbourhood of the moving contact line. The kinematic picture of the flow also suggests that the contact-line motion is only a particular case of a more general phenomenon – the process of interface formation or disappearance – and the corresponding mathematical model should be derived from first principles for this general process and then applied to wetting as well as to other relevant flows. In the present paper, the simplest theory which uses this approach is formulated and applied to the moving contact-line problem. The model describes the true kinematics of the flow so that it allows for the ‘splitting’ of the free surface at the contact line, the appearance of the surface tension gradients near the contact line and their influence upon the contact angle and the flow field. An analytical expression for the dependence of the dynamic contact angle on the contact-line speed and parameters characterizing properties of contacting media is derived and examined. The role of a ‘thin’ microscopic residual film formed by adsorbed molecules of the receding fluid is considered. The flow field in the vicinity of the contact line is analysed. The results are compared with experimental data obtained for different fluid/liquid/solid systems.

The moving contact line on a smooth solid surface

Abstract A mathematical model for the advancing contact-line motion on a smooth solid surface is proposed. It is shown that in the spreading of liquids over solid surfaces, the flow causes a surface tension gradient along the liquid-solid interface which influences the flow and, in the case of small capillary and Reynolds numbers, determines the dynamic contact angle and the force between the liquid and solid in the vicinity of the contact line. The model: (a) eliminates the shear-stress singularity of the classical model; (b) describes the fluid motion as rolling, in complete agreement with direct experimental observations; (c) determines the dynamic contact angle and the tangential force dependence on the contact-line speed; (d) explains the existence of the maximum contact angle values

Experimental evidence of nonlocal hydrodynamic influence on the dynamic contact angle

The dynamic contact angle formed when a liquid curtain impinges onto a moving solid is measured for aqueous glycerol solutions in different flow regimes. It is usually assumed that the dynamic contact angle is simply a function of the contact-line speed and the material properties of the contacting media. The new results show that this is not the case. For a given gas/liquid/solid combination and a given contact-line speed, the dynamic contact angle can be varied by varying the flow rate of the liquid and/or the curtain height, that is by varying the flow field near the contact line. The possibility of attributing this effect merely to free-surface bending and interpreting the results in terms of the so-called “apparent” contact angle is discussed and ruled out on the basis of some general qualitative arguments and analysis of the characteristic length scales involved. A probable connection between the observed effect and the physical mechanism of interface disappearance and formation incorporated in a re...

Spreading of drops on solid surfaces in a quasi-static regime

The problem of interaction of a drop with a solid boundary is formulated in the framework of a recently developed theory of the three-phase contact line motion and analyzed in the case of finite Bond and small capillary and Weber numbers. Evolution of the free-surface shape in a quasi-static regime of the drop spreading under gravity on a horizontal plane and on the surface of a rotating disk is investigated. In the considered regime, the free-surface shape deformation in time is independent of the initial conditions of the drop deposition onto the solid surface, while the three-phase contact-line motion is described by the same equations as in a general case. This feature makes the quasi-static regime informative and desirable from the point of view of investigation of the wetting phenomenon. Accuracy of the so-called “spherical cap approximation’’ often used in experimental studies of wetting is discussed. The theory describes both the “spontaneous” and “forced” regimes of the drop spreading and the transition between them. The results are compared with experimental data.

Coalescence of liquid drops: Different models versus experiment

The process of coalescence of two identical liquid drops is simulated numerically in the framework of two essentially different mathematical models, and the results are compared with experimental data on the very early stages of the coalescence process reported recently. The first model tested is the “conventional” one, where it is assumed that coalescence as the formation of a single body of fluid occurs by an instant appearance of a liquid bridge smoothly connecting the two drops, and the subsequent process is the evolution of this single body of fluid driven by capillary forces. The second model under investigation considers coalescence as a process where a section of the free surface becomes trapped between the bulk phases as the drops are pressed against each other, and it is the gradual disappearance of this “internal interface” that leads to the formation of a single body of fluid and the conventional model taking over. Using the full numerical solution of the problem in the framework of each of the two models, we show that the recently reported electrical measurements probing the very early stages of the process are better described by the interface formation/disappearance model. New theory-guided experiments are suggested that would help to further elucidate the details of the coalescence phenomenon. As a by-product of our research, the range of validity of different “scaling laws” advanced as approximate solutions to the problem formulated using the conventional model is established.

Mathematical modeling of wetting hydrodynamics

The simplest mathematical model based on a new approach to the moving contact-line problem is examined, and is shown to adequately reflect the main features of wetting phenomena revealed by experiments. A simple algebraic equation describing the dependence of the dynamic contact angle on the contact-line speed and other parameters of the problem is derived and analyzed. A preliminary quantitative comparison of the theory with experimental data is carried out.

The dynamics of liquid drops and their interaction with solids of varying wettabilities

Microdrop impact and spreading phenomena are explored as an interface formation process using a recently developed computational framework. The accuracy of the results obtained from this framework for the simulation of high deformation free-surface flows is confirmed by a comparison with previous numerical studies for the large amplitude oscillations of free liquiddrops. Our codes ability to produce high resolution benchmark calculations for dynamic wetting flows is then demonstrated by simulating microdrop impact and spreading on surfaces of greatly differing wettability. The simulations allow one to see features of the process which go beyond the resolution available to experimental analysis. Strong interfacial effects which are observed at the microfluidic scale are then harnessed by designing surfaces of varying wettability that allow new methods of flow control to be developed.

Coalescence and capillary breakup of liquid volumes

The problem of the mathematical modeling of coalescence and breakup of liquid volumes surrounded by an inviscid gas is considered. As is shown, an unphysical singularity in the known self-similar solutions of the Navier–Stokes equations intended to describe the topological transition of the flow domain arises as a consequence of the assumption that the free surface becomes smooth immediately after the onset of coalescence or remains so up to the very moment of breakup. Then the standard kinematic boundary condition prescribes that fluid particles belonging to the free surface remain there at all times and thus couples the scales for lengths and velocities in a self-similar solution leading to the singularity. An alternative approach allowing one to remove the singularity at a macroscopic level is formulated. Its key idea is that the topological transition, being a particular case of an interface formation/disappearance process, is associated with a free-surface cusp either propagating away from the point ...

A parametric study of the coalescence of liquid drops in a viscous gas

The coalescence of two liquid drops surrounded by a viscous gas is considered in the framework of the conventional model. The problem is solved numerically with particular attention paid to resolving the very initial stage of the process which only recently has become accessible both experimentally and computationally. A systematic study of the parameter space of practical interest allows the influence of the governing parameters in the system to be identified and the role of viscous gas to be determined. In particular, it is shown that the viscosity of the gas suppresses the formation of toroidal bubbles predicted in some cases by early computations where the gas’ dynamics was neglected. Focusing computations on the very initial stages of coalescence and considering the large parameter space allows us to examine the accuracy and limits of applicability of various ‘scaling laws’ proposed for different ‘regimes’ and, in doing so, reveal certain inconsistencies in recent works. A comparison with experimental data shows that the conventional model is able to reproduce many qualitative features of the initial stages of coalescence, such as a collapse of calculations onto a ‘master curve’ but, quantitatively, overpredicts the observed speed of coalescence and there are no free parameters to improve the fit. Finally, a phase diagram of parameter space, differing from previously published ones, is used to illustrate the key findings.

Viscous flow over a chemically patterned surface

The classical fluid dynamics boundary condition of no slip suggests that variation in the wettability of a solid should not affect the flow of an adjacent liquid. However, experiments and molecular dynamics simulations indicate that this is not the case. In this paper we show how flow over a solid substrate with variations of wettability can be described in a continuum framework using the interface formation theory developed earlier. Results demonstrate that a shear flow over a perfectly flat solid surface is disturbed by a change in its wettability, i.e., by a change in the chemistry of the solid substrate. The magnitude of the effect is shown to be proportional to cos theta1 - cos theta2, where theta1 and theta2, are the equilibrium contact angles that a liquid-gas free surface would form with the two chemically different parts of the solid surface.