Ak Allen Chesters

The drainage and rupture of partially-mobile films between colliding drops at constant approach velocity

A numerical study is presented of the drainage and rupture of the liquid film between two drops whose centres approach each other at constant velocity. The considerations are restricted to the partially-mobile case (in which the drop viscosity is rate-determining) and to small approach velocities. The latter restriction permits a transformation of the governing equations to a single universal form, which is solved with the help of boundary integral theory. As in the constant force case, the numerical results show the formation of a dimple but the final drainage behaviour differs considerably. Finally, the influence of van der Waals forces is investigated and the results are shown to correspond well with a simple model proposed earlier for the effective critical film-rupture thickness.

The effect of the dispersed to continuous-phase viscosity ratio on film drainage between interacting drops

Abstract The deformation and drainage of the film between colliding drops is studied numerically at small capillary numbers, small Reynolds numbers and a range of dispersed to continuous-phase viscosity ratios, λ, covering the transition from partially-mobile to immobile interfaces. Two types of collision are considered: constant approach velocity and constant interaction force. The problem is solved numerically by means of a finite difference method for the equations in the continuous phase and a boundary integral method or finite-element method in the drops. The velocity profile in the gap between the drops is the sum of a uniform and a parabolic contribution, governed respectively by viscous forces within the dispersed and the continuous phases. Solutions to date concern the limiting cases of partially-mobile or immobile interfaces, in which either the parabolic or plug contribution is negligible. A transformation of variables then results in a universal set of governing equations. In the intermediate regime a transformed viscosity ratio, λ ∗ , enters these equations. In the constant-force case, the transformed drainage rate increases monotonically with λ ∗ and the final (rate-determining) stage of drainage is well described by a power-law dependence of the minimum film thickness on time, enabling compact analytical approximations to be developed for the drainage time. These expressions reduce to those in the partially-mobile and immobile limits for λ ∗ -values outside the range 10 ∗ 3 . In the constant-velocity case the behavior is more subtle, drainage at the periphery of the film being strongly affected by the plug contribution in the adjoining outer region. This provides an explanation for the much higher final drainage rates predicted numerically under constant-velocity conditions in the partially-mobile case. From a practical point of view the most important case to model is that dividing coalescing from non-coalescing drop collisions. While the constant-force approximation is probably closest to the final interaction in this case, the sensitivity of the drainage behavior to the outer boundary conditions suggests that more realistic simulations are required which take account of the actual, time-dependent interaction force/velocity.

Film models for transport phenomena with fog formation: the classical film model

In the present analysis the classical film model (or film theory) is reviewed and extended. First, on the basis of a thorough analysis, the governing equations of diffusion, energy and momentum of a stagnant film are derived and solved. Subsequently, the well-known correction factors for the effect of suction/injection on mass, heat and momentum transfer are derived. Next, employing global balances of mass, energy and momentum, the film model is applied to channel flow. This application yields a new expression for the pressure drop and hence it is compared extensively with experimental and theoretical results of previous investigators, yielding good agreement. The onset of fog formation in a binary mixture, both in the transferring film and/or in the bulk, is explained graphically with the help of the relation between temperature and vapour mass fraction and the saturation line of the vapour.

An approximate analytical solution of the hydrodynamic problem associated with an advancing liquid-gas contact line

Abstract An alternative to perturbation techniques or full numerical solution is developed for the free-surface problem associated with an advancing liquid-gas contact line. The method, which makes use of a local-wedge approximation to obtain the free-surface pressure variation, leads to a second-order ordinary differential equation for the meniscus shape. Both analytical considerations and comparisons with available full numerical solutions for capillary tubes confirm the validity of the meniscus equation up to values of the capillary number (Ca) of order 10−1. At higher Ca the equation retains its validity in the wall region, for which an approximate analytical solution is derived. Matching of this solution with the central circular meniscus profile leads to an analytical approximation for the advancing contact angle, which compares excellently with available data (Ca up to an order 1). In contrast with preceding analyses, the classical approximations—in particular, that of no slip—are assumed to retain their validity up to the order of a molecular dimension from the wall, at which point the true contact angle is reached. While this angle is again supposed to be equal to the static value, this assumption is not critical to the dynamic angle predicted.

The influence of inter-phase mass transfer on the drainage of partially-mobile liquid films between drops undergoing a constant interaction force

Abstract The equations describing the drainage of a partially mobile liquid film separating two drops under a constant interaction force (Yiantsios and Davis, 1990; Chesters, 1991) are extended to include inter-phase solute transfer and the resulting Marangoni forces. In the limit of gentle interactions and small variations in solute concentration, a suitable transformation of variables reduces the number of parameters entering the equations to four: a transformed partition coefficient P , Peclet numbers in each phase and a Marangoni number Ma =Δ σ / σa ′2 , in which Δσ denotes the variation in interfacial tension corresponding to the difference in solute concentration between the phases and a ′ the radius of the draining film, normalized with the equivalent radius of the drops ( a ′≪1). Numerical solutions are presented for both positive and negative values of Ma (corresponding to solute transfer both to and from the drops) for fixed, physically pertinent values of the others parameters, including a large Peclet number for which the diffusion boundary layer within the drop is thin, thereby somewhat simplifying the equations to be solved. In accordance with experimental indications, the acceleration of drainage by Marangoni effects in the case of D → C transfer is found to be immense, final drainage rates rising by two orders of magnitude for concentration differences of only a few percent. These effects are associated with an intensification of the dimple. For C → D transfer, dimple formation is suppressed and initial drainage rates greatly reduced.

Interdrop coalescence with mass transfer: comparison of the approximate drainage models with numerical results

Abstract The partially mobile, plane-film model developed to describe film drainage and rupture during coalescence in liquid–liquid dispersions is extended to take account of interfacial-tension gradients generated by mass transfer. The resulting Marangoni forces are predicted to greatly accelerate film drainage (which in general corresponds to dispersed to continuous phase transfer) and to diminish film drainage in the negative case. The first model is based on the approximation of constant pressure and interfacial tensions outside the film. The predictions from this model agrees with observations and available numerical data, in the case of mass transfer from dispersed to continuous phase. While for mass transfer from continuous to dispersed phase, a second model is proposed, in this case the first model is adapted to take account of the location of the region of maximum concentration gradients, which moves radially outwards as a result of the growth of the continuous phase-concentration boundary layers. At large times, the new model predicts an asymptotic return to the drainage rate in the absence of mass transfer.

AN EXPERIMENTAL STUDY OF THE MENISCUS SHAPE ASSOCIATED WITH MOVING LIQUID-FLUID CONTACT LINES

Using a new technique involving light reflection, the interfacial curvature of a meniscus formed by a well wetting liquid steadily displacing a gas in a glass capillary has been measured down to about 50 nm from the solid. Within the domain explored, the measured meniscus curvature increases strongly as the wall is approached, in agreement with classical models which make use of the continuum approximation, no slip, etc. The inner length scale, at which such models fail, is inferred from the measurements to be of the order of a molecular dimension, suggesting that non-continuum effects dominate. A comparison of measured dynamic contact angles of liquid-liquid pairs of large viscosity ratio with a model developed earlier by the authors, incorporating such an inner length scale, suggests that the true contact angle in the advancing fluid increases significantly with line speed in one of the cases.

An approximate solution of the hydrodynamic problem associated with moving liquid-liquid contact lines

Abstract The same approach used by the authors in the context of advancing and receding contact lines is applied to the case of liquid-liquid contact lines. The result is an ordinary differential equation whose solution provides an approximate description of the shape of a moving meniscus. It is shown that a system with a viscosity ratio of 103 or more may be regarded as being a purely advancing/receding case. From the comparison of the present model with experimental results a dependence of the true contact angle on the line speed is inferred.

An approximate solution of the hydrodynamic problem associated with receding liquid-gas contact lines

Abstract An ordinary differential equation, derived previously by the authors to describe liquid-gas menisci in the context of advancing contact lines, is applied to the receding case. The existence of a critical capillary number is demonstrated above which no solution of the differential equations exists. This critical capillary number exhibits a strong dependence on the system scale and true contact angle at the wall. Comparison of critical capillary numbers, predicted by the model and obtained from experiments, suggests that at the critical capillary number the true contact angle at the wall is smaller than the (receding) static contact angle.

The influence of surfactants on the hydrodynamics of surface wetting

The hydrodynamic model of steady wetting developed by Boender et al. [Boender, W., Chesters, A.K., van der Zanden, A.J.J., 1991. Int. J. Multiphase Flow 17, 661–676] is extended to include the effect of a (non-ionic) surfactant. The approximation that the meniscus inclination becomes equal to the static contact angle at a distance from the solid of the order of a molecular dimension is extended to take account of the local surfactant concentration, making use of Youngs law. A second inner boundary condition, provided by a surfactant balance at the contact line, places a restriction on the speed at which the interface is shed, leading to surfactant accumulation and partial or almost total immobilization of the interface which reduces the wetting speed. Under certain conditions this immobilization is self-stabilizing, leading to hysteresis effects.