ALBERT

Description: The evaporating meniscus of a perfectly wetting fluid exhibits an apparent contact angle Θ that is a function of superheat. Existing theory predicts Θ and the heat flow from the contact region as part of the solution of a free-boundary problem. That theory admits the possibility that much of the heat flow occurs at the nanometre scale l(Θ) at which Θ is determined. Here, the heat flow at that scale is proved negligible in typical applications. A phenomenological model of the contact region then holds since the part of the wetting film thinner than l(Θ) can be replaced by an apparent contact line. Self-consistency arguments are used to derive conditions under which (i) the phase interface can be taken as linear with assumed contact angle Θ; (ii) the heat flux to the liquid side of the phase interface is given by Newton's law of cooling with predicted heat transfer coefficient h; and (iii) the temperature satisfies Laplace's equation within the phases. When these conditions are met, prediction of the heat flow is decoupled from the physically non-trivial problem of predicting Θ. Next, this conduction theory is used to find the heat flow from the contact region of a meniscus on a conductive slab. The solution depends on Θ, the liquid-solid conductivity ratio k = K(l)/K(s) and a Biot number B = hd/K(l) based on slab thickness d. Asymptotic and numerical analysis is used to find the temperature in the double limit B-1 → 0 and k → 0. The solution has an inner-and-outer structure, and properties of the inner region prove universal. Formulae given here for the heat flow and contact line temperature on a slab thus apply to more complex geometries. Further, the solution explains the main features seen in published simulations of evaporation from conductive solids. Near the contact line, the solid temperature varies rapidly on the scale d of the slab thickness, but varies slowly with respect to the liquid temperature. The solid temperature thus proves uniform at the scale on which Θ is determined. Lastly, the quantitative predictions of the simplified model are verified against both new and published numerical solutions of the existing theory. In typical applications, the new formulae give the heat flow and contact line temperature with an error of about 10%. This error is due to the approximations made to derive the simplified model, rather than to those made to solve it.

Description: The stationary meniscus of an evaporating, perfectly wetting system exhibits an apparent contact angle Θ which vanishes with the applied temperature difference ΔT, and is maintained for ΔT 〉 0 by a small-scale flow driven by evaporation. Existing theory predicts Θ and the heat flow q∗ from the contact region as the solution of a free-boundary problem. Though that theory admits the possibility that Θ and q∗ are determined at the same scale, we show that, in practice, a separation of scales gives the theory an inner and outer structure; Θ is determined within an inner region contributing a negligible fraction of the total evaporation, but q∗ is determined at larger scales by conduction across an outer liquid wedge subtending an angle Θ. The existence of a contact angle can thus be assumed for computing the heat flow; the problems for Θ and q∗ decouple. We analyse the inner problem to derive a formula for Θ as a function of ΔT and material properties; the formula agrees closely with numerical solutions of the existing theory. Though microphysics must be included in the model of the inner region to resolve a singularity in the hydrodynamic equations, Θ is insensitive to microphysical detail because the singularity is weak. Our analysis shows that Θ is determined chiefly by the capillary number Ca = μlVl/σ based on surface tension σ, liquid viscosity μl and a velocity scale Vl set by evaporation kinetics. To illustrate this result of our asymptotic analysis, we show that computed angles lie close to the curve Θ = 2.2Ca1/4; a small scatter of ±15% about that curve is the only hint that Θ depends on microphysics. To test our scaling relation, we use film profiles measured by Kim (1994) to determine experimental values of Θ and Ca; these are the first such values to be published for the evaporating meniscus. Agreement between theory and experiment is adequate; the difference is less than ±40% for 9 of 15 points, while the scatter within experimental values is ±25%.

Description: We consider the evaporating meniscus of a perfectly wetting liquid in a channel whose superheated walls are at common temperature. Heat flows by pure conduction from the walls to the phase interface; there, evaporation induces a small-scale liquid flow concentrated near the contact lines. Liquid is continually fed to the channel, so that the interface is stationary, but distorted by the pressure differences caused by the small-scale flow. To determine the heat flow, we make a systematic analysis of this free-boundary problem in the limit of vanishing capillary number based on the velocity of the induced flow. Because surface tension is then large, the induced flow can distort the phase interface only in a small inner region near the contact lines; the effect is to create an apparent contact angle Θ depending on capillary number. Though, in general, there can be significant heat flow within that small inner region, the presence of an additional small parameter in the problem implies that, in practice, heat flow is significant only within the large outer region where the interface shape is determined by hydrostatics and Θ. We derive a formula for the heat flow, and show that the channel geometry affects the heat flow only through the value of the interface curvature at the contact line. Consequently, the heat flow relation for a channel can be applied to other geometries.

Description: To clarify the function of gravity in the shallow-water theory of the interfacial instability in aluminium reduction cells, we analyse the existing long-wave theory of the instability in the limit of vanishing gravitational acceleration g. The flow then has an inner and outer structure, with gravity remaining essential within thin layers coating the cell walls. In those thin wall layers, the growing disturbance takes the form of a trapped magnetogravity wave propagating horizontally on the internal interface, and the growth rate σ is determined by the coupling of that edge wave to the large-scale flow in the core of the cell; that coupling is expressed as an oblique-derivative problem for the core flow. Although σ is asymptotically independent of g, gravity is essential to the long-wave instability because a correlation imposed by the magneto-gravity waves is essential for the disturbance to extract power from the mean state.

Description: Viscous fingering can occur as a three-dimensional disturbance to plane flow of a hot thermoviscous liquid in a Hele-Shaw cell with cold isothermal walls. This work assumes the principle of exchange of stabilities, and uses a temporal stability analysis to find the critical viscosity ratio and finger spacing as functions of channel length, Lc. Viscous heating is taken as negligible, so the liquid cools with distance (x) downstream. Because the base flow is spatially developing, the disturbance equations are not fully separable. They admit, however, an exact solution for a liquid whose viscosity and specific heats are arbitrary functions of temperature. This solution describes the neutral disturbances in terms of the base flow and an amplitude, A(x). The stability of a given (computed) base flow is determined by solving an eigenvalue problem for A(X), and the critical finger spacing. The theory is illustrated by using it to map the instability for variable-viscosity flow with constant specific heat. Two fingering modes are predicted, one being a turning-point instability. The preferred mode depends on Lc. Finger spacing is comparable with the thermal entry length in a long channel, and is even larger in short channels. When applied to magmatic systems, the results suggest that fingering will occur on geological scales only if the system is about freeze.

Description: Motivated by experiments showing that a sessile drop of volatile perfectly wetting liquid initially advances over the substrate, but then reverses, we formulate the problem describing the contact region at reversal. Assuming a separation of scales, so that the radial extent of this region is small compared with the instantaneous radius$a$of the apparent contact line, we show that the time scale characterizing the contact region is small compared with that on which the bulk drop is evolving. As a result, the contact region is governed by a boundary-value problem, rather than an initial-value problem: the contact region has no memory, and all its properties are determined by conditions at the instant of reversal. We conclude that the apparent contact angle$heta $is a function of the instantaneous drop radius$a$, as found in the experiments. We then non-dimensionalize the boundary-value problem, and find that its solution depends on one parameter$mathscr{L}$, a dimensionless surface tension. According to this formulation, the apparent contact angle is well-defined: at the outer edge of the contact region, the film slope approaches a limit that is independent of the curvature of bulk drop. In this, it differs from the dynamic contact angle observed during spreading of non-volatile drops. Next, we analyse the boundary-value problem assuming$mathscr{L}$to be small. Though, for arbitrary$mathscr{L}$, determining$heta $requires solving the steady diffusion equation for the vapour, there is, for small$mathscr{L}$, a further separation of scales within the contact region. As a result,$heta $is now determined by solving an ordinary differential equation. We predict that$heta $varies as${a}^{- 1/ 6} $, as found experimentally for small drops ($alt 1~mathrm{mm} $). For these drops, predicted and measured angles agree to within 10–30 %. Because the discrepancy increases with$a$, but$mathscr{L}$is a decreasing function of$a$, we infer that some process occurring outside the contact region is required to explain the observed behaviour of larger drops having$agt 1~mathrm{mm} $.

Description: During intaglio (gravure) printing, a blade wipes excess ink from the engraved plate with the object of leaving ink-filled cells defining the image to be printed. That objective is not completely attained. Capillarity draws some ink from the cell into a meniscus connecting the blade to the substrate, and the continuing motion of the engraved plate smears that ink over its surface. By examining the limit of vanishing capillary number (, based on substrate speed), we reduce the problem of determining smear volume to one of hydrostatics. Using numerical solutions of the corresponding free-boundary problem for the Stokes equations of motion, we show that the hydrostatic theory provides an upper bound to smear volume for finite . The theory explains why polishing to reduce the tip radius of the blade is an effective way to control smearing. © 2016 Cambridge University Press.