ALBERT

Beschreibung: In certain geophysical contexts such as lava lakes and mantle convection, a cold, viscous boundary layer forms over a deep pool. The following model problem investigates the buoyant instability of the layer. Beneath a shear-free horizontal boundary, a thin layer (thickness d1) of very viscous fluid overlies a deep layer of less dense, much less viscous fluid; inertia and surface tension are negligible. After the initial unstable equilibrium is perturbed, a long-wave analysis describes the growth of the disturbance, including the nonlinear effects of large amplitude. The results show that nonlinear effects greatly enhance growth, so that initial local maxima in the thickness of the viscous film grow to infinite thickness in finite time, with a timescale 8μ/∆pgd1. In the final catastrophic growth the peak thickness is inversely proportional to the remaining time. (A parallel analysis for fluids with power-law rheology shows similar catastrophic growth.) While the small-slope approximation must fail before this singular time, the failure is only local, and a similarity solution describes how the peaks become downwelling plumes as the viscous film drains away. © 1993, Cambridge University Press. All rights reserved.

Beschreibung: This work determines the pressure—velocity relation of bubble flow in polygonal capillaries. The liquid pressure drop needed to drive a long bubble at a given velocity U is solved by an integral method. In this method, the pressure drop is shown to balance the drag of the bubble, which is determined by the films at the two ends of the bubble. Using the liquid-film results of Part 1 (Wong, Radke & Morris 1995), we find that the drag scales as Ccr :lin the limit C a - 0 (Ca = fiU/a, where /i is the liquid viscosity and σ the surface tension). Thus, the pressure drop also scales as Ca2‘3. The proportionality constant for six different polygonal capillaries is roughly the same and is about a third that for the circular capillary. The liquid in a polygonal capillary flows by pushing the bubble (plug flow) and by bypassing the bubble through corner channels (corner flow). The resistance to the plug flow comes mainly from the drag of the bubble. Thus, the plug flow obeys the nonlinear pressure—velocity relation of the bubble. Corner flow, however, is chiefly unidirectional because the bubble is long. The ratio of plug to corner flow varies with liquid flow rate Q (made dimensionless by era2//t, where a is the radius of the largest inscribed sphere). The two flows are equal at a critical flow rate Qc, whose value depends strongly on capillary geometry and bubble length. For the six polygonal capillaries studied, Qc〈1 10 B. For Qc〈 Q 〈 l, the plug flow dominates, and the gradient in liquid pressure varies with Qm. For Q 〈 Qc, the corner flow dominates, and the pressure gradient varies linearly with Q. A transition at such low flow rates is unexpected and partly explains the complex rheology of foam flow in porous media. © 1995, Cambridge University Press. All rights reserved.

Beschreibung: Foam in porous media exhibits an unusually high apparent viscosity, making it useful in many industrial processes. The rheology of foam, however, is complex and not well understood. Previous pore-level models of foam are based primarily on studies of bubble flow in circular capillaries. A circular capillary, however, lacks the corners that characterize the geometry of the pores. We study the pressure-velocity relation of bubble flow in polygonal capillaries. A long bubble in a polygonal capillary acts as a leaky piston. The ‘piston’ is reluctant to move because of a large drag exerted by the capillary sidewalls. The liquid in the capillary therefore bypasses the bubble through the leaky corners at a speed an order higher than that of the bubble. Consequently, the pressure work is dissipated predominantly by the motion of the fluid and not by the motion of the bubble. This is opposite to the conclusion based on bubble flow in circular capillaries. The discovery of this new flow regime reconciles two groups of contradictory foam-flow experiments. Part 1 of this work studies the fluid films deposited on capillary walls in the limit CaO (Ca = fiU/a, where [i is the fluid viscosity, U the bubble velocity, and cr the surface tension). Part 2 (Wong et al 1995) uses the film profile at the back end to calculate the drag of the bubble. Since the bubble length is arbitrary, the film profile is determined here as a general function of the dimensionless downstream distance x. For 1 〈 x 〈 Ca-1, the film profile is frozen with a thickness of order Ca2/3 at the centre and order Ca at the sides. For x ~ Ca-1, surface tension rearranges the film at the centre into a parabolic shape while the film at the sides thins to order Ca4/3. For x Ca-1, the film is still parabolic, but the height decreases as film fluid leaks through the side constrictions. For x ~ Ca-5/3, the height of the parabola is order Ca2/3. Finally, for x 〉 Ca“5/3, the height decreases as Ca1/4x”1/4. © 1995, Cambridge University Press. All rights reserved.

Beschreibung: 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.

Beschreibung: 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%.

Beschreibung: This is an experimental and theoretical study of the slow translation of a hot sphere through a fluid at rest at infinity. The viscosity depends strongly on temperature, i.e., if AT = T0—2% is the applied temperature difference and y —| (d/dT0) In µ(T0)|, then the parameter 6—y AT is large: it is about 6.5 in the experiments and is taken as infinite in the theory. The flow is determined by two large parameters, namely the Nusselt number N and the modified viscosity ratio ε1= v∝/(voθ3). The qualitative state of the flow is observed to depend on the relation between N and e. If ε-1-〉∝ (N fixed, possibly large) previous analysis (Morris 1982) shows that all the shear occurs in a thin low-viscosity film coating the sphere; this film and the associated thermal layer separate at the equator, and a separation bubble of low-viscosity fluid trails the sphere. (ii) If N-〉∝ (ε1large but fixed) even the most viscous fluid deforms, and both the drag and heat losses are found to be controlled by this highly viscous flow. The present work maps the major asymptotic states which separate these two end-states for small ε. The drag and heat-transfer laws are determined experimentally and theoretically: in addition it is shown that separation of the thermal layer ceases when the drag is controlled by the most viscous fluid, even though the heat transfer in this case can be still controlled by the dynamics of the least-viscous fluid. The heat-transfer and drag laws are also given for a sphere moving in a spherical container of finite radius. This model is shown to give a close estimate of wall effects for a sphere moving in a cylindrical container. For state (i) the theory predicts the heat transfer to within 20% and, for the smallest ε, the drag to within 30%. In the experiments ε is small enough for all limiting states to be evident but, apart from state (i), a design flaw prevents a quantitative test of the theory. For the other states, the theory is compared with numerical results from Daly & Raefsky (1985). Although the values of ε in the calculations are not small enough for the limiting states to be achieved, the theory predicts the drag to within 8% and the heat transfer to within 10%. © 1985, Cambridge University Press. All rights reserved.

Beschreibung: 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.

Beschreibung: 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.

Beschreibung: 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.

Beschreibung: 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} $.