Macroscopic thin liquid films are entities that are important in biophysics, physics, and engineering, as well as in natural settings. They can be composed of common liquids such as water or oil, rheologically complex materials such as polymers solutions or melts, or complex mixtures of phases or components. When the films are subjected to the action of various mechanical, thermal, or structural factors, they display interesting dynamic phenomena such as wave propagation, wave steepening, and development of chaotic responses. Such films can display rupture phenomena creating holes, spreading of fronts, and the development of fingers. In this review a unified mathematical theory is presented that takes advantage of the disparity of the length scales and is based on the asymptotic procedure of reduction of the full set of governing equations and boundary conditions to a simplified, highly nonlinear, evolution equation or to a set of equations. As a result of this long-wave theory, a mathematical system is obtained that does not have the mathematical complexity of the original free-boundary problem but does preserve many of the important features of its physics. The basics of the long-wave theory are explained. If, in addition, the Reynolds number of the flow is not too large, the analogy with Reynolds's theory of lubrication can be drawn. A general nonlinear evolution equation or equations are then derived and various particular cases are considered. Each case contains a discussion of the linear stability properties of the base-state solutions and of the nonlinear spatiotemporal evolution of the interface (and other scalar variables, such as temperature or solute concentration). The cases reducing to a single highly nonlinear evolution equation are first examined. These include: (a) films with constant interfacial shear stress and constant surface tension, (b) films with constant surface tension and gravity only, (c) films with van der Waals (long-range molecular) forces and constant surface tension only, (d) films with thermocapillarity, surface tension, and body force only, (e) films with temperature-dependent physical properties, (f) evaporating/condensing films, (g) films on a thick substrate, (h) films on a horizontal cylinder, and (i) films on a rotating disc. The dynamics of the films with a spatial dependence of the base-state solution are then studied. These include the examples of nonuniform temperature or heat flux at liquid-solid boundaries. Problems which reduce to a set of nonlinear evolution equations are considered next. Those include (a) the dynamics of free liquid films, (b) bounded films with interfacial viscosity, and (c) dynamics of soluble and insoluble surfactants in bounded and free films. The spreading of drops on a solid surface and moving contact lines, including effects of heat and mass transport and van der Waals attractions, are then addressed. Several related topics such as falling films and sheets and Hele-Shaw flows are also briefly discussed. The results discussed give motivation for the development of careful experiments which can be used to test the theories and exhibit new phenomena.

Long-scale evolution of thin liquid films

This study describes the development and validation of Computational Fluid Dynamics (CFD) methodology for the simulation of dispersed two-phase flows. A two-fluid (Euler-Euler) methodology previously developed at Imperial College is adapted to high phase fractions. It employs averaged mass and momentum conservation equations to describe the time-dependent motion of both phases and, due to the averaging process, requires additional models for the inter-phase momentum transfer and turbulence for closure. The continuous phase turbulence is represented using a two-equation k − ε−turbulence model which contains additional terms to account for the effects of the dispersed on the continuous phase turbulence. The Reynolds stresses of the dispersed phase are calculated by relating them to those of the continuous phase through a turbulence response function. The inter-phase momentum transfer is determined from the instantaneous forces acting on the dispersed phase, comprising drag, lift and virtual mass. These forces are phase fraction dependent and in this work revised modelling is put forward in order to capture the phase fraction dependency of drag and lift. Furthermore, a correlation for the effect of the phase fraction on the turbulence response function is proposed. The revised modelling is based on an extensive survey of the existing literature. The conservation equations are discretised using the finite-volume method and solved in a solution procedure, which is loosely based on the PISO algorithm, adapted to the solution of the two-fluid model. Special techniques are employed to ensure the stability of the procedure when the phase fraction is high or changing rapidely. Finally, assessment of the methodology is made with reference to experimental data for gas-liquid bubbly flow in a sudden enlargement of a circular pipe and in a plane mixing layer. Additionally, Direct Numerical Simulations (DNS) are performed using an interface-capturing methodology in order to gain insight into the dynamics of free rising bubbles, with a view towards use in the longer term as an aid in the development of inter-phase momentum transfer models for the two-fluid methodology. The direct numerical simulation employs the mass and momentum conservation equations in their unaveraged form and the topology of the interface between the two phases is determined as part of the solution. A novel solution procedure, similar to that used for the two-fluid model, is used for the interface-capturing methodology, which allows calculation of air bubbles in water. Two situations are investigated: bubbles rising in a stagnant liquid and in a shear flow. Again, experimental data are used to verify the computational results.

Computational fluid dynamics of dispersed two-phase flows at high phase fractions

The growth of a vapor bubble in a superheated liquid is controlled by three factors: the inertia of the liquid, the surface tension, and the vapor pressure. As the bubble grows, evaporation takes place at the bubble boundary, and the temperature and vapor pressure in the bubble are thereby decreased. The heat inflow requirement of evaporation, however, depends on the rate of bubble growth, so that the dynamic problem is linked with a heat diffusion problem. Since the heat diffusion problem has been solved, a quantitative formulation of the dynamic problem can be given. A solution for the radius of the vapor bubble as a function of time is obtained which is valid for sufficiently large radius. This asymptotic solution covers the range of physical interest since the radius at which it becomes valid is near the lower limit of experimental observation. It shows the strong effect of heat diffusion on the rate of bubble growth. Comparison of the predicted radius‐time behavior is made with experimental observations in superheated water, and very good agreement is found.

The growth of vapor bubbles in superheated liquids. report no. 26-6

Review of two-phase flow instability

Physical analysis of explosive, magma-water interaction is complicated by several important controls: (1) the initial geometry and location of the contact between magma and water; (2) the process by which thermal energy is transferred from the magma to the water; (3) the degree to and manner by which the magma and water become intermingled prior to eruption; (4) the thermodynamic equation of state for mixtures of magma fragments and water; (5) the dynamic metastability of superheated water; and (6) the propagation of shock waves through the system. All of these controls can be analyzed while addressing aspects of tephra emplacement from the eruptive column by fallout, surge, and flow processes. An ideal thermodynamic treatment, in which the magma and external water are allowed to come to thermal equilibrium before explosive expansion, shows that the maximum system pressure and entropy are determined by the mass ratio of water and magma interacting. Explosive (thermodynamic) efficiency, measured by the ratio of maximum work potential to thermal energy of the magma, depends upon heat transfer from the pyroclasts to the vapor during the expansion stage. The adiabatic case, in which steam immediately separates from the tephra during ejection, produces lower efficiencies than does the isothermal case, in which heat is continually transferred from tephra to steam as it expands. Mechanisms by which thermal equilibrium between water and magma can be obtained require intimate mixing of the two. Interface instabilities of the Landau and Taylor type have been documented by experiments to cause fine-scale mixing prior to vapor explosion. In these cases, water is heated rapidly to a metastable state of superheat where vapor explosion occurs by spontaneous nucleation when a temperature limit is exceeded. Mixing may also be promoted by shock wave propagation. If the shock is of sufficient strength to break the magma into small pieces, thermal equilibrium and vapor production in its wake may drive the shock as a thermal detonation. Because these mechanisms of magma fragmentation allow calculation of grain size, vapor temperature and pressure, and pressure rise times, detailed emplacement models can be developed by critical field and laboratory analysis of the resulting tephra deposits. Deposits left by dense flows of tephra and wet steam as opposed to those left by dilute flows of dry steam and tephra show contrasts in median grain size, dispersal area, grain shape, grain surface chemistry, and bed form.

Explosive magma-water interactions: Thermodynamics, explosion mechanisms, and field studies

Structure of air-water bubbly flow in a vertical pipe—I. liquid mean velocity and turbulence measurements

Structure of air-water bubbly flow in a vertical pipe—II. Void fraction, bubble velocity and bubble size distribution

A free-radical polymerization with kinetic equations taking into account monomer, polymer, and solvent transfer and initiation, termination, propagation, and inhibition reactions was studied both experimentally and by simulation to determine the performance of various Kalman filters, with feasible measurements used.

S.G. Bankoff

Papers

Structure of air-water bubbly flow in a vertical pipe—I. liquid mean velocity and turbulence measurements

Structure of air-water bubbly flow in a vertical pipe—II. Void fraction, bubble velocity and bubble size distribution

Digital monitorina and estimation of polymerization reactors

Countercurrent flow of air/water and steam/water through a horizontal perforated plate

Slip velocity ratios in an air-water system under steady-state and transient conditions