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# One Dimensional Two-Phase Flow

01 Aug 1969-

About: The article was published on 1969-08-01 and is currently open access. It has received 3841 citations till now. The article focuses on the topics: Laminar flow reactor & Laminar flow.

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TL;DR: In this article, 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.

Abstract: 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.

2,689 citations

### Cites background from "One Dimensional Two-Phase Flow"

...The split in the evaporative shear stress between the bulk liquid and vapor is not readily determined and has been equally apportioned in the above equation (Wallis, 1969; Chung and Bankoff, 1980)....

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TL;DR: In this article, a two-phase mixture theory is presented which describes the deflagration-to-detonation transition (DDT) in reactive granular materials, based on the continuum theory of mixtures formulated to include the compressibility of all phases and the compaction behavior of the granular material.

Abstract: In this paper, a two-phase mixture theory is presented which describes the deflagration-to-detonation transition (DDT) in reactive granular materials. The theory is based on the continuum theory of mixtures formulated to include the compressibility of all phases and the compaction behavior of the granular material. By requiring the model to satisfy an entropy inequality, specific expressions for the exchange of mass, momentum and energy are proposed which are consistent with known empirical models. The model is applied to describe the combustion processes associated with DDT in a pressed column of HMX. Numerical results, using the method-of-lines, are obtained for a representative column of length 10 cm packed to a 70% density with an average grain size of 100 μm. The results are found to predict the transition to detonation in run distances commensurate with experimental observations. Additional calculations have been carried out to demonstrate the effect of particle size and porosity and to study bed compaction by varying the compaction viscosity of the granular explosive.

1,155 citations

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TL;DR: In this article, it was shown that for a wide varicty of both fixed-bed and suspended-particle systems, file voidage function may be expressed as ϵ−β, where the exponent β is dependent on the particle Reynolds number but independent of other system variables.

Abstract: The drag force on a particle in a fluid—multiparticle interaction system may be expressed as the product of the drag force on an unhindered particle, subject to the same volumetric flux of fluid, and a voidage function. It is demonstrated that for a wide varicty of both fixed-bed and.suspended-particle systems, file voidage function may be expressed as ϵ−β, where the exponent β is dependent on the particle Reynolds number but independent of other system variables.

1,090 citations

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01 Jan 2003

TL;DR: In this paper, the authors describe the development and validation of Computational Fluid Dynamics (CFD) methodology for the simulation of dispersed two-phase flows, which employs averaged mass and momentum conservation equations to describe the time-dependent motion of both phases.

Abstract: 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.

968 citations

### Cites background from "One Dimensional Two-Phase Flow"

...See Wallis [414] for a general introduction....

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...However, the older literature for particles, droplets and bubbles has been reviewed by Wallis [414]....

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TL;DR: In this article, the authors survey Rayleigh-Taylor instability, describing the phenomenology that occurs at a Taylor unstable interface, and reviewing attempts to understand these phenomena quantitatively, and present a survey of the literature on Rayleigh Taylor instability.

Abstract: The aim of this talk is to survey Rayleigh-Taylor instability, describing the phenomenology that occurs at a Taylor unstable interface, and reviewing attempts to understand these phenomena quantitatively.

953 citations