We review the development of diffuse-interface models of hydrodynamics and their application to a wide variety of interfacial phenomena. These models have been applied successfully to situations in which the physical phenomena of interest have a length scale commensurate with the thickness of the interfacial region (e.g. near-critical interfacial phenomena or small-scale flows such as those occurring near contact lines) and fluid flows involving large interface deformations and/or topological changes (e.g. breakup and coalescence events associated with fluid jets, droplets, and large-deformation waves). We discuss the issues involved in formulating diffuse-interface models for single-component and binary fluids. Recent applications and computations using these models are discussed in each case. Further, we address issues including sharp-interface analyses that relate these models to the classical free-boundary problem, computational approaches to describe interfacial phenomena, and models of fully miscible fluids.

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Diffuse-interface methods in fluid mechanics

We present a bibliography on moving and free boundary problems for the heatdiffusion equation, particularly regarding the Stefan and related problems. It contains 5869 titles referring to 588 scientific Journals, 122 books, 88 symposia (having at least 3 contributions on the subject), 30 collections, 59 thesis and 247 technical reports. It tries to give a comprehensive account of the western existing mathematicalphysical-engineering literature on this research field. RESUMEN Se presenta una bibliografía sobre problemas de frontera móvil y libre para la ecuación del calor-difusión, en particular sobre el problema de Stefan y problemas relacionados. Contiene 5869 títulos distribuidos en 588 revistas científicas, 122 libros, 88 simposios (teniendo al menos 3 contribuciones en el tema), 30 colecciones, 59 tesis y 247 informes técnicos o prepublicaciones. Se da un informe amplio de la bibliografía matemática, física y de las ingenierías existente en occidente sobre este tema de investigación. Primary Mathematics Subject Classification Number (*): 35R35, 80A22 Secondary Mathematics Subject Classification Number (*): 35B40, 35C05, 35C15, 35Kxx, 35R30, 46N20, 49J20, 65Mxx, 65Nxx, 76R50, 76S05, 76T05, 93C20. (*) Following the 1991 Mathematics Subject Classification compiled by Mathematical Reviews and Zentralblatt fur Mathematik. Primary key words: Enthalpy formulation or method, Filtration, Free boundary problems, Freezing, Melting, Moving boundary problems, Mushy region, Phase-change problem, Solidification, Stefan problem. Secondary key words: Continuous mechanics, Diffusion process, Functional analysis, Heat conduction, Mathematical methods, Numerical methods, Partial differential equations, Variational inequalities, Weak solutions. Palabras claves primarias: Método o formulación en entalpía, Filtración, Problemas de frontera libre, Congelación, Derretimiento, Problemas de frontera móvil, Región pastosa, Problema de cambio de fase, Solidificación, Problema de Stefan. Palabras claves secundarias: Mecánica del continuo, Procesos difusivos, Análisis funcional, Conducción del calor, Métodos matemáticos, Métodos numéricos, Ecuaciones diferenciales a derivadas parciales, Inecuaciones variacionales, Soluciones débiles. El manuscrito fue recibido y aceptado en octubre de 1999. D.A. Tarzia, A bibliography on FBP. The Stefan problem, MAT Serie A, # 2 (2000). 3

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A bibliography on moving-free boundary problems for the heat-diffusion equation. The stefan and related problems

We show that, under a suitable assumption on the pressure tensor, the mass and momentum balance equations of hydrodynamical theory, introduced in the early 1980s by many authors to describe matrix separation, yield the equations of the Madelung fluid that are equivalent to the Schrodinger-like equation with logarithmic nonlinearity. This equation has solitary-waves solutions as required by many experimental volcanic models.

Logarithmic Schrödinger-like equation as a model for magma transport

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Diffuse interface modeling of two-phase flows based on averaging: mass and momentum equations

We investigate the self-locomotion of an elongated microswimmer by virtue of unidirectional tangential surface treadmilling. We show that the propulsion could be almost frictionless, as the microswimmer is propelled forward with the speed of the backward surface motion, i.e. it moves through an almost quiescent fluid. We investigate this swimming technique using the special spheroidal coordinates and also find an explicit closed-form optimal solution for a two-dimensional treadmiller via complex-variable techniques. Slender-object surface treadmilling is a particularly efficient mode of locomotion because the viscous drag is determined by the smallest length scale of the object rather than by the largest scale, as is the usual case for low Reynolds number flow.

https://iopscience.iop.org/article/10.1088/1367-2630/9/5/145/pdf

A frictionless microswimmer

A phenomenological model of capillarity, which accounts for the structure of the liquid-vapor layer, is successively derived assuming that the entropy of a particle depends on the internal energy, the density, and the gradient of the density. Employing classical thermodynamic principles in combination with the balance of mass, momentum, and energy, a rheological expression for capillary stresses is obtained in terms of the free energy of a liquid-vapor system. It is demonstrated that this model admits potential solutions, provided that external forces are potential and the effects of viscous dissipation of energy and heat conductivity are neglected. Moreover, it is shown that a variational principle can be formulated for potential flows, which generalizes Luke's variational principle for free-boundary inviscid flow. This model can be applied to flows involving a topological change of the capillary interface, such as those associated with the spontaneous growth, coalescence and breakup of vaporous bubbles in a liquid.

Microscale theory of surface tension.

The paper addresses the mathematical modelling of the formation of a pointed drop in a four-roller mill, observed by Taylor (1934) in the Cavendish Laboratory. Since the experiments were carried out with drops of small diameter compared to the mill size, the method of matched asymptotic expansions is applicable. A two-dimensional Stokes flow generated by the rotating rollers in the mill but with no drop effect (outer problem) is computed numerically by a boundary-element method. The local expansion of that flow at the centre of the mill, where the drop is to be positioned, is used as a far field for the flow around the drop in unbounded fluid (inner problem). Employing a plane-flow model and using complex-variable techniques, the explicit solutions previously obtained by the author are adapted to the inner problem. It is proved that, with an increasing rotation rate of the rollers, the drop does develop two apparent cusps on the interface, and its shapes have striking similarities with Taylor's experiments. Response diagrams showing the drop distortion versus the elongational strain demonstrate that these are one-to-one function of each other if the drop diameter is greater than a critical value determined by the size of the mill but cease to be one-to-one otherwise. This behaviour is identified with a sudden transition from a rounded drop to a cusped one at a critical strain.

Leonid K. Antanovskii

Papers

Microscale theory of surface tension.

Formation of a pointed drop in Taylor's four-roller mill