Mathematical Problems in Engineering

Mathematical Modelling of Momentum and Energy Transport at Fluid-Fluid Interfaces

Publishing date
01 Jun 2022
Submission deadline
14 Jan 2022

1Curtin University, Perth, Australia

2Indian Institute of Technology Kanpur, Kanpur, India

This issue is now closed for submissions.

Mathematical Modelling of Momentum and Energy Transport at Fluid-Fluid Interfaces

This issue is now closed for submissions.


Gas-liquid and liquid-liquid interfaces appear frequently in industrial processes and in nature. The shape, size, and stability of these boundaries can influence significantly the transport rates of mass, momentum, and energy. The process performance is closely linked to the fluxes at such interfaces. It needs to be independently understood. Meanwhile, there is a need to mathematically model and simulate the shape, velocity, and heat fluxes at such fluid-fluid interfaces. The mathematical model should consider over and above continuum modelling of transport phenomena in bulk. Examples of gas-liquid interfaces are air-water surface waves in lakes and oceans, where the evaporative flux depends, among other factors, on the shape of the free surface, water-steam interface in boilers, and gas bubbles rising in a liquid column. Liquid-liquid interfaces appear in oil-water layers in oil spills. Crystal growth where there is an encapsulant layer is required to prevent the evaporation of one of the dopants. In wave mechanics, fluid-fluid interfaces are dealt with in terms of deformation, interfacial shear, and gravitational instability. Two configurations of recent origin where interfacial phenomena play a central role are those involving drops and bubbles.

If the applicable length scale is small, surface tension and its spatial gradients are important. If not considered, it can create a new challenge in determining interfacial fluxes. Applications such as movement of a liquid drop over a textured surface, coalescence of drops over textured surfaces, and bubble dynamics are physical problems of enormous complexity. Considering the multiplicity of length and timescales, these processes need to be thoroughly understood first from a modelling perspective. The associated analytical and numerical techniques will have to be carefully designed to extract meaningful information from the model developed. Multiphase flows in mini- and microchannels are being studied in applications such as drug delivery and thermal management of high-power electronics where power dissipation is particularly high. These configurations also require careful modelling and analysis for the development of realizable devices. Air-water flow in porous media has applications in groundwater. Recently, there has been further research discussing modelling evaporation rates and condensation at interfaces. Moreover, more research has been done for assessing three-phase contact lines in the context of desalination and water purification. The examples referred to above share the formation of gas-liquid and liquid-liquid interfaces. Occasionally, such interfaces sweep over a solid surface, creating contact lines with appreciable magnitudes of mass, momentum, and energy fluxes. A variety of forces arising from surface tension, surface tension gradients, viscosity, and pressure jointly determine the evolution of the shape of the interface as well as the accompanying fluxes of energy and passive scalars. These forces have to be determined from a carefully planned mathematical model, followed by an appropriately developed or selected numerical technique for solving the system of coupled. In these situations, nonlinear differential equations are often used.

The aim of this Special Issue is to bring together original research and review articles that focus on all aspects of mathematical modelling of fundamental phenomena. Submissions can include interfacial transport, parameter estimation, closure considerations, and numerical simulation. Moreover, we also welcome submissions discussing validation against experiments, and connection with mesoscale models. Submission of device design research is also highly encouraged.

Potential topics include but are not limited to the following:

  • Mathematical modelling of interfacial phenomena, jump conditions, and closure
  • Numerical algorithms, stability, convergence, and coupled versus segregated solvers
  • Multi-phase modelling, VOF and level sets, and phase-field method
  • Modelling at the meso-scale and continua
  • Moving contact lines on hydrophobic surfaces
  • Dropwise condensation of vapor on wires and meshes
  • Evaporation models for droplets laden with virulent matter
  • Modelling droplet impingement and splashing
  • Ab initio modelling of three-phase contact line motion
  • Kinetic theory model of evaporation from a deformed liquid interface
  • Gas-liquid interfaces in porous media and immiscible interface movement and fingering in porous media
  • Evaporation models for aqueous suspension and blood
  • Interfacial and free-surface phenomena in mini- and microchannels
  • Modelling capillarity in heat pipes and related devices
  • Equilibrium shape of liquid drops on complex surfaces and wires


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