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Multiphase and Multicomponent Flows

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The Lattice Boltzmann Method

Part of the book series: Graduate Texts in Physics ((GTP))

Abstract

After reading this chapter, you will be able to expand lattice Boltzmann simulations by including non-ideal fluids, using either the free-energy or the Shan-Chen pseudopotential method. This will allow you to simulate fluids consisting of multiple phases (e.g. liquid water and water vapour) and multiple components (e.g. oil and water). You will also learn how the surface tension between fluid phases/components and the contact angle at solid surfaces can be varied and controlled.

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Notes

  1. 1.

    This concept can easily be extended to systems with more than two components by introducing more order parameters.

  2. 2.

    In reality we cannot write ρ (1) = 0 or ρ (2) = 0 since the local density of a given component is never exactly zero. For example, in a water-oil mixture, one can always find a few water molecules in the oil-rich phase and the other way around. However, these minority densities are usually so small that we can neglect them here.

  3. 3.

    For readers unfamiliar with Gibbs and Helmholtz free energies, their descriptions can be found in most textbooks on thermodynamics, e.g. [22]. Briefly, Gibbs free energy is usually used when the system is under constant pressure and temperature, while the Helmholtz free energy is taken when the system is under constant volume and temperature.

  4. 4.

    This definition is strictly valid only for simple liquids. More generally, the energy per unit area for stretching the interface is given by Γ = γ + dγ∕dε where ε is the strain. For simple liquids we have dγ∕dε = 0 and Γ = γ.

  5. 5.

    At a given point on a surface, we can define two radii of curvature, as shown in Fig. 9.3a. The mean curvature is simply defined as the average (1∕R 1 + 1∕R 2)∕2, while the Gaussian curvature is the product 1∕(R 1 R 2). Since one of the curvature radii can be negative and the other positive (e.g. a saddle surface), the mean curvature can vanish, even for a non-planar surface.

    Fig. 9.3
    figure 3

    Schematic diagrams for (a ) the Laplace pressure and (b ) Young’s contact angle. Each point at the interface can be characterised by two independent radii of curvature that can be positive or negative. In (a ), the surface is convex and both radii are positive. The average curvature (1∕R 1 + 1∕R 2)∕2 and the surface tension γ lg are related to the pressure jump (Laplace pressure) p l − p g across the interface. In (b ), a liquid droplet is in contact with a surface and forms a contact angle θ. For this angle, all surface tension force components tangential to the surface are in mechanical equilibrium

    Full size image
  6. 6.

    For multicomponent flows, an additional equation of motion is needed to describe the evolution of the order parameter. This is usually given by the Cahn-Hilliard or Allen-Cahn equation, see e.g. Sect. 9.2.2.3.

  7. 7.

    Thermodynamic consistency is defined in Sect. 9.1.1. See also Appendix A.7 where this is shown explicitly for the Landau multiphase model.

  8. 8.

    We note that our convention here follows that of Chap. 6 In the literature, sometimes the prefactor \(\left (1 -\frac{1} {2\tau }\right )\) is included in the definition of F i itself.

  9. 9.

    In fact, the tanh profile provides a good initial interfacial profile for most multiphase and multicomponent models.

  10. 10.

    Differences usually become visible at low densities when the density is plotted logarithmically.

  11. 11.

    We could also simulate a gas bubble in a liquid.

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Authors and Affiliations

  1. School of Engineering University of Edinburgh, Edinburgh, UK

    Timm Krüger

  2. Department of Physics, Durham University, Durham, UK

    Halim Kusumaatmaja

  3. Maya Heat Transfer Technologies, Westmount, Québec, Canada

    Alexandr Kuzmin

  4. Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, USA

    Orest Shardt

  5. IDMEC/IST, University of Lisbon, Lisbon, Portugal

    Goncalo Silva

  6. Acoustics Research Centre, SINTEF ICT, Trondheim, Norway

    Erlend Magnus Viggen

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  1. Timm Krüger
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  2. Halim Kusumaatmaja
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  4. Orest Shardt
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  5. Goncalo Silva
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  6. Erlend Magnus Viggen
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Krüger, T., Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G., Viggen, E.M. (2017). Multiphase and Multicomponent Flows. In: The Lattice Boltzmann Method. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-44649-3_9

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