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Physical and Mathematical Modeling of Macroscopic Interface Dynamics

This chapter delves into the fundamental physical and mathematical principles governing macroscopic interface dynamics, covering continuity equations, momentum equations, and the behavior of incompressible flows. It discusses various forms of basic equations, including integral and derivative forms, and emphasizes the role of substantial derivatives in fluid dynamics. The chapter also explores the intricacies of interface definition, geometry in both 2D and 3D, and the mathematical representation of interfaces using Heaviside and Dirac functions. Furthermore, it addresses fluid mechanics involving interfaces, phase changes, and includes conservation laws and surface effects.

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Physical and Mathematical Modeling of Macroscopic Interface Dynamics

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  1. Chapter 03:Macroscopic interface dynamics Part A: physical and mathematical modeling of interface Xiangyu Hu Technical University of Munich

  2. Basic equations (1) • Continuity equation • Integral form • Derivative form • Form with substantial derivatives Substantial derivative

  3. Basic equations (2) • Momentum equation • Integral form • Derivative form • Form with substantial derivatives • Equation of state Stress tensor

  4. Incompressible flows (1) • Continuity equation • Momentum equation or Kinematic viscosity

  5. Incompressible flows (2) • Boundary conditions • No-slip • Finite slip Shear rate along normal direction

  6. Interface: definition and geometry • 3D: a surface separates two phases • 2D: a line

  7. Mathematical representation of a 2D interface • Implicit function • Characteristic function • H=0 in phase 1 and H=1 in phase 2 • 2D Heaviside step function • Distribution concentrated on interface • Dirac function dS normal to interface • Gradient of H • Interface motion Change volume integrals into surface integrals

  8. Fluid mechanics with interfaces (1) • Mass conservation and velocity condition • Without phase change • Velocity continuous along normal direction • Interface velocity equal to fluid velocity along normal direction • With phase change • Velocity discontinuous along normal direction • Rankine-Hugoniot condition

  9. Fluid mechanics with interfaces (2) • Momentum conservation and surface tension and Marangoni effects • Split form along normal and tangential direction Derivative of surface tension along the interface Shear rate tensor

  10. Momentum equation including surface effects (1) • Integral form • With surface integral on interface • With volume integral on fluids

  11. Momentum equation including surface effects (2) • Derivative form • With surface force • With surface stress • Usually constant surface tension considered

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