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Chapter 03: Macroscopic interface dynamics

Chapter 03: Macroscopic interface dynamics. Part A: physical and mathematical modeling of interface. Xiangyu Hu Technical University of Munich. Basic equations (1). Continuity equation Integral form Derivative form Form with substantial derivatives. Substantial derivative.

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Chapter 03: 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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