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

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  S 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 Shear rate tensor Derivative of surface tension along the interface

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