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

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

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

Incompressible flows (1) Continuity equation Momentum equation or Kinematic viscosity

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

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

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

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

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

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

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