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

0

ut

dAdVt AV

nu

0 uDt

D

()()()

utDt

DSubstantial derivative

Basic equations (2)

• Momentum equation

– Integral form

– Derivative form

– Form with substantial derivatives

• Equation of state

Tguuu

)(

t

dAdVdAdVt AVAV

Tngnuuu )(

Tgu

Dt

D

Tp uuIuT 2

1)( Stress tensor

)(pp

Incompressible flows (1)

• Continuity equation

• Momentum equation

0

u

tDt

D0 uor

uguuu 21

p

t

Kinematic viscosity

Incompressible flows (2)

• Boundary conditions

– No-slip

– Finite slip

wallUu

nkwall

u

Uu

nu

Shear rate along normal direction

Interface: definition and geometry

• 3D: a surface separates two phases• 2D: a line

12

n

t

s

n

n

t

nt

ds

dds

d

||

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

12

N

t

s

N

0),(: yxFS

0Dt

DF

SV S dsfdVf )()()( xxx 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 direct

ion– Rankine-Hugoniot condition

nunuV 21

0][ Su

VnuVnu 2211

Fluid mechanics with interfaces (2)

• Momentum conservation and surface tension and Marangoni effects

• Split form along normal and tangential direction

SSp nnDI

SS

Sp

tnDt

nDn

TuuD 2

1Shear rate tensor S

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

S SAVAV

dSdAdVdAdVdt

d nTngnuuu )(

1

2

N

t

s

N

C o n tro l Vu lo m eA

V

V SSAVAV

dVdAdVdAdVdt

d nTngnuuu )(

Momentum equation including surface effects (2)

• Derivative form– With surface force

– With surface stress

• Usually constant surface tension considered

SSpt

nuguuu 11 2

11 2uguu

up

t

S nnI SS n

0 S