Transcript Chapter 3 - Xiangyu Hu

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

dV    u  ndA

V
A
t

   u   0
t
D
   u  0
Dt
D()  ()

 u  ()
Dt
t
Substantial derivative
Basic equations (2)
• Momentum equation

udV    u(u  n)dA   gdV   n  TdA
A
V
A
t V
– Integral form
u
   ( uu)  g    T
t
– Derivative form
Du
 g    T
Dt
1
T  (  p     u ) I   u  u T
2
– Form with substantial
derivatives
• Equation of state


p  p(  )

Stress tensor
Incompressible flows (1)
• Continuity equation
• Momentum equation



D 

 u    0 or   u  0
Dt
t
u
1
 u  u   p  g  2u
t

Kinematic viscosity
Incompressible flows (2)
• Boundary conditions
– No-slip
– Finite slip
u  U wall
u  U wall  k
u
n
u
Shear rate along normal direction
n
Interface: definition and geometry
• 3D: a surface separates two phases
• 2D: a line
n
dt
 n
ds
dt
|  |
ds
   n
2
1
s
t
n
Mathematical representation of a 2D
interface
S : F ( x, y)  0
• Implicit function
• Characteristic function
N
– 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
d
V
S
(x) f (x)dV   f (x)ds
S
• Interface motion
Change volume integrals
into surface integrals
DF
0
Dt
2
1
s
t
N
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
[u]S  0
V  u1  n  u 2  n
– With phase change
• Velocity discontinuous along normal
direction
– Rankine-Hugoniot condition
1 u1  n  V  2 u2  n  V
Fluid mechanics with interfaces (2)
• Momentum conservation and surface
tension and Marangoni effects
 pI  DS  n  n  S
• Split form along normal and tangential
direction
 p  n  D  nS  
t  D  nS  t   S
D

1
u  u T
2

Shear rate tensor
 S
Derivative of surface
tension along the
interface
Momentum equation including surface
effects (1)
• Integral form
– With surface integral on interface
d
dt

V
udV    u(u  n)dA   gdV   n  TdA    n   S dS
A
V
A
S
– With volume integral on fluids
d
udV    u(u  n)dA   gdV   n  TdA    n   S d S dV
A
V
A
V
dt V
2
N
s
1
t
N
V
A
Control Vulome
Momentum equation including surface
effects (2)
• Derivative form
– With surface force
u
1
1
 u  u   p  g  2u  n   S d S
t


– With surface stress
u
1
1
 u  u   p  g  2u    
t


   I  nnd S
     n  S d S
• Usually constant surface tension considered
 S  0