Download Fluid Flow

Fluid Flow
Pressure, momentum flux and
viscosity.
How can we attack this problem?
(1) Define viscosity
Viscosity-fluid property that influences the rate of fluid flow under stress.
y
The viscosity is the ratio between the
shear stress and the velocity gradient
between the plates, or
dv( x)
  
dy
Newton’s Law of Viscosity
Units of viscosity


d v 
 x
 dy 
Pa
 m
 
s
m
Pa s
Layers of fluid particles with top layer moving
faster than bottom layer
Activation energy
energy
Position

 o  exp 
log   
log  

activation_energy 
 
log  o 
 
RT


 activation_energy   1


R

 T
log  o 
 activation_energy   1

 T
R


The top plate drags the fluid particles in the top fluid layer, which then drag
the particles in the adjacent lower layer, which then drag the particles in the
next lowest layer, and so on—thus giving rise to momentum transfer
Momentum transfer
Units on shear stress and pressure
Force/area=mass*length/time2 * 1/area
=mass * length/time * 1/time * 1/area
= mass*velocity * 1/(time * area)
= momentum/(time*area)
= momentum flux
A momentum flux (or stress) multiplied by a cross sectional area is a FORCE!
Forces balance at steady state (equilibrium)
Rate of momentum in =rate of momentum out at steady state.
2. Determine velocity profile. If flow is fully developed, the fluid velocity only depends on y.
V(x)
 yx
 
d
v( x)
dy
0
y
y  y
y
‘control volume’
z
x
Patm
y
x
x  x
High Pressure
Papplied
Low Pressure
Hydrostatic pressure varies with x while
shear stress varies with y! We only have to
consider the shear stress acting normal to
the xz plane and the hydrostatic pressure
component acting normal to the yz plane.
   yx   yx
  xz  P x - Px xyz
y  y 
 y

0
   yx   yx
  xz  P x - Px xyz
y  y 
 y

   yx   yx
  P x- P

y  y 
x x 
 y



y
x


d   d P
yx
d
y

 dx

0
0
Function of x unless
P=constant or P linear with
x.
Velocity can only depend on y
d
 yx
dy
P
L

 d  dy
 d y yx





P
L
dy
 yx
P
L
C is the integration constant from indefinite integration
y  C
If the velocity is a local maximum at y=0 (center in between plates),
then
C=0
 yx
d
  v( x)
dy
P
L

   d v ( x) d y

dy

y
v ( x)
C1
v ( x)
P
2L 
2
 y  C1
P
2L 
P
2L 
2

 2  y2




P
L
 y dy