fluid flow pressure, momentum flux and viscosity
Post on 22-Dec-2015
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Fluid Flow
Pressure, momentum flux and viscosity.
Viscosity-fluid property that influences the rate of fluid flow under stress.
The viscosity is the ratio between the shear stress and the velocity gradient between the plates, or
dy
xdv )(
y
How can we attack this problem?
(1) Define viscosity
Newton’s Law of Viscosity
yvx
d
d
Pa
m
s
m
Pa s
Units of viscosity
Layers of fluid particles with top layer moving faster than bottom layer
energy
Position
Activation energy
o expactivation_energy
RT
log log o activation_energyR
1
T
log log o activation_energy
R
1
T
Momentum transfer
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
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
Forces balance at steady state (equilibrium)
Rate of momentum in =rate of momentum out at steady state.
A momentum flux (or stress) multiplied by a cross sectional area is a FORCE!
‘control volume’
y
xz
2. Determine velocity profile. If flow is fully developed, the fluid velocity only depends on y.
Papplied
Patm
x x x
y
y y
V(x)
y
yx yv x( )d
d
0
yxy yxy y
xz P x Px x yz
0-
High Pressure 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.
yxy yxy y
xz P x Px x yz
0
yxy yxy y
y
P x Px x
x
0
y yx
d
d
xPd
d
Velocity can only depend on y
Function of x unless P=constant or P linear with
x.
y yx
d
d
PL
yy yx
d
d
d yPL
d yxPLy C
-
-
C is the integration constant from indefinite integration
C=0
yx yv x( )d
d
PLy y
yv x( )d
d
d yPLy
d
v x( )P
2L y2 C1
C1P
2L 2
v x( )P
2L 2
y2
If the velocity is a local maximum at y=0 (center in between plates), then