Transcript Lecture 6 - Aalborg Universitet

Problems
• 6.8 An incompressible viscous
fluid is placed between two large
parallel plates. The bottom plate is
fixed and the top moves with the
velocity U. Determine:
–
–
–
–
volumetric dilation rate;
rotation vector;
vorticity;
rate of angular deformation.
y
u U
b
• 6.74 Oil SAE30 at 15.6C steadily flows between fixed horizontal
parallel plates. The pressure drop per unit length is 20kPa/m
and the distance between the plates is 4mm, the flow is
laminar.
Determine the volume rate of flow per unit width; magnitude
and direction of the shearing stress on the bottom plate;
velocity along the centerline of the channel
Problems
• 7.19. One type of viscosimeter is designed as
shown in the figure. The reservoir is filled with liquid
and the time required for the liquid to drop from Hi to
Hf determined. Obtain relationship between viscosity
m and draining time t. Assume the variables involved
Hi,Hf, D and specific weight g.
• 7.50 The drag D=f(d,D,V,r). What dimensional
parameters will be used? If in experiment d=5mm,
D=12.5mm and V=0.6m/s, the drag is 6.7x10-3N.
Estimate drag on a sphere in 0.6 m diam tube where
water flowing with a velocity of 1.8 m/s and the
diameter of sphere is such that the similarity is
maintained
Viscous flow in pipes
General characteristic of Pipe flow
• pipe is completely filled with water
• main driving force is usually a pressure gradient along the
pipe, though gravity might be important as well
Pipe flow
open-channel flow
Laminar or Turbulent flow
well defined streakline, one velocity component
V  ui
Re  2100
Re  4000
velocity along the pipe is unsteady
and accompanied by random
component normal to pipe axis
V  ui + vj + wk
Laminar or Turbulent flow
• In this experiment water flows through a clear pipe with increasing speed.
Dye is injected through a small diameter tube at the left portion of the
screen. Initially, at low speed (Re <2100) the flow is laminar and the dye
stream is stationary. As the speed (Re) increases, the transitional regime
occurs and the dye stream becomes wavy (unsteady, oscillatory laminar
flow). At still higher speeds (Re>4000) the flow becomes turbulent and the
dye stream is dispersed randomly throughout the flow.
Entrance region and fully developed flow
• fluid typically enters pipe with nearly uniform velocity
• the length of entrance region depends on the Reynolds number
dimensionless
entrance length
le
 0.06 Re
D
le
1/ 6
 4.4  Re 
D
for laminar flow
for turbulent flow
Pressure and shear stress
no acceleration,
viscous forces balanced
by pressure
pressure balanced by
viscous forces and
acceleration
Fully developed laminar flow
• we will derive equation for fully developed
laminar flow in pipe using 3 approaches:
– from 2nd Newton law directly applied
– from Navier-Stokes equation
– from dimensional analysis
2nd Newton’s law directly applied
2nd Newton’s law directly applied
p1 r 2  ( p1  p) r 2   2 rl  0
p 2

l
r
doesn’t depend on radius
  Cr , at r  D / 2 stress is maximum  w wall shear stress
2 w r
4l w

and p 
D
D
2nd Newton’s law directly applied
  
for Newtonian liquid:
du
dr
 p 
   r
 2l 
 p 
du
 
r
dr
 2l 
 p  2
u  
 r  C1
 4l 
 p  2
boundary condition: u  0 at r  D / 2  C1  
D
 16  l 
2
 pD 2    2r  
u (r )  
 1    
 16  l    D  
Flow rate:
Q   udA  
D/2
0
 D4 p
u(r )2 rdr 
128l
2nd Newton’s law directly applied
• if gravity is present, it can be added to the pressure:
p   gl sin  2

l
r
p   gl sin   D 2

V
32l
  p   gl sin   D 4
Q
128l
Navier-Stokes equation applied
∇ V  0
V
p
 V  ∇V  
 g  v 2V
t

in cylindrical coordinates:
p
1   u 
  g sin   
r 
x
r r  r 
• The assumptions and the result are exactly the same as
Navier-Stokes equation is drawn from 2nd Newton law
Dimensional analysis applied
p  F V , l , D,  
Dp
 l 
 
V
D
assuming pressure drop proportional to the length:
Dp Cl

V
D

p C V
 2
l
D
( / 4C )pD 4
Q  AV 
l
Turbulent flow
• in turbulent flow the axial component of velocity fluctuates randomly,
components perpendicular to the flow axis appear
• heat and mass transfer are enhanced in turbulent flow
• in many cases reasonable results on turbulent flow can be obtained using
Bernoulli equation (Re=inf).
Fluctuation in turbulent flow
•
•
•
All parameters fluctuate in turbulent flow (velocity, pressure, shear stress,
temperature etc.) behave chaotically
flow parameters can be described as an average value + fluctuations (random
vortices)
can be characterized by turbulence intensity and time scale of fluctuation
turbulence intensity
T 
 u 
u
2
1

T

t0 T

t0

2

 u  dt 

u
2
Shear stress in turbulent flow
• Turbulent flow can often be thought of as a series of random,
3-dimensional eddy motions (swirls) ranging from large eddies
down through very small eddies
• Vortices transfer momentum, so the shear force is higher
du
compared with laminar flow:
     
  uv
lam
turb
dy
Shear stress in turbulent flow
• The turbulent nature of the flow of soup being stirred in a bowl is made
visible by use of small reflective flakes that align with the motion. The initial
stirring causes considerable small and large scale turbulence. As time
goes by, the smaller eddies dissipate, leaving the larger scale eddies.
Eventually, all of the motion dies out. The irregular, random nature of
turbulent flow is apparent.
Shear stress in turbulent flow
• Shear stress is a sum of laminar portion and a turbulent
portion
du
 
  uv   lam   turb , u  u  u
dy
positive
shear stress is larger in turbulent flow
• Alternatively:
 turb
du

dy
h – eddy viscosity
Prandtl suggested that turbulent flow is characterized by
random transfer over certain distance lm:
  lm 2
du
dy
 turb
 du 
  lm 

dy


2
2
Turbulent velocity profile
 turb  100 1000 lam
Turbulent velocity profile
• in the viscous sublayer
u
yu *

u*
v
where, y=R-r, u – time averaged x
component, u*=(t/r)½ friction velocity
valid near smooth wall:
0  yu * / v  5
function of Reynolds number
• in the turbulent layer:
1/ n
(Vc  u ) / u*  2.5ln( R / y)
or
u  r
 1  
Vc  R 
Turbulent velocity profile
•
An approximation to the velocity profile in a pipe is obtained by observing the
motion of a dye streak placed across the pipe. With a viscous oil at Reynolds
number of about 1, viscous effects dominate and it is easy to inject a relatively
straight dye streak. The resulting laminar flow profile is parabolic. With water at
Reynolds number of about 10,000, inertial effects dominate and it is difficult to
inject a straight dye streak. It is clear, however, that the turbulent velocity profile is
not parabolic, but is more nearly uniform than for laminar flow.
Dimensional analysis of pipe flow
• major loss in pipes: due to viscous flow in the straight
elements
• minor loss: due to other pipe components (junctions etc.)
Major loss:
p  F (V , D, l ,  ,  ,  )
roughness
• those 7 variables represent complete
set of parameters for the problem
 VD l  
p
 
, , 
2
1
  D D
2 V
as pressure drop is proportional to length of the tube:
p
l 

   Re, 
2
1
D 
D
2 V
Dimensional analysis of pipe flow
friction factor
p
l 



Re,


2
1

V
D
D


2
 

f    Re, 
D

pD
f 1
2
l

V
2
and
l V 2
p  f
D 2
• for fully developed laminar flow
f  64 / Re
• for fully developed steady incompressible flow (from Bernoulli eq.):
hL major
p
l V2

 f
g
D 2g
Moody chart
Friction factor as a function of Reynolds number and relative roughness for round pipes
 /D
1
2.51
 2.0 log 

 3.7 Re f
f




Colebrook formula
Non-circular ducts
• Reynolds number based on hydraulic diameter:
VDh
Reh 

4A
Dh 
P
cross-section
wetted perimeter
• Friction factor for noncircular ducts:
for fully developed laminar flow: f  C / Re
Equivalent circuit theory
flow:
electricity:
p  R  Q
V  R I
• channels connected in series
Equivalent circuit theory
• channels connected in parallel
Compliance
• compliance (hydraulic capacitance):
Q – volume V/time
flow:
Chyd
dV

dp
I – charge/time
electricity:
dq
C
dU
Equivalent circuits
Pipe networks
• Serial connection
Q1  Q2  Q3
hLAB  hL1  hL2  hL3
• Parallel connection
Q  Q1  Q2  Q3
hL1  hL2  hL3
Problems
• Ethanol solution of a dye (h=1.197
mPa·s) is used to feed a fluidic lab-onchip laser. Dimension of the channel
are L=122mm, width w=300um, height
h=10 um. Calculate pressure required
to achieve flow rate of Q=10ul/h.
• 8.7 A soft drink with properties of 10 ºC water is sucked through
a 4mm diameter 0.25m long straw at a rate of 4 cm3/s. Is the
flow at outlet laminar? Is it fully developed?
• Calculate total resistance of a
microfluidic circuit shown. Assume
that the pressure on all channels is
the same and equal Dp.