1. 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.
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

2. 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

3. Viscous flow in pipes

4. 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
open-channel flow
Pipe flow

5. Laminar or Turbulent flow
well defined streakline, one velocity component
As was first found by British scientist Osborne Reynolds flow in pipes can be laminar at low velocity or turbulent at high velocity.
If we want to correctly describe what is low and what is high we have to assign a dimensionless number.
velocity along the pipe is unsteady and accompanied by random component normal to pipe axis

6. 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.

7. 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

8. Pressure and shear stress
no acceleration, viscous forces balanced by pressure
pressure balanced by viscous forces and acceleration

9. 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

10. 2nd Newton’s law directly applied

11. 2nd Newton’s law directly applied
doesn’t depend on radius

12. 2nd Newton’s law directly applied
for Newtonian liquid:
Flow rate:

13. 2nd Newton’s law directly applied
if gravity is present, it can be added to the pressure:

14. Navier-Stokes equation applied
in cylindrical coordinates:
The assumptions and the result are exactly the same as Navier-Stokes equation is drawn from 2nd Newton law

15. Dimensional analysis applied
assuming pressure drop proportional to the length:

16. 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).

17. 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

18. 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 compared with laminar flow:

19. 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.

20. Shear stress in turbulent flow
Shear stress is a sum of laminar portion and a turbulent portion
positive
shear stress is larger in turbulent flow
Alternatively:
h – eddy viscosity
Prandtl suggested that turbulent flow is characterized by random transfer over certain distance lm:

21. Turbulent velocity profile

22. Turbulent velocity profile
in the viscous sublayer
where, y=R-r, u – time averaged x component, u*=(t/r)½ friction velocity
valid near smooth wall:
function of Reynolds number
in the turbulent layer:

23. 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.

24. 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:
roughness
those 7 variables represent complete set of parameters for the problem
as pressure drop is proportional to length of the tube:

25. Dimensional analysis of pipe flow
friction factor
for fully developed laminar flow
for fully developed steady incompressible flow (from Bernoulli eq.):

26. Moody chart
Friction factor as a function of Reynolds number and relative roughness for round pipes
Colebrook formula

27. Non-circular ducts Reynolds number based on hydraulic diameter:
cross-section
wetted perimeter
Friction factor for noncircular ducts:
for fully developed laminar flow:

28. Equivalent circuit theory
flow:
electricity:
channels connected in series

29. Equivalent circuit theory
channels connected in parallel

30. Compliance
compliance (hydraulic capacitance): Q – volume V/time I – charge/time
flow:
electricity:

31. Equivalent circuits

32. Pipe networks
Serial connection
Parallel connection

33. Problems
Ethanol solution of a dye (h=1.197 mPa·s) is used to feed a fluidic lab-on-chip 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.