1. FLUID FLOW
STREAMLINE – LAMINAR FLOW
TURBULENT FLOW
REYNOLDS NUMBER

2. Streamlines for fluid passing an obstacle streamlines v
Velocity of particle
Velocity profile for the laminar flow of a non viscous liquid -
tangent to streamline

3. Turbulent flow indicted by the swirls – eddies and vortices

6. REYNOLDS NUMBER
A British scientist Osborne Reynolds (1842 – 1912) established that the nature of the flow depends upon a dimensionless quantity, which is now called the Reynolds number Re.
Re =  v L / 
r density of fluid
v average flow velocity over the cross section of the
pipe
L dimension characterising a cross section

7. Re is a dimensionless number
Re =  v L / 
[Re]  [kg.m-3] [m.s-1][m] [Pa.s]-1
 [kg] [m-1][s-1][kg.m.s-2.m-2.s]-1 = [1]
For a fluid flowing through a pipe – as a rule of thumb
Re < ~ laminar flow
~ < Re < ~ unstable laminar / turbulent
Re > ~ turbulent

8. Re =  v L / 
Sydney Harbour Ferry

9. Re =  v L /  turbulent flow Re = (103)(5)(10) / (10-3) Re = 5x107
r = 103 kg.m-3
v = 5 m.s-1
L = 10 m
 = Pa.s
turbulent flow

10. Spermatozoa swimming
Re =  v L / 

11. Re =  v L /  laminar - just as well !!! Spermatozoa swimming
r = 103 kg.m-3
v = m.s-1
L = 10 mm
 = Pa.s
Re = (103)(10-5)(10x10-6) / (10-3)
Re = 10-4
laminar - just as well !!!

12. Household plumbing pipes
Typical household pipes are about 30 mm in diameter and water flows at about 10 m.s-1
Re ~ (10)(3010-3)(103) / (10-3) ~ 3105
The circulatory system
Speed of blood v ~ 0.2 m.s-1
Diameter of the largest blood vessel, the aorta L ~ 10 mm
Viscosity of blood probably  ~ 10-3 Pa.s (assume same as water)
Re ~ (0.2)(1010-3)(103) / 10-3) ~ 2103

13. The method of swimming is quite different for
Fish (Re ~ ) and sperm (Re ~ )
Modes of boat propulsion which work in thin liquids (water) will not work in thick liquids (glycerine)
It is possible to stir glycerine up, and then unstir it completely. You cannot do this with water.

14. EQUATION OF CONTINUITY
FLUID FLOW
IDEAL FLUID
EQUATION OF CONTINUITY
How can the blood deliver oxygen to body so successfully?
How do we model fluids flowing in streamlined motion?

15. IDEAL FLUID
Fluid motion is usually very complicated. However, by making a set of assumptions about the fluid, one can still develop useful models of fluid behaviour.
An ideal fluid is
Incompressible – the density is constant
Irrotational – the flow is smooth, no turbulence
Nonviscous – fluid has no internal friction (  = 0)
Steady flow – the velocity of the fluid at each point is constant in time.

16. where streamlines crowd together the
EQUATION OF CONTINUITY (conservation of mass)
In complicated patterns of streamline flow, the streamlines effectively define flow tubes. So the equation of continuity –
where streamlines crowd together the
flow speed must increase.

17. EQUATION OF CONTINUITY (conservation of mass)
m1 = m2
 V1 =  V2
 A1 v1 t =  A2 v2 t
A1 v1 = A2 v2
A v measures the volume of the fluid that flows past any point of the tube divided by the time interval
volume flow rate Q = dV / dt
Q = A v = constant
if A decreases then v increases
if A increases then v decreases

18. Applications
Rivers
Circulatory systems
Respiratory systems
Air conditioning systems

19. Blood flowing through our body
The radius of the aorta is ~ 10 mm and the blood flowing through it has a speed ~ 300 mm.s-1. A capillary has a radius ~ 410-3 mm but there are literally billions of them. The average speed of blood through the capillaries is ~ 510-4 m.s-1.
Calculate the effective cross sectional area of the capillaries and the approximate number of capillaries.

20. Setup
radius of aorta RA = 10 mm = 1010-3 m
radius of capillaries RC = 410-3 mm = 410-6 m
speed of blood thru. aorta vA = 300 mm.s-1 = m.s-1
speed of blood thru. capillaries RC = 510-4 m.s-1
Assume steady flow of an ideal fluid and apply the equation of continuity
Q = A v = constant  AA vA = AC vC
where AA and AC are cross sectional areas of aorta & capillaries respectively.
aorta
capillaries

21. Action
AC = AA (vA / vC) =  RA2 (vA / vC)
AC = (1010-3)2(0.300 / 510-4) m2 = m2
If N is the number of capillaries then
AC = N  RC2
N = AC / ( RC2) = 0.2 / { (410-6)2}
N = 4109