http:// web notes: lect5.ppt flow1.pdf flow2.pdf

http://www.physics.usyd.edu.au/teach_res/jp/fluids/wfluids.htm

http://www.physics.usyd.edu.au/teach_res/jp/fluids/

web notes: lect5.ppt

flow1.pdf flow2.pdf

FLUID FLOW STREAMLINE – LAMINAR FLOW TURBULENT FLOW REYNOLDS NUMBER

Velocity of particle- tangent to streamline

streamlines

Streamlines for fluid passing an obstacle

v

Velocity profile for the laminar flow of a non viscous liquid

Turbulent flow indicted by the swirls – eddies and vortices

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 /

density of fluidv average flow velocity over the cross section of the pipeL dimension characterising a cross section

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 < ~ 2000 laminar flow

~ 2000 < Re < ~ 3000 unstable laminar / turbulent

Re > ~ 2000 turbulent

Re = v L /

Sydney Harbour Ferry

Re = v L /

= 103 kg.m-3

v = 5 m.s-1

L = 10 m

= 10-3 Pa.s

Re = (103)(5)(10) / (10-3)

Re = 5x107

Re = v L /

Spermatozoa swimming

Re = v L /

Spermatozoa swimming

= 103 kg.m-3

v = 10-5 m.s-1

L = 10 m

= 10-3 Pa.s

Re = (103)(10-5)(10x10-6) / (10-3)

Re = 10-4

Household plumbing pipesTypical household pipes are about 30 mm in diameter and water flows at about 10 m.s-1

Re ~ (10)(3010-3)(103) / (10-3) ~ 3105

The circulatory systemSpeed of blood v ~ 0.2 m.s-1 Diameter of the largest blood vessel, the aorta L ~ 10 mmViscosity of blood probably ~ 10-3 Pa.s (assume same as water) Re ~ (0.2)(1010-3)(103) / 10-3) ~ 2103

The method of swimming is quite different for Fish (Re ~ 10 000) and sperm (Re ~ 0.0001)

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.

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?

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.

EQUATION OF CONTINUITY (conservation of mass)

A1

A2

v1

v2

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.

m1 = m2

V1 = V2

A1 v1 t = A2 v2 t

A1 v1 = A2 v2

EQUATION OF CONTINUITY (conservation of mass)

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 increasesif A increases then v decreases

Applications

Rivers

Circulatory systems

Respiratory systems

Air conditioning systems

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 ~ 410-3 mm but there are literally billions of them. The average speed of blood through the capillaries is ~ 510-4 m.s-1.

Calculate the effective cross sectional area of the capillaries and the approximate number of capillaries.

Setup

radius of aorta RA = 10 mm = 1010-3 mradius of capillaries RC = 410-3 mm = 410-6 mspeed of blood thru. aorta vA = 300 mm.s-1 = 0.300 m.s-1

speed of blood thru. capillaries RC = 510-4 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

Action

AC = AA (vA / vC) = RA2 (vA / vC)

AC = (1010-3)2(0.300 / 510-4) m2 = 0.20 m2

If N is the number of capillaries then

AC = N RC2

N = AC / ( RC2) = 0.2 / { (410-6)2}

N = 4109