fluid properties
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FLUID PROPERTIES
Independent variables
SCALARS
VECTORS
TENSORS
x
z
y
dy
dz
dx, u
, w
, v
REFERENCE FRAME
SCALARS
Need a single number to represent them: P, T, ρ
besttofind.com
Temperature
May vary in any dimension x, y, z, t
www.physicalgeography.net/fundamentals/7d.html
VECTORS
Have length and directionNeed three numbers to represent them: zyxx ,,
wvuV ,,
http://www.xcrysden.org/doc/vectorField.html
Unit vector = vector whose length equals 1
100
ˆ,010
ˆ,001
ˆ kji
]ˆ,ˆ,ˆ[ kji
x
y
z
i
j
k
VECTORS
In terms of the unit vector: zkyjxix ˆˆˆ
wkvjuiV ˆˆˆ
txVV ,
CONCEPTS RELATED TO VECTORS
Nabla operator:
zyx
,,
Denotes spatial variability
kz
jy
ix
ˆˆˆ
Dot Product: 321321 ,,,, vvvuuuvu
zw
yv
xuV
332211 vuvuvu
CONCEPTS RELATED TO VECTORS
CrossProduct: 321321 ,,,, vvvuuuvu
321
321
ˆˆˆ
vvvuuukji
122131132332ˆˆˆ vuvukvuvujvuvui
wvuzyxv ,,,,
wvu
zyxkji ˆˆˆ
],,[
yuxvxwzuzvyw
INDICIAL or TENSOR NOTATION
3,2,1 iAA i
Vector or First Order Tensor
iiBABA
Vector Dot Product
332211 BABABA
3,2,1;3,2,1333231
232221
131211
jiC
CCCCCCCCC
C ij
Matrix
or Second Order Tensor
txVV iii ,
INDICIAL or TENSOR NOTATION
3,2,1
ixCC
iGradient of Scalar
j
i
xBB
Gradient of Vector Second Order Tensor
jiji
ij 01
Special operator – Kronecker Delta
jkijik CC
333333323332313331
232223222222212221
131113121112111111
CCCCCCCCCCCCCCCCCC
TENSORS
x
z
y
Need nine numbers to represent them: txiijij ,
3,2,1, ji
333231
232221
131211
ij
zyxji ,,,
zzzyzx
yzyyyx
xzxyxx
ij
For a fluid at rest:
33
22
11
000000
ij
Normal (perpendicular) forces caused by pressure
MATERIAL (or SUBSTANTIAL or PARTICLE) DERIVATIVE zyxtSS ,,,
tz
zS
ty
yS
tx
xS
tS
DtDS
zSw
ySv
xSu
tS
DtDS
Fluids
Deform more easily than solids
Have no preferred shape
Deformation, or motion, is produced by a shear stress
ndeformatioofrate
zu
z
x
u
μ = molecular dynamic viscosity [Pa·s = kg/(m·s)]
Continuum Approximation
Even though matter is made of discrete particles, we can assume that matter is distributed continuously.
This is because distance between molecules << scales of variation
tx,
ψ (any property) varies continuously as a function of space and time
space and time are the independent variables
In the Continuum description, need to allow for relevant molecular processes – Diffusive Fluxes
Diffusive Fluxes
e.g. Fourier Heat Conduction law:
TkQ
z
x
t = 0
t = 1
t = 2
Continuum representation of molecular interactions
This is for a scalar (heat flux – a vector itself)
but it also applies to a vector (momentum flux)
x
z
y
dy
dz
dx
Shear stress has units of kg m-1 s-1 m s-1 m-1 = kg m-1 s-2
Shear stress is proportional to the rate of shear normal to which the stress is exerted zu
zu
at molecular scales
µ is the molecular dynamic viscosity = 10-3 kg m-1 s-1 for water is a property of the fluid
or force per unit area or pressure: kg m s-2 m-2 = kg m-1 s-2
xu
dxxu
xxu
yu
dyyu
yyu
zu
dzzu
zzu
Diffusive Fluxes (of momentum)
x
z
y
dy
dz
dx
xu
dx
xu
xxu
yu
dyyu
yyu
zu
dzzu
zzu
Net momentum flux by u
zu
zyu
yxu
x
Diffusive Fluxes (of momentum)
For a vector (momentum), the diffusion law can be written as (for an incompressible fluid):
i
j
j
iij x
uxu
2
Shear stress linearly proportional to strain rate – Newtonian Fluid (viscosity is constant)
Boundary Conditions
Zero Flux
No-Slip [u (z = 0) = 0]
z
x
u
Hydrostatics - The Hydrostatic Equation
z
g
A
z = z0
z = z0 + dzdz
p
p + (∂p/∂z ) dz
dzzpzpdzzp
z000
AdzgforceGravity
00
00
AdzgAdzzpzpAzp
z
Adzbydividing
gzp
g
dzdp
Czgp
Integrating in z:
Example – Application of the Hydrostatic Equation - 1
z
H
Find hDownward Force?
Weight of the cylinder = W
Upward Force?Pressure on the cylinder = F
aircAhgF ;
c
c
AgWh
AhgW
Same result as with Archimedes’ principle (volume displaced = h Ac) so thebuoyant force is the same as F
Pressure on the cylinder = F = W
AC
h
Example – Application of the Hydrostatic Equation - 2
z
W
D
Find force on bottom and sides of tank
On bottom?
On vertical sides?
Same force on the other side
TD ADgF x
L
AT = L W
dFx gWzdzdzWpdFx
Integrating over depth (bottom to surface)
22
2020 DgWzgWgWzdzF
DDx
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