FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.
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Transcript of FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.
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 direction
Need three numbers to represent them: zyxx ,,
wvuV ,,
http://www.xcrysden.org/doc/vectorField.html
Unit vector = vector whose length equals 1
1
0
0ˆ,
0
1
0ˆ,
0
0
1ˆ 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
z
w
y
v
x
uV
332211 vuvuvu
CONCEPTS RELATED TO VECTORS
CrossProduct: 321321 ,,,, vvvuuuvu
321
321
ˆˆˆ
vvv
uuu
kji
122131132332ˆˆˆ vuvukvuvujvuvui
wvuzyxv ,,,,
wvu
zyx
kji ˆˆˆ
]
,
,[
yuxv
xwzu
zvyw
INDICIAL or TENSOR NOTATION
3,2,1 iAA i
Vector or First Order Tensor
iiBABA
Vector Dot Product
332211 BABABA
3,2,1;3,2,1
333231
232221
131211
jiC
CCC
CCC
CCC
C ij
Matrix
or Second Order Tensor
txVV iii ,
INDICIAL or TENSOR NOTATION
3,2,1
ix
CC
i
Gradient of Scalar
j
i
x
BB
Gradient of Vector Second Order Tensor
ji
jiij 0
1Special operator – Kronecker Delta
jkijik CC
333333323332313331
232223222222212221
131113121112111111
CCCCCC
CCCCCC
CCCCCC
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
00
00
00
ij
Normal (perpendicular) forces caused by pressure
MATERIAL (or SUBSTANTIAL or PARTICLE) DERIVATIVE
zyxtSS ,,,
t
z
z
S
t
y
y
S
t
x
x
S
t
S
Dt
DS
z
Sw
y
Sv
x
Su
t
S
Dt
DS
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
dx
xu
xxu
y
u
dyyu
yyu
zu
dzzu
zzu
Diffusive Fluxes (of momentum)
x
z
y
dy
dz
dx
xu
dx
xu
xxu
y
u
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
u
x
u
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
dzzp
zpdzzpz0
00
AdzgforceGravity
00
00
AdzgAdzzp
zpAzpz
Adzbydividing
gzp
gdzdp
Czgp
Integrating in z:
Example – Application of the Hydrostatic Equation - 1
z
H
Find h
Downward Force?
Weight of the cylinder = W
Upward Force?
Pressure on the cylinder = F
aircAhgF ;
c
c
AgW
h
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 D
gWz
gWgWzdzFD
Dx