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