FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

24
FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS

Transcript of FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

Page 1: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

FLUID PROPERTIES

Independent variables

SCALARS

VECTORS

TENSORS

Page 2: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

x

z

y

dy

dz

dx, u

, w

, v

REFERENCE FRAME

Page 3: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

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

Page 4: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

VECTORS

Have length and direction

Need three numbers to represent them: zyxx ,,

wvuV ,,

http://www.xcrysden.org/doc/vectorField.html

Page 5: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

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

Page 6: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

VECTORS

In terms of the unit vector: zkyjxix ˆˆˆ

wkvjuiV ˆˆˆ

txVV ,

Page 7: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

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

Page 8: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

CONCEPTS RELATED TO VECTORS

CrossProduct: 321321 ,,,, vvvuuuvu

321

321

ˆˆˆ

vvv

uuu

kji

122131132332ˆˆˆ vuvukvuvujvuvui

wvuzyxv ,,,,

wvu

zyx

kji ˆˆˆ

]

,

,[

yuxv

xwzu

zvyw

Page 9: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

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 ,

Page 10: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

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

Page 11: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

TENSORS

x

z

y

Need nine numbers to represent them: txiijij ,

3,2,1, ji

333231

232221

131211

ij

zyxji ,,,

zzzyzx

yzyyyx

xzxyxx

ij

Page 12: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

For a fluid at rest:

33

22

11

00

00

00

ij

Normal (perpendicular) forces caused by pressure

Page 13: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

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

Page 14: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

Fluids

Deform more easily than solids

Have no preferred shape

Page 15: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

Deformation, or motion, is produced by a shear stress

ndeformatioofrate

zu

z

x

u

μ = molecular dynamic viscosity [Pa·s = kg/(m·s)]

Page 16: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

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

Page 17: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

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)

Page 18: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

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)

Page 19: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

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

Page 20: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

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)

Page 21: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

Boundary Conditions

Zero Flux

No-Slip [u (z = 0) = 0]

z

x

u

Page 22: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

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:

Page 23: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

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

Page 24: FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.

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