IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Friction (and Shear) n Gas u Origin of Viscosity u...

IIT-Madras, Momentum Transfer: July 2005-Dec 2005

Friction (and Shear)

Gas Origin of Viscosity Mix of gases

Liquid Origin of Viscosity Effect of foreign materials

Dilute vs Concentrated (sol-gel) Non-newtonian Fluids

Concentrated Effect of non-spherical dispersed materials Presence of structure


Gas

123

VY

X

Gas Kinetic Theory of gas Non polar, low density

22

33

2

d

Tm

Mean Free Path is large Molecular movement between 1 and 2 (and 2 and 1, etc) Momentum Transfer between planes ==> viscosity Increase Temp ==> Increase velocity, Viscosity

Rigid Spheres

dy

dVx


Gas

Accounting for van der Waals attractive force Lennard-Jones potential

26107.2

MT Sigma- collision dia

omega- collision integral M -molecular wt

Mix of gases2

4

12

1

2

1

\

118

1

i

j

j

i

j

iij M

M

M

M

iji

iimix x

x


Liquids Theory is not as well developed

Eyring’s Theory Inter-molecular forces cause viscosity (NOT moving molecules) Temp increase ==> more energy for molecule ==> less viscosity

Similar to reaction equilibrium


Liquid Viscosity

State

Energy A

B

C

To go from A to C, the particle should have energy EAct(Activation Energy)

Energy released is heat of reaction ERxn

ActE

RxnE

For Liquid movement EA and EC are same

Application of stress shifts A up and C down

==> Movement from left to rightState

Energy A

B

C

ActE

RxnE A’ C’

Force


Dilute solutions Assume

No interaction between particles Spherical, uncharged

Liquid velocity on particle surface = particle surface velocity

fractionvoleff .5.21 Newtonian behavior Emulsions will show lower viscosity

particles do not shear, emulsions will surface contamination will increase emulsion viscosity


Non newtonian fluids When one or more of the assumptions are violated Usually heterogenous

Higher concentration (eg 40% of blood has red blood cells in plasma) ==> interaction between particles Non spherical particles Electrically charged (not discussed here)


Non newtonian fluids High concentration (high is relative) Interaction, structure formation

Structural viscosity

Application of shear stress breaks structure over time ==> thixotropic breaks structure quickly, more stress ==> more disintegration ==> pseudoplastic alternate: cylinders, ellipses align better with flow under higher shear ==> pseudoplastic thixotropic (60 sec) --> pseudoplastic

L

D

Axis Ratio = L/D

D

Lfeff


Non newtonian fluids Dilatant: Mostly solids with some fluid in between

Low stress ==> lubrication and less viscosity higher stress ==> insufficient lubrication, more viscosity

Stress

Strain

Newtonian

Dilatant

Pseudo plastic

Bingham Plastic

Bingham Plastic Minimum yield stress Newtonian


Non newtonian Fluids: Models

dx

dVNewtonian :

n

dx

dVKDilatentticPseudoplas

:,

factoryconsistencK

indexlawpowern

dx

dVPlasticBingham 0:

stressyield0

dependenttimecThixotropielasticVisco ,


Non Newtonian Fluids:Models

Viscoelastic: usually coiled or connected structure stretched (not broken) by stress recoil after stress is released normal stress on pipe != 0 eg. Pull back after the applied force is removed

Non-newtonian != high viscosity Many polymers added to reduce friction in water


Fluid flow in a pipe Hagen-Poiseuille’s law

Momentum balance

Assumptions Laminar flow steady state no-slip incompressible

xr

r

Vols

VoldVt

dAnVVF )(.

rrxrrrxxxx dxrdxrdrrPdrrPF 2222

022. xxxxxxx

s

VrrVVrrVdAnVV Pressure drop = friction


Fluid flow in a pipe xr

r

r

Cr

dx

dPrx

1

2

0

dr

rd

dx

dPr rx

finiter ,0 2

r

dx

dPrx

Newtonian xrx

dV

dr

2

32

D

V

x

P avg

Flow Rate Average Velocity

22

14 R

rR

dx

dPVx


Non newtonian: power law fluidFluid flow in a pipe

n

dx

dVK

n

nn

nn

x rRn

n

dx

dP

KV

111

12

1


Double the pressure != double velocity

2

0 0

,R

Q V r r dr d


Non newtonian: Bingham PlasticFluid flow in a pipe


Double the pressure != double velocity

dx

dVPlasticBingham 0:

arfor

220 1

4 R

rR

dx

dPrRVx

0xV arfor

dxdP

a 02


Flow between plates

Micro fluidics Identification of DNA fragments (for example) Flow rate depends on

Viscosity Surface Tension Sample movement rate depends on affinity

Sample 1

Sample 2


Flow between plates

X

Y

Z

Steady state Incompressible Laminar flow no-slip

Element of width length X, height Y and width (or depth) of 1 unit

Vols

VoldVt

dAnVVF )(.

yyyxyyxxxx XXYPYPF

2b


Flow between plates

0

dy

d

dx

dP yx1Cy

dx

dPyx

By symmetry, at the center, shear stress =0 ydx

dPyx

Newtonian

dy

dVxyx

1

2

2

1C

y

dx

dPVx

Flow rate Average velocity

byVx @,0

2

1 2yb

dx

dPVx


Flow between plates Non newtonian : Power law fluids

n

dx

dVK

1

11

11

1

C

n

ydxdP

K

dxdPK

V

n

w

byVx @,0

0@, yw

Flow rate Average velocity


Examples Pipe flow Fluid flow ~= Current flow P = Voltage, Vavg = current avgV

D

xP

2

32

Resistance Non-newtonian fluid: non-linear relation

between P and Vavg

Newtonian fluid: easier prediction of results of changing one or more parameters


Example Non newtonian : Bingham Plastic

42

3

1

3

41

32mm

x

PDVavg PD

Lm

04 1m

smkg /2.0Pa200 3/2000 mkg

H=10 m

L=20m/5m

D=0.1m082.04 0

PD

Lm

89.0factor

smVavg /6.13

02.04 0

PD

Lm

97.0factor smVavg /4.59


Example Find the time taken to drain the tank

1

221 A

VAV

dt

dH

H=10 m

L=20m/5m

D=0.1m

V2 is a function of H

Tank will not drain completely!


Example Non newtonian : Power law fluid

n

n

avg xK

PD

n

n

xK

PDV

12

431

4

32

smVm

kgn

mskgKcmDP

avg /1,2000,5.0

,/5.2,1?,

3

2

3

MPaP 79.0

1m/s

25 m Long, 1cm dia

MPaP 12.1

If flow rate has to be doubled, pressure needed

navg PCV

1


Example Pbm. 8.2

Given, L12=22 km, L23=18 km, Q, P known

Consider this as resistance model

2

32

D

V

x

P avg

L12 L23

4

128

D

Q

2312 LLafterbefore PPPP

4

13

D

QLKP beforebefore

4

1212 D

QLKP after

4

2323 2

1

D

QLKP after

31/40

223

12

13

L

L

L

Q

Q

before

after


Viscometers Tube,Cone&Plate,Narrow gap cylinder, infinite gap

cylinder


Viscometers Cone and Plate Viscometer Ref: BSL, pbm 2B.11

0

22 0

tofromchangecan

drdr

X

TorquevelangularRGiven ,,

)2

tan(,)sin( 000

h

yrVx

YA

rV

Goal: Shear Stress, Velocity Profile, Torque

Fluid between two plates, linear profile

0

2

rV


Viscometers Shear stress vs Velocity: Spherical Co-ordinates

V

r

V

r sin

1

sin

sin

22

2

sin

cossin

sin

sin

r

VV

r

Shear Stress at cone: )tan(,2 0

0

, , ,independent of r z

Force, Torque

drdrF

rdrdrTorque

R

drdrTorque0

2

0

2

0

0

3

3

2

R

Torque

Practical 0 ~ 1o


Viscometers Cylindrical viscometer

rgapplatesparallelgapNarrow ,

r

r

r

r

V

dr

dV

rrLrFTorque 2

Vary , obtain Torque and velocity gradients for plots

Torque


Differential momentum balance:Navier-Stokes Equation

Newtonian Fluids ONLY Assumptions/applicability:

Isothermal conditions both Compressible/Incompressible both laminar/turbulent Stokes assumption for bulk-viscosity (needed for

compressible fluids)

0.,2 VifVPgDt

DV

0. VDt

D Continuity (Velocity Divergence)

0, ifPgDt

DV


Appendix

Pbm. 8.6 Given, L=8m,P=207kPa, d=.635 cm, Find velocity for no friction vs friction

22

22

11

21

22gh

Pvgh

Pv

P

VhhV

2

,0 2211

Frictional effects

LPD

V32

2

2

2.11

2322

cos

PD

L

V

V

viscous

ityVisNo

1 2


Appendix: Blood Flow in Arteries 40% red blood cells in plasma, non-newtonian Pulsating motion, varying pressure Re = 600, during exercise 6000 Blood vessel dilation , short term, long term Shear stress vs platelet activation (wound vs

stenosis); ultrasonic detection Tensile vs compressive stress; structure of blood

vessel Collapse of vessel during BP measurement Collapse near stenosis and cardiac arrest Mass & heat transport

IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Friction (and Shear) n Gas u Origin of Viscosity u...

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Transcript of IIT-Madras, Momentum Transfer: July 2005-Dec 2005 Friction (and Shear) n Gas u Origin of Viscosity u...