Friction (and Shear)

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

2dTm

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

dydVx


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

41

21

21

\

118

1

i

j

j

i

j

iij M

MMM

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 right

State

Energy A

B

CActE

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

DLfeff


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

DilatantPseudo plastic

Bingham Plastic

Bingham Plastic Minimum yield stress Newtonian


Non newtonian Fluids: Models

dxdVNewtonian :

n

dxdVKDilatentticPseudoplas

:,

factoryconsistencKindexlawpowern

dxdVPlasticBingham 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. xxxxxxxs

VrrVVrrVdAnVV Pressure drop = friction


Fluid flow in a pipe xr

r

rCr

dxdP

rx1

2

0

drrd

dxdPr rx

finiter ,0 2r

dxdP

rx

Newtonian xrx

dVdr

2

32DV

xP avg

Flow Rate Average Velocity

22

14 R

rRdxdPVx


Non newtonian: power law fluidFluid flow in a pipe

n

dxdVK

n

nn

nn

x rRnn

dxdP

KV

111

121


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

dxdVPlasticBingham 0:

arfor

220 1

4 RrR

dxdPrRVx

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

dyd

dxdP yx

1CydxdP

yx

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

yx

Newtonian

dydVx

yx

1

2

21 Cy

dxdPVx

Flow rate Average velocity

byVx @,0

2

1 2ybdxdPVx


Flow between plates Non newtonian : Power law fluids

n

dxdVK

1

11

11

1

C

n

ydxdP

K

dxdPKV

n

w

byVx @,0

0@, yw

Flow rate Average velocity


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

DxP 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

31

341

32mm

xPDVavg PD

Lm

04 1m

smkg /2.0Pa200 3/2000 mkg

H=10 m

L=20m/5m

D=0.1m082.04 0

PDLm

89.0factor

smVavg /6.13

02.04 0

PDLm

97.0factor smVavg /4.59


Example Find the time taken to drain the tank

1

221 A

VAVdtdH

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

nn

avg xKPD

nn

xKPDV

12

4314

32

smVmkgn

mskgKcmDP

avg /1,2000,5.0

,/5.2,1?,

3

23

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

32DV

xP avg

L12 L23

4

128DQ

2312 LLafterbefore PPPP

413

DQLK

P beforebefore 4

1212 D

QLKP after 4

2323 2

1DQLK

P after

31/40

223

12

13

LL

LQQ

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

hyrVx

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

Vr

Vr sin

1sin

sin

22

2

sincossin

sinsin

rVV

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

32

RTorque

Practical 0 ~ 1o


Viscometers Cylindrical viscometer

rgapplatesparallelgapNarrow ,

r

rr

rV

drdV

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 VifVPgDtDV

0. VDtD Continuity (Velocity Divergence)

0, ifPgDtDV


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

22

22

11

21

22ghPvghPv

PVhhV

2,0 2211

Frictional effects

LPDV

32

2

2

2.112322

cos

PD

LV

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

Friction (and Shear)

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