5lecture slip noslip

8/2/2019 5Lecture Slip NoSlip

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Indian Institute of Technology, Kharagpur, IndiaIndian Institute of Technology, Kharagpur, India -- 721302721302

Boundary Conditions in FluidBoundary Conditions in Fluid

Mechanics: Slip or No Slip?Mechanics: Slip or No Slip?

March, 2011

Suman

Chakraborty

Professor

Mechanical Engineering Department

Indian Institute of Technology (IIT) Kharagpur, IndiaE-mail: [email protected]

http://www.stanford.edu/~sumancha/
mailto:[email protected]:[email protected]


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Macroscopic vs. Microscopic

Viewpoint: the Continuum

Hypothesis

Molecular Approach:

Direct analysis of dynamics of individual molecules

Microscopic Approach:

Statistically-averaged

behavior of many

molecules

Macroscopic Approach: Gross or averaged effect of many molecules

that can be captured by direct measuring instruments (treats the

fluid as a

continuous medium disregarding the discontinuity in the underlyingmolecular picture)

Continuum Hypothesis

works when: (a) there are sufficiently large

numbers of molecules in chosen elemental volumes so that statistical

uncertainties with regard to their respective positions and velocities do notperceptibly influence the averaged fluid/ flow property predictions, as well

as the predictions in the local gradients of properties through well-known

rules of differential calculus, and (b) the system is not significantly deviated

from local thermodynamic equilibrium.


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Assessment of Continuum Considerations: Gas FlowsAssessment of Continuum Considerations: Gas Flows

When gas molecules collide with a solid boundary, those aretemporarily adsorbed on the wall and are subsequently ejected. Thisallows a partial transfer of momentum and energy of the walls to

the

gas molecules.

If the frequency of collisions is very large, the momentum and

energy exchange is virtually complete and there may be no relativetangential momentum between the fluid and the solid boundary. Thisis known as No-Slip

Boundary Condition.

However, in a less-dense system, deviations from such idealization

are significantly more ominous. The extent of this deviation is notmerely dictated by the mean free path () in an absolute sense, butalso its comparability with the characteristic system length scale (L)that describes the relative importance of rarefaction in the system.

The ratio of these two, known as the Knudsen number (Kn= /L),appears to the single important decisive parameter that determinesthe applicability of a particular flow modeling strategy as against theextent of rarefaction of the flow medium.


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Gas Flows: Slip vs. No slip

The notion underlying the no-slip boundary condition is that there cannot be anyfinite velocity or temperature discontinuities within the fluid

Such discontinuities would result in infinite velocity/temperature gradients and

hence infinite stress and heat flux thereby destroying the discontinuities in no time.

Thus, the fluid velocity must be zero at the wall and also the temperature of thefluid must be the same as that of the wall.

However, the above boundary conditions are valid only if the fluid adjacent to the

solid wall is in thermodynamic equilibrium.

The achievement of thermodynamic equilibrium requires an infinitely large

number of collisions between the fluid molecules and the solid surface.

The no-slip condition holds good so long as Kn


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Maxwell First Order Slip Model : A SimpleMaxwell First Order Slip Model : A Simple

DerivationDerivation

i r

i w

diffusespecularfrom top layer

from bottom layer

1 11

2 2

g wU U U U

2ww

dUU U

dn

U

Ug

Uw

: tangentialmomentum

Subscript: iincident

rreflectedwwall

=1diffusereflection

=0specularreflection

Eliminating U

, it follows:2 3

4g w

w wgas

dUU

dn

T

TU

s

(Tangential

momentumaccomodation

coefficient)

Add term in presence

of temp grad


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Summary: Slip Behavior of GasesSummary: Slip Behavior of Gases

The first set of fluid molecules comes in contactwith the plate, these molecules tend to stick to

the solid.

Molecules of a fluid next to a solid surface are adsorbed onto the surface fora short period of time, and are then desorbed and ejected into the fluid.

This process slows down the fluid and renders the tangential component of

the fluid velocity equal to the corresponding component of the boundary

velocity. However, this consideration remains valid only if the fluid adjacent tothe solid wall is in thermodynamic equilibrium.

Deviation from thermodynamic equilibrium may result in a slip

between fluid

and the solid boundary in small channels where the mean free path may be

of comparable order as that of the channel dimension.

This phenomenon may be more aggravated by the presence of strong

local

gradients of temperature and/or density, because of which the molecules

tending to slip

on the walls experience a net driving force. Such phenomena

are usually termed as thermophoresis

and diffusophoresis, respectively.


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Slip Boundary Condition for Liquids??Slip Boundary Condition for Liquids??

Because of sufficient intermolecular forces of attraction between themolecules of the solid surface and a dense medium such as theliquid, it is expected that the liquid molecules would remainstationary relative to the solid boundary at their points of contact.

Only at very high shear rates (typically realizable only in extremelynarrow confinements of size roughly a few molecular diameters), thestraining may be sufficient enough in moving the fluid molecules

adhering to the solid by overcoming the van der Waals

forces of

attraction.

Another theory argues that the no-slip boundary condition arises dueto microscopic boundary roughness, since fluid elements may getlocally trapped within the surface asperities. If the fluid is a

liquidthen it may not be possible for the molecules to escape from thattrapping, because of an otherwise compact molecular packing.

Following this argument, it may be conjectured that a molecularlysmooth boundary would allow the liquid to slip, because of the non-

existence of the surface asperity barriers.


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Continuum picture Molecular picture

No-Slip Boundary Condition, A Paradigm

0slipv

0slip

v

?

n

Slip or No-Slip?


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What Happens for CarbonWhat Happens for Carbon NanotubesNanotubes??

Researchers have demonstrated that the rate of liquid flow througha membrane composed of an array of aligned carbon nanotubes

might turn out to be four to five orders of magnitude faster than thatpredicted from classical fluid-flow analysis. They attributed this phenomenon to an apparently frictionlessinterfacial condition at the carbon-nanotube

wall.

Such observations were contrary to the common consensus thatfluid flow through nano-pores having chemical selectivity is ratherslow.

However, from fundamental physical considerations, water is likelyto be able to flow fast through hydrophobic single-walled carbonnanotubes; the primary reason being the fact that the processcreates ordered hydrogen bonds between the water molecules.Accordingly, ordered hydrogen bonds between water molecules andthe weak attraction between the water and smooth carbon nanotube

graphite sheets, as well as the rapid diffusion of hydrocarbons arequalitatively attributed to the fundamental scientific origin of

reducedfrictional resistances encountered in such systems.


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: slip length, from nano- to micrometerPractically, no slip in macroscopic flows

sslip

lv

0// RlUv sslip

:shear rate at solid surfacesl

RU /

(1823)Slip Boundary Condition

Need to address:

1.

Apparent Violation

seen from

the moving/slippingcontact line

2.

Infinite Energy Dissipation

(unphysical singularity)


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Slip at High Shear RatesSlip at High Shear Rates


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Liquid Slip: Role of Surface Characteristics

ManufacturingProcess

SurfaceCharacteristics

Fluid Flow

?

LETS TRY TO ANSWER

Recent studies have demonstrated that the intuitive assumption of no slip

at the boundary

can fail greatly not only when the fluidic substrates are

sufficiently smooth, but also when they are sufficiently rough!!


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Linking Through the Friction Factor..

For fluid flows through microchannel the friction factor (f)has been experimentally obtained

Higher

W. Peiyi et al(Cryogenics 23, 273,1983); C. Y. Yang et al(Int. J. Heat

Mass Transfer39, 791,1996); W. Qu et al(Int. J. Heat Mass Transfer43, 353, 2000);

D. Pfund et al(AIChE J. 46, 1496, 2000)

Lower

B. X. Wang et al(Int. J. Heat Mass Transfer37, 73,1994); X. F. Peng et al

(Int. J. Heat Mass Transfer39, 2599,1996); S. B. Choi et al(ASME-DSC 32,

123,1991)

than the classically predicted value

(fRe~24 for parallel plate)

THIS CHALLENGES THE CLASSICAL THEORY WHICH

STATES THAT THE PRODUCT OF FRICTION FACTOR ANDREYNOLDS NUMBER IS A CONSTANT FOR FULLY

DEVELOPED LAMINAR FLOW, INDEPENDENT OF THE

SURFACE ROUGHNESS CHARACTERISTICS

[Ref: S. Chakraborty

and K. D. Anand, Physics of Fluids 20, 043602 (2008)]


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Preliminary Investigations with Simple Experiments!Preliminary Investigations with Simple Experiments!


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2D & 3D profile of 102D & 3D profile of 10

mm

1010 m scan aream scan areaof microchannelof microchannel

surface, processed bysurface, processed by

Up milling at 12Up milling at 12

mm/min feed ratemm/min feed rate

577.08 nm

0.00 nm

AFM Imaging and PSD Analysis

Due to the multi-scale

nature of roughness , a

surface profile is

considered to be

composed of asuperposition of spatial

waves of increasing

frequency.

L

dxexZfP

Lfxi

022 ))((

)(

P(f)

is the power of surface roughness wave of

frequency fZ(x)

is the height variation function

L

is the total scan length

X

is the spatial variable


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NefPnACF

LN

Lf

Nfni

2/

/2

)/(2

)()(

2

1.

),( NZYXZN

mn

mna

The correlation length,The correlation length, ll, also known as the, also known as the

independent length, was determined as theindependent length, was determined as thedistance at which the ACF fell to 1/e timesdistance at which the ACF fell to 1/e times

its maximum valueits maximum value.

Auto CoAuto Co--Relation Function (ACF)Relation Function (ACF)

was deducedwas deducedfrom the Inverse Fourierfrom the Inverse Fourier

Transform of P(f)Transform of P(f)

Auto Correlation Function and Arithmetic Roughness

Arithmetic SurfaceArithmetic Surface

Roughness:Roughness:Z = Mean pixelated height from Scan profile= Mean pixelated height from Scan profile

Correlation lengths were found to beCorrelation lengths were found to be

independent of the length ofindependent of the length ofscanningscanningaa

fractal nature of thefractal nature of the

surface?surface?


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0,0 a

0,1 fD

)10( 3 OD

D

f

h

a

)]()(1[0exp fh

at DD

CC

)ln(h

a

D

A

qfDB )1(

A Mod i f ied Con sider a t ion on Po iseu i l l e Num ber f o r

M ic rochanne ls

The fittingfunctions can bechosen as,

&Also,

Dependence on

average relative

surface roughness

Dependence on

surface topology

)(2

])2([ 0*

WH

WDWHDD

ff

f

])1(96.7

)ln(

241[ 5.1*0exp f

h

at D

D

CC


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Exploring the Science: Revisiting the

Continuum ConsiderationsContinuum Hypothesis assumes:

The local properties such as density and velocity are

defined as averages over elements large compared tothe microscopic structure but small in comparison with

the scale of the flow to permit the use of differential

calculus.

The flow must not be too far from thermodynamic

equilibrium.

The former condition is more commonly satisfied. It isusually the later one which restricts the validity of the

model


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Hydrodynamics over Slipping Surfaces[Ref: S. Chakraborty, Phys. Rev. Lett. 99, 094504 (2007)]

Confining rough surfaces made of

hydrophobic materials may trigger theformation of tiny bubbles adhering to the walls

in tiny channels at certain locations. This

incipient vapor layer acts as an effective

smoothening blanket, by disallowing the liquidon the top of it to be directly exposed to the

rough surface asperities. In such cases, the

liquid is not likely to feel the presence of the

wall directly and may smoothly sail over the

intervening vapor layer shield. Thus, instead of

sticking

to a rough channel surface, the liquid

may effectively slip

on the same.

There may also be effective stick-slip motion due to the random surface

inhomogeneities

that are directly exposed to the liquid being transported,

over the remaining fraction of the interface. Relative contribution of

these two effects is stochastic, due to uncertainties in surface

characteristics and thermodynamic conditions


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Apparent Slip and Stick

2

,eff

0

2sh dK

l S K K

l

Ex: For the surface with a

Gaussian form, i.e., of the form2

2exp( )x

l

2exp 4KS K

Nature Materials

2,

221227 (2003)

,effw s

w

uu l

y


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A Generalized Proposition For The Friction

Factor

Generalized slipGeneralized slip--

based stochasticbased stochasticboundary conditionboundary conditionUncertaintiesUncertainties

associated with theassociated with the

production, densityproduction, density

and sizeand size

distribution of thedistribution of the

nanobubblesnanobubbles

Implications ofImplications of

surface roughnesssurface roughness

elements andelements and

hydrophobicity,hydrophobicity,

within awithin a

continuumcontinuum--basedbased

frameworkframework

Stochastic formalism of the Navier-Stokes equations

Generalized treatment of surface conditions for

microchannel liquid flows


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Quantitative fitting of Simulation data on Friction Factor

StokesStokes

flow scaling forflow scaling for

thethe nanobubblenanobubble--

dispersed layerdispersed layer

2 33 2

0 0 1 0 2 0 3

1~h h h h

l

dpuD a z a z D a z D a D

dx

2 3

0 1 2 3

4Re ~f

a a a a wherewhere

hDz 0

StickStick--slip scaling forslip scaling for

liquids directlyliquids directly

exposed to the surfaceexposed to the surface

roughness elementsroughness elements

0 0

1Re ~

s

h

fl

c dD

Th F i ti F t A W i ht d A d C bi ti


23/31


2

3 18

481

Re

h

h

H K

D

D lf f

3

21 3 33 3

)2537.09564.07012.19467.13553.11( 5432 aaaaaf KH

v

l

0

h

z

D

HartnettHartnett--Kostic polynomial correction factor that accommodatesKostic polynomial correction factor that accommodates

a nona non--infinite extent of the rectangular microchannel into accountinfinite extent of the rectangular microchannel into account

is the mean nanobubble surface layer thicknessis the mean nanobubble surface layer thickness0z

The Friction Factor: A Weighted Averaged Combination

Key inputs to the model:

Average relative surface roughness

Surface correlation length

Average relative thickness of thenanobubble

layer

Fractional surface occupancy of

nanobubbles

()

[Ref: S. Chakraborty, App. Phys. Lett. 90, 034108 (2007), S. Chakraborty

et al.,

J. App. Phys. 102, 104907 (2007)]

The Mesoscopic physics of Superfluidic Transport in Narrow fluidic


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The Mesoscopic

physics of Superfluidic

Transport in Narrow fluidic

Confinements: The Rough makes it Smooth!

Nanobubbles

are formed when the driving force required to minimize the

area of liquid-vapor interface is smaller than the forces that pin the contactline of the substrate.

Rough surfaces made of hydrophobic materials and narrow confinementstrigger the nanobubble

formation

Thermal fluctuations lead to nanobubbles

of sizes with an order governed

by the surface free energy scale

The above leads to a decrement in viscosity near the wall. With the bulkphase viscosity still being employed for the continuum fluid flowcalculations; this decrement in effective viscosity needs to be compensatedwith a consequent enhancement in the local shear strain rate, in

order to

achieve continuity in the shear stress (rate of momentum transport). Thiscan be well captured by a phase field model

Relative contribution of stick-slip is stochastic, due to uncertainties insurface characteristics and thermodynamic conditions

The EDL electrodynamics amplifies this tendency of slippage to a largeextent, by pumping the layer of fluid even more effectively along with themovable charges.

Ref: S. Chakraborty, Phys. Rev. Lett. 100, 097801 (2008)


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Confinement induced HydrodynamicConfinement induced Hydrodynamic

Interactions: The Basic PhysicsInteractions: The Basic Physics

Nanobubbles

may be nucleated when the driving force required to minimizethe area of liquid-vapor interface is smaller than the forces that pin the contact

line of the substrate.

Hydrophobic units are not thermodynamically favored to form hydrogenbonds with water molecules. Hence, these give rise to excluded volume

regions encompassing the locations characterized with sharply diminishingnumber density of water molecules. Loss of hydrogen bonds close to any suchhydrophobic surface effectively repels liquid molecules, thereby

favoring the

formation of liquid-depleted regions.

Close to small hydrophobic units, water molecules can structurally change andreorganize without sacrificing their hydrogen bonds. However, close to largerhydrophobic units, persistence of a hydrogen bond network is virtuallyimpossible, thereby forming persistent vapor layers. Such interfacialfluctuations can destabilize the liquid further away from the solid walls, leadingto a pressure imbalance. This effectively gives rise to an attractive potentialbetween the two surfaces.

In confined fluids, long-ranged interactions can also trigger separation-inducedphase transitions. Such separation-induced cavitation

physically originates

from an increase in the local molecular field due to the replacement ofpolarizable

fluids by solid walls.

Can the effect of slip be captured by ExtendedCan the effect of slip be captured by Extended NavierNavier


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Can the effect of slip be captured by ExtendedCan the effect of slip be captured by Extended NavierNavier

Stokes Equation, despite using NoStokes Equation, despite using No--Slip BoundarySlip BoundaryCondition?Condition?

Hypothesis based on:

continuum-based interpretation of experimentally observed thermophoretic

motion

reordering of Burnett terms in Chapman-Enskog

expansion of the viscous stress

the velocity/thermal creep coefficients introduced by Maxwell

deviatoric j i kv v vij ij

i j k

U U U

x x x

Fluids mass velocity Fluids volume velocity

(featured in

continuity equation)(volumetric flux density)

Brenners modification //i i v

i

Tq k P U

x

Fourier law in compressible limit

Modified Newtons viscosity lawSubscript v refers to volume-velocity (which is physically an Eulerian

flux density of

volume, as a combined consequence of the local advective

and diffusive mechanisms)


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ChakrabortyChakraborty--Durst HypothesisDurst HypothesisSalient Features

Constitutive relationship based on the upscaled

analogy of molecular transport

Transport coefficients for diffusion are gross volume-averaged manifestations

of the transport phenomena sub-continuum length scales

Local density and temperature gradients give rise to an additional diffusive

transport of massAssumptions:

compressible flow of ideal gases

Prandtl

and Schmidt numbers close to unity

strong local gradients in density and temperature

continuum

hypothesis remains valid

Phoretic

mass flux:

(analogy from the kinetic theory of gases)

1

fp

i

i fp i

uu C

x u x

fpu is the statistical averaged fluid particle velocity

C1

= -D

where D is the self diffusion coefficient


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Constitutive viscous behaviour

of the fluid in terms of the advective

and phoretic

fluxes

ijIII

ijII

ijij 1

1 1

2

jI

ij j

i i i

U TU

x x T x

II iij i j

j

UU u

x

1 kIIIij ij

k

U

x

Term Value Physical meaning

the transfer of molecular

momentum originating

from the interaction between

the normalized advective

flux

components Uj

exchange of momentum between

the

j-th

component

of the phoretic

velocity and the i-

th

component of the normalized

bulk advective

flux density

volumetric dilation of the fluid

elements

I

ij

II

ij

III

ij


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Using Stokes Hypothesis and necessary manipulation, stress tensor can be obtained a

, , ,net j net i net k

ij ij

i j k

u u u

x x x

wherenet u U u

netu represents a sum of the advective

and phoretic

flux densities

associated with the transport of linear momentum

Net heat flux can be represented as a sum of conductive and phoretic

components as:

//1 1

2i p

i i i

T Tq k C T D

x x T x

Using 11

2

i

i i

Tu C

x T x

and with the help of ideal gas equation one obtains:

// p

i i

i

CTq k p u

x R

Towards Extended Constitutive Relationships

Summary: Slip or No Slip?Summary: Slip or No Slip?


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Summary: Slip or No Slip?Summary: Slip or No Slip?

Slip or no-slip is just a paradigm that needs to be justified from flow physics/ scaleissues

For gases, if the frequency of collisions between the molecules and the solid

boundary is very large, the momentum and energy exchange is virtually completeand there may be no relative tangential momentum between the fluid and the solidboundary, giving rise to a no-slip boundary condition. However, in rarefied systems(high Kn) such collisions may be infrequent, giving rise to interfacial discontinuities invelocity or temperature. Such discontinuities may be aggravated by strong local

gradients of density or temperature.

Because of compact molecular arrangements, slip in liquids may occur only at highshear rates or on ultra-smooth surfaces.

However, there may be apparent slip of liquid on a solid substrate because of theformation of an intermediate vapor layer of nanometer scale. The

vapor layer, in

effect, acts like a shield, preventing the liquid from being directly exposed to thesurface irregularities. In such cases, the liquid is not likely to feel the presence of thewall directly and may smoothly sail over the intervening vapor layers, instead ofbeing in direct contact with the wall roughness elements.

Such conditions could be termed as apparent slip, since the no-slip boundarycondition still remained to be a valid proposition at the walls.

It is only the apparent

inability to capture and resolve the steep velocity gradients within the ultra-thin vaporlayers that prompts an analyzer to extrapolate the velocity profiles obtained in theliquid layer above the vapor blanket, to mark an apparent deviation from the no-slipboundary condition at the wall.


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THANK YOU FOR YOUR PATIENT

HEARING WITH MINIMAL ADHERENCETO SLEEP BOUNDARY CONDITION!!

5lecture slip noslip

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