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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/
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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
8/2/2019 5Lecture Slip NoSlip
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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!
8/2/2019 5Lecture Slip NoSlip
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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
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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!!
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