slip boundary condition in fluid mechanics.
Let us just briefly recapitulate what we did.
We first gave a bit of a perspective or motivation
on why we need to investigate on the possibility
of slip boundary conditions.
Then, we discussed about continuum hypothesis
and how does it relate to the thermodynamic
equilibrium conditions at the fluid-solid
interface with regard to the boundary condition
at the fluid-solid interface.
Then we assessed the possibility of slip boundary
condition for gasses and for gasses our general
conclusion was that like if the number of
collisions or the frequency of collisions
between the gas molecules and solid boundary
is sufficiently large, then there is almost
a perfect exchange of momentum and energy
between the gas molecules and the solid boundary.
So if the solid boundary is at rest, the gas
molecule in vicinity of the solid boundary
will also come to rest and that is known as
no-slip boundary condition.
However, if the frequency of collisions is
not sufficiently large, then this may not
hold true and the issue of largeness or smallness
of the frequency of collision is well parameterized
by the non-dimensional number called as the
Knudsen number.
So if the Knudsen number, which is the ratio
of the molecular mean free path with respect
to the characteristic system length scale.
If the Knudsen number is larger and larger
or it gets larger and larger, that means the
molecules traverse a large distance before
another collision, that means on an average
the number of collisions is less.
So that is the hallmark of a relatively rarified
system.
So now, you can for the same molecular mean
free path, that is for the same level of rarification,
your Knudsen number will actually be larger
and larger if the characteristics length scale
becomes smaller and smaller.
Therefore if a gas is not very highly rarified,
but it is put in a confinement that equivalent
defect can be a strong rarification in a larger
scale system.
So in a smaller scale system, the effect of
verification is magnified because of the reduction
in the length scale of the system and that
makes the possibility of infrequent collisions
more for gases in a micro or nano confined
system than for gases in confinements, which
are of larger dimensions.
So based on this we discussed about some classification
of flow regimes of gases based on the Knudsen
number and we made a remark that the no-slip
boundary condition holds good so long as the
Knudsen number is typically less than the
order of 10 to the power -3, but again this
is just a rough estimate and this is not a
hard-and-fast number.
Then we discussed about a model which was
introduced by Maxwell and it is a very classical
model known as Maxwell's first order slip
model.
So for that we introduced a parameter which
is called as tangential momentum accommodation
coefficient and the tangential momentum a
common accommodation coefficient is basically
defined as tau i-tau r where tau is the tangential
momentum, i is the incident, and r is the
reflected.
So tau i-tau r in the numerator and tau i-tau
wall in the denominator, where i is for the
incident molecules tangent tau i is the tangential
momentum of the incident molecules and tau
wall is the tangential momentum of the wall.
So you can see that this value of sigma 
is something between 0-1 because there are
limiting cases of specular deflection.
Specular deflection is when tau i=tau r that
is the incident tangential momentum is same
as the deflected tangential momentum and then
sigma=0.
On the other hand for diffuse reflection,
the reflected molecules assume the tangential
momentum of the wall.
So tau r=tau wall which makes sigma=1.
So in reality, it is neither fully specular
deflection nor fully diffuse reflection because
the relative proportion of these depend on
the condition of the surface.
So in reality it is neither of these two,
but something in between.
So keeping that sigma as a parameter, we arrived
at a boundary condition, which is the Maxwell's
first order slip boundary condition ,which
is the difference in the velocity of gas molecule
at one molecular mean free path from the wall
with respect to the wall molecules.
So 
that particular boundary condition was augmented
later on by Smoluchowski who introduced the
concept of thermal creep that is in addition
to the velocity slip, you can also have a
sort of temperature slip.
I mean it is not a correct term but just to
give you an analogy a slip due to thermal
effects and so the idea is that because of
tangential temper gradients, there can be
a physical slip of gas molecules over the
solid boundary and for that Smoluchowski added
a term in the presence of a temperature gradient.
This is tangential temperature gradient not
normal temperature gradient and this term
you can see in the last expression of the
view graph that is presented.
One important point to mention in this context
is that you can use this boundary condition,
provided you know what is the value of the
parameter sigma.
Now again this is like a mathematical exercise
that you can substitute any value of sigma
between 0-1 and you can get some sort of behavior
of the system.
But in reality how do you get the value of
Sigma.
See we are converging to a situation when
we will realize that it is eventually molecular
simulations that can only give us the right
picture.
So to understand what is the value of sigma,
sigma is a parameter, is an upscaled parameter
to represent the behavior but where from we
will get the parameter.
So for that we have to appeal to molecular
simulations, sometimes statistical.
I mean molecular dynamics simulation depends
on the platform at hand and depends on the
length scales and other physical scales that
are being addressed, but we have to keep in
mind that so far as continuum level calculation
sigma is like a fuzzy parameter, but this
parameter can actually be obtained from smaller
scale simulations that needs to be carefully
understood.
So now another point that we discussed in
the previous lecture is that what is Navier’s
slip boundary condition and how can we fit
the Navier slip boundary condition in the
context of Maxwell's first order slip.
Then we introduced something called as slip
length and the slip length again for Maxwell's
first order slip boundary condition is you
can see very clearly 2 – sigma/sigma*lambda.
The expression that is given in the last equation.
So in that we are not fully confident with
lambda what you know I mean if there are if
there is some possibility by which you know
the Knudsen number of the system, then you
can have an idea about the average molecular
mean free path if you know what is sigma by
some molecular simulation or from some other
source of information then you can treat the
problem like a continuum problem with a modification
in the boundary condition so long as the paradigm
of continuum is valid.
Otherwise when continuum hypothesis loser
is validity altogether you cannot use the
Navier-Stokes boundary equation with whatever
boundary condition.
Then when continuum hypothesis is valid just
remember this very simple thing when the continuum
hypothesis is not valid, then forget about
the Navier-Stokes equations.
The Navier-Stokes equation cannot give you
the right picture when the continuum hypothesis
is not valuable.
However it may be possible to develop a strategy
by which you can use the Navier-Stokes equation
to get some information, provided you make
some additions and alterations in the Navier-Stokes
equation, but that is beyond the scope of
this particular course and we are not going
to discuss that.
Now the more interesting issue is about the
possibility of slip boundary condition for
liquids.
This is an interesting issue because of several
points.
The first and foremost point is that for gases
slip boundary condition is something, which
may not be very intuitive but relatively more
intuitive than liquids the reason is that
the gases are not as compact as liquids in
terms of molecular arrangement.
So if you apply a force and if the gas molecules
are located on a solid boundary then it is
much easy to dislodge the molecules from the
solid boundary, because the gases are not
very compact and the intermolecular force
of attraction is not considered to be that
strong.
On the other hand, liquids are highly compact
systems.
So they have strong intermolecular forces
of attraction, despite that is it a possibility
that liquids can slip on a solid boundary.
That is the question that we would like to
answer in today's discussion.
So slip boundary conditions for liquids, is
it a possibility?
Because of sufficient intermolecular forces
of attraction between the molecules of the
solid boundary and the liquid, it is expected
that the liquid molecule should remain stationary
relative to the solid boundary at their points
of contact.
So this is something which is expected.
As I repeat this ideology almost all the time
that in science this is something, which is
very important that all of us have certain
intuitions.
It is not bad to try to explain most of the
things through intuitions.
The intuitions may be correct the intuitions
may not be correct, but we always start with
intuition.
So because of a sufficient intermolecular
force of attraction in the liquid, it is intuitively
expected that liquid molecules would remain
stationary relative to the solid boundary,
but is it a possibility that the liquid molecules
may be dislodged from the solid boundary?
Yes, if we apply a very high shear force right.
If you apply a very high shear force, then
with the help of the shear force, now how
high that is a big question.
So if you apply a very high shear force, then
with that very high shear force the liquid
molecules may be physically dislodged from
the solid boundary, even if the boundary is
smooth or the boundaries rough, whatever if
the more if the molecules were stationary
and they were not moving by virtue of their
interaction with the surface number one.
Number two their own compact intermolecular
arrangement but some other effect may be a
strong shear force is trying to dislodge the
liquid molecules from the solid boundary and
when they are trying to dislodge the liquid
molecules from the solid boundary, then there
could be a possibility of slip.
In such cases the straining may be sufficient
to move the molecules by overcoming the Van
der Waals forces of attraction.
See by the molecules adhering to the solid
boundary will try to adhere to the solid boundary
by virtue of the Van der Waals forces of attraction.
Now if we will discuss more about the Van
der Waals forces of attraction, when we discuss
about nano fluidics.
Now when the force on a molecule is sufficient
to overcome that attraction, then the molecule
will start moving.
So that is a possibility that it may require
a very high strain, but that possibility cannot
be ruled out.
There is another theory which argues that
the no-slip boundary condition arises due
to microscopic boundary roughness.
So to understand that what the theory talks
about let us briefly go to the board and try
to understand.
So let us say that you have a flat boundary
like this.
I am not going for a curved boundary.
Let us consider a flat boundary.
So this flat boundary is what it is a boundary,
which is macroscopically flat.
Now to analyze this macroscopically flat boundary,
what we are doing is we are magnifying a portion
of the solid boundary.
So when we are magnifying a portion of the
solid boundary.
So how do you magnify it.
You put it under a very powerful microscope
and then what you see is not something which
is microscopically flat.
You see something which is as jagged as may
be possible and because of the artifacts with
the manufacturing process it is very likely
that except for a very few exceptionally flat
surfaces made out of very controlled manufacturing
processes over certain length scales, this
is what is commonly expected rather than something
which under the microscope will also look
flat.
Now what happens is.
This is just a proposition that when you have
molecules, the molecules are trapped in the
asperities.
They are also at their peaks and they are
also at the valleys.
Now something there if there is a force acting
on these molecules that force finds it hard
to dislodge this, because the already trapped
molecules have good intermolecular forces
of attraction binding them together.
So this roughness elements on the surface
are expected to create a hindrance.
This is what we are discussing in a static
condition.
Even in a dynamic condition what can happen.
Even in a dynamic condition if you have say
a blob body and if you have fluid flow past
it, there will be a drag force.
This is from the basic consideration of fluid
mechanics of flow past blob bodies.
Now roughness elements are like small, small
blob bodies.
So if you have more roughness elements, it
is intuitively expected that it will increase
the drag.
We will see later on whether that intuitive
expectation can always be justified or not,
but this is what is very common that roughness
elements can play some role.
So going back to the description of the theory
that argues that the no-slip boundary condition
arises due to microscopic boundary roughness,
since the fluid elements get may get locally
trapped within the surface asperities.
If the fluid is liquid, then it may not be
possible for the molecules to escape from
the trapping, because of an otherwise compact
molecular packing.
Following this argument, it may be conjectured
however that molecule early smooth boundary
conditions should allow slip.
If roughness be the sole consideration or
if roughness be the driving factor for triggering
no-slip boundary condition for liquid, then
in those cases those may be relatively hypothetical,
but cannot be ruled out in those cases where
the surface is very, very smooth in those
cases, you could possibly have slip in the
liquid.
So we have to consider from this consideration
that no-slip boundary condition is just a
paradigm.
So we should even for liquids, it is a paradigm
which has commonly been encountered by engineers
and scientists in practical scenarios involving
larger length scales, but that does not mean
that it is something, which must be the case.
So the reason is that in this continuum picture,
see if you look at this view graph, in the
continuum picture you have velocity vectors.
These velocity vectors are gross manifestations
of what is happening on the molecular level
and we have significant questions with respect
to what is happening in the molecular level
and the answers to these questions depend
on the nature of the surface, the amount the
magnitude of the shear force that is being
applied and so many other things.
So until and unless those issues are resolved
at the molecular scale, we cannot confidently
tell that no-slip boundary condition is something
that will occur in reality.
Now one of the big deviations from the no-slip
boundary condition, see deviations from the
no-slip boundary condition for gases do these
have been observed for years.
Deviation from no-slip boundary condition
for liquids have not been observed so well.
I mean they have been inferred from some experiments.
I am again repeating inferred, that means
it is not that somebody is actually seeing
slip of liquids.
But what is happening is that from the description
of the flow rate or from the description of
the velocity profile, some inferences are
coming which is triggering the conclusion
that there is a possibility of slip at the
wall.
Now one such inferring example for liquid
slip is a classical example.
I mean it has now become classical but of
course a few years back when the carbon nanotubes
first came, it was a relatively new idea that
what happens for carbon nanotubes.
So researchers have demonstrated that the
rate of liquid flow through a membrane composed
of an array of carbon nanotubes may turn out
to be four orders of magnitude faster 4-5
orders.
This is just an estimate than that predicted
from classical fluid flow analysis.
So that means if you make a membrane made
of aligned carbon nanotubes and then you measure
the fluid flow through that arrangement, then
experimentally you get fluid flow which is
4-5 times more than what you get through a
no-slip boundary condition prediction.
So this phenomena now every research finding
has to be explained and sometimes this is
just like I would say that like in a light
mood I am saying don't take it seriously.
I have encountered researchers who say that
like I always tell my students, I mean that
researcher is saying that you give me a graph,
in the graph Y is increasing with X, I can
explain.
If Y is decreasing with X that also I can
explain.
So every phenomenon has this kind of explanation
and this is this can be a joking remark.
But this is something I mean which talks about
little bit of philosophy of the kind of mind
set with which people have to do research.
In every aspect, there are some aiding forcing
parameters and some opposing forcing parameters.
So it is possible that in some experiment
under certain conditions the aiding parameters
dominate over the opposing parameters, in
some other experiment the other parameters
dominate.
So it is not something which is you know philosophically
that can be ruled out.
So I am not saying that this phenomenon was
also explained in that way, but that is something
which I thought that is not bad to just tell
to make the discussion a little bit light.
So these researchers attributed this phenomenon
to an apparently frictionless interfacial
condition at the carbon nanotube wall.
Such observations were contrary to common
consensus that fluid flow through nanopores
having chemicals selectivity is rather slow.
You expect that it through to nano pores and
nano channels, these are very narrow passages.
So we expect the fluid flow through these
narrow passages to be very small, that is
what is the common intuition, but if you find
that through nano channels and nanopores fluid
is flowing remarkably fast then that is something
which can be used for technology.
So that is something which is remarkable.
So from fundamental physical consideration
what happens.
Water is likely to be able to flow fast through
hydrophobic single walled carbon nanotubes
one of the reasons is that this process creates
ordered hydrogen bond between the water molecules,
but that also occurs for many other cases
additional things happen.
What ordered hydrogen bonds between water
molecules and weak attraction between the
water and the smooth carbon nanotubes graphite
sheets that is what is important.
That is water being very nicely bonded with
hydrogen bond and the very weak bonding between
the water and the solid boundary.
Why it is happening, because the smooth graphite
sheets of the carbon nanotubes, those are
very weakly bonded with the water.
So not only that the rapid diffusion of carbons,
these are qualitatively attributed to the
fundamental scientific origin of reduced frictional
resistance in such systems.
So people gave some explanation, but you know
that in microfluidics and nanofluidics we
are not always working with materials like
carbon nanotubes.
So we have to search for examples where we
can find out slip as a common phenomenon in
microfluidics and nanofluidics and that also
with liquids.
So to understand this particular mechanism
we can just revisit the slip boundary condition,
which we discussed in the previous lecture
because we will be using this boundary condition
to explain some phenomenon.
So you can see that this is just like a magnification
of the situation close to the wall.
The dotted line is a tangent to the velocity
profile at the interface and when it is extrapolated
then it meets the zero velocity at a length
at a distance slip length LS from the solid
boundary.
That is the definition of the slip length
and as I told you in the previous lecture
that this was introduced in an era by Navier
when there was no hype on nanotechnology that
was in 1823.
In those days nobody has heard of what is
nanotechnology.
But nowadays this is having its most use in
the context of nanotechnology.
Now I am going to discuss about something
which is interesting and important.
Now whether it is slip or no-slip that also
depends on something which is how small a
distance you can resolve.
Like think about a scenario.
Let us say that you are capturing the velocity
profile experimentally.
So experimentally if you are capturing the
velocity vector distribution, let us say that
you are using one of the very sophisticated
techniques in microfluidics say micro scale
PIV or micro particle image velocimetry.
So if you are using that, then what is your
resolution.
Your resolution is not really to the molecular
level.
It is not to a few nanometers.
So that kind of very small resolution is not
there.
So let us say that you have a situation when
there is a less dense space.
i will explain you that when that can be possible.
There is a less dense phase say this there
is a liquid water above this dotted line and
there is let us say water vapour just as an
example below the dotted line.
Now how far is the dotted line from the wall.
Let us say that the dotted line is within
10-20 nanometers from the wall.
Now that cannot be resolved even by the most
sophisticated modern day flow measuring device.
That layer can be resolved just as a layer
in a static environment, but what fluid flow
is taking place in that layer the wear of
construction of velocity vectors and all those
things this is still an impossibility even
with so much of advancement in technology.
So what the velocity measuring or what the
velocity vector capturing arrangement, what
it is giving us.
This is giving us the velocity vectors for
simplicity, we have drawn it like a straight
line.
It is not necessary, you may just consider
it to be an arbitrary line.
This velocity vector above the dotted line.
Below the dotted line something happens, but
this line which is drawn below the dotted
line that line is not well resolved.
So because that line is not well resolved,
then what happens.
Because that line is not well resolved something
interesting can happen.
So let me draw a picture in the board to explain
that.
Let us say this is the dotted line.
So you have a velocity profile here and in
the less dense phase you have a velocity profile
like this.
Now you cannot really resolve this because
this is a very thin layer.
So what you do is that you are basically seeing
only the black velocity profile.
So what you do is you extrapolate this velocity
profile and this extrapolated velocity profile
meets the solid boundary with a velocity u
wall, which is we call as apparent slip velocity.
This terminology apparent slip is what is
very important.
You see this is an illusion, why this is an
illusion.
This is an illusion because the phenomenon
is going beyond the resolution of the probing
arrangement to get the velocity profile in
this thin interfacial layer.
So you extrapolate the velocity profile in
the outer layer, which is the bulk liquid
and that is meeting the wall at a velocity,
which is different from 0 velocity.
So this is called as apparent slip.
On the other hand, had the phenomenon be like
this, then this is something what we could
call as true slip.
Now when there is a vapor layer or gas layer,
it is also possible that you can have a true
slip in that condition, because you know that
for certain rarified for certain Knudsen number
and so on, it is possible that even this green
line can exhibit a slip boundary condition.
But I am not focusing on that because our
main objective of this discussion is not to
complicate that but to bring out the fact
what is the difference between true slip and
apparent slip.
So let us summarize this that in true slip
the velocity of the moving fluid literally
extrapolates to zero at a notional distance
inside the wall and is finite when it crosses
the wall.
This is common for flow over mica sheets.
This is something which has been observed
for flow over mica sheets.
In apparent slip the low viscosity component
in the fluid facilitates the flow because
it segregates near the surface.
So it is like a two-phase type of system with
one phase separated or less dense phase separated
at the wall and a more dense phase in the
outer region.
The velocity gradient is larger near the surface
because the viscosity is smaller and a classical
example is flow over super hydrophobic surfaces
like lotus leaf.
Two fluid model for apparent slip.
Now let us go to the board to explain these
two fluid model.
This is a very simple model I mean this model
can be followed by anybody who has done undergraduate
fluid mechanics.
So let us say, this is the dotted line which
is the boundary between the two layers.
Let us say this is that channel centerline
and this is the wall, half height of the channel
is H. This distance let us say is delta.
So this distance is H – delta.
The axial direction is x and we set up our
y-axis such that, this is y=zero.
The interface between the let us say this
is liquid and this is vapour.
Instead of vapor it can also be a low density
system.
So our governing equations, we are assuming
low Reynolds number flow.
So we will quickly write the governing equations
0 = -minus dp/dx + mu L. Assume everything
as a function of Y only, l for liquid and
v for vapor.
So if you integrate it dul/dy = 1/mu L dp/dx
* + c1 that means ul = 1/mu L dp/dx * y square/2
+ c1y + c2.
Similarly this equation will give you V = 1/mu
V dp/dx y square/2 * c3y + c4.
Yes this is dp/dx.
You are discussing about slip, this is slip
of writing.
So we have four constants of integration and
we need to find out these constants of integration
using four boundary conditions.
So what are the boundary conditions.
Number one we will write the boundary conditions
in one corner and use the boundary conditions
to find out the other constants.
So at boundary conditions at y is equal to
what is the wall y = -delta , uv = 0, this
is no-slip.
If the gas layer is at a high Knudsen number
or in a high Knudsen number regime, then it
can also be slip there, but we are assuming
that it is no slip.
At y=0 what are the boundary conditions, continuity
of velocity and continuity of shear stress.
So at y=zero ul = uv at y = zero mu l dul/dy
= mu v * duv/dy.
This is continuity of shear stress and at
y=H - delta what is the boundary condition
dul/dy = 0.
These are centerline symmetry.
The center line symmetry will hold only when
everything is physically and geometrically
symmetrical that means on the other wall also
if it is a parallel plate channel.
See that channel is something like this.
On this wall also you have a thickness of
low-density layer of thickness delta.
Now things change in the other wall, then
that symmetry boundary condition will not
hold true, but I am assuming just for simplicity
in algebra that that is what is the case.
So now we substitute the boundary conditions
at y = - delta uv = 0, so 0 = 1/mu v dp/dx
* delta square/2 + c3 delta + c4 at y = 0
ul = uv that is c2 = c4 at y = 0 mu L * dul/dy
* that is mu L * c1 = mu v *into duv/dy that
is mu v * c3 and at y = H - delta dul/dy = 0.
So 1/mu L dp/dx at y = H – delta.
So H - delta + c1 = 0zero.
If there is a mistake you please in the first
condition.
This is minus delta, right so minus c3 delta
at y = - delta.
Others are okay.
So the strategy is very simple.
This equation will give you what is c1.
So c1 = 1/mu L dp/dx * H - delta with a minus
sign, then this equation will give you what
is c3, mu L by mu vc1.
Then if you substitute c3 in the first equation,
you will get what is c4 and c4 = c2.
So very straight forward.
Now so you can get the velocity profile but
the interesting part of the story is that
you are not actually able to resolve this.
So what you are doing, in experiments you
are drawing a velocity profile like this and
then extrapolating this velocity profile and
using a condition that you are not able to
resolve this.
So you are not even extrapolating to the wall.
So what you are doing is your resolution maybe
ends here.
You are not able to resolve.
So the wall may be here but for your experimental
measurement purpose, there is no distinction
between these and these I mean we can distinguish
this and this for the sake of theoretical
discussion, but experimentally there is no
difference between these and these so it is
as if this is like the wall.
As if this is like the wall and as if there
is a slip velocity at the wall because clearly
in this arrangement if you draw the velocity
profile, the velocity here is non-zero.
The velocity is zero only here and it may
be interesting to draw the velocity profile
properly.
So you must draw it in such a way that it
clearly reflects the proper physics.
See here the velocity is continuous mu du/dy
is continuous.
This is liquid, this is vapor which mu is
more liquid mu or vapor mu.
Liquid mu is more.
So liquid du/dy must be less.
So the slope that you are visualizing here
is not du/dy but dy/du because u is being
plotted here and y is being plotted here.
So with all these considerations, you should
draw this velocity profile carefully so that
qualitative physics is being explained by
the velocity profile, but we are just going
for a quantification here.
So I will not bother you so much on that at
this moment.
So the coming back to the point so if we can
now explain with c1, c2, c3, c4 you have got
a velocity profile u as a function of y.
This u as a function of y includes the vapor
layer and the liquid layer.
So now if we can say that u at y = 0, you
can find out what is du/dy at y = 0 that also
we can find out because we know all the constants
of integration.
So now if we write u at y = 0 = LS * du/dy
at y = 0.
Then we can get what is this LS.
You understand the strategy.
So what we are doing, we are getting the velocity
profile from this model by getting the constant
c1, c2, c3, c4.
So we know what is u at y = 0 what is y = 0,
this cross location is y = 0.
What is u at y = 0.
It is not definitely 0.
We get what is du/dy at y = 0.
So this y = 0 we should write at the bottom
du/dy at y = 0.
So u at y = 0 = LS *du/dy at y = 0, if we
replace this equivalent by a slip length based
boundary condition.
So this y = 0 is like a hypothetical wall.
It is as good as the wall, but we do not know
actually where the wall is located.
So you can find out LS.
LS will be a function of what, what are the
parameters on which LS will depend, mu L,
mu v and delta.
So you can non-dimensionalize LS with respect
to the height of the channel LS/H. So LS/H
will depend on mu L/mu V and delta/H and I
will show you the next view graph which is
displayed in the screen that will highlight
that will give the expression of LS/H which
is written as L/H in this view graph.
So L is LS, so you can make a note of this
final expression which is displayed in the
view graph and I leave it on you as a homework
to show that this is the expression, the steps
I have clearly explained it is just a simple
algebra that you have to do.
So you can say that it is a function of mu
L/mu V and delta/H. So clearly when delta/H
tends to 0, that is when there is no near
wall layer then the slip length is 0, then
it becomes no-slip boundary condition.
So it is sort of a manifestation of the near
wall variation.
Now this is something which is apparent slip.
True slip can be realized as what I have discussed
earlier that it can be realized if the shear
is very high.
If the shear is very high, then liquid molecules
despite their compactness in molecular arrangement
can be dislodged from the solid boundary.
So if you look at the view graph, this is
a very widely cited paper by Thomson and Troian
reported in the famous journal Nature in 1998.
That if you have a slip length as a function
of the shear rate.
See this is the molecular dynamics simulation
based prediction because very high shear rates
may be able to dislodge the liquid molecules
from the solid boundary, but for that such
a shear rate you need a very small confinement
because shear rate is what.
Shear rate scales with the shear velocity
divided by the length that the gap between
the top and the bottom plates, if you consider
a Couette flow.
You have two parallel plates and you are considering
the gap between the two plates.
So it is basically the relative velocity divided
by the gap.
So if the gap is until unless the gap is very
small going to a few nanometers the shear
rate may not be sufficient to dislodge the
molecules and over that few nanometer gap,
it is very difficult to do actually physical
experiments.
So people started with molecular dynamics.
These are also called as synthetic experiments.
These are simulated experiments.
So if you look at this graph you can see that
beyond the critical shear rate the slip length
starts increasing phenomenon.
So the slip length as a function of shear
rate beyond a critical shear rate because
the liquid molecules start getting dislodged
from the solid boundary, you can get phenomenal
slip.
There have been controversies associated with
this model there have been several corrections
put to this elementary understanding, but
I will not come into the details of that at
this level.
What could be the other factors that could
dictate the slip.
There are two important factors that I will
summarize.
One is the surface roughness and we have discussed
that why surface roughness is important that
if you have a manufacturer process, then the
manufacturing process will result in some
surface roughness characteristics and that
surface roughness characteristics may in turn
dictate the fluid flow.
So that is why it is very important in microfluidics
to relate the manufacturing process with the
fluid flow, because the manufacturing process
of the surface dictates the nature of the
surface and that in turn dictates the flow
in a very significant manner.
It is unlike the case of a laminar fully developed
flow where the friction factor versus Reynolds
number characteristics are independent of
the surface roughness in classical fluid mechanics.
The other factor is surface wettability that
is we have discussed about some we have made
passing remarks on super hydrophobic surfaces
and all these.
So you can nowadays engineer the wettability
of a substrate.
In engineering this is possible that you can
make the surface as wettable or as non-wettable
as you want.
You can pattern the surface with wettability
gradients.
So many things you can do.
So you can play with the roughness, you can
play with the wettability and combination
of roughness and wettability can control the
apparent slip behavior.
How can it control the apparent slip behavior.
This is something that we what see with significantly
in our research group and this is something
which can give rise to some non-intuitive
finding.
So our common idea is that if we have a rough
surface, the rough surface will hinder fluid
flow.
It will not allow fluid flow past the surface
quite easily and the reason is quite obvious
that if you have a rough surface.
The rough surface in fluid mechanics will
act rough at the surface more it will act
like a blob body and that will have a significant
drag force.
So that will increase the friction.
This is something what is intuition, nothing
wrong with it.
But what can also be possible is that if you
have a rough hydrophobic surface, then in
a highly confined environment the roughness
and hydrophobicity together can trigger the
formation of a less dense layer.
It can be nanoscale bubbles.
These nanoscale bubbles see bubble formation
in fluid mechanics can be very interesting.
So I can just draw a small schematic to explain.
That if you have a channel where you have
bubbles of this size then that is something
which may not be desirable because it can
block the flow in engineering that is very
disturbing.
But if instead of a bubble like this, if you
have bubbles like this.
Let us say that this is highly stable and
this is around 10 nanometer roughly length
scale wise and the bulk length scale is significantly
greater than this length scale.
So then what happens is that see how we are
this created by interaction between three
parameters, roughness, wettability, and confinement.
These three important parameters and interplay
of that.
It is a very involved physics that comes into
the picture here and believe me or not it
is truly beyond my capacity to explain this
to you whatever research we have done in this
area in an elementary class of micro fluidics.
So you can refer to the papers from our group
published in various Journals including Physical
Review Letters on like what are the mechanisms
by which this can form, but I am not going
into the mechanism.
I am just trying to cut the story short and
say that this layer of bubbles or even if
it is not a bubble, it can be a low density
layer forget about the bubble to generalize
it can be a low density layer.
So the liquid which is flowing on the top
of this low-density layer what is happening
to it.
It is not feeling the effect of the solid
boundary.
It may be very rough here, but this roughness
is not exposed to the liquid so the liquid
as if is smoothly sailing on the top of a
vapor cushion or a low density cushion.
So what has given rise to this low density
cushion, one of the factors is roughness.
That is why I say that it is rough that makes
it smooth.
So roughness in small scale fluid mechanics
can be utilized as a blessing in disguise,
but the whole problem is that although this
phenomenon now is well known, one of the factors
that is hindering its application in common
engineering practice is that it still a stochastic
phenomena.
So controllability, repeatability there are
so many other issues that are involved.
So there are still very open-ended areas of
research on this particular topic.
(refer time: 01:00:12)
So we have discussed about the slip and the
no-slip boundary conditions and we have discussed
that for gases the no-slip boundary condition
may be more common.
For liquids, you can have mostly apparent
slip, but not true slip, true slip only at
abnormally high shear rates which may not
be realizable in experimental practice, but
in experimental practice the apparent slip
may be common.
So thank you very much for your patient hearing
with minimum adherence to slip boundary condition.
Thank you very much.
