WEBVTT
Kind: captions
Language: en
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In
this lecture , we will understand surface
00:00:18.990 --> 00:00:26.739
tension and it is implications in driving
flows in micro channels . This is one of the
00:00:26.739 --> 00:00:37.510
very important aspects of microfluidics . And
the reason is that , you know surface tension
00:00:37.510 --> 00:00:40.440
is a force that acts at the interface.
00:00:40.440 --> 00:00:48.300
And microfluidic devices have very small
volume as compared to a large surface area.
00:00:48.300 --> 00:00:56.429
So, area by volume ratio is very large, and
that makes these devices amenable to manipulation
00:00:56.429 --> 00:01:05.290
by using surface effects .
So, to be brief in telling you what is surface
00:01:05.290 --> 00:01:07.280
tension.
00:01:07.280 --> 00:01:18.270
Let us say that there is a container . In
this container there is a liquid , and there
00:01:18.270 --> 00:01:20.990
is a vapor or a gas at the top.
00:01:20.990 --> 00:01:26.040
So, if you have some molecule in the liquid.
00:01:26.040 --> 00:01:33.000
This molecule is being attracted from all
it is sides by other molecules.
00:01:33.000 --> 00:01:44.840
And in this process this molecule is in equilibrium
; however, if you consider a molecule at the
00:01:44.840 --> 00:01:48.530
interface.
00:01:48.530 --> 00:01:56.990
If you consider a molecule at the interface
, then this molecule is acted upon by different
00:01:56.990 --> 00:02:04.659
forces . What are the different forces ? The
different forces are due to the fact that
00:02:04.659 --> 00:02:11.680
from the top, you have the force by the vapor
molecules.
00:02:11.680 --> 00:02:14.780
And from the bottom you have force by the
liquid molecules.
00:02:14.780 --> 00:02:22.980
Liquid being much more dense than vapor, the
force of attraction of liquid molecules on
00:02:22.980 --> 00:02:25.640
the interfacial molecule will be much more.
00:02:25.640 --> 00:02:30.380
And that will try to bring this molecule towards
the inside.
00:02:30.380 --> 00:02:34.920
But this molecule does not go towards the
inside.
00:02:34.920 --> 00:02:41.960
Because if it goes to the inside then the
interface will not be stable.
00:02:41.960 --> 00:02:48.410
But the interface remains it is composure;
that means, the interface has some energy
00:02:48.410 --> 00:02:55.590
which overcomes this differential force and
keeps the interface as it is.
00:02:55.590 --> 00:02:58.540
And this energy is called a surface energy.
00:02:58.540 --> 00:03:04.840
And because of this differential force it
is therefore, can if therefore, can be perceived
00:03:04.840 --> 00:03:10.840
that the interface is in some kind of tension
. And that is called as interfacial tension
00:03:10.840 --> 00:03:13.510
or surface tension.
00:03:13.510 --> 00:03:20.710
So, surface tension accordingly is expressed
as either force per unit length, that is newton
00:03:20.710 --> 00:03:33.330
per meter . Or surface energy per unit area
. So, that is also newton per meter.
00:03:33.330 --> 00:03:41.070
Now the question is that what can surface
tension do in microfluidics.
00:03:41.070 --> 00:03:45.900
Surface tension is a beautiful force in nature.
00:03:45.900 --> 00:03:53.120
You will see think of tall trees; so, when
there is a water that is suck from the ground,
00:03:53.120 --> 00:03:58.090
the water reaches the topmost branches of
the tree without any pump.
00:03:58.090 --> 00:04:04.570
So, this movement of water which is called
as a scent of sap, from the ground to the
00:04:04.570 --> 00:04:09.790
topmost level is because of the capillary
effect; which is nothing but a surface tension
00:04:09.790 --> 00:04:10.790
driven flow.
00:04:10.790 --> 00:04:16.150
So, we will understand in this lecture how
that is possible, and how that can be applied
00:04:16.150 --> 00:04:17.840
in microfluidics.
00:04:17.840 --> 00:04:25.770
But given the significant volume of
contents, I will try to go through this using
00:04:25.770 --> 00:04:33.289
the slides . So, the first point is how can
surface tension actuate or control fluid flow.
00:04:33.289 --> 00:04:38.860
One is you can use surface tension as it
is.
00:04:38.860 --> 00:04:45.400
The other is you can generate gradients in
surface tension by using gradients in temperature,
00:04:45.400 --> 00:04:52.960
gradients in concentration, electric field
or playing with the weightability of the
00:04:52.960 --> 00:04:58.789
surface making a hydrophobic hydrophilic patch
and a combination of all these effects.
00:04:58.789 --> 00:05:04.980
But these are essentially not the primary
effect the primary effect is the surface tension
00:05:04.980 --> 00:05:08.550
itself rather than the gradients.
00:05:08.550 --> 00:05:14.930
And that we will learn through some of the
basic facets of surface tension.
00:05:14.930 --> 00:05:22.650
So, first is young Laplace equation . So,
look into this figure.
00:05:22.650 --> 00:05:29.270
So, here there is a membrane, which is at
configuration 1.
00:05:29.270 --> 00:05:35.499
By virtue of a pressure difference across
this membrane, this membrane attains a
00:05:35.499 --> 00:05:43.840
configuration to where the dimension of one
side x, these was x now this has become x
00:05:43.840 --> 00:05:45.569
plus delta x.
00:05:45.569 --> 00:05:51.810
And the lateral side the other side which
was having y dimension it has become y plus
00:05:51.810 --> 00:05:57.840
delta y .
The radius of curvature of this surface, the
00:05:57.840 --> 00:06:08.310
this curve is 1 R 1 and the radius of curvature
of this surface is R 2, ok.
00:06:08.310 --> 00:06:16.300
So now, you can see that we have a membrane,
this membrane is stretched because of the
00:06:16.300 --> 00:06:22.249
pressure difference, and when the once the
membrane is stretched it by virtue of the
00:06:22.249 --> 00:06:27.569
pressure difference there is a work done,
and this work actually supplies the necessary
00:06:27.569 --> 00:06:30.979
surface energy for allowing it to stretch.
00:06:30.979 --> 00:06:33.740
That is the basic physics that is taking place.
00:06:33.740 --> 00:06:36.550
So, there is a work energy interaction.
00:06:36.550 --> 00:06:38.879
So, what is the work energy interaction?
00:06:38.879 --> 00:06:46.469
You look into this equation . So, the pressure
difference if it is delta p across the 2 sides,
00:06:46.469 --> 00:06:50.900
then delta p into x into y is the force.
00:06:50.900 --> 00:06:57.690
Right, if delta p is the pressure difference
that times the area x into y is the force,
00:06:57.690 --> 00:07:00.240
times the displacement delta z that is the
work.
00:07:00.240 --> 00:07:04.930
And the surface energy surface tension times
the change in area.
00:07:04.930 --> 00:07:11.969
So, the change in area is x plus delta x into
y plus delta y minus x y.
00:07:11.969 --> 00:07:16.759
And that becomes x delta y plus y delta x
.
00:07:16.759 --> 00:07:19.990
Now, from similarity geometric similarity.
00:07:19.990 --> 00:07:27.340
You can write R 1 divided by R 1 plus delta
z is equal to x by x plus delta x.
00:07:27.340 --> 00:07:31.841
So, that means, R 1 is delta z into x by delta
x.
00:07:31.841 --> 00:07:41.360
Similarly, R 2 from the same geometric principle,
it is delta z into y by delta y.
00:07:41.360 --> 00:07:49.340
So, here in this equation if you divide it
by xy delta z, and then use this expression
00:07:49.340 --> 00:07:55.930
of R 1 and R 2 you will simply get delta p
is equal to sigma into 1 by R 1 plus 1 by
00:07:55.930 --> 00:07:58.879
R 2; where sigma is the surface tension coefficient.
00:07:58.879 --> 00:08:01.669
So, what does this equation relate to?
00:08:01.669 --> 00:08:07.189
Of course, this equation is a very famous
equation again in elementary physics, most
00:08:07.189 --> 00:08:12.510
of you have studied this equation, this is
called as young Laplace equation.
00:08:12.510 --> 00:08:21.310
This equation essentially relates the pressure
difference across the 2 sides of an interface
00:08:21.310 --> 00:08:24.779
with the surface tension and the radius of
curvature.
00:08:24.779 --> 00:08:32.180
So, the question is, what will be the pressure
difference if the interface is flat, if the
00:08:32.180 --> 00:08:35.900
interface is flat then radius of curvature
is what?
00:08:35.900 --> 00:08:42.090
Radius of curvature is infinity . So, 1 by
R 1 and 1 by R 2 in this equation will become
00:08:42.090 --> 00:08:43.090
0.
00:08:43.090 --> 00:08:45.580
So, delta p will be equal to 0.
00:08:45.580 --> 00:08:52.810
So, on the other hand for a spherical droplet
R 1 and R 2 will be equal to R. So, delta
00:08:52.810 --> 00:08:56.180
p equal to 2 sigma by R , ok.
00:08:56.180 --> 00:09:00.340
So, this is the young Laplace equation.
00:09:00.340 --> 00:09:07.380
There is another very famous equation called
as youngs equation . And this equation defines
00:09:07.380 --> 00:09:13.040
certain parameters and one of the very important
parameter is the contact angle.
00:09:13.040 --> 00:09:20.240
So, to understand what is the contact angle
let let us refer to this figure . So, in this
00:09:20.240 --> 00:09:28.480
figure, you have interface so, this is
liquid on on on the other side this is vapor
00:09:28.480 --> 00:09:31.410
and the surface is a solid surface.
00:09:31.410 --> 00:09:39.330
So, when we say surface tension, actually
it is a property between 2 phases.
00:09:39.330 --> 00:09:47.010
So, when we write the surface tension here,
at the solid between the solid and the liquid,
00:09:47.010 --> 00:09:48.750
it is sigma sl.
00:09:48.750 --> 00:09:54.140
So, the surface tension coefficient for a
solid liquid interface.
00:09:54.140 --> 00:10:00.770
Similarly, for solid vapor it is sigma sv,
and for liquid vapor it is sigma lv.
00:10:00.770 --> 00:10:05.390
And this angle between these 2 is defined
as the contact angle theta.
00:10:05.390 --> 00:10:13.280
So, for equilibrium, the forces must be balanced,
in the horizontal direction otherwise equilibrium
00:10:13.280 --> 00:10:14.280
is disturbed.
00:10:14.280 --> 00:10:21.430
So, you can write sigma lv cos theta plus
sigma sl minus sigma sv equal to 0.
00:10:21.430 --> 00:10:23.550
So, simple force balance equation.
00:10:23.550 --> 00:10:30.140
And that will give you this youngs equation
cos theta in terms of the surface tension
00:10:30.140 --> 00:10:36.520
coefficients . So, of course, this is a
very simple derivation.
00:10:36.520 --> 00:10:44.540
But you can give a more rigorous derivation
by using energy minimization principle.
00:10:44.540 --> 00:10:49.080
But I am not going to get into that because
our objective in the biomicrofluidics is not
00:10:49.080 --> 00:10:55.420
to enter into all the possible depths of microfluidics.
00:10:55.420 --> 00:11:02.410
If you want to get into that please refer
to my NPTEL video lectures on microfluidics.
00:11:02.410 --> 00:11:09.110
But here in a nutshell we will try to understand
the basics and very quickly get into biological
00:11:09.110 --> 00:11:13.550
applications, which are the prime focus of
these biomicrofluidics .
00:11:13.550 --> 00:11:21.500
So, if you look into this slide, you will
see that we are writing a force balance in
00:11:21.500 --> 00:11:22.800
the horizontal direction.
00:11:22.800 --> 00:11:24.670
What about the vertical direction?
00:11:24.670 --> 00:11:29.551
The in the vertical direction, it appears
that there is an unbalanced force, but that
00:11:29.551 --> 00:11:34.610
force is balanced by a normal reaction from
the surface.
00:11:34.610 --> 00:11:41.330
The other important point is that the young
Laplace equation and young equation are necessary
00:11:41.330 --> 00:11:45.720
conditions for equilibrium, but are not sufficient
conditions.
00:11:45.720 --> 00:11:51.890
That means, even if these equations are satisfied,
then also if there can be a deviation from
00:11:51.890 --> 00:11:55.250
equilibrium ok.
00:11:55.250 --> 00:12:05.490
So, the next thing that we will learn is capillary
rise, which is a very basic phenomenon . And
00:12:05.490 --> 00:12:25.450
we will try to learn that by a simple calculation
. So, let us say there is a capillary tube
00:12:25.450 --> 00:12:37.051
. And this is a cylindrical capillary . You
have an interface like this, and this is the
00:12:37.051 --> 00:12:47.740
free surface . So, if on this side you have
pressure p 1 and on this side you have pressure
00:12:47.740 --> 00:12:57.860
p 2 , from the interface interfacial forces
see surface tension will be acting in this
00:12:57.860 --> 00:13:11.760
way . So, if this is p 1 and this is p 2 , you
can write p 1 plus so, p 1 is equal to p 2
00:13:11.760 --> 00:13:18.490
plus some pressure due to surface tension
. The vertical component of this .
00:13:18.490 --> 00:13:40.100
So, we can conclude that p 1 is greater than
p 2, right . So, we can write here is that
00:13:40.100 --> 00:13:47.740
now if the pressure here is p 1 . Here
also the pressure will be p 1 . And if you
00:13:47.740 --> 00:13:49.890
take an average height of this meniscus.
00:13:49.890 --> 00:13:53.340
See this is where practically engineering
comes into picture.
00:13:53.340 --> 00:14:00.630
This is actually curved, but this being such
a narrow tube that curvature whatever it is
00:14:00.630 --> 00:14:04.540
it can be represented effectively by an equivalent
flat length.
00:14:04.540 --> 00:14:20.970
So, if that is h then you can write p 1 is
equal to p 2 plus h rho g ok.
00:14:20.970 --> 00:14:34.980
Now, by the young Laplace equation you can
write p 2 p 1 minus p 2 is equal to 2 sigma
00:14:34.980 --> 00:14:46.590
by R . So, this R . So, let us put a different
R here . This let us call is as R 1 . So,
00:14:46.590 --> 00:14:53.420
what is this R ? So, if we assume this to
be a spherical interface see this is small.
00:14:53.420 --> 00:15:00.300
So, assuming it to be a part of a sphere is
not a very wrong assumption.
00:15:00.300 --> 00:15:10.610
So, if you consider this to be a part of a
sphere, then p 1 minus p 2 is 2 sigma by R
00:15:10.610 --> 00:15:28.640
1 where R 1 is the radius of the sphere .
So, that is so, R 1 is not the radius of the
00:15:28.640 --> 00:15:34.720
capillary tube, radius of the capillarity
tube is R , and if this is the contact angle
00:15:34.720 --> 00:15:43.270
theta . So, you can write from the right angle
triangle, that R 1 is equal to R by cos theta
00:15:43.270 --> 00:16:00.570
. So, p 1 minus p 2 is equal to 2 sigma cos
theta by R . So, 2 sigma cos theta by R is
00:16:00.570 --> 00:16:14.350
equal to h rho g . So, that means, h is equal
to 2 sigma cos theta by rho gR ok.
00:16:14.350 --> 00:16:19.670
So, this h is called as capillary rise.
00:16:19.670 --> 00:16:23.710
See this rise is a generic term, it could
also be fall.
00:16:23.710 --> 00:16:30.960
It depends on theta i, cos theta i is less
than if cos theta is greater than 0 it
00:16:30.960 --> 00:16:34.830
is rise , cos theta is less than 0 it is fall.
00:16:34.830 --> 00:16:36.010
So,
00:16:36.010 --> 00:16:43.130
So, cos theta whether it is greater than 0
or less than 0 it depends on whether adhesive
00:16:43.130 --> 00:16:47.430
forces are overcoming the cohesive forces
or cohesive forces are overcoming the adhesive
00:16:47.430 --> 00:16:48.430
forces.
00:16:48.430 --> 00:17:00.570
So, the most important aspect of these scaling
law is h is proportional to 1 by R; that means,
00:17:00.570 --> 00:17:07.630
again if you have a one-meter diameter pipe
and if you convert it to a one-micron capillary,
00:17:07.630 --> 00:17:13.020
then the capillary rise becomes 10 to the
power 6 times.
00:17:13.020 --> 00:17:19.730
So, in a pipe there is surface tension, but
that is not manifested as an effective force
00:17:19.730 --> 00:17:20.980
to drive the flow.
00:17:20.980 --> 00:17:26.860
But in a narrow capillary that can be an effective
force to make the fluid move from one height
00:17:26.860 --> 00:17:34.110
to the other . So, this capillary rise is
important as the formula, but it is also important
00:17:34.110 --> 00:17:39.179
to understand that this is just the description
of statics.
00:17:39.179 --> 00:17:46.409
But how fast, or how quickly this capillary
is being filled that is called as capillary
00:17:46.409 --> 00:17:48.010
filling dynamics.
00:17:48.010 --> 00:17:54.690
And why it is important in biomicrofluidics
is because you may have let us say a micro
00:17:54.690 --> 00:18:01.210
needle, or you may have a medical diagnostic
device through which a blood sample is moving.
00:18:01.210 --> 00:18:07.820
The there is a reaction site or there is a
target site to which the blood will reach.
00:18:07.820 --> 00:18:12.419
Faster the blood will reach more rapid will
be the performance of your device.
00:18:12.419 --> 00:18:18.760
So, to understand first of all how surface
tension drives the blood flow in your device,
00:18:18.760 --> 00:18:23.980
and how quickly does it take the blood sample
from one point to the other, it is important
00:18:23.980 --> 00:18:25.470
to understand the dynamics.
00:18:25.470 --> 00:18:32.440
So, we will look into a very simple model
of understanding the dynamics, and that is
00:18:32.440 --> 00:19:03.720
called as a Lucas Washburn model . So, in
this model, what we will simply see?
00:19:03.720 --> 00:19:11.549
Simply assume is that when the fluid is moving
we are considering it as a lumped mass . And
00:19:11.549 --> 00:19:18.710
therefore, it is just like a point mass moving,
and it is acted upon by 2 forces which are
00:19:18.710 --> 00:19:20.639
viscous and surface tension.
00:19:20.639 --> 00:19:22.059
And they are balancing each other.
00:19:22.059 --> 00:19:28.990
So, we will consider we will not consider
gravity as such and in in the in this model
00:19:28.990 --> 00:19:37.950
therefore, we are considering either a horizontal
channel or even if the channel is not horizontal,
00:19:37.950 --> 00:19:43.679
because it is a microscale phenomenon the
volumetric effects are much less than surface
00:19:43.679 --> 00:19:44.679
effects.
00:19:44.679 --> 00:19:49.070
So, we are neglecting the gravity effect all
together .
00:19:49.070 --> 00:19:55.630
So, the forces which are balancing our surface
tension and viscous forces . See here we have
00:19:55.630 --> 00:20:00.110
not considered viscous forces because it is
a statics problem.
00:20:00.110 --> 00:20:05.539
Viscous forces come into picture when there
is a relative motion between the fluid and
00:20:05.539 --> 00:20:08.549
the various fluid layers or fluid and the
solid bounded.
00:20:08.549 --> 00:20:12.220
So, surface tension plus viscous equal to
0.
00:20:12.220 --> 00:20:15.139
So, how will you calculate the surface tension
force?
00:20:15.139 --> 00:20:18.119
So, I will show you something very interesting.
00:20:18.119 --> 00:20:29.399
So, you multiply both sides of this equation
by pi R square . So, what does it become?
00:20:29.399 --> 00:20:33.950
Pi R square h rho g, what is pi R square h?
00:20:33.950 --> 00:20:41.400
It is the volume of this fluid, that times
rho into g is the weight of this fluid right.
00:20:41.400 --> 00:20:47.330
So, this is the weight of the liquid column,
and this is sustained by the surface tension
00:20:47.330 --> 00:20:50.179
force . What is the surface tension force?
00:20:50.179 --> 00:20:52.369
So, that is the left hand side.
00:20:52.369 --> 00:21:01.149
So, that is 2 sigma into 2 pi R cos theta
.
00:21:01.149 --> 00:21:07.470
So, for any geometry it is sigma into perimeter
of the cross section which is 2 pi R into
00:21:07.470 --> 00:21:08.669
cos theta.
00:21:08.669 --> 00:21:21.480
So, f surface tension we can write .
How will we write the viscous force?
00:21:21.480 --> 00:21:27.940
So, for viscous force we will refer we will
assume that it is a fully developed flow through
00:21:27.940 --> 00:21:29.320
a circular tube.
00:21:29.320 --> 00:21:32.809
And we will use the Hagen Poiseuille equation.
00:21:32.809 --> 00:21:42.890
So, for the viscous force .
So, let us draw the channel here, because
00:21:42.890 --> 00:21:57.529
will write some equations below it . So,
if we write if the flow is taking place
00:21:57.529 --> 00:22:08.840
in this direction, then the viscous force
is tau into 2 pi R into the instantaneous
00:22:08.840 --> 00:22:19.820
length l up to which the capillaries filled
. Tau is the shear stress times 2 pi R is
00:22:19.820 --> 00:22:24.960
the 2 pi Rl is the surface over which the
shear stress is acting.
00:22:24.960 --> 00:22:27.840
So, this is the total shear force.
00:22:27.840 --> 00:22:33.399
So, tau this is tau where tau at the wall.
00:22:33.399 --> 00:22:44.670
So, tau at the wall is what minus mu dV z
dr at small R equal to capital R , which is
00:22:44.670 --> 00:22:53.539
the wall . So, Vz by V average is equal to
2 into 1 minus R square by this formula we
00:22:53.539 --> 00:23:09.190
derived in the previous lecture . So, dV z
dr is equal to minus , this becomes 4 V average
00:23:09.190 --> 00:23:22.139
small r by R square . So, dV z dr at small
r is equal to capital R is 4 V minus 4 V average
00:23:22.139 --> 00:23:37.120
by R.
So, this becomes this total expression . It
00:23:37.120 --> 00:23:54.799
becomes 4 into 2 8 mu 8 there is a pi here
, mu V average into l . This division by R
00:23:54.799 --> 00:23:57.340
and multiplied by R they get cancelled out.
00:23:57.340 --> 00:24:07.320
So, a viscous will be minus 8 pi mu Vl ok.
00:24:07.320 --> 00:24:12.809
So, why it is minus?
00:24:12.809 --> 00:24:15.200
The reason is very straightforward.
00:24:15.200 --> 00:24:21.019
It is minus because the viscous force resist
the surface tension force.
00:24:21.019 --> 00:24:27.460
So, the surface tension force will try to
drive the flow along the capillary the viscous
00:24:27.460 --> 00:24:29.919
force will try to resist the flow.
00:24:29.919 --> 00:24:35.769
There are cases when surface tension actually
hinders the flow instead of driving the flow,
00:24:35.769 --> 00:24:38.570
but that is not this kind of contact angle.
00:24:38.570 --> 00:24:41.309
There the contact angle is greater than 90
degree.
00:24:41.309 --> 00:24:45.480
So, here in this diagram, we have shown contact
angle less than 90 degree.
00:24:45.480 --> 00:24:50.169
So, here the capillary action is driving the
flow.
00:24:50.169 --> 00:24:59.809
So now just a small trick here in place of
V average we can write dl dt . V is the rate
00:24:59.809 --> 00:25:04.179
of change of displacement is the velocity.
00:25:04.179 --> 00:25:18.830
So, you have 8 pi mu l dl dt is equal to sigma
into 2 pi R cos theta ok.
00:25:18.830 --> 00:25:32.399
So, from here you can write l dl dt is equal
to some k where k is sigma 2 pi R cos theta
00:25:32.399 --> 00:25:33.789
divided by 8 pi mu.
00:25:33.789 --> 00:25:43.350
So, if you integrate it, you have l square
by 2 is equal to kdt , sorry, k kt not
00:25:43.350 --> 00:25:48.330
k integral dt plus some constant c 1.
00:25:48.330 --> 00:25:58.950
So, at t equal to 0 if l equal to 0, then
you can see that this c 1 will be 0 . And
00:25:58.950 --> 00:26:05.610
then you get a very interesting relationship
between l and t, we will get l is proportional
00:26:05.610 --> 00:26:09.600
to square root of t ok.
00:26:09.600 --> 00:26:15.120
So, this is the hallmark of the Lucas Washburn
model.
00:26:15.120 --> 00:26:21.789
I could have discussed about many other models
given then opportunity of time; which you
00:26:21.789 --> 00:26:26.519
can look into my detail NPTEL lecture in
on microfluidics.
00:26:26.519 --> 00:26:30.570
But I feel that the Lucas Washburn model is
the most fundamental model.
00:26:30.570 --> 00:26:39.169
Because it captures the essential force balance,
and you will see that no matter how what kind
00:26:39.169 --> 00:26:43.010
of fluidity is you can cast it in this very
simple model.
00:26:43.010 --> 00:26:48.549
So, if for example, instead of water blood
is filling into the into the capillary then
00:26:48.549 --> 00:26:50.320
what will happen to this model.
00:26:50.320 --> 00:26:57.399
So, what will happen to this model if blood
is filling the capillary is the fact that
00:26:57.399 --> 00:27:02.590
instead of this expression for velocity profile
there will be a different expression for velocity
00:27:02.590 --> 00:27:03.590
profile.
00:27:03.590 --> 00:27:11.200
And not only that the wall shear stress instead
of using the newton's law of viscosity it
00:27:11.200 --> 00:27:12.740
will be something else.
00:27:12.740 --> 00:27:18.389
So, this viscous force mathematically expression
will change, but the physics the concept the
00:27:18.389 --> 00:27:25.919
concept still remains the same, surface tension
and viscous together they make a net 0 force
00:27:25.919 --> 00:27:37.889
in a inertial free regime .
So, with this understanding I will try
00:27:37.889 --> 00:27:45.660
to give you a case study on capillary dynamics
of blood flow through microfluidic channels.
00:27:45.660 --> 00:27:49.250
This is one of the very interesting case studies
in biomicrofluidics.
00:27:49.250 --> 00:27:56.710
So, if you look into this equation so now,
I will show you an equation; where so, you
00:27:56.710 --> 00:28:04.519
have a surface tension force, you have a viscous
drag force, and you also have an inertial
00:28:04.519 --> 00:28:05.519
effect.
00:28:05.519 --> 00:28:11.999
So, this inertial effect is kept here with
an understanding that you know this is a more
00:28:11.999 --> 00:28:14.580
general model than the Lucas Washburn model.
00:28:14.580 --> 00:28:19.539
The Lucas Washburn model the left hand side
that is the ddt of M v this is essentially
00:28:19.539 --> 00:28:20.869
the newton's second law.
00:28:20.869 --> 00:28:26.889
So, the left hand side is ddt of M v rate
of change of linear momentum and right hand
00:28:26.889 --> 00:28:27.889
side is force.
00:28:27.889 --> 00:28:31.629
In the Lucas Washburn equation these term
is taken as 0.
00:28:31.629 --> 00:28:34.049
So, that is called as inertia free regime.
00:28:34.049 --> 00:28:41.309
But if inertial effects are there so, in the
mass you will have a fluid mass, that is ready
00:28:41.309 --> 00:28:46.119
to enter the capillary, which is called as
added mass.
00:28:46.119 --> 00:28:57.711
Added mass is the so, if you for example,
consider that this is a capillary and some
00:28:57.711 --> 00:28:59.370
fluid is entering into it.
00:28:59.370 --> 00:29:04.509
So, this additional mass which is outside
the capillary, but an integral part of the
00:29:04.509 --> 00:29:05.509
system.
00:29:05.509 --> 00:29:11.929
So, that is called as an added mass .
So, on the top of the added mass, you have
00:29:11.929 --> 00:29:15.350
this mass within the capillary.
00:29:15.350 --> 00:29:20.799
So, this total mass ddt of M v is the resultant
force.
00:29:20.799 --> 00:29:25.960
The shear stress depends on how you model
blood.
00:29:25.960 --> 00:29:33.809
And here I will try to describe a little bit
of hematological issues that can crop into
00:29:33.809 --> 00:29:37.600
picture when we are modeling blood.
00:29:37.600 --> 00:29:44.470
So, you will see that blood can be modeled
in very various ways like there are blood
00:29:44.470 --> 00:29:50.039
is of course, intrinsically a viscoelastic
type of material, where it is partly viscous
00:29:50.039 --> 00:29:55.360
and partly elastic because of the blood cells
which are there inside.
00:29:55.360 --> 00:29:59.720
So, the blood constitutive model if you look
into this equation the shear stress.
00:29:59.720 --> 00:30:06.830
Yeah, this is the stress tensor so called
it is k into the rate of deformation tensor
00:30:06.830 --> 00:30:08.070
to the power n.
00:30:08.070 --> 00:30:10.570
So, this is a power law model.
00:30:10.570 --> 00:30:15.049
There are other models like Kasson model
or there are different types of models.
00:30:15.049 --> 00:30:17.340
So, if you have this power law model.
00:30:17.340 --> 00:30:23.159
This k and n these are not constants, but
these are functions of certain parameters.
00:30:23.159 --> 00:30:29.100
And those parameters are considered within
this c 1, c 2, c 3.
00:30:29.100 --> 00:30:34.669
These are constants, but constants for certain
individuals these are different constants.
00:30:34.669 --> 00:30:40.809
And then on the top of that you have a very
important parameter in hematology which is
00:30:40.809 --> 00:30:41.960
called as Hematocrit.
00:30:41.960 --> 00:30:47.700
So, Hematocrit is the volume fraction of the
red blood cells or the packed cell volume.
00:30:47.700 --> 00:30:54.570
So, Hematocrit or the packed cell volume is
a very, very important hematological parameter.
00:30:54.570 --> 00:31:01.539
And you will see that this is commonly
used even to monitor certain diseases.
00:31:01.539 --> 00:31:04.299
For example, after dengue fever.
00:31:04.299 --> 00:31:10.080
People start monitoring not just the platelets
count, but also the packed cell volume or
00:31:10.080 --> 00:31:18.119
the Hematocrit .
So, blood essentially you know blood is a
00:31:18.119 --> 00:31:19.340
very complex fluid.
00:31:19.340 --> 00:31:23.370
And that is why there is a great challenge
in modeling blood.
00:31:23.370 --> 00:31:33.159
But blood fundamentally is you know is a collection
of cells in plasma.
00:31:33.159 --> 00:31:38.869
But at the same time we have to understand
that is plasma just like water?
00:31:38.869 --> 00:31:44.640
Plasma is not just like water, because plasma
contains certain proteins.
00:31:44.640 --> 00:31:49.809
These proteins are primarily albumin, globulin
and fibrinogen.
00:31:49.809 --> 00:31:53.359
And these proteins form their own network
type of structure.
00:31:53.359 --> 00:31:59.750
And that makes the rheology not exactly the
same as that that of water, although it looks
00:31:59.750 --> 00:32:01.190
like water.
00:32:01.190 --> 00:32:10.410
So, the blood contains primarily the red
blood cells, the white blood cells and the
00:32:10.410 --> 00:32:11.529
platelets.
00:32:11.529 --> 00:32:14.730
These are the 3 different types of cells.
00:32:14.730 --> 00:32:23.729
And you can see in this view graph the
shape of the red blood cells.
00:32:23.729 --> 00:32:25.700
And these are like biconcave disks.
00:32:25.700 --> 00:32:31.999
So, under a shear this biconcave disk shape
is disruptive.
00:32:31.999 --> 00:32:38.100
And there is a shape variation of the red
blood cells . And that can alter the dynamics
00:32:38.100 --> 00:32:39.109
very interesting.
00:32:39.109 --> 00:32:46.550
And therefore, the rheology of blood is that
it is it is shear is altering the effective
00:32:46.550 --> 00:32:50.609
viscosity effective viscosity is also a function
of shear in the blood .
00:32:50.609 --> 00:32:58.609
So, unlike the red blood cell which is
commonly affecting the flow dynamic.
00:32:58.609 --> 00:33:04.940
So, if somebody asks you that what aspect
of the blood affects the dynamics of flow.
00:33:04.940 --> 00:33:12.159
So, the most important aspect is the inclusion
of the red blood cells.
00:33:12.159 --> 00:33:19.720
But the other factors are also important like
the white blood cells which protect
00:33:19.720 --> 00:33:27.769
the body from diseases and platelets which
are not normally influencing the flow.
00:33:27.769 --> 00:33:33.409
But platelets are important in forming
blood clots, and they may severely interfere
00:33:33.409 --> 00:33:41.510
with the flow, if blood clot formation is
either enhanced or resisted.
00:33:41.510 --> 00:33:48.820
So, platelets may not be directly you know
influencing the rheology, but in case the
00:33:48.820 --> 00:33:55.020
platelet count goes below a certain limit
or beyond a certain limit I mean these can
00:33:55.020 --> 00:34:00.989
be affecting the fluid dynamic to a significant
extent.
00:34:00.989 --> 00:34:09.960
So, as I told you now all these factors, they
come into this parameter c 1, c 2 and c 3
00:34:09.960 --> 00:34:13.060
which I have mentioned in this slide.
00:34:13.060 --> 00:34:16.250
So, this c 1, c 2 and c 3.
00:34:16.250 --> 00:34:20.940
So, c 1 depends on the plasma globulin concentration
and Hematocrit.
00:34:20.940 --> 00:34:28.530
C 1 increases with a parameter which is called
as TPMA total protein minus the albumin fraction.
00:34:28.530 --> 00:34:33.169
Similarly, c 2 also increases directly with
TPMA.
00:34:33.169 --> 00:34:37.289
C 3 is a relatively independent parameter.
00:34:37.289 --> 00:34:42.961
In a sense that it does not depend so much
on the plasma chemistry and appears to be
00:34:42.961 --> 00:34:46.800
remarkably constant for a given animal species.
00:34:46.800 --> 00:34:53.490
And for a given blood sample, how will you
know what are c 1, c 2, c 3?
00:34:53.490 --> 00:34:55.470
That is a big question.
00:34:55.470 --> 00:35:03.050
And these I mean these depend on
the training of your model with a given
00:35:03.050 --> 00:35:08.730
data set for from a large number of patient
samples.
00:35:08.730 --> 00:35:14.220
So, that your model knows that what are the
parameters based on which c 1, c 2 and c 3
00:35:14.220 --> 00:35:16.280
depend and accordingly they can fit the model.
00:35:16.280 --> 00:35:22.510
So, c 1 c 2 c 3 for a given blood sample depend
on chemistry of plasma.
00:35:22.510 --> 00:35:27.619
Chemistry of RBC specifically it is hemoglobin
concentration.
00:35:27.619 --> 00:35:35.770
Number of the red blood cells their size
and degree of cellular aggregation.
00:35:35.770 --> 00:35:43.549
The red blood cell shape geometry and deformability,
mass density of the red blood cell plasma
00:35:43.549 --> 00:35:44.930
and composite whole blood.
00:35:44.930 --> 00:35:50.430
So, you will see that you know these are these
appear to be parameters, but these parameters
00:35:50.430 --> 00:35:56.589
are so strongly dependent on the composition
of the blood.
00:35:56.589 --> 00:36:06.630
Now, hydrodynamics of blood as it is flowing
through a microfluidic channel, it may be
00:36:06.630 --> 00:36:09.390
affected by certain diseased conditions.
00:36:09.390 --> 00:36:17.910
But even without that a very important aspect
which I want to highlight is the far use Lindquist
00:36:17.910 --> 00:36:18.910
effect.
00:36:18.910 --> 00:36:21.049
So, what is this effect?
00:36:21.049 --> 00:36:26.510
If you look into this graph the top graph
in the figure you will see, that as you reduce
00:36:26.510 --> 00:36:28.910
the diameter of the blood vessel.
00:36:28.910 --> 00:36:34.600
The apparent viscosity instead of increasing
it actually reduces.
00:36:34.600 --> 00:36:41.020
So, in a very small channel, how blood can
remarkably flow by surface tension is because
00:36:41.020 --> 00:36:43.329
of that following 2 things.
00:36:43.329 --> 00:36:50.970
One is that if the channel is narrow, then
surface tension drives the blood flow in a
00:36:50.970 --> 00:36:57.160
remarkable way; the second is that if the
channel is very narrow the viscous effects
00:36:57.160 --> 00:36:59.109
also become less.
00:36:59.109 --> 00:37:00.109
And why the viscous effects become less?
00:37:00.109 --> 00:37:09.230
Is because at the wall the red blood cells
form a wall free a cell free layer.
00:37:09.230 --> 00:37:14.700
So, cell free layer means actually the red
blood cells, come out of the wall and join
00:37:14.700 --> 00:37:15.740
the centerline.
00:37:15.740 --> 00:37:23.230
So, thus wall of the blood vessels, the wall
of the micro channel, this wall becomes devoid
00:37:23.230 --> 00:37:24.859
of the red blood cells.
00:37:24.859 --> 00:37:30.220
Had the red blood cells being placed at the
wall that would have enhanced the viscosity.
00:37:30.220 --> 00:37:35.330
But now the red the walls are devoid of the
red blood cells.
00:37:35.330 --> 00:37:40.990
And that makes the situation such that the
effective viscosity is less.
00:37:40.990 --> 00:37:46.970
So, this in medical science is a very remarkable
phenomenon and called as Fahraeus Lindquist
00:37:46.970 --> 00:37:47.970
effect.
00:37:47.970 --> 00:37:54.869
So, with all these things it is possible to
make our mathematical model which shows that
00:37:54.869 --> 00:38:01.990
how in a microfluidic channel, you can transmit
blood by using surface tension.
00:38:01.990 --> 00:38:11.300
So, if we know this, this becomes one of the
basic premises of designing microneedles and
00:38:11.300 --> 00:38:16.510
designing medical diagnostic devices; which
primarily work on capillary action.
00:38:16.510 --> 00:38:22.410
For example, medical diagnostic devices on
paper substrates on which capillary action
00:38:22.410 --> 00:38:24.480
only drives the sample.
00:38:24.480 --> 00:38:29.599
So, if you are interested to learn about the
details of the mathematical modeling.
00:38:29.599 --> 00:38:32.780
I have referred to a couple of my papers.
00:38:32.780 --> 00:38:35.980
One is published in lab on a chip.
00:38:35.980 --> 00:38:38.260
And another is in analytical Timmy character.
00:38:38.260 --> 00:38:41.349
These are published in 2005 and 2007.
00:38:41.349 --> 00:38:45.920
And I have given the citations of these
2 papers.
00:38:45.920 --> 00:38:54.109
And I believe that if you go through this
you will get an insight on how capillary transport
00:38:54.109 --> 00:38:59.750
of blood in a microfluidic channel can be
modeled in a very simple way.
00:38:59.750 --> 00:39:05.150
So, with this let us draw a conclusion to
this particular lecture
00:39:05.150 --> 00:39:11.890
So, in this lecture we have studied; what
is surface tension, what is capillary rise
00:39:11.890 --> 00:39:18.240
and how surface tension can drive a flow in
a micro channel, and how that modeling can
00:39:18.240 --> 00:39:25.299
be enhanced in terms of physical and mathematical
understanding when it is blood instead of
00:39:25.299 --> 00:39:26.299
a Newtonian fluid.
00:39:26.299 --> 00:39:26.789
Thank you very much .