Transport Phenomena

By Prof Sunando DasGupta

We have seen in the last class how to treat unsteady flow behavior. While treating unsteady flow
behavior, we have also seen that for the very simple case of a plate sub set in motion in an infinite
body of fluid the penetration depth of the effect of motion of the plate is something which can be
expressed in terms of an error function. Beyond that point the effect of the motion of the plate is
simply nonexistent. So any velocity gradient, that you would expect is only going to be between the
point and the penetration distance because in these layers of the fluid, the velocity varies from that of
the solid plate to zero because the fluid flow is stagnant. So any velocity gradient that would exist in
this distance only and beyond that distance the gradient is essentially zero.
Now, that is interesting because unless and until you have a velocity gradient, you would not have
transport of momentum because the viscous transport of momentum is only mode by which
momentum gets transported in laminar flow, it can work only when there is a velocity gradient. If
there is no velocity gradient, there is no transport of momentum. So all the transport phenomenon that
you can think of in terms of momentum transferred, is limited in a layer close to the plate and nothing
happens beyond that point.
So with that understanding, from our previous problem we now embark on a journey to understand
what happens near a solid-liquid, solid-fluid interface. This is extremely important with applications
in a multitude of problems that we encounter every day, in the design of a fast moving vehicle, to the
design of a rocket to the sports, everywhere you will see what we are going to discuss in the four or
five classes from now onwards. And I will give you examples of that.
But it’s time for a brief lesson in history. This theoretical hydrodynamics was existed for a long time.
The concept of viscosity and the Navier Stokes equation came a bit late. So in the early 1800, the
major industry at that point was to design ships, how good the design of your ship is, how effortlessly
you can move that ship at a high velocity in water, that was the big thing at that instant of time. So the
Euler’s equation was available at that time. Now the problem that the designers of ships at that time,
is that the existing equations do not tell you much about the force which is needed to move the ship
in sea water, for example. So the reason that it’s the effect of viscosity, so to say the effect of drag
was not incorporated into their calculations. And then came the Navier - Stokes equation and all
which took into account the viscosity, and therefore the drag of the fluid on the surrounding fluid, on
the moving ship. But the problem is not solved as it is. First of all, there is a huge departure between
the theoretical results predicted from Euler’s equation and the practical drag that is experienced by
the moving ship. Even when the concept of Navier- Stokes equation came, it was almost impossible
to solve. It’s a very complicated equation and if it’s a two dimensional flow case then a solution of
that Navier -Stokes equation in the entire flow domain surrounding the moving ship was impossible
at that time. So then came someone with a bright idea. His name was Prandtl. What he conceptualized
and demonstrated is that all these viscous transport of momentum is taking place in a layer very close
to the surface of the ship. So if a ship moves in water, only in a layer very close to that of the ship, the
effect of viscous transport of momentum is important or in other words the Navier- Stokes equation is
applicable.
Viscosity is important, the velocity varies from that of the ship to the surrounding fluid to the sea
water whose velocity is, say zero. So the velocity varies from that of the ship to that of the sea in a
region very close to the surface of the ship. So, all the velocity gradients that you can think of is
confined to a very narrow region close to the ship. You need to solve the complete Navier- Stokes
equation for that thin region. And once you understand and you define the region to be very thin
where viscous forces are important there would be some other approximations which can then be
introduced to make our life simpler. Any point beyond that region the fluid can be treated as inviscid
and therefore Euler’s equation, can safely be used for that region. So the entire flow domain, around
the ship can now be defined as consisting of a very thin region and any region beyond that is inviscid
flow. So this was the missing link between theory and experiment which Prandtl provided with the
concept which we called as boundary layers.
So boundary layer is that layer, that thickness of the fluid in which there is a gradient in velocity and
when we talk about the thermal boundary layer, there is going to be a gradient in temperature. When
we talk about the mass boundary layer, the concentration boundary layer there is a concentration
gradient. So, velocity gradient for hydrodynamic boundary layer, temperature gradient for thermal
boundary layer and concentration gradient for the concentration or mass transfer boundary layer these
three boundary layers are different. So this is the concept which is extremely important, which gave
rise to the ideas of hydrodynamic boundary layer, in which momentum transport takes place, a
thermal boundary layer in which the the transport of heat is taking place, and the concentration
boundary layer in which the transport of a species is taking place. In essence these boundary layers
combine the collective heat, mass and momentum transport process which are taking place near a
solid liquid interface and which is physically the most important region that one should examine to
predict what would be the total transport of heat, mass and momentum or what can be done as an
engineer to ensure that you have higher heat transport, mass transport or momentum transport from
that point.
And where it is going to be used? All of you have probably seen the racing cars. The racing cars
when they are all aligned and start to move you would rarely see they all follow a pattern. There
would be one car in the front, there would be another car very close to the rear end of the front car
and its going to stay as long as possible in this location and there would be a third car which is going
to follow the second car and its going to be close to the end of the second car. So each car would like
to have its nose at the end of the car just in front of it. Now why does that happen? Whenever a car
moves in at a high speed, a boundary layer is formed on the car. Each car is streamlined so as to
reduce the drag as much as possible. So when it is moving at a very fast speed it’s going to encounter
the air over it. So the air flows from the boundary layer, comes to the back end point, and then it goes
away. The air molecules which are at the front, they have a certain momentum associated with it. But
at some point the boundary layer which was attached to the surface, it is going to detach itself from
the point which is called the boundary layer detachment and will form the wake.
All of us know what wakes are. When anything moves at a very fast velocity, there would be a region
at the back end which is a wake and which is essentially the low pressure region. So, in racing cars
you would see something which is called as spoiler. So you have a spoiler which is something like a
projected part near the end. The only purpose of that is to break the wakes which are formed on such
surfaces. Because if this is the low pressure at the back side and the high pressure at the front side,
due to the stagnation pressure, since the air is going to come, heat it and will probably at some point it
will come to a zero velocity, the pressure drag is going to slow this car down.
So pressure drag and fiction drag these are the two major drags, but the pressure drag is going to
create a low pressure region in it. So if I am in the second car and if I have studied my transport
phenomenon, and I am doing intelligently, I would always like to keep the front end of my car in the
wake which is formed by the first car. So the wake region over here is low pressure, so my front end
does not experience a high pressure, it would experience an artificially low pressure created by the
wakes formed by the first car.
So the second car would always follow the first car keeping its nose in the wake of the first car, and
the third car, fourth car and so on. So if you do it this way, then the wear and tear on your tyres, on
your engine, on your fuel consumption, everything would be less and you try to overtake and take the
lead position only during the last lap where you would like to come to the front. So by the time the
front car it has endured enough of high pressure, which will slow it down in the final lap. So you try
to overtake it as late as possible and take the lead position and you are fresh because your front end
all this time has been exposed to a low pressure.
So thats an example from the design of streamlining of cars, the shapes that you see in modern cars ,
in buses, in trains , in planes, in space shuttle, in all of them the outer surface is designed in such a
way to reduce drag, to improve fuel efficiency, and so on.
In fact the re-entry of rockets back into the earth’s atmosphere, the velocity gradient is so large near
the surface that it will create such a huge friction, that it is going to blow, the pictures of which we
have seen. In the field of sports, the use of boundary layers is extremely interesting. Towards the end
of this part of this course, I will tell you that. Many of you are probably interested in cricket and you
know that when the fast bowler bowls at you, it may start to swing. What exactly is swing? That is
the ball is coming straight towards you, you have taken a stance to go into the line of the ball and to
play, but suddenly in mid air the ball changes its direction. Either it moves away from you which is
out swing, or comes towards you which is in swing. But at that point of time, you are already
committed to play in a certain way. You have already picked the line of the ball. In the last moment
the ball starts to deviate from its line, then you are bound to make a mistake. So the seam bowlers
always do this, and as their name suggests, the seam, they use the seam of the cricket ball to move the
ball in the air either out swing or in swing. They are essentially trying to control the boundary layers
on both sides of the ball. So they would purposefully try to keep one side of the ball under laminar
flow conditions, the other side of the ball in turbulent flow conditions. If you have two sides of the
ball having two different roughnesses and you are using the seam to disturb the flow, then something
interesting happens. So we will see mathematically what it is later on. But always remember now
when you watch a cricket game, if you see the ball moving in air, you know that its due to the
formation of different types of boundary layers on two different surfaces. It is also the reason that you
would see the fielders and the bowlers always trying to keep one end of the ball shining. They will
never do the same thing on the other side. So purposefully they would like to have one surface
roughened and other surface smooth. If a surface is smooth, it is more likely that the laminar
condition will prevail and obviously a rough surface will initiate turbulent flow. So the bowler
always try to rub the ball, keep the shine of the ball on one side and use it, let the other side gets
rough.
So from automobiles, to aero planes, to cricket balls, to basket ball, even to golf, you would see
applications of boundary layers. You have seen the shape of the golf balls. The golf balls are never
smooth. They have dimples on it. If you take a golf ball in your hand you will see that they have
dimples in it. When you hit the golf ball with a high velocity, the dimples present on the surface of
the ball is going to disturb the boundary layers on it and it would reduce the formation of the wake,
therefore reduce the drag. So if you keep two balls identical in size, shape and weight, only one is a
golf ball and the other is a ball whose surface is very smooth, hit it with equal force in the same
direction, our normal understanding would be that the golf ball with dimples on it will not go further
and the smooth ball will go further. But it’s just the reverse. The golf ball with dimples on it will
cover a larger distance as compared to the very smooth ball which is due to the presence of the
dimples, how they affect the growth of the boundary layer and the formation of the wake.
So the applications and the possibilities are endless. I have just talked about the hydrodynamic part of
the boundary layer, so there exists a thermal boundary layer, a concentration boundary layer, and
situations in which all these three boundary layers are present.
So think of a hot object, let us say bullet, which is moving in air. It is going to have a thin
hydrodynamic boundary layer, in which the velocity of the air varies from that of the object to the
velocity of the air well above it. So if it is moving in still air the velocity is zero, but the velocity
varies from that of the bullet to that of the air far from it. But, when I say far from it, it’s essentially a
very thin region because all boundary layers are very, very thin. All your transport phenomena are
taking place in a thin layer. Now the bullet is let us say is hot and the air is cooler. So the temperature
of the air close to the bullet will vary from that of the bullet, since the temperature has to be equal at
the solid liquid interface, to that of the fluid. That is called the extent to which the effect of
temperature has penetrated, the thermal boundary layer. Now think that the bullet is made of
naphthalene right now. So if a naphthalene bullet travels through air, the naphthalene is going through
the sublimation process and the concentration of naphthalene very close to the bullet is going to be
maximum and as you move away, the concentration of naphthalene is going to fall to a value equal to
zero, because the air does not contain any naphthalene. As a result of which, a mass transfer boundary
layer will form around the moving naphthalene bullet.
So you can see that three different types of boundary layers are possible. One is a hydrodynamic
boundary layer where we deal with velocity, the second is a thermal boundary layer where we deal
with the temperature, and the third is the concentration boundary layer, where we speak about the
species concentration as a function of its distance from the moving object. The thickness of all these
three layers can be and in most cases will be different. So we will have different governing equations
as well. The approximations which are used to reduce to simplify those equations are fundamentally
similar in nature.
So the combined application of thermal, hydrodynamic and concentration boundary layer, the field is
enormous. It is well researched field, but still we do not have all the answers. So far we are restricting
ourselves to laminar flow, but the boundary layer is never going to be laminar. Beyond certain
distance, the disturbance of the moving object would be such that the flow inside the boundary layer,
will change itself from laminar to turbulent layer. The moment it becomes turbulent, the amount of
momentum transfer, heat transfer or mass transfer will increase significantly. The layer close to the
surface of the solid which is called the boundary layer, the thickness of that boundary layer in a
laminar flow grows slowly. But the moment it becomes turbulent, it starts to grow rapidly. So the
behavior of the transport, the thickness of the boundary layer, all are going to be different when we
go for the transition from laminar to turbulent flow.
So that is something which we have to keep in mind. And we would to see that it is not possible
always to get an analytical solution, we will have to resort to numerical techniques and especially in
the case of turbulent flow, we will have to use some approximations. Instead of the differential
analysis of motion, which would give you the velocity at every point in the flow field, at certain point
we would have to resort to other techniques, which are known as integral techniques to deal with flow
where you are more interested in finding out the averages, not the values at every point. So all those
will come into our discussion of boundary layers.
(Refer Slide Time: 24:57)

But let’s look at this figure first, we have drawn an air foil which is moving in air. So the upstream
velocity, you can think of this as a relative velocity, U ∞ ,i with which the air approach the air foil.

There is the stagnation point. There are streamlines of the liquid, which form around the air foil. And
the point where it hits the air foil is called the stagnation point. The stagnation point would give you
the highest pressure and at this point the boundary layer starts to form. It is first going to form along
the red dotted line over here, initially it will remain laminar, so its laminar boundary layer and at
some point it will become turbulent and the thickness will rise rapidly and it may even detach itself
from the surface of the air foil which is going to give rise to viscous wakes that we talked about.
As a result of these layers the air foil experiences a net force as a result of shear and pressure forces
acting on its surfaces. We all know how the aero planes lifts from the ground, and the forces which
are experienced by the air foil, because of its shape, there is going to be a component parallel to the
flow which is called the drag force, and there will be a component perpendicular to it up to infinity
which is known as the lift. So the drag force also has two components, one which is called the
pressure drag and the second which is called the viscous drag or the shear drag. So these are
something which we are going to discuss in over here.
(Refer Slide Time: 27:26)

Now the situation here is complicated as you can see. There are so many things we have to keep in
mind. Let’s see the simplest possible case: I have a solid plate over which some fluid is approaching.
The y direction is perpendicular to the plate and the x direction is along the flow. So on the solid plate
the velocity is going to be zero. But if I move slightly up, there is going to be a velocity which is not
going to be equal to the velocity of the approach velocity, V. So if I draw the contour of the point at
which I can sense the velocity, then this is known as the boundary layer thickness. So the velocity is
going to be zero over the solid plate and the velocity is going to be equal to the approach velocity at
the boundary line of the layer. The imaginary layer is known as the boundary layer. So if I simply
magnify this region the velocity is going to be increasing from zero to the approach velocity V. So the
profile looks something like a curved line as shown in the figure above.
So the velocity changes from value equal to zero to that of the approach velocity for flow over a flat
plate over certain distance and this distance over which the velocity changes from zero to V is called
the boundary layer thickness or the disturbance thickness. Beyond this point, the velocity does not
vary at all. So the location at which the velocity reaches the free stream velocity is called the
boundary layer thickness. The point beyond that, there is no change in velocity and the flow here is
inviscid. In the boundary layer the flow is viscous. But generally it is not the attainment of the
approach velocity which is used to demarcate the boundary layer. It’s when we say that vx is about
99% of V, 99% of the approach velocity for a flat plate. This y location where vx = 0.99V is called

the boundary layer thickness δ. So the boundary layer thickness δ is defined as the y location, where
the velocity reaches 99% of the free stream velocity. Why is it called free string velocity, because its
free from the effects of viscosity and for the case of a flat plate, this approach velocity and free stream
velocities are equal. In general free string velocity is defined as U ∞ . So whether you write 0.99V or

0.99U ∞ it really does not matter for for flat plate, because V and U ∞ are same. But on a curved

surface, U ∞ can be different from V. So I think the correct definition would be 0.99U ∞ . We will

discuss it once again in the next class but what I have done here is introduced the concept of
boundary layers from a historical perspective, discussed some of the interesting applications of it and
started to give you the definition of what is a boundary layer thickness. There are so many things to
cover in this part.
So we will start with the thickness of the boundary layer, the approach velocity, free stream velocity
and we would see that it is very difficult to experimentally measure what is a boundary layer
thickness because it varies slowly and merges asymptotically with the free stream. So if it is varying
slowly and merging asymptotically with the free stream, it is difficult to pin point the exact location
where the velocity inside the boundary layer becomes equal to 99% of the free stream velocity. So
different methods have been suggested to address this problem where we can say with some
confidence that whatever we call as the thickness of the boundary layer, it is correct. There could be
large experimental errors to decide the location where the velocity inside the boundary layer is 99%
of the free stream velocity. So those more detailed description of boundary layers, both descriptive as
well as mathematical one, we will take up in the next class.