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Welcome to this second lecture on this course
on fluid mechanics, designed for chemical
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engineering undergraduate students. In the
1st lecture, I laid out the motivations as
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to why fluid mechanics is very important in
chemical process industries by showing various
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examples.
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For example, we will try to show, that fluid
mechanics
in chemical process industries. It can come
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in various forms, so there are some cases
in which the role of fluid mechanics is direct,
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so let me say direct role, as in the case
of pumping of fluids, that is there in all
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process industries; secondly, you have pumping
and transportation. Secondly, you have flow
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measurement. Thirdly, we saw, that mixing
an agitation of reactors
in chemical reactors. Fourthly, we saw, that
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the design of packed and fluidized beds. These
are some examples where fluid mechanics plays
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a direct role in the design of chemical process
equipment, as well as, various unit operations,
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that involve in detail the nature of fluid
flow.
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There are cases where fluid mechanics is important,
but plays an indirect role where the goal
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is not fluid for, per say, in the sense, that
there are, there are some other unit operations,
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that are happening. For example, heat transfer
in heat exchanger equipment or for example,
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you could have a bubble column reactor, where
reaction is happening between the gas phase
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and liquid phase there.
The role of fluid mechanics is important,
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but it is slightly indirect because there,
the role of the fluid is to bring the reactant
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from a place to a point of interest. For example,
if you have a catalyst particle, we saw in
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the last lecture, and you, this could be,
for example, a catalyst particle in a packed
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bed reactor.
If you take an individual catalyst particle,
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you have fluid flow around it like this. So,
the role of fluid flow is to bring the reactant
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from this point close to the surface, so that
reaction can take place. Here, the detailed
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nature of flow is very important and if the
flow is slightly different, for example, like,
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so if there are recirculating zones in the
end of the cylinder or, or surface, so this
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really determines whether the product is taken
away and how fast the product is taken away
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as compared to this case. So, the detailed
nature of flow is important in most unit operations;
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is important in most unit operations.
The fundamental reason is that in chemical
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industry most of the operations are carried
out with fluid as the carrying medium, that
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is, that is primarily because handling fluids
and pumping fluids is easier and mixing fluids
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is easier and diffusion rates are much faster
in a fluid. So, for these reasons, for these
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fundamental reasons, most processing happens
typically in the fluid phase, even though
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the final product need not be in the fluid
form. Final product could be a powder of a
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pharmaceutical, but the processing itself
happens largely in the fluid phase.
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So, fluid mechanics, as we saw in the last
lecture and as I am reviewing here, plays
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the fundamental role in many chemical processing
industries and equipment.
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And we saw, that there are two ways in which
we will analyze fluid mechanics problems,
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one is the macro-level approach and the other
is a micro-level approach. The macro level
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approach will involve questions, like what
is the power required to mix a fluid in a
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tank using an impeller, or what is the pumping
cost, that is required to move fluid from
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one destination to another. These are macro-scale,
macro level approach questions because it
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is not required, as we will see, as we go
long, to know the detailed flow, that is happening
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in the mixing tank, or for example, in the
pipe. To answer these questions, also having
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detailed information will not hurt, but it
is not necessary to have detailed information.
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And we will also see that in engineering problems,
it is not often possible to have detailed
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information about flow fluids. So, there are
cases where we can live with or we can use
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the macro level approaches successfully in
engineering design, whereas micro level approach
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will involve detailed structure of the flow.
For example, we saw the example in case of
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heat exchanging equipment, where you have
hot fluid and cold fluid flowing in two adjacent
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pipes, for example, concentric pipes and the
nature of heat transfer will be crucially
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dependant on whether the hot fluid flows parallely
or it flows like so; whether there are secondary
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recirculating regions, which will enhance
heat transfer across the stream lines. So,
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there are cases where detailed structure of
flow has an impact on the many processes in,
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in chemical process industries.
So, we will in this course discuss both, macro
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level approaches, as well as, micro level
approaches in equal measure. And both have
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their own successes and both have their own
limitations, as we will have opportunity to
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point out all through this course.
There is another important approach, that,
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that sort of complements the, these two approaches,
that is, the third approach, which complements
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the macro and micro level approaches. This
is experimentation, experimental observation.
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This is because it is not often possible to
analyze all problems, that occur in engineering
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applications using a fundamental approach,
either microscopic or macro level approaches.
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In many cases we have to do systematic experiments
to understand a given process equipment, for
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example, like a packed bed or a fluidized
bed and so on, where it is not really possible
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to understand everything from 1st principles.
So, experiments play a major role in engineering
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design, especially in process industries.
And here we will show, that by judicious use
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of dimensional analysis, experiments can be
made much more, doing experiments can be made
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in a very rational way and we can collect
data and present them in a very, very economical
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way. And dimensional analysis has also very,
very important applications in scale up of
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process equipment, as we will have opportunities
to discuss later. So, this is roughly the,
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these are the kind of approaches, that we
are going to take as, as we go long in this
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course, but as an engineer it is important
to have a balance mix of all three.
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So, ideally, we would like to solve all problems
as fundamentally as possible, using 1st principles
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approach, that is, the micro-scale approach,
where we compute the detailed flow fills everywhere
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in the flow. But where it is not possible
to have such an approach, we have to settle
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for a macroscopic approach where we will see,
that we need a lot of experimental input.
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So, the macroscopic approach and the experimental
approach go hand in hand, where some parts
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of this analysis can be done using the macro
approach, but certain inputs are required
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from experimental data in the macroscopic
approach. So, we will have opportunities to
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discuss, opportunity to discuss all the three
approaches as we go along.
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So, this brings us to the detailed contents
of the course or outline of the course. So,
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having motivated what this course is about
and why it is relevant to chemical engineering,
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it is now the right time to tell you in detail
what are the topics that we are going to cover
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as we go along.
So, the first topic is continuum approximation.
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This is the framework in which most problems
in fluid mechanics are addressed or solved.
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So, continuum approximation and introduction
to what a fluid is, this is the 1st topic,
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that we will discuss. Second topic is fluid
statics; before discussing fluid mechanics
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it is fluid dynamics, that is, the motion
of flow. We will first discuss the case of
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no-flow, a static fluid and there are several
problems of engineering interest that come
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within the purview of fluid statics. We will
address them as well.
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Then, we will discuss, we will introduce kinematics.
Any mechanics, any topic in mechanics is divided
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into two aspects, one is kinematics, which
relates to description of motion without reference
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to what forces, that caused this motion and
then the motion itself, which is driven by
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certain applied forces. So, we will first
discuss kinematics, introduce kinematics and
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the notion of kinematics is very, very important
in description of fluid flow.
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Then, we will come to integral or what are
called macroscopic balances, so of mass, momentum
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and energy. So, as I was just pointing out
few minutes back, there are two approaches
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that are largely taken in addressing fluid
flow problems, one is the macro-scale approach
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where the detailed flow structure is not required
and the integral balances are one way of addressing
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such problems using integral balance, using
macroscopic balance of mass, momentum and
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energy. And then, we will go to differential
balances or microscopic balances of mass,
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momentum and energy. As the name suggests,
integral balances will involve integrals of
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various quantities, such as mass, momentum,
energy and how they change in a flow while
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differential balances will involve differential
equations of these quantities, like mass,
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momentum and energy. And finally, these differential
equations will be valid at each and every
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point in the fluid and if you can solve them,
this will be the most, you know, detailed
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information, that one can have for a fluid
flow.
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After we do, do that we will do dimensional
analysis
and then apply this to pipe flows and fittings.
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And then we will do, we will see, that fluid
flow is normally characterized by a single
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parameter. Once we do dimensional analysis,
it is a parameter called Reynolds number,
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which will recur throughout this course after
we, we are done with the basics. So, the Reynolds
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number, when it is small the flow speeds are
small; when the Reynolds number, when it is
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large the flow speeds are very large. So,
when the flow speeds are very large we will
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deal with what are called potential flows.
These are some approximations of the full
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microscopic equations, differential equations
of flow.
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And we will discuss, that when potential flow
fails we have to invoke, what is called, boundary
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layer theory. This is a very important topic
in fluid mechanics. Once we are done with
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this, this roughly concludes the basic aspects
of fluid mechanics.
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So, then we will go to fluid solid systems,
this involves chemical engineering applications
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such as settling, sedimentation, and so on.
Then, we will proceed to the analysis of packed
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beds, fluidized beds
and filtration. We will then proceed to mixing
an agitation
in chemical, in chemical industries. So, these
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are some of the primary applications of fluid
mechanics in chemical processing industries.
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And then we will find that finally, we will
go to slightly fundamental, but advanced topics,
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which are unique to chemical, chemical industries.
One is usually the flows, that we encounter
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in chemical industries, are not very simple.
For example, the flow will not happen at a
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very slow pace, it will happen at a very rapid
flow rate. In such cases the flow will become
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turbulent, so we will give a brief introduction
to turbulence and there are some basic equations
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of turbulent flows, so that will be the next
part.
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And finally, we will discuss non-Newtonian
flows; I will discuss what these are as we
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go along. Simple fluids like water and air
called Newtonian fluids because they behave
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in a particular way, whereas many complex
fluids that are, many fluids, that are encountered
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in chemical industry, for example they may
be slurries or suspensions of one particle
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in another or dispersions of one liquid in
another and so on. These are very complex
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systems and the way in which they behave in
there flow is very different from how water
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and air behave. So, these are non-Newtonian
fluids or since they exhibit some elasticity
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in additions to viscous effects, they are
called visco-elastic fluids. So, we will give
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a brief introduction to non-Newtonian fluids.
So, this is the agenda for this course and
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I have already told you the texts that are
suggested for additional readings. Although
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all these material will be inherently self-contained,
so I have already mentioned this to you, that
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the basic parts of this course up to these
will, for example you can follow Fox and McDonald
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for additional reading or White. Fluid mechanics
for these, in addition to the lecture notes,
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you can follow Mccabe, Smith and Harriet.
For these, you can follow Bird, Stewart and
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Lightfoot transport phenomena. So, this is
the rough agenda for this course.
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Now, the very first topic that I am going
to discuss is units and dimensions of physical
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properties. So, in engineering and in all
physical applications we have to measure quantities,
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for example like mass and then we have to
quantify things, like flow rate and then,
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we have to measure forces and so on. So, every
physical property in, in science or in physical
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sciences, it has a characteristic dimension
and the following quantities are said to be
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the fundamental dimensions. Every physical
property has a characteristic dimension associated
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with it.
The fundamental dimensions are conventionally
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assumed to be mass, which is denoted by the
letter M, the dimension of mass is denoted
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by M; length, the dimension of which is denoted
by L and time, the dimension is denoted by
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T. These are the conventions assumed to be
fundamental dimensions. And if you are interested
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in heat transfer, where transfer of energy
through non-mechanical means is involved,
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you can consider heat content or internal
energy as, roughly denoted as heat, so this
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is characterized by temperature. This is an
additional dimension, sometimes it is denoted
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by capital theta. So, this is, these are,
these three are the fundamental dimensions
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in mechanical settings, like fluid mechanics,
while if you are interested in heat transfer
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and thermodynamics, you will have to invoke
this as an additional fundamental dimension.
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So, while these are the fundamental dimensions,
the dimensions of all other quantities can
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be obtained from the basic definitions of
various quantities. For example, if you are
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worried about velocity, because in fluid mechanics
we will have to worry about velocity a lot,
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it is a rate at which fluid flows per unit
time, rate at which displacement of a fluid
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changes per unit time. So, it has dimensions
of L, length over time, LT to the minus 1.
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So, acceleration of a fluid is the rate at
which velocity changes, so velocity is rate
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at which displacement changes per time, acceleration
is rate of velocity. So, it is velocity by
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time, so it becomes LT to the minus 2 and
then force is, from Newton’s 2nd law, mass
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times acceleration, mass times acceleration,
so MLT to the minus 2.
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So, one can derive units of density and so
on, density is mass divided by volume, it
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is mass of a, mass of a fluid per unit volume,
it is M by L cube, it is, volume has a dimensions
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of length cube, so it is ML to the minus 3.
And so, one could derive pressure in, as is
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defined as the force, that is acting normally
per unit area. So, we will divide force MLT
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to the minus 2 by L square, area has dimensions
of length square, so it is ML to the minus
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1 T to the minus 2 and so on. So, these are
dimensions, these are fundamentally ascribed
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to all physical quantities there are.
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These are very unique, but units are human
constructs, so whenever you measure a physical
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quantity you have to accept a convention.
For example, if you measure length you have
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to use meters or centimeters or feet, or whatever
you want. So, different people can measure
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the same dimension differently and they will
report the number. For example, length is
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reported as 5 meters, these are the units
and the numbers specify the numerical value
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with which length is the amount of length,
that is present in that particular measurement
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in terms of the unit that has been chosen.
For example, 5 meters, so the dimensions of
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all physical, of various physical quantities
are unique, but units are not, units are basically
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means of characterizing or measuring various
physical quantities, and as long as, even
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if one uses different, for example, somebody
else measures lengths in feet and somebody
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else measure lengths in, in, in inches or
so on, as long as you can convert from one
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to another in a unique way. If you have unique
conversion rows that mean that you can compare
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measurements, various measurements.
So, units are not fundamental, as fundamental
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as dimensions; units are merely a convention
that everybody agrees to work on. So, if a
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certain group of people agree to work with
some set of units and some other group decide
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to work on some other set of units, as long
as, these two groups agree on the conversion
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between one’s, one units to another, there
both the measurements can be compared. You
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can be convert, you can, it can be converted.
So, there must be a unique relation between
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one set of units to another.
For example, you can measure lengths in terms
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of meters or centimeters and with the conversion
rule, that 100 centimeter is 1 meter and so
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on. So, time is measured, time can be measured
in seconds, minutes or hours; seconds or minutes
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or hours. With the unique conversion rule,
that 60 seconds is 1 minute, 60 minutes is
197
00:24:26,750 --> 00:24:33,750
1 hour and so on. Therefore, 3600 seconds
is 1 hour. However, you cannot measure time
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in months or report time in months because
there is no unique conversion between months
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00:24:50,630 --> 00:24:56,590
and all these fundamental units, like minutes
or hours or seconds. So, cannot report time
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in months, that is not correct because there
is no precise conversion factor between months
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00:25:00,850 --> 00:25:07,850
and, and the fundamental unit, like second,
minutes or hours.
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00:25:08,890 --> 00:25:15,890
The, as I said units are conventions, that
everybody accepts to follow, the accepted
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convention in modern scientific usage is the
SI units where the three fundamental dimensions
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are measured using, length is measured using
meters, time is measured in seconds. So, the
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00:25:40,850 --> 00:25:47,850
short form for meter is m, short form for
second is s and mass is measured in, sorry,
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00:25:53,590 --> 00:26:00,590
kilograms, this is kilogram; this is kg. So,
this is a metric system and if you want to
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00:26:12,860 --> 00:26:19,860
report 1 millimeter, mm is 10 to the minus
3 meter or else, 1 kilometer. So, the suffix
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00:26:26,429 --> 00:26:33,429
milli refers to the 10 to the minus 3, suffix
kilo refers to the plus 3. So, likewise, one
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00:26:36,080 --> 00:26:43,080
can use for kilograms, grams and so on. So,
these are the fundamental units.
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00:26:44,590 --> 00:26:51,590
The derived units of various quantities of
interest are as follows. So, velocity is length
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00:26:56,549 --> 00:27:03,549
per time, it is meter per second. Acceleration
is meter per second squared; force is mass
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00:27:06,929 --> 00:27:13,100
time acceleration, so it is kilogram meter
per second square. So, 1 kilogram per meter
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00:27:13,100 --> 00:27:20,100
per second square is defined as 1 Newton or
1 N. So, pressure is force by unit area, so
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00:27:28,809 --> 00:27:35,809
it is 1 kg per meter second square, it is
defined as 1 Pascal, Pascal, or shorthand
215
00:27:42,850 --> 00:27:49,580
as 1 Pa.
And work, these are some of the quantities
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00:27:49,580 --> 00:27:56,580
that we will encounter as we go along, work
is force times distance, so you have 1 kg
217
00:28:00,960 --> 00:28:07,960
meter square plus second square, it is defined
as 1 Joule or one J in short. And power is
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00:28:12,750 --> 00:28:19,750
rate at which work is being done; it is therefore
1 kg per meter squared second cube is 1 Watt
219
00:28:23,100 --> 00:28:30,100
and so on.
And temperature is measured in Kelvin that
220
00:28:37,210 --> 00:28:43,200
is unique to SI units, it is not measured
in centigrade, it is measured in degree Kelvin,
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denoted only by the letter K, K, Kelvin. So,
if you report room temperature as 300 K for
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00:28:49,519 --> 00:28:54,730
example.
So, you could convert all these various quantities
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to other unit system. So, in this course we
will be using SI units.
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00:28:59,539 --> 00:29:06,539
But traditionally, in, in Britain and northern
America, the FPS system is used, which is
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00:29:07,659 --> 00:29:14,659
also called the English units. So, here, length
is measured using a foot or feet and mass
226
00:29:24,710 --> 00:29:31,710
in pounds. So, it is denoted as ft, a pound
mass is denoted by lb subscript m and seconds,
227
00:29:36,580 --> 00:29:43,580
time is denote measured with seconds, which
is the same as in SI. And traditionally, people
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00:29:47,269 --> 00:29:54,269
have also used centimeter-gram-second, where
length is in centimeter, mass is in gram and
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00:29:58,059 --> 00:30:05,010
time in seconds.
So, if you look at various text books or some
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00:30:05,010 --> 00:30:09,880
other tables, they will tell you how to convert
from one system of unit to others, although
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00:30:09,880 --> 00:30:15,159
in this course we will follow only the SI
units. It is sometimes useful if you read
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00:30:15,159 --> 00:30:20,429
literature or text books where they follow
some other units; you should be able to convert
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00:30:20,429 --> 00:30:27,429
those units to the SI units. So, for example,
1 meter, so I am just giving you some examples,
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00:30:28,070 --> 00:30:35,070
conversion, so SI to FPS, so 1 meter is 3.281
feet and so on. You have this conversion and
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00:30:43,830 --> 00:30:50,830
1 Newton is 0.2248, the unit for force is
pound force lb f and 1 Pascal is 1.450 into
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00:30:55,019 --> 00:31:02,019
10 to the minus 4 lb f per inch square and
so on. So, for example, 1 meter is 100 centimeter,
237
00:31:04,490 --> 00:31:11,490
if you were to convert this to CGS units;
1 Newton force is 10 to the power 5 dynes,
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00:31:14,409 --> 00:31:20,690
this is equal to; 1 Pascal is 10 dynes per
centimeter squared and so on.
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00:31:20,690 --> 00:31:27,690
So, one should be able to convert from one
form of units to other by looking up a text
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00:31:29,010 --> 00:31:34,559
books or tables, but in this course we will
follow only SI units; that, is the accepted
241
00:31:34,559 --> 00:31:35,740
system of units.
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00:31:35,740 --> 00:31:42,480
So, then, so having discussed the fundamental
definition of dimensions of various quantities,
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00:31:42,480 --> 00:31:48,500
we are ready to understand what a fluid is.
So, firstly, before understanding what a fluid
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00:31:48,500 --> 00:31:53,970
is we have to introduce the framework with
which we are going to understand fluid flows,
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00:31:53,970 --> 00:32:00,970
the framework with which fluid flows are normally
understood, it is called the continuum approximation.
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00:32:03,039 --> 00:32:10,039
So, let me describe in detail what this approximation
is. Many of you must have dealt with or read
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00:32:13,909 --> 00:32:20,429
courses, had courses on mechanics of point
particles in your physics classes. There,
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00:32:20,429 --> 00:32:27,159
of course you are interested, for example,
in motion of a point particle under the influence
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00:32:27,159 --> 00:32:34,159
of some forces. So, the fundamental equation,
that describes the motion of point particles
250
00:32:35,850 --> 00:32:42,740
is the Newton’s 2nd law of motion, which
says, that the mass of a particle times it
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00:32:42,740 --> 00:32:49,740
acceleration is the sum of forces
Remember, if x is the, suppose you have a
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00:32:50,730 --> 00:32:57,730
co-ordinate system x, y, z, then let x vector
of t be the position of a particle and that
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00:33:03,120 --> 00:33:10,120
particle is moving as a result of some forces
with time. So, the acceleration of a particle
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00:33:10,389 --> 00:33:17,360
is, velocity of a particle is rate of change
of its position or its displacement, acceleration
255
00:33:17,360 --> 00:33:24,360
a, is rate of change of velocity. Therefore,
it is the second derivative of, sorry, the
256
00:33:28,299 --> 00:33:32,760
position.
So, if you know what are the forces, that
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00:33:32,760 --> 00:33:37,630
are acting on a particle, which could be gravitational
force or some other force, whatever be it,
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00:33:37,630 --> 00:33:44,630
you can solve this differential equation subject
to the condition, that a time t equal to 0.
259
00:33:46,200 --> 00:33:51,250
You know the position of the particle and
a time t equal to 0. You know the velocity
260
00:33:51,250 --> 00:33:58,250
of the particle; the velocity is basically
the 1st derivative of the position. If you
261
00:34:05,220 --> 00:34:12,220
know these two conditions, these are called
initial conditions, you can solve this differential
262
00:34:13,520 --> 00:34:19,530
equation and get, suppose here is where the
particle was a time, so let us use some other
263
00:34:19,530 --> 00:34:25,200
color for the particle, suppose a particle
was here at time t equal to 0 and you know
264
00:34:25,200 --> 00:34:32,200
what its velocity is by solving this equation,
you can find the, what is called, the trajectory
265
00:34:32,630 --> 00:34:39,630
of the particle.
So, sometimes it is simply called particle
266
00:34:41,580 --> 00:34:48,580
trajectory. This is simply the positions of
the particle at various time instances as
267
00:34:49,620 --> 00:34:55,960
you follow the particle and this is a time
t equal to 0; it is a time t. So, this is
268
00:34:55,960 --> 00:35:02,410
the position of the particle at time t. So,
this is how one understands or one tries to
269
00:35:02,410 --> 00:35:07,450
understand motion of point particles in Newtonian
mechanics.
270
00:35:07,450 --> 00:35:12,910
But even if you have a collection, not just,
if you do not have one particle, if you have
271
00:35:12,910 --> 00:35:17,070
few particles, let us say 10 particles that
are all moving about and interacting due to
272
00:35:17,070 --> 00:35:24,070
some forces, you can still apply the same
equation. m i d squared x i by d t square
273
00:35:25,640 --> 00:35:30,980
is sum of all forces acting on particle i
and you could compute the trajectories of
274
00:35:30,980 --> 00:35:37,980
various particles, if not by analytical methods,
using paper and pencil, you could use a computer
275
00:35:38,280 --> 00:35:44,680
to solve it so, and so on.
So, this is how one does, one understands
276
00:35:44,680 --> 00:35:51,680
motion in normal particle mechanics, but if
you consider a fluid, a chunk of fluid, if
277
00:35:54,140 --> 00:36:01,140
you take a glass of water or fluid flowing
in a pipe fluid, it appears smooth and continuous
278
00:36:03,930 --> 00:36:10,930
towards, you know, unaided senses. For example,
unless you, we use a sophisticated microscope,
279
00:36:14,150 --> 00:36:20,590
like an electron microscope or something,
we cannot discern the fact, that a fluid has
280
00:36:20,590 --> 00:36:26,000
discrete molecules or atoms present in it.
A fluid, normally, to our naked eyes appears
281
00:36:26,000 --> 00:36:33,000
smooth, continuous; it appears smooth and
continuous. But, the ultimate reality is a
282
00:36:38,030 --> 00:36:45,030
fluid is comprised of discrete entities, which
could be molecules or atoms. This is a reality,
283
00:36:51,330 --> 00:36:57,010
but we do not perceive this reality using
our normal senses. We do see a fluid as the
284
00:36:57,010 --> 00:37:04,010
smooth and continuous medium.
Now, as I told you in the beginning, fluid
285
00:37:06,170 --> 00:37:12,500
mechanics is concerned with forces, that occur
in flowing fluids for prescribed motion or
286
00:37:12,500 --> 00:37:19,500
we are interested in calculating the motion
for prescribed forces. If that is the case,
287
00:37:20,590 --> 00:37:26,700
how do we go about calculating forces if the
reality of the fluid is actually discrete
288
00:37:26,700 --> 00:37:30,910
and not continuous?
So, a fluid is, complied, comprised of discrete
289
00:37:30,910 --> 00:37:37,910
molecules and a whole number of them and the
force is, for example, a solid object, phase,
290
00:37:38,020 --> 00:37:43,200
phases because of the forces, that are exerted
fundamentally by all these molecules on the
291
00:37:43,200 --> 00:37:47,660
solid surface.
So, while that is the reality it is not really
292
00:37:47,660 --> 00:37:52,670
possible for us to find the forces that are
exerted by all these innumerable number of
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00:37:52,670 --> 00:37:59,670
molecules of a fluid on a solid surface. So,
a detailed molecular approach of understanding
294
00:38:04,930 --> 00:38:11,930
fluid flow is very, very difficult, that would
amount to solving. For example, if you look
295
00:38:17,660 --> 00:38:24,660
at this equation, which is essentially Newton’s
laws for whole bunch of molecules or particles,
296
00:38:25,460 --> 00:38:31,750
here the, the number of particles i goes from
1 to n. If n is a total number of molecules
297
00:38:31,750 --> 00:38:36,510
or particles and this n is of the order of
20, 10 to the 23, Avogadro number of molecules.
298
00:38:36,510 --> 00:38:41,060
Even if you consider 1 mole of liquid, like
water, there is 10 to the 23 molecules. So,
299
00:38:41,060 --> 00:38:46,100
there are huge number of molecules, which
are just, it is not just possible to solve
300
00:38:46,100 --> 00:38:51,060
the Newton’s equations even if we know all
the forces accurately, that are acting at
301
00:38:51,060 --> 00:38:56,410
a molecular level. So, a detailed molecular
approach or solution to the problem is not
302
00:38:56,410 --> 00:39:02,080
easy, it is very, very difficult.
And secondly, I will argue, that it is not
303
00:39:02,080 --> 00:39:09,080
necessary to have a detailed molecular approach
to understanding many engineering fluid flow
304
00:39:10,100 --> 00:39:15,510
problems. For example, if you are interested
in the force that is experienced by a plate,
305
00:39:15,510 --> 00:39:21,330
this is one of the most common problems in
fluid flows. So, you take a rigid plate and
306
00:39:21,330 --> 00:39:25,660
fluid is flowing in some fashion and you are
interested in what is the force that is exerted
307
00:39:25,660 --> 00:39:32,660
by the fluid on the solid surface? This is
the solid surface; this is the fluid. Ultimately,
308
00:39:34,310 --> 00:39:39,430
we will all agree, that if you take and blow
up this region, there are molecules that come
309
00:39:39,430 --> 00:39:45,410
and collide and so on and, and forces are
ultimately due to these interactions.
310
00:39:45,410 --> 00:39:51,520
But if you have to collect data for all these
10 to the 23 molecules, that are hitting the
311
00:39:51,520 --> 00:39:56,510
solid, it is just not practically possible
to compute the force even if you have a fully
312
00:39:56,510 --> 00:40:01,620
molecular detailed approach or detailed data
or information available with you. It is not
313
00:40:01,620 --> 00:40:07,460
feasible to compute forces using such large
number of data. So, it is just not possible,
314
00:40:07,460 --> 00:40:09,210
so it is just not.
315
00:40:09,210 --> 00:40:15,030
And so, even if you have a detailed molecular
picture it is not necessary, as I will show
316
00:40:15,030 --> 00:40:22,030
in the next slide, because ultimately, what
we have measured using devices, we measure
317
00:40:23,880 --> 00:40:28,440
only averages, specifically we measure time
averages.
318
00:40:28,440 --> 00:40:34,620
Even if you consider at a molecular level,
a whole lot of molecules, that are coming
319
00:40:34,620 --> 00:40:41,620
and hitting a solid surface and that results
in the force. No macroscopic measuring device
320
00:40:42,460 --> 00:40:47,920
will measure or resolve, the force is exerted
by individual molecules. Instead, we are going
321
00:40:47,920 --> 00:40:54,920
to measure only averages of various quantities.
So, this again goes to reinforce notion, that
322
00:40:59,000 --> 00:41:05,690
a detailed molecular picture or detailed molecular
approach is not necessary from the context
323
00:41:05,690 --> 00:41:11,110
of understanding what is the force exerted
by a fluid on a solid surface because that
324
00:41:11,110 --> 00:41:16,450
is simply not necessary.
So, what the continuum, I first, I will first
325
00:41:16,450 --> 00:41:22,280
state what the continuum approximation means
and then, I am going to justify when continuum
326
00:41:22,280 --> 00:41:27,630
approximation works and finally, I will also
point out context where the continuum approximation
327
00:41:27,630 --> 00:41:34,630
may fail. So, what is the continuum hypothesis?
The continuum hypothesis assumes the fluid
328
00:41:45,240 --> 00:41:52,240
to be a smooth, continuous medium. And various
quantities of the fluid, various properties
329
00:42:02,950 --> 00:42:09,950
of the fluid for example, density, which is
denoted by the letter rho and pressure, pressure
330
00:42:10,880 --> 00:42:16,720
is nothing, but the force, that is exerted
by the fluid per area of any surface. And
331
00:42:16,720 --> 00:42:22,660
velocity of the fluid, velocity is normally
denoted by the vector v, velocity is a quantity,
332
00:42:22,660 --> 00:42:26,070
is a vectorial quantity, it has both magnitude
and direction.
333
00:42:26,070 --> 00:42:33,070
So, we will in this course, the notation for
vector, vectors will be denoted by an underscore.
334
00:42:41,290 --> 00:42:48,230
So, v underscore means it is a vector.
335
00:42:48,230 --> 00:42:55,230
So, there are various quantities in a fluid,
so the various quantities in a fluid, like
336
00:42:59,100 --> 00:43:06,100
density, pressure, velocity, all these are
assumed to be smooth functions of spatial
337
00:43:11,050 --> 00:43:17,120
coordinates. For example, x, y, z because
you want to analyze the flow with respect
338
00:43:17,120 --> 00:43:21,990
to a coordinate system, the simplest is a
rectangular coordinate or cartision coordinate,
339
00:43:21,990 --> 00:43:27,290
where you have three mutually perpendicular
axis. So, all these quantities, density, pressure,
340
00:43:27,290 --> 00:43:34,290
velocity are assumed or postulated to be smooth
functions of x, y, z and time t.
341
00:43:42,490 --> 00:43:49,120
So, this is the essential statement of continuum
hypothesis, that at each and every point in
342
00:43:49,120 --> 00:43:55,170
the fluid, each and every point in the, with
respect to the coordinate system, we can ascribe
343
00:43:55,170 --> 00:44:02,170
properties such as density, velocity, pressure
temperature unique values and these are functions,
344
00:44:02,460 --> 00:44:04,100
which are smoothly varying with space and
time.
345
00:44:04,100 --> 00:44:11,100
Now, I am going to illustrate why this hypothesis
is powerful and that will also point us to
346
00:44:11,910 --> 00:44:14,180
the special cases where this hypothesis may
fail.
347
00:44:14,180 --> 00:44:21,180
So, I am going to do this in two contexts.
Let us first consider a simple quantity, such
348
00:44:21,430 --> 00:44:28,430
as density. Density of a fluid is simply,
mass per unit volume. So, the way you will
349
00:44:29,290 --> 00:44:36,290
do density measurement or the way you will
characterize density is by taking a volume,
350
00:44:36,400 --> 00:44:43,400
for simplicity let me take a cubic volume,
I will take a cube of fluid with a volume
351
00:44:48,340 --> 00:44:55,340
delta v. Once I take this volume, then I will
measure the mass, so density rho, is divided,
352
00:44:55,410 --> 00:45:00,760
is defined as delta m by delta v, this is
the general definition.
353
00:45:00,760 --> 00:45:06,950
Now, if I were to ascribe density to each
and every point with respect to a coordinate
354
00:45:06,950 --> 00:45:11,370
system, I cannot obviously take a very large
volume, I will have to take a sufficiently
355
00:45:11,370 --> 00:45:18,370
small volume. So, if I were to, if density
has to be ascribed to each point in a fluid
356
00:45:25,130 --> 00:45:32,130
flow, then what I would do is I would take
a point and around that point I will construct
357
00:45:35,900 --> 00:45:42,900
a cubic volume element of volume delta v and
I will define density at that point x. x vector
358
00:45:47,210 --> 00:45:52,900
is defined as rho, this is a short form for
rho, as a function of x, y, z.
359
00:45:52,900 --> 00:45:59,610
So, I will denote a point with this by labeling
it with its position vector, x is defined
360
00:45:59,610 --> 00:46:05,900
as limit density around, suppose this point
is denoted by x with respect to a coordinate
361
00:46:05,900 --> 00:46:12,900
system. So, let us say this is the point x,
this is the position vector of the point x,
362
00:46:17,090 --> 00:46:24,090
then density is defined as, the fundamental
definition of density is limit delta v. As
363
00:46:24,770 --> 00:46:31,770
I shrink that volume to that point, I will
measure, sorry, we erase this
and I will keep this, so limit delta v going
364
00:46:42,230 --> 00:46:48,710
to 0 delta m.
So, just to differentiate, since the symbol
365
00:46:48,710 --> 00:46:55,710
v is used for both velocity and volume, I
am going to, as far as possible, denote volumes
366
00:46:56,760 --> 00:47:01,930
with the v cross because this will remove
the confusion between volume and velocity
367
00:47:01,930 --> 00:47:08,630
because we, we use v the symbol capital V
for velocity and small v for, sorry, capital
368
00:47:08,630 --> 00:47:15,020
V for volume and small v for velocity, so
I will use v cross delta m by delta v. This
369
00:47:15,020 --> 00:47:19,280
is mass per unit volume as you shrink the
volume to a point.
370
00:47:19,280 --> 00:47:26,280
Now, then the next question comes, when can
we naturally shrink the volume to a point?
371
00:47:26,490 --> 00:47:32,300
Can we shrink that volume arbitrarily to a
point? Well, you cannot naturally shrink this
372
00:47:32,300 --> 00:47:38,270
volume to 0 because you need some finite volume
to get some finite mass. So, we have to be
373
00:47:38,270 --> 00:47:40,080
a little careful in doing this.
374
00:47:40,080 --> 00:47:47,080
So, let us consider what I am going to, let
us do a experiment, wherein we are going to
375
00:47:49,190 --> 00:47:56,190
measure the density as limit delta v going
to 0 delta m by delta v. As a function, let
376
00:48:03,030 --> 00:48:07,050
us say, since we are going to span a wide
ranges of volumes I am going to plot it as
377
00:48:07,050 --> 00:48:13,120
a function of logarithm of the volume. I am
going to plot in a hypothetical experiment
378
00:48:13,120 --> 00:48:18,800
where you measure density of a fluid by varying
the size of the probe volume, the cubic volume,
379
00:48:18,800 --> 00:48:23,900
that I drew in the last slide. So, how will
this look?
380
00:48:23,900 --> 00:48:30,900
Now, if your volume is very, very small, if
delta v, suppose delta v is l cube, delta
381
00:48:31,580 --> 00:48:38,580
v is the volume of a cube of length or side
l. If l is comparable to molecular dimensions,
382
00:48:42,340 --> 00:48:49,340
let us say you consider a fluid, a liquid,
the diameter of this liquid molecules, let
383
00:48:51,570 --> 00:48:58,570
us say a, if l is comparable to a, then if
you take a tiny volume that is comparable
384
00:48:58,820 --> 00:49:05,820
to a cube. If then at a, this volume is comparable
to molecular dimensions, dimensions of molecular
385
00:49:07,720 --> 00:49:12,850
volume, at a molecular level.
You know, that even in a, from basic physical
386
00:49:12,850 --> 00:49:18,630
chemistry or physics we will know, that molecules
are undergoing very, very violent and strong
387
00:49:18,630 --> 00:49:25,630
thermal motion. So, it is very, it is a very,
very probabilistic thing to find a volume,
388
00:49:25,940 --> 00:49:32,940
is to find a molecule of diameter a in a volume
of diameter roughly a cube proportional to
389
00:49:34,000 --> 00:49:39,010
a cube because these molecules will be undergoing
very, very strong thermal motion.
390
00:49:39,010 --> 00:49:46,010
So, when your volume is very, very small,
your density will highly fluctuate with respect
391
00:49:46,790 --> 00:49:52,600
to given spatial point because at given spatial
point may either have a molecule or not and
392
00:49:52,600 --> 00:49:57,720
even if this has two molecules, the number
of molecules will fluctuate wildly. So, initially
393
00:49:57,720 --> 00:50:02,550
what you will find, when, when the volume
is very, very small, you will find, that the
394
00:50:02,550 --> 00:50:09,550
density fluctuates a lot, but eventually we
will find, that the density settles down.
395
00:50:10,770 --> 00:50:17,770
Now, typically, this is about of the order
of 10 to the minus 9 millimeter cube, that
396
00:50:20,230 --> 00:50:27,110
is, when your size of the cube is of the order
of 1 micron, the density would have nicely
397
00:50:27,110 --> 00:50:34,110
settled down to a flat value, a constant value
and from this point on, regardless of how
398
00:50:34,810 --> 00:50:39,910
you choose the volume to be, the value of
the density that you will get will remain
399
00:50:39,910 --> 00:50:44,630
the same. So, this is the continuum value
of density.
400
00:50:44,630 --> 00:50:51,630
So, this is what we mean by density at a point
because in these huge ranges of volume, the
401
00:50:54,690 --> 00:50:59,970
density is a well defined quantity. Now, eventually,
if you keep increasing the size of the cube,
402
00:50:59,970 --> 00:51:05,030
you will start seeing density differences
that are due to genuine macroscopic variations
403
00:51:05,030 --> 00:51:10,440
in density. So, clearly, you cannot take this
as a point value.
404
00:51:10,440 --> 00:51:16,310
So, if the size of the cube in which you are
using, which you are probing the density or
405
00:51:16,310 --> 00:51:21,250
measuring the density, if the size of the
cube is large compared to molecular dimensions,
406
00:51:21,250 --> 00:51:28,250
a is the diameter of a molecule for example,
and it is small compared to the system dimension
407
00:51:30,940 --> 00:51:35,790
for example, it could be the diameter of a
pipe in which the fluid is flowing for example,
408
00:51:35,790 --> 00:51:42,790
or the diameter of a tank in which the fluid
is present, then the density at a point makes
409
00:51:43,370 --> 00:51:50,370
sense. And even if you change the, the probe
volume, you will find a unique value, that
410
00:51:51,550 --> 00:51:56,370
is ascribed to the density at a point and
you can repeat this for different points in
411
00:51:56,370 --> 00:52:01,600
a fluid. And therefore, this is remember,
density as a function of this probe volume.
412
00:52:01,600 --> 00:52:07,630
If you repeat this, then you can get density
as a function of x or something like that.
413
00:52:07,630 --> 00:52:13,540
So, density could vary smoothly as a function
of x where at each and every point you have
414
00:52:13,540 --> 00:52:18,290
taken this probe and measured the density
by measuring the mass and regardless of the
415
00:52:18,290 --> 00:52:23,350
size of the probe volume you will get a unique
value in your experiment. So, this is the
416
00:52:23,350 --> 00:52:30,350
notion in which density, for example, is defined
in a continuum approximation. So, in order
417
00:52:30,780 --> 00:52:37,780
for the continuum description of a quantity
to work, you should have a separation of scales,
418
00:52:39,830 --> 00:52:46,830
separation of length scales.
The continuum approximation works when your
419
00:52:48,770 --> 00:52:55,770
length scales of interest is very, very small
compared to the molecular dimensions and it
420
00:52:57,900 --> 00:53:01,490
is the, when the length scale of interest
is large compared to molecular dimensions,
421
00:53:01,490 --> 00:53:06,370
I am sorry, and it is small compared to a
macroscopic dimensions, like the diameter
422
00:53:06,370 --> 00:53:12,830
of the cube in which the fluid is present,
in which case you can plot the density from
423
00:53:12,830 --> 00:53:19,830
0 to d as a function of a position variable,
such as x and you will find the smooth behavior.
424
00:53:22,260 --> 00:53:27,480
So, you can describe density at each and every
point in this sense, in the continuum approximation.
425
00:53:27,480 --> 00:53:33,240
Now, this also tells you when the continuum
approximation is going to fail. When this
426
00:53:33,240 --> 00:53:40,240
length scale separation, if is it is not there,
then continuum approximation will fail, is
427
00:53:44,170 --> 00:53:51,070
likely to fail because you can no longer safely
define, ascribe density and unique value of
428
00:53:51,070 --> 00:53:55,900
density to each and every point in a fluid.
So, continuum approximation is most likely
429
00:53:55,900 --> 00:54:02,900
to be not valid. Now, the same thing can be
extended to quantities, like pressure for
430
00:54:04,690 --> 00:54:06,090
example.
431
00:54:06,090 --> 00:54:13,090
Let us take the case of pressure where you
take a rectangular surface and this is a solid
432
00:54:14,060 --> 00:54:21,060
surface a flat surface and which, let us say,
that are fluid molecules, gas molecules, that
433
00:54:21,980 --> 00:54:28,980
are coming and colliding and bouncing back,
and so on. So, the pressure is fundamentally
434
00:54:30,460 --> 00:54:37,460
defined as the normal force per unit area
of the solid. By a normal force I mean the
435
00:54:40,370 --> 00:54:44,960
force that is perpendicular to the plain of
the solid. If n is the unit normal to the
436
00:54:44,960 --> 00:54:51,040
solid, it is the force that is exerted by
the fluid. So, if you have a fluid, the force
437
00:54:51,040 --> 00:54:58,030
exerted by the fluid on the solid in the direction
of the normal to the solid per area of the
438
00:54:58,030 --> 00:55:02,610
solid. So, this is the fundamental definition
of pressure.
439
00:55:02,610 --> 00:55:07,170
But why does pressure happen at a molecular
level? It happens because of molecular collisions.
440
00:55:07,170 --> 00:55:14,170
Whenever molecules come and collide, the rate
of change of momentum of these molecules appears
441
00:55:15,710 --> 00:55:21,170
as a force on the surface and that force per
unit area is a pressure. So, the pressure
442
00:55:21,170 --> 00:55:28,170
is fundamentally defined as limit as delta
a goes to 0 delta f by delta a.
443
00:55:31,580 --> 00:55:36,000
So, the continuum hypothesis says, that it
is not necessary to worry about individual
444
00:55:36,000 --> 00:55:41,100
molecules. The force is exerted by individual
molecules because they are too huge in number.
445
00:55:41,100 --> 00:55:48,100
So, it is all and any macroscopic probing
device will measure only averages. So, even
446
00:55:48,480 --> 00:55:55,480
if you consider a very, very tiny area of
the order of 1 micron by 1 micron, one can
447
00:55:55,860 --> 00:56:01,450
use simple kinetic theory, that one’s read
in physical chemistry. So, this area is 10
448
00:56:01,450 --> 00:56:07,580
to the minus 12 meter square. It is very,
very small, but the number of molecular collisions
449
00:56:07,580 --> 00:56:11,520
is huge.
If you consider r at room temperature that
450
00:56:11,520 --> 00:56:18,520
collide per unit time per second, is from
kinetic theory of gases, that one reads in
451
00:56:21,190 --> 00:56:28,190
physical chemistry, is of the order of 10
to the 7 molecules per second. So, it is a
452
00:56:28,490 --> 00:56:33,240
huge number. So, even within a second, any
probing device will not measure individual
453
00:56:33,240 --> 00:56:40,240
collisions. So, devices will measure only
averages.
454
00:56:41,510 --> 00:56:48,510
We will stop here and we will continue from
the next lecture.