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Background Review
Now that you have gathered what fluid mechanics is all about, let us present some
tools that we use investigate fluid flow behavior. Since fluids are mostly invisible, it
is difficult to visualize what goes inside a fluid medium. Historically, the study of
fluid mechanics started even before people had even conceived of some modern flow
visualization tools. Developing a solid mathematical basis can assist understanding
and measurements of various flow quantities.
There are two ways to investigate fluid flow phenomena. If we consider a fluid
medium from a macroscopic standpoint, fluid particles appear as a smooth medium
in which individual particles cannot be distinguished. On the other hand, if we
investigate fluid mechanics from a microscopic standpoint, there is no continuous
medium at all, and flow phenomena occur as a continuous interaction between
discrete fluid particles. This latter approach is a lot more chaotic, as you are aware
from physical chemistry and Brownian motion. It is considerably more difficult to
analyze also, since any such flow models could not be analyzed before the advent of
supercomputers. Therefore historically people chose the macroscopic approach to
analyze fluid mechanics. In this class we shall adopt this approach also.
The benefits of the macroscopic model are several. It is simpler to model using
calculus and applied mathematics. The models have a rational means and therefore
are both verifiable and, in most instances, completely solvable.
Before we develop the macroscopic outlook of fluids, let us consider when such an
outlook is justifiable. For very obvious reasons, this approach will fail whenever
fluid particles are not very closely packed. For example, if we travel to a very high
altitude earth’s atmosphere is very thin. Air particles are very lightly packed there.
One may find only a few air particles in several cubic feet of air collected there. On
the other hand, if we look at a tankful of water at sea level, the water medium
definitely can qualify to be a smooth, continuous medium. We shall commonly use
the term characteristic length to represent a typical dimension in a fluid flow
medium. For example, if we are investigating internal flows of air in a room, any of
the length, L, width, W, or, height, H, can be chosen as a characteristic length
depending on the direction of the flow. Similarly, if we are exploring external flows
over a circular cylinder, the diameter, D, or, the length, L can be chosen as the
length scale depending on the flow direction.
What allows us to treat fluid medium as a continuum? Remember, the continuum
requirement must be satisfied to study fluid media in the macroscopic view. Our
requirement also must be met from a mathematical standpoint. We simply want to
record changes in fluid media by creating derivatives and integrals of fluid flow
variables. As you recall from calculus, any function must be treated continuous if we
wish to work with its derivatives. Given below is a simple hypothesis that allows us
to treat fluid as a continuous medium.
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Continuum Hypothesis: A fluid medium can be considered a continuum (with
all the associated mathematical benefits) if and only if the minimum characteristic
length, L in that medium can be considered to be much larger than the mean free
path (the distance a fluid particle travels freely before colliding with the next fluid
particle in a Brownian motion), i.e., L >> m.f.p.
Thus the above hypothesis can act as our crude looking glass through which any
fluid media can appear as a continuum. Note that any time this hypothesis is
invalidated in an application, we may not be able to apply any mathematical
equation that we develop in this class. Fortunately most fluid flow applications can
be treated successfully because they do satisfy this requirement.
Once we can consider fluid as a continuum, we may think of every point in the fluid
medium occupied by a certain fluid particle. This is a big advantage in our model
since no part of our fluid medium can be considered empty any more. Furthermore,
instead of writing motion of individual fluid particles, we’ll always analyze a
complete flow field by writing all fluid properties as continuous functions of space
and time variables, e.g., r(x,y,z,t), V(x,y,z,t), T(x,y,z,t), etc. Simply by going to a
particular location of the flow field, we can determine what is the motion of a
specific fluid particle that is in that location of the flow field at some instant.
As with any study of mechanics, we begin our study of fluid mechanics with the
study of flow kinematics. Fluid flow motions are commonly described by two
popular approaches - (i) the Lagrangian approach, and, (ii) the Eulerian approach.
The first approach is based on tracking a specific fluid particle at all times. For
example, a path line tracks the locus of all points in the fluid medium a particle
travels through at different times. Similarly, a streak line tracks all particles that
pass through a specific location in the fluid medium at different times. This latter
Eulerian approach is the more popular one since it can focus on a control volume at
all times by passing fluid particles through it at different times. However the
Lagrangian approach, or, the system approach, have become popular more
recently, with the advent of supercomputers. In this approach a collection of fluid
particles are studied at all times while they travel through the fluid medium.
Although this collection of fluid particles can rotate, distort, or move relative to one
another, they are not allowed to leave the system boundaries.
Reynolds was the first to relate the fluid property changes in a fluid system
(Lagrangian approach) to those in a fluid control volume (Eulerian approach). The
theorem presented below is an extremely powerful tool to understand almost all
conservation laws in fluid mechanics.
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Reynolds Transport Theorem
This theorem was developed in general terms using two fluid variables  and N
only. This way, we could define them to represent specific variables and develop
different conservation laws in fluid mechanics. Let represent the intensive (mass
independent) property and N represent the corresponding extensive (mass
dependent) property corresponding to in a control volume. According to these
definitions, N can be related to as follows: For a small mass dm,
dN = dm =  dV. or,
With this definition, the rate of change of N in the system can be related to the rate
of change of N in the control volume as
The right hand side has two terms. The first term represents the local rate of change
of the property N inside the control volume, and the second term represents the rate
of change of property N inside the control volume due to the flux of N through the
control surfaces.
Note that the control surface of any control volume represents the surface area
completely enclosing the control volume from all directions. For example, the
control surface of a parallelepiped can be represented by 6 surfaces. As long as
there exist a net inflow and outflow through those surfaces, the net change of
property in the stationary control volume may be affected. The local rate of change
of N, represented by the first term in the above equation, is totally independent of
the second term. This term may be set to zero for a steady flow, when all property
changes occur only due to influx and outflux of quantities, without any property
addition or reduction locally.
With this single theorem given above, all basic conservation laws of fluid mechanics
may be derived. For example, if we wish to derive the mass-conservation law
(continuity equation), we set  = 1, and therefore N = M. Similarly for the
momentum equation and the energy equation we may set  = V, and  = e (specific
energy), respectively. The equations resulting from those two equations require
additional use of the Newton’s second law of motion, and the first law of
thermodynamics to attain the final forms given below. We summarize all three
conservation laws resulting from the use of the Reynolds Transport Theorem below.
Mass Conservation:
Momentum
Conservation:
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Energy Conservation:
In the above equations, each conservation law has a local component and a
convective component. The left-hand sides of the second and the third equation
represent external influences on the control volume. For example, in the momentum
equation, the external forces are of the body force and surface force type,
represented by FB and FS. Similarly, on the left-hand side of the energy equation,
the external influences are the rate of heat addition to the control volume and the
rate of work done on the control volume.
Using the control volume forms in problem solving is often called the integral
approach or, the control volume approach (since the laws represent summations of
local and convective effects on the control volume and the control surface). This
approach is extremely useful for engineers whenever you need to determine the
overall effects of flow motions on complete systems. It does not however provide us a
detailed point-by-point picture of the flow field. To be able to determine the forms
of the above equation at each point of the fluid flow, we must shrink the size of the
control volume to an infinitesimally small size. In the limit when this happens we are
able to obtain the differential equations of fluid motion (resulting in the differential
approach).
The solution of the differential equations of fluid motions is the basis of many of the
successful flow-solving computer programs available in the market today. If you
need to understand the solution at every point in a flow field in these programs, you
must have a thorough knowledge of the differential approach of fluid motion. The
assumptions and simplifications in this approach require working closely with
calculus and the fluid flow conservation laws. You should develop a very systematic
approach to problem solving if you wish to master this area. Solved problems in
these notes offer examples of this methodology. Compare with your background
knowledge and add to it as you advance.
The topics in the realm of fluid dynamics studied in this course can be categorized
based upon some flow variables. For example, if the flow is categorized based upon
density , we may talk about incompressible vs. compressible flows. Most of the
liquid flows can be treated as incompressible (i.e.,  = constant). However some
gaseous flows may also be treated like incompressible flows. For example, if the flow
of air is at Mach No. M (=V/c, where, V = speed of flow, c = speed of sound in the air
medium) < 0.3, it may be treated as an incompressible flow. Similarly flows may be
categorized as irrotational ( = 0) vs. rotational, or, inviscid ( = 0) vs. viscous.
The study of inviscid flows is important, since it adds much to our understanding of
fluid dynamics. Though most of the real fluid flows are viscous, there are some
viscous flows that may be treated like an inviscid flow. This will become clear as you
progress through the flow kinematics. To classify viscous flows, we use a non-
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dimensional variable called the Reynolds Number, Re (= VL/, where V and L
represent fluid characteristic speed and length scale). If the flow’s Re > ReCritical, the
flow is considered a turbulent flow. Otherwise the flow is considered laminar. In this
class we shall consider applications from both kinds of flows. Likewise, you will see
applications of both internal and external flows.
Most of our problems will involve multi-dimensional fluid flows. Thus it is
important to recall what we mean by dimension in a fluid flow. For example, if the
velocity vector in a continuous fluid medium can be expressed as V = 10 x i + 30 x2 j
+ 5 x3 k, is it a 3-dimensional flow? Although it has all three components of velocity
present (u = 10 x, v = 30 x2, and, w = 5 x3) it turns out to be a one-dimensional flow.
Thus dimensionality does not depend on how many components of fluid velocity is
present, but what are the total number of independent (in this case, Cartesian) space
variables upon which the velocity vector depends. Note that in this example, u(x),
v(x) and w(x) only, without any explicit dependence on the variables y and z.
Similarly, since the flow velocity vector does not explicitly involve the time variable,
this flow would be considered steady.
Let us begin with the task of deriving the mass conservation law (continuity
equation) in the differential approach.
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