Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasmas) and the forces on them. Originally applied to water (hydromechanics), it found applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical, and biomedical engineering, as well as geophysics, oceanography, meteorology, astrophysics, and biology. It can be divided into fluid statics, the study of various fluids at rest, and fluid dynamics. Fluid statics, also known as hydrostatics, is the study of fluids at rest, specifically when there's no relative motion between fluid particles. It focuses on the conditions under which fluids are in stable equilibrium and doesn't involve fluid motion. Fluid kinematics is the branch of fluid mechanics that focuses on describing and analyzing the motion of fluids, such as liquids and gases, without considering the forces that cause the motion. It deals with the geometrical and temporal aspects of fluid flow, including velocity and acceleration. Fluid dynamics, on the other hand, considers the forces acting on the fluid. Fluid dynamics is the study of the effect of forces on fluid motion.  It is a branch of continuum mechanics, a subject which models matter without using the information that it is made out of atoms; that is, it models matter from a macroscopic viewpoint rather than from microscopic. Fluid mechanics, especially fluid dynamics, is an active field of research, typically mathematically complex. Many problems are partly or wholly unsolved and are best addressed by numerical methods, typically using computers. A modern discipline, called computational fluid dynamics (CFD), is devoted to this approach. Particle image velocimetry, an experimental method for visualizing and analyzing fluid flow, also takes advantage of the highly visual nature of fluid flow. Fundamentally, every fluid mechanical system is assumed to obey the basic laws : Conservation of mass Conservation of energy Conservation of momentum The continuum assumption For example, the assumption that mass is conserved means that for any fixed control volume (for example, a spherical volume)—enclosed by a control surface—the rate of change of the mass contained in that volume is equal to the rate at which mass is passing through the surface from outside to inside, minus the rate at which mass is passing from inside to outside. This can be expressed as an equation in integral form over the control volume. The continuum assumption is an idealization of continuum mechanics under which fluids can be treated as continuous, even though, on a microscopic scale, they are composed of molecules. Under the continuum assumption, macroscopic (observed/measurable) properties such as density, pressure, temperature, and bulk velocity are taken to be well-defined at "infinitesimal" volume elements—small in comparison to the characteristic length scale of the system, but large in comparison to molecular length scale

2.  Fluids have a tendency to move or flow even if there exists a very small shear stress.
 The study of velocity and acceleration of a flowing fluid and the description and
visualization of the fluid motion are dealt with fluid kinematics.
 Two Methods:
1.Eulerian Method:
 concerned with field of flow.
 The properties (pressure, velocity, density etc.) are expressed as function of space and time.
F = f(x, y, z, t)
2.Lagrangian Method:
 follows an individual particle( or given mass) flowing through the flow field and thus
determines the fluid properties of individual particles or masses as a function of time like
p(t), v(t) etc.
 A fluid consists of large number of particles, therefore it is convenient to use the
Eulerian method for describing the fluid motion.

3.  SYSTEM :
 It refers to a fixed, identifiable quantity of mass in space:
 Three types of system: open system, closed system, isolated system.
 SURROUNDING:
 All other matter around the system is called surrounding
 BOUNDARY:
 The system boundaries forming a closed surface separates the system from
surrounding.
 The system boundary may be fixed or movable but there is no mass transfer occurs
across the system boundaries.
Example:
Consider a piston cylinder assembly:
•If the assembly is heated from out side, the
gas expands and the piston moves.
•Alternatively the gas in the cylinder can be
compressed by moving the piston.
•Although heat and work have transferred
through the boundary but there is no mass
transfer across the boundary.

4.  In case of fluid flow in pipes, channels, nozzles etc. it is difficult to focus attention
on fixed identifiable quantity of mass.
 It is convenient to focus attention on a volume, in a flow field, through which the
fluid flows. This volume is called control volume.
 Control volume:
 May be defined as an arbitrary volume in a flow field through which the fluid
flows
 The geometric boundary of the control volume is called control surface, that
may be real or imaginary or may beat rest or in motion.
Example:
Control volume for fluid flow in a pipe:
• The inside surface of pipe forms real
control surface.
• The vertical portions of control volume
are imaginary.
Points to remember:
• The size and shape of control volume can
be chosen arbitrarily.
• The location of control surface affects the
computational procedure

7.  While dealing with fluid flow problems, it is often advantageous to obtain a
visual representation of a flow field.
 The pattern of flow can be visualized in many different ways.
 Photographs
 Colored dyes
 flakes
 Hydrogen bubbles etc.
Path lines are the trajectories that individual fluid particles follow. These can be thought of as "recording" the
path of a fluid element in the flow over a certain period. The direction the path takes will be determined by the
streamlines of the fluid at each moment in time.
Timelines are the lines formed by a set of fluid particles that were marked at a previous instant in time,
creating a line or a curve that is displaced in time as the particles move.
Streamlines are a family of curves that are
instantaneously tangent to the velocity vector
of the flow. These show the direction a
massless fluid element will travel in at any
point in time.
There can be no flow across a streamline
Streak lines are the locus of points of all the
fluid particles that have passed continuously
through a particular spatial point in the past.
Dye steadily injected into the fluid at a fixed
point extends along a streak line.

8.  Stream tube:
 A typical set of neighboring streamlines that forms a passage through which
the fluid flows is called stream tube.
 No flow is possible across a stream tube except through its ends.
 The stream tube need not be a solid and are fluid surface.
 Stream filament:
 It is a stream tube with its cross section sufficiently small so that variation of
velocity over it may be considered negligible.

11. Streamline patterns and corresponding convective accelerations.

13. Therefore, ………………(1)
………………(2)
………………(3)
So, the total acceleration is obtained by combining the acceleration given by the above three equations
………………(4)
………………(5)

14. • The principle of mass conservation is valid for a flowing fluid.
• In a fixed region of flow constituting a control volume, mass of fluid entering the volume per unit time
must be equal to the mass of fluid leaving the control volume per unit time plus (or minus) the increase
(or decrease) of mass of fluid in the control volume per unit time.
• But for steady flow there can not be any change in mass within the control volume
Consider a stream tube of varying cross section

15. ………………………….……………………(1)
………………………….……………………(2)
………………………….……………………(3)

16. Now total rate of mass flow to the control volume
………….……………………(4)
…………………………(5)
This equation is called the continuity equation in Cartesian coordinates.
In vector form
…………………………(6)
In cylindrical polar coordinates the continuity equation can be written as
…………………………(7)

17. …………….………….(8)
…………….………….(9)
…………….………….(10)

18. Consider a transparent plane parallel and at a unit distance from the board.

20. The partial derivative of the stream function with respect to any direction gives the
velocity component at 90o
anticlockwise to that direction.
The stream function is only defined for two dimensional in compressible flow, for which the continuity
equation can be written as
Putting the values of u and v from equation 1 and 2, we will get
This shows that the stream function always satisfies the continuity equation.
In polar coordinates
…………….(3)
…………….(4)

21. An infinitesimal element of fluid may move in fluid such
that the elements undergoes following basic
displacements.
1. Translation:
The elements moves bodily without being rotated or
deformed
2. Rotation:
Planes( top, base and sides of element) and medians as
well as diagonal of the element may change as a result of
their rotation about any one (or all three) of the
coordinate axes.

22. 4. Linear deformation:
Shapes of the elements gets changed without change
in orientation of the element. The planes ( top, base
and sides of the element) as well as the median lines
of the elements are displaced while remaining parallel
to their original position.
3. Angular deformation:
It involves a distortion of the fluid element in which planes
that were initially perpendicular to each other are no longer
so.

23. • Counter clock wise rotation is considered to be positive.
• If the fluid particles in a flow region have rotation about any axis the flow is called
rotational flow or vortex flow.
• If the fluid particles in a flow region do not have rotation about any axis the flow is called
irrotational flow
………………………(A)

24. Consider the motion of a fluid particle in xy-plane
Derivation of expression for Rotation
The angular velocity of the line OA can be written as
and

25. Now, the rotation of fluid element about z-axis a per definition of rotation can be written as
Similarly the rotation of fluid element
about x and y axis can be found out as
………………………….(1)
………………………….(2)
………………………….(3)
Hence from equation A
………………………….(B)
In vector form
………………………….(C)

26. Another measure of rotation of fluid element is vorticity which is equal to twice the rotation
………………………….(4)
………………………….(5)
………………………….(6)
………………………….(D)

27. here
And,
So,

28. For closed curve OACB
Or,
Or,
This means circulation per unit area is equal to the vorticity or twice the rotation

29. The circulation around the closed curve ABPCA is given by:
Or,
Where,
vs is the component of velocity along ds
Or,
Or,

30. Or,
Substituting the value of u, v and w in continuity equation we will get
Now vorticity,
Since,

31. • To check the squareness of a grid one can join
the diagonals of all the squares, the diagonals
also should result into a grid of squares

32. The magnitude or velocity components of the
resultant motion are given by the algebraic sum
of those for the constituent motion.

33. Example:
Q1

36. Tutorial Question

37.  Reference: Fluid Flow in pipes and channels by GL ASAWA