1. FLUID MECHANICS – 1
Semester 1 2011 - 2012
Week – 5
Class – 1
Kinematics of Fluids
Compiled and modified
by
Sharma, Adam
2. Objectives
• Description of Fluid flow
• Steady and unsteady flow
• Uniform and non uniform flow
• Dimensions of flow
• Material derivative and acceleration
• Differentiate between streamlines, pathlines and
streaklines
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3. Fluid Description
• Kinematics: The study of motion.
• Fluid kinematics: The study of how fluids flow and how to
describe fluid motion.
• There are distinct ways to describe motion of fluid
particles:
a) Lagrangian
b) Eulerian
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4. Lagrangian description
• This method requires us to track the position and velocity
of each individual fluid particle.
• If the number of objects is small, such as billiard balls on
a pool table, individual objects can be tracked.
However, if a fluid lump
changes its shape, size
and state as its moves with
time, it is difficult to trace
the lump in Lagrangian
description.
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5. Eulerian Description
• To describe the fluid flow, a flow domain
of a finite volume or control volume is
defined, through which fluid flows in and
out of control volume.
• Instead of tracking individual fluid particles, we define field
variables such as velocity, pressure as functions of
space and time, within the control volume.
• The field variable at a particular location at a particular time
is the value of the variable for whichever fluid particle
happens to occupy that location at that time.
• Eulerian description is often more convenient for fluid
mechanics applications. Experimental measurements are
generally more suited with Eulerian approach. 5
6. Variation Of Flow Parameters
Steady and Unsteady flow
• Steady flow is defined as the flow in which pressure and
density do not change with time in a control volume
• In the Lagrangian approach, time is inherent in
describing the trajectory of any particle. But in steady
flow, the velocities of all particles passing through any
fixed point at different times will be same.
Uniform and Non-uniform flow
• When velocity and other hydrodynamic parameters at
any instant of time do not change from point to point in a
flow field, the flow is said to be uniform. Hence for a
uniform flow, the velocity is a function of time only.
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7. Streamlines and Streamtubes
Streamline: A curve that is
everywhere tangent to the
instantaneous local velocity
vector.
Stream line at any instant can be
defined as an imaginary
curve or line in the flow field,
so that the tangent to the
curve at any point, represents
the direction of the
instantaneous velocity at that
point.
Streamlines are useful as
indicators of the
instantaneous direction of
fluid motion throughout the
flow field.
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8. Properties of stream lines
• The component of velocity, normal to a streamline is zero,
there can be no flow across a streamline.
• Since the instantaneous velocity at a point in a fluid flow
must be unique in magnitude and direction, the same point
cannot belong to more than one streamline.
• In other words, a streamline cannot intersect itself nor can
any streamline intersect another streamline.
• In a steady flow, the orientation or the pattern of
streamlines will be fixed.
• In an unsteady flow where the velocity vector changes with
time, the pattern of streamlines also changes from instant
to instant.
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9. Equation Of Stream Lines
• Consider a streamline in a plane flow in the x-y plane.
By definition, the velocity vector U at a point P must be
tangential to the streamline at that point. It follows that
dy v
dx
= tan θ =
u
; u dy − v dx = 0
where u and v are velocity components along x and y
directions respectively. The velocity vector is expressed
as
U =U ( s , t )
This shows that the velocity may vary along a streamline
direction as well as with the passage of time.
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10. EQUATION OF STREAM LINE
• Consider an elementary displacement element along a
general streamline where the velocity U such that
U = ui + vj + wk δ s = ( δ x) i + ( δ y ) j + ( δ z ) k
i j k
U xδs =0 or
u v w =0
dx dy d z
i.e., ( vδ z - wδ y ) i − ( uδ z − wδ x ) j + ( uδ y − vδ x ) k = 0
dx dy dz
= =
u v w
which is the equation of a streamline. 10
11. STREAM TUBE
• Stream tube: A bundle of neighboring stream lines may
be imagined to form a passage through which the fluid
flows. This passage (non necessarily circular in cross-
section) is known as a stream tube.
• Since a stream tube is bounded on all sides by
streamlines, velocity does not exist across a streamline,
no fluid may enter or leave a stream tube except through
its ends.
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12. A stream tube consists of a
bundle of streamlines much
like a communications
cable consists of a bundle
of fiber-optic cables.
Both streamlines
and stream tubes
are instantaneous
quantities, defined
at a particular
instant in time
according to the In an incompressible flow field, a stream tube
velocity field at (a) decreases in diameter as the flow
that instant. accelerates or converges and (b) increases in
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diameter as the flow decelerates or diverges.
13. Dimensions Of Flow
• In general, fluid flow is three dimensional. This means
that the flow parameters like velocity, pressure vary in all
the three coordinate directions.
• Sometimes simplification is made in the analysis of
different fluid flow problems by selecting the coordinate
directions so that appreciable variation of the
hydrodynamic parameters take place in only two
directions or even in only one.
• So in one dimensional flow, all the flow parameters are
expressed as functions of time and one space
coordinate only. This single coordinate is usually the
distance measure along the centre line of some conduit
In which the fluid is flowing.
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15. Material Derivative
The material derivative D/Dt is defined by following a fluid
particle as it moves throughout the flow field. In this
illustrations, the fluid particle is accelerating to the right
as it moves up and to the right.
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17. Material Derivative
• The total derivative operator d/dt in this equation is given
a special name, the material derivative; it is assigned a
special notation, D/Dt, in order to emphasize that it is
formed by following a fluid particle as it moves through
the flow field.
• Other names for the material derivative include total,
particle, Lagrangian, Eulerian, and substantial derivative.
Even under steady flow,
a fluid particle can be
accelerated as in the
flow through a nozzle
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