Download Chapter 2

Introduction to Fluid Mechanics
Chapter 2
Fundamental Concepts
© Fox, McDonald & Pritchard
Main Topics
Fluid as a Continuum
Velocity Field
Stress Field
Viscosity
Surface Tension
Description and Classification of
Fluid Motions
© Fox, McDonald & Pritchard
Fluid as a Continuum
Fluid as a Continuum – disregard the behavior of individual
molecules. (A less rigid assumption than that in solid mechanics – in
some cases such as rarefied gas flow, we have to abandon this
concept)
As a consequence of the continuum assumption, each fluid property
is assumed to have a definite value at every point in space
(The
concept
of
a
continuum breaks down
whenever the mean free
path of the molecules
becomes the same order of
magnitude as the smallest
significant characteristic
dimension of the problem
– rarefied gas in the upper
reaches
of
the
atmosphere).
Velocity Field
© Fox, McDonald & Pritchard
Velocity Field
Consider also
Steady and Unsteady Flows
1D, 2D, and 3D Flows
Timelines, Pathlines, and
Streaklines
© Fox, McDonald & Pritchard
One-, Two-, and Three-Dimensional Flows
Everything is inherently three-dimensional – to simplify the problem
– fewer dimensions are frequently considered.
Fundamental Concepts
Timeline -- the flow lines (structures, or fields) at a given instant (for
a transient problem).
Pathline – the path or trajectory traced out by a moving fluid particle
(using dye or smoke).
Streakline – The line joining the fluid particles, which had, at some
time, passed through the given location in space.
Streamline – lines drawn in the flow field so that at a given instant
they are tangent to the direction of flow at every point in the flow
field.
(Since the streamlines are tangent to the velocity vector at every
point in the flow field, there can be no flow across a streamline.)
In a steady flow, pathlines, streaklines, and streamlines are identical
lines in the flow field (Example 2.1)
The Use of Streamlines (Google images)
Stress Conditions on a Surface
Stress condition on a surface
passing through a given point
under a coordinate system-the
use of a coordinate to distinguish
individual points.
Stress Conditions at a Point (Stress Field)
C
Since there could be numerous
surfaces passing through a
point, the stress condition on a
single surface may be
inadequate to describe the stress
condition at the point. However,
it has been proven that the
stress conditions on any three
mutually perpendicular
surfaces passing through the
given point could be adequate to
describe the stress condition at
that point. Or the stress
conditions on the surfaces of an
infinitesimal rectangular cuboid
at that point may be used to
represent the stress conditions
at that point.
Stress condition at a given point
The first subscript is
related to the direction
of the surface, while
the second is related
to the direction of the
stress. Every stress is
marked in the positive
direction.
Viscosity
 Newtonian Fluids
• Most of the common fluids (water, air, oil, etc.)
• “Linear” fluids
Although any shear stress could
produce a fluid deformation, a
higher deformation rate requires a
larger shear stress (or shear
force). Therefore, a fluid still has
the resistive characteristics
against an applied shear stress in
terms of deformation rate. This
resistive characteristics may be
considered as a property of a
given fluid and can be measured
experimentally in term of the
concept of viscosity.
© Fox, McDonald & Pritchard
Viscosity
With different forces
applied, the deformation
rate is different.
The velocity
difference between
the top and bottom
plates is du
l / y  tan   
ut  y
Newtonian Fluids
For Newtonian fluids (most common fluids, such as water and
air), experiments have found that the shear stress as applied
is directly proportional to the rate of deformation:
Viscosity
 Non-Newtonian Fluids
• Special fluids (e.g., most biological fluids,
toothpaste, some paints, etc.)
• “Non-linear” fluids
© Fox, McDonald & Pritchard
Surface Tension
© Fox, McDonald & Pritchard
Description and Classification
of Fluid Motions
© Fox, McDonald & Pritchard
Laminar Flow
Turbulent flows
Fluid particles rapidly mix as they move along due to random
three-dimensional velocity fluctuations. Semi-empirical theories
in conjunction with experimental data are the common approach
for a turbulent flow. Computational solutions are also available
through the use of some empirical parameters, however.
http://www.google.com/images
Turbulent flows
http://www.google.com/images