Scaling
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Scaling

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eBook - PDF

Scaling

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About This Book

Many phenomena in nature, engineering or society when seen at an intermediate distance, in space or time, exhibit the remarkable property of self-similarity: they reproduce themselves as scales change, subject to so-called scaling laws. It's crucial to know the details of these laws, so that mathematical models can be properly formulated and analysed, and the phenomena in question can be more deeply understood. In this 2003 book, the author describes and teaches the art of discovering scaling laws, starting from dimensional analysis and physical similarity, which are here given a modern treatment. He demonstrates the concepts of intermediate asymptotics and the renormalisation group as natural attributes of self-similarity and shows how and when these notions and tools can be used to tackle the task at hand, and when they cannot. Based on courses taught to undergraduate and graduate students, the book can also be used for self-study by biologists, chemists, astronomers, engineers and geoscientists.

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Yes, you can access Scaling by Grigory Isaakovich Barenblatt in PDF and/or ePUB format, as well as other popular books in Mathematics & Mathematics General. We have over one million books available in our catalogue for you to explore.

Information

Publisher
Cambridge University Press
Year
2003
ISBN
9781107299498
Topic
Mathematics
Subtopic
Mathematics General
Index
Mathematics
34
1
Dimensional
analysis
and
physical
similarity
Figure
1.7.
When
a
punch
is
pressed
into
a
block
of
fused
silica,
a
conical
crack
is
formed
(Benbow
1960).
experimental
work
required
of
the
physico-chemists
would
have
been
reduced
by
a
large
factor.
Problem
5
.
Derive,
using
dimensional
analysis,
the
relation
between
the
di-
ameter
of
the
conical
crack
formed
in
a
brittle
block
under
a
punch
and
the
applied
load.
In
the
remarkable
experiments
of
Roesler
(1956)
and
Benbow
(1960)
a
punch
with
a
small
ļ¬‚at
point
was
pressed
slowly
into
the
face
of
a
cubic
block
of
transparent
brittle
material
(fused
silica,
see
Figure
1.7
).
A
perfect
conical
crack
was
formed
under
the
punch
and,
as
the
load
increased,
the
crack
increased
in
size.
The
diameter
of
the
base
of
the
conical
crack
rapidly
became
much
larger
than
the
diameter
of
the
ļ¬‚at
point
of
the
punch.
Solution
.
The
conical
crack
under
the
punch
is
in
a
state
of
mobile
equilibrium
:
any
increase
in
the
load
leads
to
its
extension.
It
is
known
from
the
theory
of
elasticity
(see
Broberg
1999)
that
the
stress
directly
underneath
the
crack
tip,

Table of contents

  1. Cover
  2. Title
  3. Copyright
  4. Dedication
  5. Contents
  6. Foreword
  7. Preface
  8. Introduction
  9. Chapter 1 Dimensional analysis and physical similarity
  10. Chapter 2 Self-similarity and intermediate asymptotics
  11. Chapter 3 Scaling laws and self-similar solutions that cannot be obtained by dimensional analysis
  12. Chapter 4 Complete and incomplete similarity. Self-similar solutions of the first and second kind
  13. Chapter 5 Scaling and transformation groups. Renormalization group
  14. Chapter 6 Self-similar phenomena and travelling waves
  15. Chapter 7 Scaling laws and fractals
  16. Chapter 8 Scaling laws for turbulent wall-bounded shear flows at very large Reynolds numbers
  17. References
  18. Index