P M V Subbarao

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Vector Notation for Viscous Fluid Flow

• P M V Subbarao
• Professor
• Mechanical Engineering Department
• I I T Delhi

Powerful tool For Effective study and to
Understand Flow Devices
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Importance

• The tensor analysis is a powerful tool that
  enables the reader to study and to understand
  more effectively the fundamentals of fluid
  mechanics.
• Enables to derive all conservation laws of fluid
  mechanics without memorizing any single equation.
• The quantities encountered in fluid dynamics are
  tensors.
• A physical quantity which has a definite
  magnitude but not a definite direction exhibits a
  zeroth-order tensor, which is a special category
  of tensors.
• A first-order tensor encompasses physical
• quantities with a definite magnitude with 3
  components and a definite direction that can be
  decomposed in 3 directions.
• A second order tensor is a quantity, which has 9
  definite components and 9 definite directions.

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A Vector in Euclidian Space
According to Einstein's summation convention, it
can be written as
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Einstein Notation

• Range convention Whenever a subscript appears
  only once in a term, the subscript takes all
  possible values. E.g. in 3D space

Summation convention Whenever a subscript
appears twice in the same term the repeated index
is summed over the index parameter space. E.g. in
3D space
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Scalar Product The Base of Civilization
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Scalar Product Work Energy

• Scalar or dot product of two vectors results in a
  scalar quantity .
• Apply the Einstein's summation convention to work
  or energy scalars.

Rearrange the unit vectors and the components
separately
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Kronecker delta

• In Cartesian coordinate system, the scalar
  product of two unit vectors is called Kronecker
  delta, which is

Using the Kronecker delta,
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Thomas Savery

• As an English army officer, Thomas Savery was
  once ejected from the Lord of the Admiralty's
  office as a lunatic because he proposed a ship
  that could be propelled by side-mounted wheels
  rather than by wind or oars.
• He exhibited great fondness for mechanics, and
  for mathematicians natural philosophy and gave
  much time to experimenting, to the contriving of
  various kinds of apparatus, and to invention.
• July 2, 1698, patented the design of the first
  engine which had the most important advance in
  actual construction.

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Watt's Double-Acting Engine, 1784
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Vector or Cross Product Creation of Torque

• The vector product of two vectors is a vector
  that is perpendicular to the plane described by
  those two vectors.

Apply the index notation
With ?ijk as the permutation symbol with the
following definition
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Using the above definition, the vector product is
given by
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Non repeated subscripts
Non repeated subscripts remain fixed during the
summation. E.g. in 3D space
one for each i 1, 2, 3 and j is the dummy index.
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Special Notes

• To avoid confusion between fixed and repeated
  indices or different repeated indices, etc,
  special notes are proposed.
• Note 1 No index can be repeated more than
  twice.
• Note 2 Number of free indices shows how many
  quantities are represented by a single term.
• Note 3 If the equation looks like this