P M V Subbarao

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Vector Analysis Applications to Fluid Mechanics

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

Scalar-Vector Interaction for better Life
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Vector Calculus Natural to Fluid Mechanics
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Human Capability to Imagine Geometry
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Coordinate systems Cylindrical (polar)
An arbitrary vector
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Coordinate systems Spherical
An intersection of a sphere of radius r
A plane that makes an angle ? to the x axis,
A cone that makes an angle ? to the z axis.
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Properties of Coordinate systems Spherical
Properties
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System conversions
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Differential relations for vectors
Gradient of a scalar field is a vector field
which points in the direction of the greatest
rate of increase of the scalar field, and whose
magnitude is the greatest rate of change.
Two equipotential surfaces with potentials V and
V?V. Select 3 points such that distances
between them P1P2 ? P1P3, i.e. ?n ? ?l.
Assume that separation between surfaces is small
Projection of the gradient in the ul direction
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System conversions for Differential relations
Gradient in different coordinate systems
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Properties of Gradient Operations
Collect more of such relations, relevant to Fluid
Mechanics.
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Divergence of a vector field
Divergence of a vector field
Divergence is an operator that measures the
magnitude of a vector field's source or sink at a
given point.
In different coordinate systems
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Divergence Rules
Some divergence rules
Divergence (Gausss) theorem
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What is divergence?

• Think of a vector field as a velocity field for a
  moving fluid.
• The divergence measures the expansion or
  contraction of the fluid.
• A vector field with constant positive or negative
  value of divergence.

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Meaning of the Divergence Theorem

• The divergence theorem says is that the expansion
  or contraction (divergence or convergence) of
  material inside a volume is equal to what goes
  out or comes in across the boundary.
• The divergence theorem is primarily used
• to convert a surface integral into a volume
  integral.
• to convert a volume integral to a surface
  integral.

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Further Use of Gradient for Human Welfare

• Assume we insert small paddle wheels in a flowing
  river.
• The flow is higher close to the center and slower
  at the edges.
• Therefore, a wheel close to the center (of a
  river) will not rotate since velocity of water is
  the same on both sides of the wheel.
• Wheels close to the edges will rotate due to
  difference in velocities.
• The curl operation determines the direction and
  the magnitude of rotation.

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Curl of a vector field
Curl of a vector field
Curl is a vector field with magnitude equal to
the maximum "circulation" at each point and
oriented perpendicularly to this plane of
circulation for each point. More precisely, the
magnitude of curl is the limiting value of
circulation per unit area.
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The Natural Genius The Art of Generating Lift
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Hydrodynamics of Prey Predators
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The Art of C-Start
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The Art of Complex Swimming
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Development of an Ultimate Fluid machine
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Fascinating Vortex Phenomena Kutta-Joukowski
Theorem
The Joukowsky transformation is a very useful way
to generate interesting airfoil shapes. However
the range of shapes that can be generated is
limited by range available for the parameters
that define the transformation.
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The Curl in different coordinate systems
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Repeated vector operations
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The Laplacian Operator
Cartesian
Cylindrical
Spherical
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Operator grad div curl Laplacian
is a vector a scalar a vector a scalar (resp. a vector)
concerns a scalar field a vector field a vector field a scalar field (resp. a vector field)
Definition
resp.
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Classification of Vector Fields

• A vector field (fluid flow) is characterized by
  its divergence and curl