P%20M%20V%20Subbarao

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Field Mathematics for Fluid Mechanics

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

Quantification of the Fundamental Distributed
source of Actions
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The Special Vector fields

• There are special vector fields that can be
  related to a scalar field.
• There is a very real advantage in doing so
  because scalar fields are far less complicated to
  work with than vector fields.
• A vector field may be derived from a scalar field
  any time the vector field is conservative.
• A conservative vector field is required to
  have a zero path integral over any closed path .

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Non Cyclic Integrals of the Special Vector Field

• There are integrals called path integrals which
  have quite different properties.
• In general, a path integral does not define a
  function because the integral will depend on the
  path.
• For different paths the integral will return
  different results.
• In order to define a vector field, the integral
  must depend only on the end points.
• Then, a scalar field ? will be related to the
  vector field F by

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The Birth of A Special Operator
In order to justify the Cartesian system of
description, the fundamental Lemma states that
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The Fertility Operator
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Flow Fields Creating Complex Mechanical Force
Systems

• Laminar Absolutely Deterministic
• The structured tensors.
• The relations among tensors, vectors Scalars
• Advanced Vector calculus
• Turbulent Statistically Deterministic.
• Concept of ensemble/temporal/spatial averaging.
• Creation of more tensors to develop deterministic
  approach
• The issue of closure.

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Preliminary Vector Mathematics

• Vector and Tensor Analysis, Applications to Fluid
  Mechanics
• Tensors in Three-Dimensional Euclidean Space
• Index Notation
• Vector Operations Scalar, Vector and Tensor
  Products
• Contraction of Tensors
• Differential Operators in Fluid Mechanics
• Substantial Derivatives
• Differential Operator
• Operator Applied to Different Functions

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Real Fluids A Resource of Gradients

• At the end of the 1640s, Pascal temporarily
  focused his experiments on the physical sciences.
  
• Following in Evangelista Torricellis footsteps,
  Pascal experimented with how atmospheric pressure
  could be estimated in terms of weight.

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Hydrostatics

• A Field variable Recognized by the Pascal.
• Even based on pedagogical principle, to start
  with simple matters and turn later to the
  complicated ones, Fluid Mechanics traditionally
  starts with hydrostatics.

These are the usually desired results picturing
the connection between pressure p, conservative
external force field potential ? and density ?.
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Vector Calculus to Describe Characteristics of
Fluid Mechanics
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Differential Operators in Fluid Mechanics

• In fluid mechanics, the particles of the working
  medium undergo a time dependent or unsteady
  motion.
• The flow quantities such as the velocity V and
  the thermodynamic properties of the working
  substance such as pressure p, temperature T,
  density ? or any arbitrary flow quantity Q are
  generally functions of space and time.

During the flow process, these quantities
generally change with respect to time and space.
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Variety of Pumps to Create Gradients for Human
Welfare
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Turbo-Machines GEOMETRIES
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Deformation of Fluid Element
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Projections of Fluid Deformations
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The Fertility (Gradient) of A vector
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Mathematically Understood Flows