conversion of creative ideas into a number series…… p m v subbarao professor mechanical...
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Conversion of Creative Ideas into A Number Series……
P M V SubbaraoProfessor
Mechanical Engineering Department
I I T Delhi
Basic Mathematical Framework for Analysis of Viscous Fluid Flows
What should be Accounted for ????
• Renaissance period of Leonardo da Vinci in particular should be recalled.
• Popularly he is well known as a splendid artist, but he was an excellent scientist, too.
• Leonardo da Vinci (1452 - 1519) correctly deduced the conservation of mass equation for incompressible, one dimensional flows.
• Leonardo also pioneered the flow visualization genre close to 500 years ago.
Need for Accounting of Forces
• Systems only due to Body Forces.
• Systems due to only normal surface Forces.
• Systems due to both normal and tangential surface Forces.
– Only mechanical forces.
– Thermo-dynamic Effects (Buoyancy forces …. )
– Only electrical forces.
– Electro-kinetic forces.
– Physico-Chemical/concentration based forces (Environmental /Bio Fluid Mechanics
Major Flow Systems due to Mechanical Forces : Level 1
• Incompressible – A vector dominated…..
• Compressible – Both vector and scalar ….
Preliminary Mathematical Concepts
• 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
The question we need to answer is how can a force
occur without any countable finite bodies &
apparent contact between them?
How to Create Force???
• Newton developed the theories of gravitation in 1666, when he was only 23 years old.
• Some twenty years later, in 1686, he presented his three laws of motion in the "Principia Mathematica Philosophiae Naturalis.“
The Mother of Vector
• Let's focus on Newton's thinking.
• Consider an apple starting from rest and accelerating freely under the influence of gravity.
• The force of the earth's attraction causes the apple to fall, but how specifically?
• Until the apple hits the ground, the earth does not touch the apple so how does the earth place a force on the apple?
• Something must go from the earth to the apple to cause it to fall.
Thus Spake Newton
• The earth must exude something that places a force on the apple.
• This something exuded by the earth was called as the gravitational field.
• We can start by investigating the concept called field.
The Concept of Field
• Something must happen in the fluid to generate/carry the force, and we'll call it the field.
• Few basic properties along with surroundings must be responsible for the occurrence of this field.
• Let this field be .
• "Now that we have found this field, what force would this field place upon my system.“
• What properties must the fields have, and how do we describe these field?
Fields & Properties
• The fields are sometimes scalar and sometimes vector in nature.
• 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.
• We need to use the calculus as well as vector calculus.
• Study of the physical properties of vector fields is the first step to attain ability to use Viscous Fluid Flow Analysis.
Define mother by Studying the Child
• Start from path integral Work:
ldFW
.
• Conservative Vector Field
0. ldF
The energy of a Flow system is conserved when the work done around all closed paths is zero.
Mathematical Model for Field
• For a function g whose derivative is G:
Gdx
dg
the fundamental lemma of calculus states that
ab
x
x
xgxgGdxb
a
where g(x) represents a well-defined function whose derivative exists.
The mother of 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 for a path integral to become mother of a vector field it must depend only on the end points.
• Then, a scalar field will be related to the vector field F by
2
1
.12 ldF
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