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