Conversion of Creative Ideas into A Number Series…… P M V Subbarao Professor 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 ( ) 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: Conservative Vector Field 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: the fundamental lemma of calculus states that 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