Selection of A Suitable Branch of Mathematics for Thermofluids P M V Subbarao Professor Mechanical Engineering Department I I T Delhi The Right Tool for Comprehensive Description of Thermofluids….

General Mathematical Formulation for Thermofluids Let f(x, t) may be a general continuously differentiable scalar or vector variable. V(t) is a measurable, simple connected fluid volume moving with the fluid. t stands for time. It is very important to know that there is a closed surface (real or imaginary) over V is denoted by

Gauss Divergence Theorem Divergence (Gauss’s) Theorem: The volume integral of the divergence of a vector field over the volume enclosed by surface S is equal to the flux of that vector field taken over that surface S. The divergence theorem is primarily used to convert a surface integral into a volume integral. to convert a volume integral to a surface integral.

Common Fundamental Equations : Derived from Laws of Nature Conservation of Mass. Conservation of Momentum. Conservation of Energy. Conservation of Species. Use of mathematical analysis of a thermofluid field with above variables helped in Creation of Actions for Human Development.

Existence of Field Variable w/o Thermofluid ??? 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 question we need to answer is: How can a force occur without any countable finite bodies & apparent contact between them? "action at a distance" has stymied many of the great minds

Think about Field Let's focus on Newton's thinking. Consider an apple starting from rest and accelerating freely …... 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. The force of the earth's attraction causes the apple to fall, but how specifically? The earth must exude something that makes acceleration of apple. This something exuded by the earth was called as the gravitational field. Applied mathematics for Thermofluids is all about investigation of the concept called “FIELD”. There are forcing agents in all Thermofluid Fields, but how specifically are they present in these fields?

Better Utilization of Field Nature of Thermofluids Air Ship Wright Brothers Airfoil

The Earth : A Group of Material & Energy Fields for Survival of Life Morowitz [1992]

An Important Field by Nature

Vector Field due to Earth Topography

ONE TIME RESOURCE INCOMING RESOURCE FOSSIL FUEL SOLAR RADIATION WINDS SOLAR ENERGY INCOMING RESOURCE CO2 + H2O PHTOSYNTHESIS SOLAR RADIATION WINDS VEGETATION Heating of OCEAN S CLOUDS RAINS CHEMICAL ENERGY Hydrology FOSSILIZATION COAL FOSSIL FUEL PETROLEUM NATURAL GAS

ONE TIME RESOURCE INCOMING RESOURCE THERMAL FOSSIL FUEL SOLAR SOLAR ENERGY INCOMING RESOURCE CO2 + H2O PHTOSYNTHESIS SOLAR RADIATION WINDS VEGETATION OCEAN THERMAL ENERGY CLOUDS WIND ENERGY VELOCITY THERMAL WAVE RAINS CHEMICAL ENERGY HYDRO ENERGY FOSSILIZATION COAL FOSSIL FUEL PETROLEUM NATURAL GAS BIoenergy

The Concept of Field in Thermofluids 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/action 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 described using scalar or/and vector variables in thermofluids. 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/comprehesive to work with than vector fields. We need to use the calculus as well as vector calculus to describe fields generated by thermofluids.

Infusion of Mathematics in Thermofluids Start from path integral called 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 Conservative 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

Cartesian Perspective of Field Description

The Birth of A Special Operator In order to justify the Cartesian system of description, the fundamental Lemma states that;

The Gradient Operator