Field Mathematics for Fluid Mechanics P M V

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……

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.

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

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

The Fertility Operator

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.

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

Real Fluids : A Resource of Gradients • At the end of the 1640 s, Pascal temporarily focused his experiments on the physical sciences. • Following in Evangelista Torricelli’s footsteps, Pascal experimented with how atmospheric pressure could be estimated in terms of weight.

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 .

Vector Calculus to Describe Characteristics of Fluid Mechanics

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 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.

Variety of Pumps to Create Gradients for Human Welfare

Turbo-Machines & GEOMETRIES

Deformation of Fluid Element

Projections of Fluid Deformations

The Fertility (Gradient) of A vector

Mathematically Understood Flows