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Simple DEs for Conservative Thermofluids

Simple DEs for Conservative Thermofluids. P M V Subbarao Professor Mechanical Engineering Department I I T Delhi. Mathematical Description of Natural Thermofluids….

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Simple DEs for Conservative Thermofluids

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  1. Simple DEs for Conservative Thermofluids P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Mathematical Description of Natural Thermofluids….

  2. Now I think hydrodynamics is to be the root of allphysical science, and is at present second to none inthe beauty of its mathematics. (William Thomson (Lord Kelvin)1824 - 1907)

  3. Most Eligible Functions to describe A Field • For a function gwhose derivative G is expressed as: the fundamental lemma of calculus states that where g(x) represents a well-defined function whose derivative exists.

  4. Infusion of Mathematics in Thermofluids • Start from thermodynamic path integral called Work: • A Conservative Vector Field is defined as: The energy of a thermofluid system is conserved when the work done around all closed paths is zero. The three-dimensional field above the photosphere Many Natural Thermofluid systems in Universe are Conservative …..

  5. 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

  6. Selection of A Perspective for Field Description René DescartesAcademic, Philosopher, Mathematician, Scientist (1596–1650) • Discourse on; • The Method of Rightly Conducting the Reason and Seeking Truth in the Sciences. • Published in 1637.

  7. Human Perspective is Defined As Cartesian Perspective

  8. Change In Perspective on Imagination Polar to Cartesian Mapping Cylinder in Polar Coordinates New Object in Cartesian Coordinates

  9. The Birth of A Special OperatorJ. Willard Gibbs In order to justify the Cartesian system of description, the fundamental Lemma states that;

  10. Conservation Scalar Field Variable & The Gradient Operator

  11. The First & Foremost Field Variable for Thermofluids • Compared to solids, fluids seem almost alive, magical. • They flow, change form to accommodate the surroundings. • Produce gurgling sounds, and refract light to produce shimmer. • What causes (Primarily) fluids to flow? • As with solids, motions can only be produced by unbalanced forces so what is the nature of the forces in a fluid?

  12. Fluids : A Resource of Gradients • At the end of the 1640s, Pascal temporarily focused his experiments on the physical sciences. • Following in Evangelista Torricelli’s footsteps, Pascal studied superimposition of gravitational field and material property field. • This is Density Field.

  13. Hydrostatics : Constant Density field • A Filed variable Recognized by the Pascal. • Gravitational field on earth • For a constant value of g=gc These are the usually desired results picturing the connection between pressure p, conservative external force field potential and density .

  14. It is Essential to know Solutions of First Order ODEs to Solve Preliminary Thermofluid Problems

  15. First Order Differential Equations for Thermofluids • The general first-order differential equation for the function y = y(x) is written as • where f (x, y) can be any function of the independent variable x and the dependent variable y. • It is not always possible to find an analytical solution.

  16. Separable First Order Differential Equations • A first-order ode is separable if it can be written in the form where the function v(y) is independent of x and u(x) is independent of y. Homogeneous first order ODE: Non-homogeneous first order ODE:

  17. Solution of Separable Homogeneous First ODE Divide through by v(y) to obtain Proceed to integrate both sides of this equation with respect to x, to get • Variables are separated, because • the left-hand side contains only the variable y and • the right-hand side contains only the variable x. • It can be tried to integrate each side separately. • If required integration is actually performed, it is possible to obtain a relationship between y and x.

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