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Vector & Tensor Analysis for Fluid Mechanics

Vector & Tensor Analysis for Fluid Mechanics. P M V Subbarao Professor Mechanical Engineering Department I I T Delhi. Innovative Vector Actions animating but non-alive Fields …… Scalar-Vector Interaction for better Life ……. Cartesian Vectors. Concept of Force Vector.

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Vector & Tensor Analysis for Fluid Mechanics

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  1. Vector & Tensor Analysis for Fluid Mechanics P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Innovative Vector Actions animating but non-alive Fields …… Scalar-Vector Interaction for better Life ……

  2. Cartesian Vectors

  3. Concept of Force Vector In order to justify the Cartesian system of description, the fundamental Lemma states that;

  4. Scalar Product & Orthoganoality Early Experience of Scalar Product Early Experience of Orthagonality:

  5. The Engine : An agent to generate Force

  6. Dot Product to Vector or Cross Product • The dot product of two vectors is a scalar, which was easily felt by every common engineer at early stages. • The vector product of two vectors is a vector that is perpendicular to the plane described by those two vectors. The components of the torque vector are:

  7. Creation of Motive Power: Dangerous & Inhuman Technologies by copying Wild Solutions to Extrasomatism

  8. Thomas Savery • As an English army officer, Thomas Savery was once ejected from the Lord of the Admiralty's office as a lunatic because he proposed a ship that could be propelled by side-mounted wheels rather than by wind or oars. • He exhibited great fondness for mechanics, and for mathematicians natural philosophy and gave much time to experimenting, to the contriving of various kinds of apparatus, and to invention. • July 2, 1698, patented the design of the first engine which had the most important advance in actual construction.

  9. The Family of Steam Engines A Direct Hardware Creations due to the Cross Product…..

  10. Watt's Double-Acting Engine, 1784

  11. Vector Product: An Intellectual Imagination Think of Nine Product of two Vectors

  12. Formalization of Vector Product

  13. Coordinate systems: Cylindrical (polar) Diff. length: An intersection of a cylinder and 2 planes r Diff. area: Diff. volume: r An arbitrary vector:

  14. Coordinate systems: Spherical An intersection of a sphere of radius r A plane that makes an angle  to the x axis, A cone that makes an angle  to the z axis.

  15. Properties of Coordinate systems: Spherical Properties: Diff. length: Diff. area: Diff. volume: An arbitrary vector:

  16. System conversions 1. Cartesian to Cylindrical: 2. Cartesian to Spherical: 3. Cylindrical to Cartesian: 4. Spherical to Cartesian:

  17. Gradient in Arbitrary Coordinate System Gradient of a scalar function: Gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.

  18. Fluid Mechanical Significance of Gradient of A Scalar

  19. Gradient of a Vector

  20. System conversions for Gradient Gradient in different coordinate systems: • Cartesian : • Cylindrical: • Spherical:

  21. Properties of Gradient Operations Collect more of such relations, relevant to Thermo-Fluid Sciences.

  22. 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 the 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.

  23. Fluid Mechanical Interpretation of Divergence

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