2. Scalar-Vector Interaction for better Life …… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Vector Analysis : Applications to Fluid Mechanics

3. Vector Calculus Natural to Fluid Mechanics

4. Human Capability to Imagine Geometry

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

6. 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.

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

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

9. Differential relations for vectors 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. Gradient of a scalar function: Two equipotential surfaces with potentials V and V+  V. Select 3 points such that distances between them P 1 P 2  P 1 P 3, i.e.  n   l. Assume that separation between surfaces is small: Projection of the gradient in the u l direction:

10. System conversions for Differential relations Gradient in different coordinate systems:

11. Properties of Gradient Operations Collect more of such relations, relevant to Fluid Mechanics.

12. Divergence of a vector field Divergence of a vector field: Divergence is an operator that measures the magnitude of a vector field's source or sink at a given point. In different coordinate systems:

13. Divergence Rules Some “divergence rules”: Divergence (Gauss’s) theorem:

14. What is divergence? Think of a vector field as a velocity field for a moving fluid. The divergence measures the expansion or contraction of the fluid. A vector field with constant positive or negative value of divergence.

15. Meaning of the Divergence Theorem The divergence theorem says is that the expansion or contraction (divergence or convergence) of material inside a volume is equal to what goes out or comes in across the boundary. The divergence theorem is primarily used –to convert a surface integral into a volume integral. –to convert a volume integral to a surface integral.

16. Further Use of Gradient for Human Welfare V V Assume we insert small paddle wheels in a flowing river. The flow is higher close to the center and slower at the edges. Therefore, a wheel close to the center (of a river) will not rotate since velocity of water is the same on both sides of the wheel. Wheels close to the edges will rotate due to difference in velocities. The curl operation determines the direction and the magnitude of rotation.

17. Curl of a vector field: Curl is a vector field with magnitude equal to the maximum "circulation" at each point and oriented perpendicularly to this plane of circulation for each point. More precisely, the magnitude of curl is the limiting value of circulation per unit area.

18. The Natural Genius & The Art of Generating Lift

19. Hydrodynamics of Prey & Predators

20. The Art of C-Start

22. The Art of Complex Swimming

23. Development of an Ultimate Fluid machine

24. Fascinating Vortex Phenomena : Kutta-Joukowski Theorem The Joukowsky transformation is a very useful way to generate interesting airfoil shapes. However the range of shapes that can be generated is limited by range available for the parameters that define the transformation.

25. The Curl in different coordinate systems:

26. Repeated vector operations

27. The Laplacian Operator Cartesian Cylindrical Spherical

28. OperatorgraddivcurlLaplacian is a vectora scalara vector a scalar (resp. a vector) concerns a scalar field a vector field a scalar field (resp. a vector field) Definition resp.

29. Classification of Vector Fields A vector field (fluid flow) is characterized by its divergence and curl