1. Fluid Flows Described by Vector Calculus
P M V Subbarao
Professor
Mechanical Engineering Department
I I T Delhi
ABC Type of Flows……

2. Flow through an Arbitrary Volume

3. Flow through an Arbitrary Volume

4. Local Net Mass Flow rate

5. Accounting for Mass Flux Thru a Differential Volume

6. Divergence (Gauss’s) theorem:
Divergence Theorem
Divergence (Gauss’s) 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.

7. The Divergence Effect in A Fluid flow
A velocity field is the major vector field essential for description of a moving fluid.
The divergence measures the expansion or contraction of the fluid.
A vector field with constant positive or negative value of divergence.
A vector field whose divergence vanishes identically is called as solenoidal Field.

8. Non-Solenoid Steady Flows
Incompressible – A vector dominated…..
Compressible – Both vector and scalar ….

9. Divergence of Velocity in Various Coordinate Systems
In different coordinate systems:
Cartesian :
Cylindrical:
Spherical:

10. Divergence Rules
Some “divergence rules”:

11. Measurement of Flow Rate for Human Welfare

12. Simple Methods for Measurement of Flow Rate
Insertion of small paddle wheels in a flowing river.
A wheel close to the center (of a river) was not not rotating or rotating slowly.
Wheels close to the edges were rotating fast.
A mystery of mere engineering……

13. Disclosure of Mystery

14. Scientific Flow Measurement in a River

15. Curl of a vector field: Curl of a vector field:
Circulation is the amount of force that pushes along a closed boundary or path. It's the total "push" received while going along a path, such as a circle.
Curl is simply circulation per unit area, circulation density, or rate of rotation (amount of twisting at a single point)
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.

16. The Curl of Velocity Field
Define the vorticity vector as being the curl of the velocity
vorticity vector in cylindrical co-ordinates:
vorticity vector in spherical co-ordinates:

17. The Atmospheric Flow : a mild to Dangerous Rotational Flow