1. Vector Calculus for Primitive Thermofluids
P M V Subbarao
Professor
Mechanical Engineering Department
I I T Delhi
Mathematical Description of Fundamental Processes in Thermofluids….

2. Mathematical Operations on A Single Field Variable
Vector Calculus

3. Conservation Scalar Field Variable & The Gradient Operator

4. The First & Foremost Field Variable for Thermofluid
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?

5. 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.

6. Hydrostatics : Constant Density field
A Filed variable Recognized by the Pascal.
These are the usually desired results picturing the connection between pressure p, conservative external force field potential  and density .

7. Calculus of Single Vector Field Variable???

8. Actions inside a Differential Themofluid Volume

9. Cartesian Fluid Element

10. Visualization of Higher Order Field Variables

11. Tensor Field Variables

12. Scalar Inventions turned into Tensors !!!!!
Temperature is a scalar quantity.
Heat flux is defined with direction and Magnitude : A Vector.
Mathematically it is possible to have:
Using the principles of vector calculus applied to Fourier Law of conduction:

13. Further Physical Description
Why k must be same in all directions?
Will k be same in all directions?
Why k cannot be different each direction?
Why k cannot be a vector?
Will mathematics approve this ?
What is the most general acceptable behavior of k, approved by both physics and mathematics?

14. Most General form of Fourier Law of ConductionIf
We are at cross roads !!!!!

15. Physically & mathematically Feasible Model
Taking both physics and mathematics into consideration, the most feasible model for Fourier’s Law of conduction is:
Thermal conductivity of a general material is a tensor.

16. Surprising Inventions !!!

17. Fire Resistant Wood
Among the assessment properties of wood composite of structural members in building construction, fire performance is important and getting more attention nowadays.
A new composite called molded carbon phenolic spheres (CPS), a mixture of sugi wood charcoal powders and phenol formaldehyde resin molded with a hot press is developed by a research group in Japan.
The heat due to a fire accident should be thrown out fast outside the building.

18. Spread of Fire in A Room

19. Micro-structure of CPS

20. Thermal Performance

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

22. 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.