1. Rajai1 y b

2. 2 APPLICATIONS v Heat and mass transfer rates are enhanced by the oscillation of the surrounding fluid. Useful in combustion, drying and the passage of sound waves through particulate systems. v Particle-laden flows, Brownian motion, suspension rheometry, colloidal suspension, and particle motion in filters.

3. Rajai3 MODES OF HEAT TRANSFER v RADIATION v CONDUCTION v CONVECTION (forced & free)

4. Rajai4 GOVERNING EQUATIONS Conservation of momentum where

5. Rajai5 First law of thermodynamics Conservation of mass

6. Rajai6 VECTOR RELATIONS

7. Rajai7 Define:

8. Rajai8 then,

9. Rajai9 and,

10. Rajai10 a g SPHERICAL COORDINATES SCALE FACTORS r

11. Rajai11 VARIABLES

12. Rajai12 DIMENSIONLESS NUMBERS kinematic viscosity thermal diffusivity volumetric expansion coef. frequency of oscillation surface temperature far-field temperature

13. Rajai13 USING THE SPHERICAL COORDINATE SYSTEM, THE EQUATIONS REDUCE TO: (1) (2) (3)

14. Rajai14 BOUNDARY CONDITIONS

15. Rajai15 Equations are two dimensional, time-dependent, nonlinear, coupled, and of infinite domain. No explicit boundary conditions for vorticity on the surface. Difficulties with finite- differences such as, indeterminate forms, etc..

16. Rajai16

17. Rajai17 THE METHOD OF SOLUTION

18. Rajai18 HOW ?!

19. Rajai19 INTEGRALS NEEDED !! and in general,

20. Rajai20 where After Mavromatis & Alassar:

21. Rajai21 where

22. Rajai22 Saalschutz’s Theorem

23. Rajai23 3-j SYMBOLS Represent the probability amplitude that three angular momentum j1, j2, and j3 with projections m1, m2, and m3 are coupled to yield zero angular momentum. They are related to Clebsch-Gordan coefficients (C) by:

24. Rajai24 where Clebsch-Gordan Coefficients

25. Rajai25 (1) (2) (3)

26. Rajai26 where,

27. Rajai27 MODES BOUNDARY CONDITIONS

28. Rajai28 NUMERICAL METHOD CRANK-NICOLSON F.D. SCHEME

29. Rajai29

30. Rajai30 SPECIALIZED STEP-BY-STEP METHOD

31. Rajai31

32. Rajai32

33. Rajai33 PHYSICAL PARAMETERS Nusselt Number

34. Rajai34

35. Rajai35

36. Rajai36 Drag Coefficient

37. Rajai37 Pressure Coefficient

38. Rajai38 VALIDATION

39. Rajai39 SOME RESULTS

40. Rajai40

41. Rajai41

42. Rajai42

43. Rajai43

44. Rajai44

45. Rajai45

46. Rajai46 0.2300 0.0000 0.2500

47. Rajai47 0.2625 0.2725

48. Rajai48 0.2750 0.3125

49. Rajai49 0.5000 0.5550 0.5650

50. Rajai50 0.60000.7250 0.7500

51. Rajai51 FUTURE RESEARCH Heat transfer from a sphere in a spinning infinite fluid. Heat transfer from other geometries such as oblate and prolate spheroids.

52. Rajai52 SPHEROIDS

53. Rajai53 Stream Function Vorticity

54. Rajai54 Boundary Conditions Energy

55. Rajai55 A NOTE ON THE PROBLEM OF SPINNING STREAM

56. Rajai56