Computational Fluid Dynamics Seminar.pptx

PhD final defence

1. Mathematical Analysis of Ferrofluid Naseer Muhammad khan Supervised by Prof. Dr. Pan Kejia School of Mathematics and Statistics, CSU Changsha 410083, Hunan, P. R. China.

2. Layout of the presentation Definitions Objective Mathematical formulation Graphical results Conclusion 3

3. Ferrite nanoparticles Ferrofluids (invented in 1963 by NASA's Steve P.) are colloidal liquids made of ferrite nanoparticles suspended in a carrier fluid. A ferrofluid is a liquid that becomes strongly magnetized in the presence of magnetic field. Rosensweig evolved a new branch of fluid mechanics termed ferrohydrodynamics . Ferrofluid Ferrite nanoparticles or iron oxide nanoparticles (iron oxides in crystal structure of maghemite or magnetite) are the most explored magnetic nanoparticles up to date. They exhibit their magnetic behaviour only when an external magnetic field is applied. With the external magnetic field switched off, the remanence falls back to zero. Just like non-magnetic oxide nanoparticles, the surface of ferrite nanoparticles is often modified by surfactants, silica, silicones or phosphoric acid derivatives to increase their stability in solution. 4

4. Curie temperature The temperature at which certain materials lose their permanent magnetic properties. Ferromagnetic nanofluids lose their magnetic properties at sufficiently high temperatures, known as the Curie temperature. The Curie temperature is named after Pierre Curie (1895), who showed that magnetism was lost at a critical temperature. 5

5. Magnetic behavior of ferrofluid In the absence of a magnetic field, the magnetic moments of the particles are randomly distributed, and the fluid has no net magnetization. When a magnetic field is applied to a ferrofluid, the magnetic moments of the particles orient along the field lines almost instantly. The magnetization of the ferrofluid responds immediately to the changes in the applied magnetic field and when the applied field is removed, the moments randomize quickly. In a gradient field, the whole fluid responds as a homogeneous magnetic liquid which moves to the region of highest flux. This means that ferrofluids can be precisely positioned and controlled by an external magnetic field. The forces holding the magnetic fluid in place are proportional to the gradient of the external field and the magnetization value of the fluid. This means that the retention force of a ferrofluid can be adjusted by changing either the magnetization of the fluid or the magnetic field in the region. 6

6. Application of ferromagnetic nanofluids Ferrofluid enables audio speakers to function more efficiently, with improved audio response and better power handling. Ferrofluids are commonly used in loudspeakers to remove heat from the voice coil, and to passively damp the movement of the cone. They reside in what would normally be the air gap around the voice coil, held in place by the speaker's magnet. Since ferrofluids are paramagnetic, they obey Curie's law and thus become less magnetic at higher temperatures. A strong magnet placed near the voice coil (which produces heat) will attract cold ferrofluid more than hot ferrofluid thus forcing the heated ferrofluid away from the electric voice coil and toward a heat sink. This is a relatively efficient cooling method which requires no additional energy input. A typical ferrofluid may contain by volume 5% magnetic solid, 10% surfactant and 85% carrier liquid. 7

7. Application of ferromagnetic nanofluids Fig. 1.1 Loudspeaker structure. 8

8. Mathematical formulation Law of conservation of mass = 0   (1.1) (1.2) Law of conservation of momentum     (1.3)   (1.4) 9

9. Law of conservation of temperature   (1.5)   (1.6) Velocity field for the flow is given as follows:   (1.7) 10

10. (1.8) (1.10) ,   Magnetic dipole The flow of ferrofluid is affected by the magnetic field due to the magnetic dipole whose magnetic scalar potential is given by ,   The components for the magnetic field are   ,   (1.9) 11

11. (1.11) (1.13) ,   Since the magnetic body force is proportional to the gradient of the magnitude of , we obtain   ,   ,   (1.12) Making use of Eqs . (14) and (15) in Eq. (16), after having expanded in powers of and retained terms up to order , then from the resulting function we have   12

12. (1.14)   Assuming that the applied field is sufficiently strong to saturate the ferrofluid and the variation of magnetization with temperature can be approximated by a linear equation of state,   13

13. Chapter 02 Heat transfer analysis of a thermally stratified ferromagnetic fluid Heat transfer Steady and incompressible Viscous Ferrofluid Magnetic dipole Stretching sheet Stagnation point 14 Fig. 2.1 Geometry of flow

14. (2.1) (2.2) (2.3) Conservation laws are reduced to the forms = 0,   = -   + =   Subject to the Boundary conditions = , = , ,   = Q , = ,   (2.4) 15

15. Transformation (2.5) ,   , ,       (2.6) (2.7) , = ,   ,   (2.8) 16   (2.9)

16. Dimensionless parameters ,   , = , = .   (2.9) 17

17. Skin friction coefficient and local Nusselt number ,   .   In dimensionless form we have,   (2.10) (2.11) (2.12) 18

18. Homotopic Solutions       (2.13) 19 , ,    

19. Fig. 2.3 20 Fig. 2.4 Fig. 2.5 Fig. 2.6

20. Fig. 2.7 21 Fig. 2.8 Fig. 2.9 Fig. 2.10

21. Table 4: Comparison of the Nusselt number in presence of magnetic dipole. Pr Chen [01] OHAM results BVPh2-midpoint 0.72 1.0885 1 . 088461 1 . 0885521 1.0 1.3333 1 . 333250 1 . 3333720 2.0 2.0210 2 . 021910 2 . 0210619 3.0 2.5097 2 . 509692 2 . 5097472 4.0 --- 2 . 903481 2 . 9030492 10.0 4.7968 --- 4 . 7968310 [01] Chen, C.H., Laminar mixed convection adjacent to vertical continuously stretching sheets, Heat Mass Transf. 33,(1998): 471-476 . 22

22. Chapter 03 Evaluation of Fourier's law in a ferrofluid in porous medium Heat transfer Steady and incompressible Jeffrey Ferrofluid Magnetic dipole Stretching sheet Stagnation point 23

23. (3.1) (3.2) (3.3) Conservation laws are reduced to the forms = 0,   = - u,   + = ,   Subject to the boundary conditions = , = , ,   = Q , = ,   (3.4) 24

24. Transformation (3.5) ,   , ,       (3.6) (3.7) , = ,   ,   (3.8) 25   (3.9)

25. Dimensionless parameters , ,   , , .   (3.10) 26

26. Skin friction coefficient and local Nusselt number ,   .   In dimensionless form we have,   (3.11) (3.12) (3.13) 27

27. Homotopic Solutions       (3.14) 28 , ,    

28. Fig. 3.3 29 Fig. 3.4 Fig. 3.5 Fig. 3.6

29. Fig. 3.7 30 Fig. 3.8 Fig. 3.9 Fig. 3.10

30. Fig. 3.11 31 Fig. 3.13 Fig. 3.12

31. Table 4: Comparison of the Nusselt number in presence of magnetic dipole. Pr Zeeshan and Majeed [02] OHAM results BVPh2-Midpoint 0.72 1.088534 1 . 088542 1 . 0882302 1.0 1.333347 1 . 333341 1 . 3332183 2.0 --- 2 . 021082 2 . 0215192 3.0 2.509729 2 . 509783 2 . 5097533 4.0 --- 2 . 903042 2 . 9034172 10.0 4.796874 --- 4 . 7868615 32 [02] Majeed et al. Unsteady ferromagnetic liquid flow and heat transfer analysis over a stretching sheet with the effect of dipole and prescribed heat flux. J. Mol. Liq. 223 (2016): 528-533.

32. Chapter 04 Analysis of friction drag and heat transfer in a ferrofluid Heat transfer Second grade Ferrofluid Magnetic dipole Stretching sheet Thermal stratification 33

33. (4.1) (4.2) (4.3) Conservation laws are reduced to the forms = 0,   - ,   + ,   Subject to the boundary conditions = , , ,   = Q , ,   (4.4) 34

34. Transformation (2.5) ,   , ,       (2.6) (2.7) , = ,   ,   (2.8) 35   (2.9)

35. Dimensionless parameters , ,   , , .   (2.9) 36

36. Skin friction coefficient and local Nusselt number ,   .   In dimensionless form we have,   (2.10) (2.11) (2.12) 37

37. Homotopic Solutions       (2.13) 38 , ,    

38. Fig. 4.1 39 Fig. 4.2 Fig. 4.3 Fig. 4.4

39. Fig. 4.5 40 Fig. 4.6 Fig. 4.7 Fig. 4.8

40. Table 4: Comparison of the Nusselt number in presence of magnetic dipole. Pr Chen [03] Abel et al. [01] Optimal HAM 0.72 1 . 0885 1 . 0885 1 . 088521 1.0 1 . 3333 1 . 3333 1 . 333306 2.0 2 . 0210 --- 2 . 021092 3.0 2 . 5097 --- 2 . 509762 4.0 --- --- 2 . 903051 10.0 4 . 7968 4 . 7968 --- 41 [03] Chen: Laminar mixed convection adjacent to vertical, continuously stretching sheets. Heat and Mass Transfer 33, no. 5-6 (1998): 471-476.

41. Table 4: Comparison of the Nusselt number in presence of magnetic dipole. Pr 1.0 1.0 0.2 0.2 1 . 0590 1 . 2923 2.0 1 . 0329 2 . 0928 3.0 1 . 0289 2 . 3828 2.0 1.0 0.2 0.2 1 . 2976 2 . 4426 2.0 1 . 5032 2 . 3213 3.0 1 . 6121 2 . 3172 2.0 1.0 0.2 0.2 1 . 1248 . 80757 0.4 1 . 0868 . 90595 1.0 1 . 0263 . 99595 2.0 1.0 0.2 0.2 1 . 1248 . 80757 0.4 1 . 15317 . 90386 0.6 1 . 35607 . 99310 Pr 1.0 1.0 0.2 0.2 1 . 0590 1 . 2923 2.0 1 . 0329 2 . 0928 3.0 1 . 0289 2 . 3828 2.0 1.0 0.2 0.2 1 . 2976 2 . 4426 2.0 1 . 5032 2 . 3213 3.0 1 . 6121 2 . 3172 2.0 1.0 0.2 0.2 1 . 1248 . 80757 0.4 1 . 0868 . 90595 1.0 1 . 0263 . 99595 2.0 1.0 0.2 0.2 1 . 1248 . 80757 0.4 1 . 15317 . 90386 0.6 1 . 35607 . 99310 3442

42. Chapter 05 Hybrid isothermal model for the ferrohydrodynamic chemically reactive species Chemically reactive species Viscous Ferrofluid Magnetic dipole Stretching sheet Newtonian Heating 43

43. (5.1) (5.2) (5.3) Conservation laws are reduced to the forms = 0,   - ,   + ,   Subject to the boundary conditions = , , , , ,   , ,   (5.4) 44 - ,   - ,   - ,   (5.5) (5.6) (5.7)

44. Transformation (5.8) ,   , , = ,       (5.9) (5.10) (5.11) 45   (5.12)       (5.13) (5.14)

45. Dimensionless parameters (5.16) 46     , = ,   ,   , at = 0.   (5.15) Boundary conditions are

46. The chemically reactive species A, B, and C are considered to be of the same size, due to this assumption the diffusions species coefficients , , and are equivalent i.e., , we thus have   ,   .     (5.17) (5.18) (5.19) 47 Through Eqs . (5.12-5.14), we obtain the following equation

47. Skin friction coefficient and local Nusselt number ,   .   In dimensionless form we have,   (5.20) (5.21) (5.22) 48

48. Homotopic Solutions ,       (5.23) 49 , ,    

49. Fig. 5.1 50 Fig. 5.2 Fig. 5.3 Fig. 5.4

50. Fig. 5.5 51 Fig. 5.6 Fig. 5.7

51. Fig. 5.8 52 Fig. 5.9 Fig. 5.10

52. Table 1: Skin friction coefficient. Pr Sc  HAM 3.0 1.2 1.5 0.3 1 . 48031 1 . 480322 3.5 1 . 46416 1 . 464137 4.5 1 . 30295 1 . 302969 3.0 1.2 1.5 0.3 1 . 48031 1 . 480322 1.5 1 . 30052 1 . 300951 2.0 1 . 21052 1 . 210570 3.0 1.2 1.5 0.3 1 . 48031 1 . 480322 1.8 1 . 49372 1 . 493739 2.0 1 . 65091 1 . 650963 3.0 1.2 1.5 0.3 1 . 48031 1 . 480322 0.5 1 . 34271 1 . 342738 0.7 . 50641 . 506447 Pr Sc  3.0 1.2 1.5 0.3 1 . 48031 1 . 480322 3.5 1 . 46416 1 . 464137 4.5 1 . 30295 1 . 302969 3.0 1.2 1.5 0.3 1 . 48031 1 . 480322 1.5 1 . 30052 1 . 300951 2.0 1 . 21052 1 . 210570 3.0 1.2 1.5 0.3 1 . 48031 1 . 480322 1.8 1 . 49372 1 . 493739 2.0 1 . 65091 1 . 650963 3.0 1.2 1.5 0.3 1 . 48031 1 . 480322 0.5 1 . 34271 1 . 342738 0.7 . 50641 . 506447 53

53. Chapter 06 Ferrite nanoparticles , and in flow of ferromagnetic nanofluid   Ferrite nanoparticles , , Viscous Ferrofluid Ethylene glycol Stretching sheet Heat transfer   54

54. (6.1) (6.2) (6.3) Conservation laws are reduced to the forms = 0,   =   + =   Subject to the boundary conditions = , , ,   0, ,   (6.4) 55

55. Transformation ,   , ,       , = ,   ,   56   (6.5) (6.6) (6.7) (6.8) (6.9)

56. Dimensionless parameters (6.11) 57 (6.10) A= , B= , C=   ,   , = .  

57. Table 1: Thermo-physical properties of ethylene glycol, Nickel zinc ferrite, manganese zinc ferrite, and magnetite ferrite. Ethylene glycol 1116.6 2382 0.249 204 Nickel zinc ferrite 4800 710 6.3 --- Manganese zinc ferrite 4700 1050 3.9 --- Magnetite ferrite 5180 670 9.7 --- Ethylene glycol 1116.6 2382 0.249 204 Nickel zinc ferrite 4800 710 6.3 --- Manganese zinc ferrite 4700 1050 3.9 --- Magnetite ferrite 5180 670 9.7 --- , .   58 (28)

58. Skin friction coefficient and local Nusselt number ,   .   In dimensionless form we have,   (6.12) (6.13) (6.14) 59

59. Fig. 6.1 60 Fig. 6.2 Fig. 6.3 Fig. 6.4

60. Fig. 6.5 61 Fig. 6.6 Fig. 6.7 Fig. 6.8

61. Table 2: Comparison of the Nusselt number. Pr Rashidi et al. [04] Optimal HAM BVPh2-Midpoint 1.0 1 . 000000 1 . 000000 1 . 000000 3.0 1 . 923682 1 . 923690 1 . 923672 4.0 --- 2 . 003170 2 . 003162 5.0 --- 2 . 329810 2 . 329871 8.0 2 . 509430 --- 2 . 541990 62 [04] Rashidi et al. Influences of an effective Prandtl number model on nano boundary layer flow of γ Al2O3-H2O and γ Al2O3-C2H6O2 over a vertical stretching sheet, Int. J Heat Mass Transf. 98,(2016): 616-623.

62. Chapter 07 Impacts of ferrite nanoparticles in viscous ferromagnetic nanofluid Ferrite nanoparticles , , Viscous Ferrofluid Water , Ethylene glycol Porous medium Stretching sheet Newtonian Heating   63

63. (7.1) (7.2) (7.3) Conservation laws are reduced to the forms = 0,   = u   + =   Subject to the boundary conditions = , , ,   0, ,   (7.4) 64

64. Transformation ,   , ,       , = ,   ,   65   (7.5) (7.6) (7.7) (7.8) (7.9)

65. Dimensionless parameters (6.11) 66 (6.10) A= , B= , C=   ,   , = .  

66. Table 1: Thermo-physical properties of ethylene glycol, Nickel zinc ferrite, manganese zinc ferrite, and magnetite ferrite. Ethylene glycol 1116.6 2382 0.249 204 water 998.3 4182 0.60 6.96 Nickel zinc ferrite 4800 710 6.3 --- Manganese zinc ferrite 4700 1050 3.9 --- Magnetite ferrite 5180 670 9.7 --- Ethylene glycol 1116.6 2382 0.249 204 water 998.3 4182 0.60 6.96 Nickel zinc ferrite 4800 710 6.3 --- Manganese zinc ferrite 4700 1050 3.9 --- Magnetite ferrite 5180 670 9.7 --- , .   67 (28)

67. Skin friction coefficient and local Nusselt number ,   .   In dimensionless form we have,   (6.12) (6.13) (6.14) 68

68. Fig. 7.1 69 Fig. 7.2 Fig. 7.3 Fig. 7.4

69. Fig. 7.5 70 Fig. 7.6 Fig. 7.7

70. Fig. 7.8 71 Fig. 7.9 Fig. 7.10 Fig. 7.11

71. Table 2: Comparison of the Nusselt number. Pr Rashidi et al. [04] Optimal HAM BVPh2-Midpoint 1.0 1 . 000000 1 . 000000 1 . 000000 3.0 1 . 923682 1 . 923690 1 . 923672 4.0 --- 2 . 003170 2 . 003162 5.0 --- 2 . 329810 2 . 329871 8.0 2 . 509430 --- 2 . 541990 72 [04] Rashidi et al. Influences of an effective Prandtl number model on nano boundary layer flow of γ Al2O3-H2O and γ Al2O3-C2H6O2 over a vertical stretching sheet, Int. J Heat Mass Transf. 98,(2016): 616-623.

72. Conclusion 73 Heat transfer analysis of a thermally stratified ferromagnetic fluid Evaluation of Fourier's law in a ferrofluid in porous medium Analysis of friction drag and heat transfer in a ferrofluid Hybrid isothermal model for the ferrohydrodynamic chemically reactive species Ferrite nanoparticles , and in flow of ferromagnetic nanofluid Impacts of ferrite nanoparticles in viscous ferromagnetic nanofluid  

73. References 74 Choi, S.U.S, Enhancing thermal conductivity of fluids with nanoparticles, Developments and applications of non-Newtonian flows, D.A. Siginer and H.P. Wang, eds , 231(66), ASME, New York,(1995): 99-105. Buongiorno , J., Convective transport in nano uids , ASME J. Heat Transfer 128,(2006): 240-250. Neuringer , J. L. Some viscous flows of a saturated ferrofluid under the combined influence of thermal and magnetic field gradients. Int. J. of Non-Linear Mech. 2(1) (1966): 123-137. Andersson , H.I., Valnes O.A., Flow of a heated ferrofluid over a stretching sheet in the presence of a magnetic dipole, Acta Mech. 2(128),(1998): 39-47. Neuringer , J.L., Rosensweig , R.E., Ferrohydrodynamics , Phys. Fluids. 12(7), (1964): 1927-1937. Merkin , J.H., Natural-convection boundary-layer flow on a vertical surface with Newtonian heating, Int. J. Heat Fluid Flow 15(5),(1994): 392-398. Merkin , J.H., A model for isothermal homogeneous-heterogeneous reactions in boundary-layer flow, Math. Comput . Model. 24(8),(1996): 125-136. Liao, S., Homotopy Analysis Method in Non-Linear Differential Equations, Springer and Higher Education Press, Heidelberg (2012).

74. 75 THANK YOU