1. IV. Kinematics of Fluid Motion

2. Contents 1. Specification of Fluid Motion 2. Material Derivatives 3. Geometric Representation of Flow 4. Terminology 5. Motion and Deformation of Fluid Element 6. Rotational and Potential Flows 7. Continuity Equation

3. 1. Specification of Fluid Motion

4. Lagrangian View Study fluid motion by tracing the motion of fluid particles

5. Identify a representative fluid particle Determine its position instantaneously Determine the velocity and acceleration Determine other physical quantities

8. Eulerian View Study fluid motion by investigating the temporal and spatial variation of the flow field

10. 2. Material Derivatives

11. Definition The rate of change one observed when following the motion of a fluid particle

14. Local / Temporal Advective / Spatial Material Derivative

15. Acceleration of Fluid particles

16. 3. Geometric Representation of Flow

17. Pathline

18. A pathline is the trajectory of a fluid particle

19. Mathematical representation

20. Streamline

21. A streamline is a line whose tangent always represents the direction of velocity

22. Mathematical representation

23. Example Find the pathline and streamline of the following flow field:

24. Pathline

25. Streamline

26. Streamline is identical to pathline if the velocity is invariable with time In general, streamlines will not intercross and will not end at a solid wall, etc.

27. 4. Terminology

28. Discharge and Mass flux

29. Streamtube, Stream filament, Total flow

33. Fluid system and Control volume

35. Steady flow and Unsteady flow

37. Uniform flow and Non-uniform flow

38. The streamlines of a uniform flow is necessarily straight lines and parallel to each other

39. Gradually-varying flow and Rapidly-varying flow Curvature of all streamlines are small Curvature of all streamlines are small Streamlines are nearly parallel Streamlines are nearly parallel

40. 5. Motion and Deformation of Fluid Elements

42. Motion of a fluid element can be decomposed into Translation Translation Rotation Rotation Deformation Deformation

43. The translation is described by

44. The rotation is described by

46. The angular velocity

47. The deformation is described by

49. Rate of strain

50. Helmholtz’s theorem of velocity decomposition

51. Translation Rotation Deformation

52. 6. Rotational and Potential Flows

54. Physical Interpretation

55. Example

56. Velocity Potential Irrotational flowPotential flow

57. 7. Continuity Equation

58. Conservation of Mass: Mass in a closed system is invariant

60. Net outflow of mass through the surface of the control volume

61. Decrease of mass within the control volume

62. Mass Conservation

63. For incompressible fluid Bulk expansion

64. Continuity Equation for Steady Total Flows SoSo SeSe

66. Continuity Equation for Potential Flows