Equations of Continuity

1. Equations of Continuity

2. Outline Time Derivatives & Vector Notation Differential Equations of Continuity Momentum Transfer Equations

3. Introduction In order to calculate forces exerted by a moving fluid as well as the consequent transport effects, the dynamics of flow must be described mathematically (kinematics). FLUID Continuous medium Infinitesimal pieces of fluid

4. Fluid system: finite piece of the fluid material (Lagrangian) Fluid particle: differentially small finite piece of the fluid material (Lagrangian) Perspectives of Fluid Motion Eulerian Perspective – the flow as seen at fixed locations in space, or over fixed volumes of space (the perspective of most analysis) LagrangianPerspective – the flow as seen by the fluid material (the perspective of the laws of motion) Control volume: finite fixed region of space (Eulerian) Coordinate: fixed point in space (Eulerian)

5. Lagrangian Perspective The motion of a fluid particle is relative to a specific initial position in space at an initial time.

6. Lagrangian Perspective z Lagrangian coordinate system pathline position vector y partial (local) time derivatives x

7. Lagrangian Perspective Consider a small fluid element with a mass concentration moving through Cartesian space: y y t = t1 t = t2 x x z z

8. Lagrangian Perspective Consider a small fluid element with a mass concentration moving through Cartesian space: y y t = t1 t = t2 x x z z

9. Lagrangian Perspective Total change in the mass concentration with respect to time: If the timeframe is infinitesimally small:

10. Lagrangian Perspective Total Time Derivative Substantial Time Derivative local derivative convective derivative

11. Lagrangian Perspective stream velocity vector notation gradient

12. Lagrangian Perspective Problem with the Lagrangian Perspective The concept is pretty straightforward but very difficult to implement (since to describe the whole fluid motion, kinematics must be applied to ALL of the moving particles), often would produce more information than necessary, and is not often applicable to systems defined in fluid mechanics.

13. Eulerian Perspective z Motion of a fluid as a continuum flow Fixed spatial position is being observed rather than the position of a moving fluid particle (x,y,z). y x

14. Eulerian Perspective z Motion of a fluid as a continuum flow Velocity expressed as a function of time tand spatial position (x, y, z) y Eulerian coordinate system x

15. Eulerian Perspective Difference from the Lagrangian approach: Lagrangian Eulerian

16. Eulerian Perspective Difference from the Lagrangian approach: Eulerian Lagrangian

17. Outline Time Derivatives & Vector Notation Differential Equations of Continuity Momentum Transfer Equations

18. Equation of Continuity differential control volume:

19. Differential Mass Balance mass balance:

20. Differential Mass Balance Substituting: Rearranging:

21. Differential Equation of Continuity Dividing everything by ΔV: Taking the limit as ∆x, ∆y and ∆z 0:

22. Differential Equation of Continuity divergence of mass velocity vector (v) Partial differentiation:

23. Differential Equation of Continuity Rearranging: substantial time derivative If fluid is incompressible:

24. Differential Equation of Continuity In cylindrical coordinates: If fluid is incompressible:

25. Outline Time Derivatives & Vector Notation Differential Equations of Continuity Momentum Transfer Equations

26. Differential Equations of Motion

27. Control Volume Fluid is flowing in 3 directions For 1D fluid flow, momentum transport occurs in 3 directions Momentum transport is fully defined by 3 equations of motion

28. Momentum Balance Consider the x-component of the momentum transport:

29. Momentum Balance Due to convective transport:

30. Momentum Balance Due to molecular transport:

31. Momentum Balance Consider the x-component of the momentum transport:

32. Momentum Balance Consider the x-component of the momentum transport:

33. Differential Momentum Balance Substituting:

34. Differential Momentum Balance Dividing everything by ΔV:

35. Differential Equation of Motion Taking the limit as ∆x, ∆y and ∆z 0: Rearranging:

36. Differential Momentum Balance For the convective terms: For the accumulation term:

37. Differential Equation of Motion Substituting:

38. Differential Equation of Motion Substituting: EQUATION OF MOTION FOR THE x-COMPONENT

39. Differential Equation of Motion Substituting: EQUATION OF MOTION FOR THE y-COMPONENT

40. Differential Equation of Motion Substituting: EQUATION OF MOTION FOR THE z-COMPONENT

41. Differential Equation of Motion Substantial time derivatives:

42. Differential Equation of Motion In vector-matrix notation:

43. Differential Equation of Motion • Cauchy momentum equation • Equation of motion for a pure fluid • Valid for any continuous medium (Eulerian) • In order to determine velocity distributions, shear stress must be expressed in terms of velocity gradients and fluid properties (e.g. Newton’s law)

44. Cauchy Stress Tensor Stress distribution:

45. Cauchy Stress Tensor Stokes relations (based on Stokes’ hypothesis)

46. Navier-Stokes Equations

47. Assumptions Newtonian fluid Obeys Stokes’ hypothesis Continuum Isotropic viscosity Constant density Divergence of the stream velocity is zero

48. Navier-Stokes Equations Applying the Stokes relations per component:

49. Navier-Stokes Equations Navier-Stokes equations in rectangular coordinates

50. Cylindrical Coordinates