1. Equations of Continuity

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

3. Infinitesimal pieces of fluid
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
Fluid is considered to be a continuous medium (or continuum) made up of very small individual pieces of the fluid substance that experiences deformation and displacement (by translation and rotation) during the flow. Each infinitesimal piece of fluid material is considered as a moving “particle”, whose position and/or velocity must be mathematically defined using the principles of kinematics. However, because there are an infinite number of “particles” present in a flowing fluid, a system must be developed to describe the fluid motion.
Continuous
medium
Infinitesimal pieces of fluid

4. 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)
Lagrangian Perspective – the flow as seen by the fluid material (the perspective of the laws of motion)
Fluid system: finite piece of the fluid material (Lagrangian)
Control volume: finite fixed region of space (Eulerian)
Fluid particle: differentially small finite piece of the fluid material (Lagrangian)
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.
In the Lagrangian description of fluid flow, individual fluid particles are "marked," and their positions, velocities, etc. are described as a function of time. In the example shown, particles A and B have been identified. Position vectors and velocity vectors are shown at one instant of time for each of these marked particles. As the particles move in the flow field, their positions and velocities change with time, as seen in the animated diagram. The physical laws, such as Newton's laws and conservation of mass and energy, apply directly to each particle. 

6. Lagrangian Perspectivez
Lagrangian coordinate system
pathline
position vector
The position of each particle (a finite parcel of the fluid) is defined (hence the subscript label p) and tracked. 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
Because the particle is a “fluid”, it is subject to deformation during displacement.

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
Also called the Lagrangian derivative, material derivative, or particle derivative, the velocity terms follow the path of motion of the fluid particle.
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 flow Motion of a fluid as a continuumz
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).
In the Eulerian perspective, the system is a fixed volume in space where the fluid material flows, rather than following the property changes in a small piece of the flowing fluid. y x

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

15. Eulerian Perspective Difference from the Lagrangian approach: Eulerian
In the Eulerian perspective, since the infinitesimal control volume is at a fixed location, the analysis is no longer dependent on just one specific fluid element; instead, any changes in the property defining the control volume would be attributed to the difference between the properties of the incoming and outgoing fluid material, as well as other external factors that may interact with the system.
Also, since the control volume is fixed, deformation occurs without a net change in volume.

16. Eulerian Perspective Difference from the Lagrangian approach: Eulerian

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

18. Equation of Continuity
differential control volume:

19. Differential 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
Recall: a unidirectional flow induces momentum transport in 3 directions (2 normal to the direction of the flow + 1 parallel to the direction of the flow)
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

51. Applications of Navier-Stokes Equations

52. Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
Steady state flow

53. Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
Unidirectional flow

54. Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
Unidirectional flow

55. Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
Constant fluid properties
Isotropy (independent of position/direction)
Independent with temperature and pressure

56. Application Euler’s Equation
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
No viscous dissipation (INVISCID FLOW)
Euler’s Equation

57. Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
No external forces acting on the system

58. Application
The Navier-Stokes equations may be reduced using the following simplifying assumptions:
Laminar flow

59. Example
Derive the equation giving the velocity distribution at steady state for laminar flow of a constant-density fluid with constant viscosity which is flowing between two flat and parallel plates. The velocity profile desired is at a point far from the inlet or outlet of the channel. The two plates will be considered to be fixed and of infinite width, with flow driven by the pressure gradient in the x-direction.
Then afterwards, the solution may be applied to other configurations previously discussed during shell balance calculations.

60. Example
Derive the equation giving the velocity distribution at steady state for laminar, downward flow in a vertical pipe of a constant-density fluid with constant viscosity which is flowing between two flat and parallel plates.
Then afterwards, the solution may be applied to other configurations previously discussed during shell balance calculations.