1. Chapter 2 Reynolds Transport Theorem (RTT) 2.1 The Reynolds Transport Theorem 2.2 Continuity Equation 2.3 The Linear Momentum Equation 2.4 Conservation of Energy

2. 2.1 The Reynolds Transport Theorem (1)

3. 2.1 The Reynolds Transport Theorem (2)

4. 2.1 The Reynolds Transport Theorem (3)  Special Case 1: Steady Flow  Special Case 2: One-Dimensional Flow

5. 2.2 Continuity Equation (1)  An Application: The Continuity Equation

6. 2.3 The Linear Momentum Equation (1) ..

7. 2.3 The Linear Momentum Equation (2)

8. 2.3 The Linear Momentum Equation (3)  Special Cases

9. 2.3 The Linear Momentum Equation (4)

10. 2.4 Conservation of Energy

11. Chapter 3 Flow Kinematics 3.1Conservation of Mass 3.2 Stream Function for Two-Dimensional Incompressible Flow 3.3 Fluid Kinematics 3.4 Momentum Equation

12. 3.1 Conservation of mass Rectangular coordinate system x y z dx dy dz o u v w

13. x y z dx dy dz o u v w

14. x y z dx dy dz o u v w

15. dx dy dz o u v w x y z

16. Net Rate of Mass Flux

17. Net Rate of Mass Flux Rate of mass change inside the control volume

18. Continuity Equation

19. 3.2 Stream Function for Two- Dimensional Incompressible Flow A single mathematical function  (x,y,t) to represent the two velocity components, u(x,y,t) and  (x,y,t). A continuous function  (x,y,t) is defined such that The continuity equation is satisfied exactly

20.  Equation of Streamline Lines drawn in the flow field at a given instant that are tangent to the flow direction at every point in the flow field. Along a streamline

21.  Volume flow rate between streamlines u v x y Flow across AB Along AB, x = constant, and

22.  Volume flow rate between streamlines u v x y Flow across BC, Along BC, y = constant, and

23.  Stream Function for Flow in a Corner Consider a two-dimensional flow field

24.  Motion of a Fluid Element Translation x y z Rotation Angular deformation Linear deformation 3.3 Flow Kinematics

25.  Fluid Translation x y z Fluid particle path At t At t+dt

26.  Scalar component of fluid acceleration

27.  Fluid acceleration in cylindrical coordinates

28.  Fluid Rotation x y a a' b b' o xx yy

29. a a' b b' o xx yy

30. a a' b b' o xx yy Similarily, considering the rotation of pairs of perpendicular line segments in yz and xz planes, one can obtain

31.  Fluid particle angular velocity Vorticity: A measure of fluid element rotation Vorticity in cylindrical coordinates

32.  Fluid Circulation,  c y x o b a Circulation around a close contour =Total vorticity enclosed Around the close contour oacb,

33.  Fluid Angular Deformation x y a a' b b' o xx yy 

34.  Fluid Linear Deformation x y a a' b b' o xx yy

35. a a' b b' o xx yy

37. Rate of shearing strain (Angular deformation)  Rate of Strain Rate of normal strain

38. 3.4 Momentum Equation

39. x y z

40. Forces acting on a fluid particle x y z x-direction + +

41. Forces acting on a fluid particle x-direction + +

42. Components of Forces acting on a fluid element x-direction y-direction z-direction

43. Differential Momentum Equation

44. Momentum Equation:Vector form is treated as a momentum flux

45. Stress and Strain Relation for a Newtonian Fluid Newtonian fluid  viscous stress  rate of shearing strain

46. Surface Forces

47. Momentum Equation:Navier-Stokes Equations

48. Navier-Stokes Equations For flow with  =constant and  =constant

49. 3.5 Conservation of Energy

50.  Summary of Basic Equations