1. Introduction to Fluid Mechanics
Chapter 5
Introduction to Differential Analysis of Fluid Motion

2. Main Topics قانون بقای جرم تابع جریان برای جریان دو بعدی تراکم ناپذیر
سینماتیک
معادله مومنتوم

3. قانون بقای جرم:
Basic Law for a System

4. Conservation of Mass Differential CV and Taylor series
Infinitesimal control volume of dimensions dx, dy, dz
Area of right face = dy dz
Mass flow rate through the right face of the control volume

5. Net mass flow rate into CV:
Net mass flow rate out of CV:

6. Or, if we apply the definition of the divergence of a vector
After substitution,
Dividing through by volume dxdydz
Or, if we apply the definition of the divergence of a vector

7. Conservation of Mass
Rectangular Coordinate System

8. Conservation of Mass
Rectangular Coordinate System
“Del” Operator

9. Conservation of Mass
Rectangular Coordinate System

10. Conservation of Mass Cylindrical coordinates

11. Conservation of Mass Special Cases
Steady compressible flow
Cartesian
Cylindrical

12. Conservation of Mass Rectangular Coordinate System
Incompressible Fluid:
Steady Flow:

13. EXE :A two-dimensional converging duct is being designed for a high-speed wind tunnel. The bottom wall of the duct is to be flat and horizontal, and the top wall is to be curved in such a way that the axial wind speed u increases approximately linearly from u1 " 100 m/s at section (1) to u2 " 300 m/s at section (2) .Meanwhile, the air density ρ is to decrease approximately
linearly from ρ1 " 1.2 kg/m3 TO ρ2 " The converging duct is 2.0 m long and is 2.0 m high at section
(1). (a) Predict the y-component of velocity, v(x, y), in the duct. (b) Plot the
approximate shape of the duct, ignoring friction on the walls. (c) How high
should the duct be at section (2), the exit of the duct?

16. EXE 2:

17. Conservation of Mass
Cylindrical Coordinate System

18. Conservation of Mass
Cylindrical Coordinate System

19. Conservation of Mass
Cylindrical Coordinate System
“Del” Operator

20. Conservation of Mass
Cylindrical Coordinate System

21. Conservation of Mass Cylindrical Coordinate System
Incompressible Fluid:
Steady Flow:

22. تابع جریان برای جریان تراکم ناپذیر 2 بعدی
Two-Dimensional Flow
یاد آوری
خطوط جریان
دو مولفه سرعت را ببریم
در دل مفهوم ψ
Stream Function y

23. به درد به دست آوردن دبی حجمی
در طول خط جریان
تابع ψ در طول خط جریان ثابت است
Hence we can specify individual streamlines by their stream function values: ψ 5 0, 1, 2, etc. What is the significance of
the ψ values?
به درد به دست آوردن دبی حجمی

24. the flow rate across AB is

25. Stream Function for Two-Dimensional Incompressible Flow
Cylindrical Coordinates
Stream Function y(r,q)

27. Motion of a Fluid Particle (Kinematics)
Fluid Translation: Acceleration of a Fluid Particle in a Velocity Field
Fluid Rotation
Fluid Deformation
Angular Deformation
Linear Deformation

28. کدام یک از حرکت ها تولید تنش برشی میکند؟

29. y t1 t0
particle path x

30. Fluid Translation: Acceleration of a Fluid Particle in a Velocity Field
مشتق اصلی یا مشتق مادی

31. شتاب جابه جایی
شتاب محلی

32. Motion of a Fluid Particle (Kinematics)
Fluid Translation: Acceleration of a Fluid Particle in a Velocity Field

33. Fluid Translation: Acceleration of a Fluid Particle in a Velocity Field (Cylindrical)

34. چرخش سیال Fluid Rotation کی چرخش داریم؟
u3(x1)
u1(x3)
کی چرخش داریم؟
باید گشتاور داشته باشیم. نیروهای عمود بر سطح باعث شتاب و تغییر شکل طولی میشن و گشتاور ایجاد نمیکنند. نیروها وتنشهای برشی لازمه که فقط برای جریان ویسکوز معنی میده x1

35. Rotation Definition of rotation b Time = t + dt Time=t y Dy a Dx x
Concept of rotation: Because of shear in the fluid, during flow, an element may not only get translated, but also ‘rotated’. The rotation is given by the definition in this page. Dx x
Assume Vy|x < Vy|x+Dx
and Vx|y > Vx|y+Dy

36. Fluid Rotation

37. Ψ=0 if x=0 , 180 or r=R

40. Motion of a Fluid Particle (Kinematics)
Fluid Deformation:
Angular Deformation

41. Motion of a Fluid Particle (Kinematics)
Fluid Deformation:
Angular Deformation

42. Motion of a Fluid Particle (Kinematics)
Fluid Deformation:
Linear Deformation

43. Momentum Equation
Newton’s Second Law

44. Momentum Equation
Forces Acting on a Fluid Particle

45. Momentum Equation
Forces Acting on a Fluid Particle

46. Momentum Equation
Differential Momentum Equation

47. Momentum Equation
Newtonian Fluid: Navier-Stokes Equations

48. Momentum Equation
Special Case: Euler’s Equation

49. Computational Fluid Dynamics
Some Applications

50. Computational Fluid Dynamics
Discretization