2. Continuity Equation

3. Continuity Equation
Net outflow in x direction

4. Continuity Equation
net out flow in y direction,

5. Continuity Equation
Net out flow in z direction

6. Net mass flow out of the element

7. Continuity Equation Time rate of mass decrease in the element
Net mass flow out of the element =
Time rate of mass decrease in the control volume

8. The above equation is a partial differential equation form of the continuity equation. Since the element is fixed in space, this form of equation is called conservation form.

9. If the density is constant

10. This is the continuity equation for incompressible fluid

12. [NAVIER STOKES EQUATION]
MOMENTUM EQUATION
[NAVIER STOKES EQUATION]
Momentum equation is derived from the fundamental physical principle of
Newton second law
Fx = m a = Fg + Fp + Fv
Fg is the gravity force
Fp is the pressure force
Fv is the viscous force
Since force is a vectar, all these forces will have three components.   
First we will go one component by next component than
we will assemble all the components to get full Navier – Stokes Equation.

13. Fx – Inertial Force Inertial Force = Mass X Acceleration derivative.
Fx – Inertial Force
Inertial Force = Mass X Acceleration derivative.
Inertial Force in x direction = m X
represents instantaneous time rate of change of velocity of the fluid element as it moves through point through space.

14. Is called Material derivative or Substantial derivative or
Acceleration derivative
‘u’ is variable
Inertial force per unit volume in x direction =

15. Inertial force / volume in x direction
Inertial force / volume in y direction
Inertial force / volume in z direction

16. Body force per unit volume
Body forces act directly on the volumetric mass of the fluid element.
The examples for the body forces are
Eg: gravitational
Electric
Magnetic forces.
Body force =
Body force in y direction
Body force in z direction

17. Pressure forces per unit volume
Pressure on left hand face of the element
Pressure on right hand face of the element
Net pressure force in X direction is
Net pressure force per unit volume in X direction

18. Net pressure force per unit volume in X direction
Net pressure force per unit volume in Y direction
Net pressure force per unit volume in Z direction
Net pressure force in all direction
Net pressure force in 3 direction

19. Viscous forces

20. Resolving in the X direction
Net viscous forces

21. Net viscous force per unit volume in X direction
Net viscous force per unit volume in Y direction
Net viscous force per unit volume in Z direction

22. UNDERSTANDING VISCOUS STRESSES

31. LINEAR STRESSES = ELASTIC CONSTANT X STRAIN RATE

32. Linear strain in X direction
Volumetric strain

33. Three dimensional form of Newton’s law of viscosity for compressible flows involves two constants of proportionality.
1. dynamic viscosity.
2. relate stresses to volumetric deformation.

34. In this the second component is negligible
[ Effect of viscosity ‘ ’ is small in practice.
For gases a good working approximation can be obtained taking
Liquids are incompressible. div V = 0]

35. SHEAR STRESSES = ELASTIC
CONSTANT X STRAIN RATE

38. Having derived equations for inertial force per unit volume, pressure force per unit volume body force per unit volume, and viscous force per unit volume now it is time to assemble together the subcomponents.

39. Assembly of all the components
X direction:-
Y direction:-
Z direction:-

40. X direction:-

41. Y direction:-

42. Z direction:-

43. +

44. Divergence of the product of scalar times a vector.
CONVERTING NON CONSERVATION FORM ON
N-S EQUATION TO CONSERVATION FORM
 Navier-stokes equation in the X direction is given by
Divergence of the product of scalar times a vector.

45. Taking RHS of N-S Equation we have
Taking RHS of N-S Equation we have

46. since
Is equal to zero

48. CONSERVATION FORM:-

49. SIMPLICATION OF NAVIER STOKES EQUATION
If is constant

52. For Incompressible flow

53. Energy Equation Energy is not a vector
So we will be having only one energy equation which includes the energy in all the direction.
The rate of Energy = Force X velocity
Energy equation can be got by multiplying the momentum equation with the corresponding component of velocity

54. dQ = dE + dW
dE = dQ - dW = dQ + dW [Work done is negative] because work is done on the system.
Work done is given by dot product of viscous force and velocity vector.
for Xdirection

55. for Y direction
for Z direction

56. Body force is given by

58. Total work done
Net Heat flux into element = Volumetric Heating + Heat transfer across surface.
Volumetric heating

59. Heat transfer in X direction=
Heating of fluid element

60. dQ = B =
dQ = B

62. Energy Equation Nonconservation form

63. Non conservation:-

64. Conservation:-

65. Momentum Equation Non conservation form X direction Y direction
X direction
Y direction
Z direction

66. Momentum Equation Conservation form
X direction
Y direction
Z direction

67. Energy Equation
   Non conservation form

68. Energy equation
Conservation form

69. FORMS OF THE GOVERNING EQUATIONS PARTICULARLY SUITED FOR CFD

70. Solution vectar

71. Variation in x direction

72. Variation in y direction

73. Variation in z direction

74. Source vectar

75. Time marching Types of time marching 1. Implicite time marching
2. Explicite time marching

76. Explicit FDM

77. Implicit FDM

78. Crank-Nicolson FDM

79. Space marching