Continuity Equation

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 decreasein 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. 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. 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. 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

46. since Is equal to zero

48. CONSERVATION FORM:-

49. SIMPLICATION OF NAVIER STOKES EQUATION Ifis constant