1. Today’s Lecture Objectives:
Conservation equation
Mass
Momentum
Energy

2. From the previous classes
Pay attention to the orientation of the coordinate system

3. From the previous class: Continuity equation -conservation of mass
Mass flow in and out of fluid element
Infinitely small volume
Change of Mas =
= Σ(Mass in) - Σ(Mass out)
……………….
General

4. Shear and Normal stress
τyx

5. From the previous class Momentum equation –Newton’s second law
dimensions of
fluid particle
Stress components in x direction
forces
per unit of volume
in direction x
………………..
………………
…………….
total
derivative

6. Momentum equation Sum of all forces in x direction Internal source
y direction
z direction

7. Newtonian fluids
Viscous stress are proportional to the rate of deformation (e)
Elongation:
Shearing deformation:
For incompressible flow
Viscous stress:
viscosity

8. Momentum equations for Newtonian fluids
After substitution:
x direction:
y direction:
z direction:

9. Momentum equations for Newtonian fluids
Same like previous slide with some rearrangements:
x direction:
y direction:
z direction:

10. Momentum equations for Newtonian fluids
Finally:
x direction:
y direction:
z direction:

11. Energy equations for Newtonian fluids
Conservation of energy for the
fluid particle:
Energy
(heat) flux
source
dissipation :

12. Application Example: 2D flow
gravitation
Natural convection at vertical surface - steady boundaries:
-ρg 0
Twall
Boussinesq
approximation ∞
Twall>T∞ 0 0

13. Application Example: 2D flow
Driving force for
natural convection

14. Conservation Equations

15. Turbulence
Forced convection on flat plate