1. Today’s Lecture Objectives:
Review (Learn) fluid dynamics
Conservation equation
Mass
Momentum
Energy

2. Terminology Fluid particle Scalar vs. Vector Boundary conditions
Steady State vs. Unsteady State
Stream lines
Gradient Lines
Incompressible vs. Compressible fluids ……
Examples

3. Stream Lines and Gradient lines
Wind
Gradient Lines
Temperature
Jet airspeed

4. Steady-state vs. Unsteady-stat

5. Conservation equations
Fluid Dynamics:
Conservation equations

6. Fluid particle vs Control Volume
Pay attention to the orientation of the coordinate system

7. Important operations
Total derivative for fluid particle which is moving: V z
any scalar y
Vector and scalar operators: x
scalar
vector

8. Total Derivative
Shows how variable  change during time while traveling from A to B
Change in  are introduced by changing boundary conditions over
time and by moving from part A to B B  V z A
mathematical definition y
Total derivative allows all arguments vary x
Physical interpretation …

9. Gradient of a scalar and Divergence of a vector
Where is the gradient of
temperature largest?
- nabla operator
Divergence of a vector
Vector filed
You can also write that

10. Notation
Density () in our problem change is so small that we can assume constant (most of the time) Book (handouts) vs. Class notes We are going to use these interchangeably
Vx ≡ u
Vy ≡ v
Vz ≡ w

11. Continuity equation -conservation of mass
Mass flow in and out of fluid element
Infinitely small volume
Volume V = δxδyδz
Volume sides:
Ax = δyδz
Ay = δxδz
Az = δxδy
Change of density in volume =
= Σ(Mass in) - Σ(Mass out)
……………….
same
See equation 2.2
in notes
= ….

12. Shear and Normal stress
τyx

13. 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

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

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

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

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

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