1. GLOBAL CONSERVATION EQUATIONS
Definition of Advective Fluxes
: amount of property crossing an area per unit time A

2. MASS CONSERVATION Control Volume No sources or sinks of mass
Flux in
Flux out
No sources or sinks of mass
Flux in = Flux out or Change of mass = difference between flux in – flux out

3. Change of mass = difference between flux in – flux out
Total amount of fluid (MASS) in the control volume = ?
Total flux into the control volume = ?
If density is constant and volume is fixed, then the sum of fluxes = 0, i.e., the volume is conserved

4. EXAMPLES FOR GLOBAL MASS CONSERVATION
Channel with changing area A y A2
ρ2 U2
Mass Flux: A1
ρ1 U1 x
→ knowing A1, A2 and U1, we can predict U2

5. EXAMPLES FOR GLOBAL MASS CONSERVATIONy
ρ3 U3 A2 U2
ρh Uh A1 x

6. EXAMPLES FOR GLOBAL MASS CONSERVATION
Control Volume from Winant’s notes
Flow in a basin
Mass in basin:
Flux into basin:
Mass conservation:
Can solve for u (x)

7. CONSERVATION OF MOMENTUM
Momentum Theorem
The time rate of change of momentum contained within a fixed control volume plus the net flux of momentum out of the surfaces is equal to the sum of all forces acting on the volume.
Normal (pressure) and tangential (shear) forces

8. EXAMPLES FOR GLOBAL MOMENTUM CONSERVATIONy A2 A1
ρ U2
ρ U1
The net flux of momentum is: x
Ignoring friction, the only forces in ‘x’ are caused by pressure
Over an infinitesimal portion δx, the momentum theorem can be written as:

9. And integrating along x:
divide over volume:
And integrating along x:
Bernoulli’s equation A1 A2
ρ U2
ρ U1

10. EXAMPLES FOR GLOBAL MOMENTUM CONSERVATION
Force exerted by the pipe on the fluid to cause it to bend?
ρ, U and A are constant
If flow is steady, momentum theorem: y θ x

11. EXAMPLES FOR GLOBAL MOMENTUM CONSERVATION
Drag on an object submerged in a flow x y U0 2L 2B
Mass conservation:
Flux of x momentum:
If B is large, u in 2nd integral = U0 and equating the sum of fluxes to the Drag:

12. Mass conservation
Momentum Theorem