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

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

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

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

EXAMPLES FOR GLOBAL MASS CONSERVATION y ρ3 U3 A2 U2 ρh Uh A1 x

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)

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

EXAMPLES FOR GLOBAL MOMENTUM CONSERVATION y 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:

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

EXAMPLES FOR GLOBAL MOMENTUM CONSERVATION Force exerted by the pipe on the fluid to cause it to bend? http://www.sterlingengineering.co.uk/folding-pipe-bending.php ρ, U and A are constant If flow is steady, momentum theorem: y θ x

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:

Mass conservation Momentum Theorem