ME 362
Conservation Laws
Page 1 of 2
Conservation of Mass:
dm
  min   mout
dt CV
Rate of change of mass in C.V. = Mass flow rate into C.V. – Mass flow rate out of C.V.
n̂


m    V  nˆ dA   AVn
Mass flow rate:
A
dA
V
Vn  V  nˆ
where A is the surface area of C.V. that flow passing through
Vn is the component of flow velocity ( V ) normal to area A
Conservation of Momentum:
 
d mV
     mV 
  mV
dt
in
out
F
CV
where
F  F
applied
 Fpressure  Fshear  Fgravity
Rate of change of momentum in C.V. = Momentum flow rate into C.V. – Momentum
flow rate out of C.V. + Forces applied to C.V. (e.g., gravity force, friction force etc.)
Momentum flow rate:
mV
Note: m is a scalar (magnitude) and V is a vector (magnitude and direction)
x-momentum flow rate = m    u 

y-momentum flow rate = m    v 
+ or – depends on the flow direction
and the coordinate system
For example,
V
Mass flow rate: m   AV
x-mom flow rate = mVx  m V cos 
Vx
Vy

y-mom flow rate = mVy  m  V sin  
y
x
Note: the velocity is in the negative
y-direction
ME 362
Conservation Laws
Page 2 of 2
Assumptions:

Incompressible


applies to liquid flows or low speed gas flows
 and  are constant

Steady flow

d 
 0 (properties inside the C.V. do not change in time)
dt

Neglect gravity

Fgravity  0

Neglect viscous

Fshear  0

Neglect pressure 
Fpressure  0
(applicable when entire C.V. is exposed to
atmospheric pressure only)
Apply above assumptions to the governing equations of mass and momentum yields,
Conservation of mass:
 m in   m out



 V ) out  (m
 V ) in
Conservation of momentum:  F  (m