Ghosh - 550
Page 1
2/6/2016
Differential Continuity Equation
Let us derive the mass conservation law (continuity equation) in the differential
approach. Whenever we need to derive an equation in the differential approach, we
shall begin with the control volume form of the equation and apply it on an
infinitesimally small control volume. The Reynold's Transport Theorem for mass
conservation law is reviewed first.
REYNOLDS TRANSPORT THEOREM
The relationship between the rate of change of any arbitrary extensive property, N, of a
system and the variations of the property within a control volume is given by the
following equation, known as the Reynolds Transport Theorem.
dN
dt

SYSTEM

 dV  CS V  d A
t CV
where,
 = Intensive Property
-Property Independent of Mass
N = Extensive Property
-Property Dependent on Mass



 dV
t CV
-Local Rate of Change of N within Control Volume (CV)
CS V  d A
-Convective Rate of Change of N through Control Surface (CS)
CASE 1:
dM
dt
 = 1, N = M

SYSTEM

CV dV  CS V  d A
t
Note: The Conservation of Mass states that the mass of a system is constant, therefore
dM
dt
0
SYSTEM
Ghosh - 550
Page 2
2/6/2016
Derivation
Now, let us derive the mass conservation law for an infinitesimally small control
volume V around the point located at (x, y, z).
Z
y
z
(x,y,z)
x
Y
X
V  x  y  z
CS V  d A  2 x  surfaces V  d A  2 y  surfaces V  d A  2 z  surfaces V  d A
For a trial workout, look at integral term over the 2-y faces:
Z
y
z
(x,y,z)
x
Y
X
Ghosh - 550
V  d A
A
y
y
2
Page 3
y
y
2
2/6/2016
y
y

 

   x, y 
, z , t V  x, y 
, z, t    A
2
2

 

 xz ˆj
V  d A
V  d A
y
2
y
y

 

   x, y 
, z , t v  x , y 
, z , t xz
2
2

 

y
y
2
y
y

 

   x, y 
, z , t v x, y  , z , t xz
2
2

 

y
Using Taylor series expansion of f(x+h), (h<<x)
h2
h3
f ( x  h)  f ( x)  hf ( x) 
f ( x) 
f ( x)  ...
2!
3!
For the blue y face (located at y + h, where):
 All derivatives are taken with respect to y
 h = y/2
y 2
) 2
y
y 
 
2
 ( y  )   ( x, y , z , t ) 
( x, y , z , t ) 
( x, y, z , t )  ...
2
2 y
2! y 2
(
y
y  1  y   2 
( y  )   
  
 ...
2
2 y 2  2  y 2
2


Taylor's Theorem is also used on the y-component of velocity, v (since V  dA.= + or, vxz on them)
V  d A
y
y
2
 V  d A
y
y
2
2 2


  y v 1  y  2  2 v

y  1  y   




...
v



...
 
 


 xz
2
2
2

y
2
2
2

y
2
2

y

y
 
 

 

2

  y v 1  y  2  2 v

y  1  y   2 
  
  

...
v



...
 

 xz
2
2
2

y
2
2
2

y
2
2

y

y
 
 

 

Ghosh - 550
Page 4
2/6/2016
Ghosh - 550
Page 5
2/6/2016
Multiplying terms:
2
2


y v   y   2 v
y  v  y   2 
  v  
  

v


....
 
 xz
2
2
2

y
2
2
2

y
2
2

y

y
 
 


2
2


y v   y   2 v
y  v  y   2 
  v  
  

v


....


 xz
2
2
2

y
2
2
2

y
2
2

y

y






Simplifying:


y v
y 
  2 
 2v
 .... xz
2 y
2 y




v

    v
 .... xzy
y
y


2 y  surfaces V  d A 
 ( v)
xyz
y
From performing the same analysis on the x faces and z faces, the final result for the
convective term is:
 V  d A 
CS
 ( u )
 ( v)
 ( w)
xyz 
xyz 
xyz
x
y
z
The control volume (local) term can be written for the small V by neglecting the
integral sign:



( V ) 
V 
xyz
t
t
t
Putting the results found into the Reynolds Transport Equation for mass:
0


 ( u )
 ( v)
 ( w)
xyz 
xyz 
xyz 
xyz
CV dV  CS V  d A 
t
t
x
y
z
Ghosh - 550
Page 6
2/6/2016
Simplifying the above expression gives the Differential Equation for Conservation of
Mass (Continuity Equation):
  ( u )  ( v)  ( w)



0
t
x
y
z
In this equation, note:
 Independent Variables: x , y, z, t
 Dependent Variables: u, v, w, 
Thus, by itself, it is not a solvable equation since there are 4 unknowns in one equation.

t
- Local Term
- local rate of change of mass through the Control Surfaces per unit volume.
 ( u )  ( v)  ( w)


x
y
z
-
Convective Term
Convective rate of change of Mass through the
Control Surface per unit volume.
Exercise: Interpret Physically:
 ( u ) Answer: Convective rate of change of mass in the x direction
x (between two x-surfaces)
Simplifications of the Continuity Equation:
1. If the flow is steady – i.e., flow doesn’t change with time.
 ( u )  ( v)  ( w)


0
x
y
z
2. For Incompressible Flow – Density,  is Constant.
u v w
 
0
x y y

 V  0
where, the  Operator in Cartesian Coordinates is defined as:

 ˆ  ˆ  ˆ
i
j k
x
y
z
Continue