The Continuity Principle
by
Eka Oktariyanto Nugroho
Basic Equation 2
Eka O. N.
2.1. CONTINUITY EQUATION
The principle of continuity expresses the conservation of mass in a given space occupied by a fluid.
The simplest, well-known form of the continuity relationship in elementary fluid mechanics
expresses that the discharge for steady flow in a pipe is constant; that is, VA  constant, where A
is the cross-sectional area of the pipe and V is the mean velocity'.
In the case of an incompressible fluid (ρ = constant) in a uniform pipe (A = constant), the continuity
relationship becomes simply: V = constant. If the X axis is taken as the axis of the pipe, then
V  u , and the continuity principle expressed in differential form becomes:
dV u

0
dx x
Since no fluid is being added or subtracted during the motion, the quantity of fluid involved is
constant. This may be expressed mathematically in the case of two dimensional incompressible
motions. The development of the mathematics follows.
Consider a rectangular element in two-dimensional fluid motion as shown in Fig. 2.1. The
rectangular boundaries have sides of length a and b and are considered to be fixed with respect to
the axes. It is not a moving fluid element.
Figure 2. 1 Rectangular element of an incompressible fluid.
The volume of fluid entering the left-hand boundary by unit of time is au1 , and at the same instant,
the amount leaving the right-hand boundary is
is thus:
a  u2  u1   au .
au2 . The difference in amount in the OX direction
Similarly, the difference in amount in the OY direction is:
b  v2  v1   bv
Since the total mass of fluid within the boundaries is constant, the total loss must be zero:
au  bv  0 ;  u b    v a   0 .
In the limit, when b and a approach zero, one obtains
 u x    v y   0 . This differential form
is permitted because of the assumption of a continuous fluid. It should be noted that
u x and
v y are the rates of linear deformation of a fluid particle; hence, in an incompressible fluid, the
total sum of linear deformation is nil, as has been previously noted.
The Continuity Principle
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Basic Equation 2
Eka O. N.
2.2. THE CONTINUITY RELATIONSHIP IN THE GENERAL CASE
w
v
u
C
z
G
H
D
z
u
F
B
E
A
x
x-
2
 u
dx
x
y
x
y
u 
x
x+
x
2
Figure 2. 2 Coordinate system for continuity equation.
Conservation of Mass at a Point in for a Molecularly Averaged Continuum
Mass Can neither Appear nor Disappear
Mass stored in a CV = Mass of fluid into the CV - Mass of fluid out of the CV
By applying a fixed CV (Eulerian analysis) as shown in figure 2.2 with geometries properties of the
CV, result:

Volume of CV  dV  dx dy dz

Area Face ACBD = dy dz

Area Face EFGH = dy dz



Area Face AEHD = dx dz
Area Face BFGC = dx dz
Area Face AEFB = dx dy

Area Face DHGC = dx dy
Assume a hypothetical continuum with properties that average out the molecular scales, as follows
 , describes mass per volume down to a point in space

Density,

Velocities: u, v, w, describe motion of small parcels of fluid with molecular level motions
averaged out
Consider a fixed volume of fluid of which the edges dx, dy , dz are parallel to the axes
OX , OY , OZ respectively (Fig. 2.2). The continuity relationship is obtained by considering that the
change of fluid mass inside the volume dx dy dz during the time dt is equal to the difference
between the rates of influx into and efflux out of the considered volume during the same interval of
time (conservation of mass).
Mass of the CV can be described by :
dM   dV 
The Continuity Principle
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Basic Equation 2
Eka O. N.
The change of mass stored within the CV.
  dM 
t

   dV 
t
Expanding this equation
  dM 
t
 dV
  dV 


t
t
Since the boundary of the control volume is fixed,
  dM 
t
 dV
   dV 
t
 0 , and

t
Density with respect to time
The fluid mass at the time
t is:  dx dy dz
After a time dt , because of the change of density with respect to time, the quantity of fluid mass
becomes
 

dt  dxdydz
 
t 

Hence the change of fluid mass in a time


 dxdydz    
dt is
 

dt  dxdydz  
dtdxdydz
t 
t
On the other hand, if one takes into account the change in velocity and in density with respect to
space coordinates, the quantity of fluid mass entering through the section ABCD during a time dt ,
parallel to the OX axis, is the product  u times the area perpendicular to OX (ABCD) and the
time dt . Since, ABCD = dy dz , the quantity of fluid mass entering is
 u dy dz dt . The derivative
of u along AB and AD with respect to dz and dy is of an infinitely small order and can be
neglected. Now the quantity of fluid mass coming out during the same interval of time through the
section EFGH is:

  u  
dx  dydzdt
 u 

x


Velocities with respect to time
In the general case, both the density p and velocity u are assumed to be changed along
Hence the difference is
dx .

  u  
  u 
dx  dydzdt  udydzdt 
dxdydzdt
 u 
x
x


Similarly, the differences due to the components of motion parallel to the OY and OZ axes are,
respectively,
  v 
dxdydzdt due to the difference of discharge across the sections BFGC and AEHD
y
 dx dz 
The Continuity Principle
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Basic Equation 2
Eka O. N.
   w
z
 dx dy 
dxdydzdt due to the difference of discharge across the sections AEFB and DHGC
The total change of mass contained within the elementary region during the time
dt is
    u     v     w 



 dxdydzdt
y
z 
 x
Equating Density and Velocities
Equating density and velocity equationas yields

   u     v     w 

dxdydzdt  


 dxdydzdt
t

x

y

z


Dividing both sides by dx dy dz dt yields
   u     v     w



0
t
x
y
z
Derivation continuity according to Taylor’s Series
We assume that the variables may be expanded using Taylor series, as shown in Figure 2.2.
Mass flux into face ACBD at x  0 :
 u  dy dz

  u 

x
Mass flux out of face EFGH at x  dx :   u 

Mass flux into face at AEFB z  0 :
2  u 
x
2
 dx 

 ...  dy dz


2
  w dx dy

   w

z
Mass flux out of face DHGC at z  dz :   w 

Mass flux into face at AEHD y  0 :
dx 
dz 
 2   w
z
2
 dz 
2

 ...  dx dy


 v  dx dz

  v

y
Mass flux out of face BFGC at y  dy :   v 

dy 
2  v 
y
2
 dy 
2

 ...  dx dz


Apply the Conservation of Mass Law in a time rate form
Time rate of change of mass stored in a CV = Net rate of mass flux into the CV in the x, y & z directions
dV


  u 
 2  u 

2
   u  dy dz    u 
dx 
dx   ...  dy dz

2


t
x
x




   w
 2   w
2
dz 
dz

...
  w  dx dy    w 


 dx dy
z
z 2




  v 
2  v
2
dy 
dy

...
  v  dx dz    v 


 dx dz
y
y 2


The Continuity Principle
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Basic Equation 2
Eka O. N.
Collecting terms:
  2  u 

  u 
  v 
   w
2  v 
 2   w

dV 
dV 
dV 
dV  
dx

dy

dz  ...  dV  0
2
2
2
 x

t
x
y
z
y
z


Factoring out dV
2

2  v 
 2   w
    u     v     w      u 




dx

dy

dz

...
  0
 x 2
t
x
y
z
y 2
z 2


Letting the CV shrink to a point such that dx0, dy0, dz0 and thus dV0, lead to the
Conservation of Mass Equation at a point space and time
   u     v     w



0
t
x
y
z
Since     u   x 
  u x   u   x  ,
and similarly for the terms
   v  y and
   w z , the continuity relationship becomes
 u v w 




u
v
w
  
0
t
x
y
z
 x y z 
 u v w 
D
  
0
Dt
 x y z 
 du dv dw 

d
d
d
  
v
w
0
u
t
dx
dy
dz
 dx dy dz 

  divV+V  grad  0
t
or
D
   V=0
Dt
Where V  uiˆ  vjˆ  wkˆ and

 ˆ  ˆ  ˆ
i
j k
x
y
z
The first term, Dρ/Dt, is the derivative of the density with time at a given point. This term is nil in the
case of (1) incompressible fluid, since p is a constant and (2) a steady motion of a compressible
fluid. This term has to be considered when sound, water hammer, shock waves, etc. are studied.
2.3. APPLICATION OF MASS CONSERVATION TO AN INCOMPRESSIBLE FLUID
It can be shown that the rate of volumetric dilatation for a fluid is
1 D  dV  u v w

 
0
dV Dt
x y z
Thus for an incompressible fluid, the continuity equation is valid:
u v w
 
0
x y z
The Continuity Principle
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Basic Equation 2
Eka O. N.
This equation describe conservation of volume for the fluid which the CV must remain immersed
within the fluid
In vector notation, the continuity equation at a point is written as:
 V=0
An incompressible fluid does not imply that density is contant; the conservation of mass equation
reduces to
D
=0
Dt
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