Inertia Forces
by
Eka Oktariyanto Nugroho
Basic Equation 3
Eka O. N.
3.1.
MASS, INERTIA, AND ACCELERATION
3.1.1.
The Newton Equation
To cause the motion of a constant mass M , or, more generally, to change the state of an existing
motion, it is necessary to apply to this mass a force F , which causes an acceleration dV dt such
that
F  M  dV dt  . This is a vector relationship, i.e., true for both magnitude and direction. The
M  dV dt  is the inertia force, which characterizes the natural resistance of matter to any
product
change in its state of motion.
The considered mass M is the mass of a unit of volume of fluid
M     unit of volume   
 is the density. Hence the fundamental equation of momentum has the
form F    dV dt  . Its three components along the three coordinate axes OX , OY , OZ
where
are 
3.1.2.
 du dt  ,   dv dt  , and   dw dt  , respectively.
Relationships between the Elementary Motions of a Fluid Particle and the Inertia Terms
To each kind of motion of the fluid particles (chapter 1) there corresponds an inertia force. The
relationship between the kind of motion described and the corresponding inertia force is
straightforward.
The elementary components of velocity of a fluid particle as given in chapter 1 are, in the case of a
two-dimensional motion,
Translation
u, v
Dilatational deformation
u
dx
x
Shear deformation
1  u v 
   dy
2  y x 
1  u v 
   dx
2  y x 
Rotation
1  v u 
    dy
2  x y 
1  v u 
   dx
2  x y 
v
dy
y
To each of these velocity components corresponds a component of acceleration, which multiplied
by  , yields a component of inertia force.
Two types of inertia forces may be distinguished, depending on the type of acceleration or
elementary motion considered. These are:
1. Local acceleration-corresponding to a variation of the velocity of translation or the
derivative of velocity with respect to time.
2. Convective acceleration-corresponding to a variation of velocity of deformation and rotation
or derivative of velocity with respect to space.
The physical meaning of these accelerations and the corresponding inertia forces is first examined:
then their mathematical expression is demonstrated. Chapter 4 deals with the applied forces F
which have to be equated to these inertia forces to obtain the momentum equation.
Inertia Forces
Page
- 15
Basic Equation 3
Eka O. N.
3.2.
LOCAL ACCELERATION
Local acceleration characterizes any unsteady motion, i.e., motion where the velocity at a given
point changes with respect to time. Local acceleration results from a change in the translatory
motion of a fluid particle imposed by external forces F.
Mathematical Expression of Local Inertia
The mathematical expression of the inertia forces caused by a local acceleration is given by the
change in the velocity of the translatory motion with respect to time only. The corresponding inertia
  V t 
force is equal to
respectively: 
of which the components along the three axes are,
 u t  ,   v t  , and   w t  . The derivatives with respect to space are not
taken into account.
3.3.
CONVECTIVE ACCELERATION
Convective acceleration characterizes any nonuniform flow, i.e., when the velocity at a given time
changes with respect to distance. It is sometimes called field acceleration. Convective acceleration
results from any linear or angular deformation, or from a change in the rotation of fluid particles,
imposed by external forces F.
3.3.1.
The Case of Linear Deformation
In a convergent pipe, it has been seen that the velocity of a fluid particle, although constant with
time at a fixed location, tends to increase along the converging streamlines. The velocity of the fluid
particle increases with respect to space. This is a positive convective acceleration. The fluid tends
to resist this acceleration by convective inertia.
In a divergent conduit, the velocity decreases and the fluid tends to continue its motion with the
same velocity because of its inertia. The applied forces cause a negative convective acceleration.
Expansion or contraction of a compressible fluid is the sum of linear deformations and also results
in corresponding inertia forces.
It has been seen that the linear deformation velocity components are those given in Equation 3-1.
u
dx
x
v
dy
y
w
dz
z
(3-1)
Two-dimensional motion
Three-dimensional motion
The expressions u x , v
corresponding acceleration is
y and w z are given time, as seen in section 1.3.1 The
d  u  u dx
dx 
dt  x  x dt
Two similar expressions result for
w and u . If u 
dx
dy
dz
, v
, and w 
are substituted in
dt
dt
dt
these expressions and the result is multiplied by the density, the inertia forces are obtained. They
are:
u
 
2
u 1  u
 
x 2
x
 
2
v 1  v
v  
y 2
y
Inertia Forces
Page
- 16
Basic Equation 3
Eka O. N.
w
w 1   w
 
z 2
z
2

It should be notice that the last group of expressions may be written as   x 
 u 2  . This
2
shows that the inertia force is equal to the derivative of the kinetic energy with respect to space
along the three direction axes OX , OY , and OZ , respectively.
3.3.2.
The Case of Shear Deformation
In a bend, where the fluid particles are angularly deformed, the fluid paths are curved and because
of its inertia, the fluid tends to continue along a straight line. This causes a centrifugal force
proportional to the change of direction which is imposed by the applied forces.
It is possible for the velocity of a fluid particle to keep the same magnitude along its path, but with a
change in direction. This is the case of free vortex motion.
It has been seen that the velocity components of angular deformation for a two-dimensional motion
are
1  u v 
   dy
2  y x 
1  u v 
   dx
2  y x 
Hence, as in the previous case, using the substitutions u 
dx
dy
, v
, the corresponding inertia
dt
dt
forces become:
1  u v 
v   
2  y x 
1  u v 
u   
2  y x 
3.3.3.
The Case of a Change of Rotation
In the entrance to a pipe (Fig. 3.1), because of the change in friction forces, there is a variation of
rotation of the fluid particles. Hence there are inertia forces corresponding to the natural resistance
of the fluid to change its rotational motion. In a uniform pipe, the rotation of particles exists but
there is no change in rotational magnitude and the corresponding acceleration is zero.
Figure 3. 1 Zone of acceleration of rotation.
Inertia Forces
Page
- 17
Basic Equation 3
Eka O. N.
As in the two previous cases, since
1  v u 
    dy
2  x y 
1  v u 
   dx
2  x y 
are the velocities of the components of rotation in a two dimensional motion, the corresponding
inertia forces obtained are
1  v u 
 v   
2  x y 
1  v u 
u   
2  x y 
It has been shown that it is possible to assume that the motion is irrotational when friction effects
are negligible. It is evident that the same conditions lead to neglect of rotational inertia forces.
3.4.
GENERAL MATHEMATICAL EXPRESSIONS OF INERTIA FORCES
3.4.1.
Local and Convective Acceleration
In the general case both local acceleration and convective acceleration occur at the same time. A
simple example is when a fluid oscillates in a nonuniform curved pipe. Hence, in the general case,
V and its components u, v, and w are functions of both time and space coordinates. For example,
u  x, y, z, t  . The total differential of u is
du 
u
u
u
u
dt  dx  dy  dz
t
x
y
z
The acceleration in the
x direction is thus given by the total differential of u , with respect to time:
du u u dx u dy u dz




dt t x dt y dt z dt
Similar expressions occur for
dv dt and dw dt .
Substituting u : dxldt, u : dyldt, and w : dzldt, and multiplying by the density p, the inertia forces
given by Equation 3-2 are obtained.
 u
u
u
u 
 u v w 
x
y
z 
 t

 v
v
v
v 
 u v w 
x
y
z 
 t

 w
 t

 u
Local
acceleration
terms
3.4.2.
(3-2)
w
w
w 
v
w

x
y
z 
Convective accleration terms
Elementary Acceleration Components
Following a procedure similar to that used in the study of the elementary motions of fluid particles
(Section 1.5.2), that is, adding and subtracting
1
1
 v  v x  and  w  w x  to the first line
2
2
above, gives Equation 3-3, which emphasize the previous physical considerations. Similar forms
can be obtained for the y and z components of the forces.
Inertia Forces
Page
- 18
Basic Equation 3
Eka O. N.
 u
  t
 u

1  v u  1  u w  1  u w  1  v u   (3-3)
v     w 
  w 
  v   
2  x y  2  z x  2  z x  2  x y  
Acceleration
in linear
deformation
Local
Acceleration
resulting in
a change in
translatory
motion
3.4.3.
u
x
Acceleration in angular deformation
Acceleration in rotation
Separation of Rotational Terms
It is often useful to transform the acceleration terms to a form which emphasizes both the kinetic
energy terms and the rotational terms. Adding and subtracting
 v  v x   w  w x   to
the
first line, gives the following expression, valid along the OX axis:
 u  u
v
w   u v 
 u w  
  u  v  w   v     w 

x
x   y x 
 z x  
 t  x

But
u
u
v
w 1  2 2
 V2
v w

u  v  w2 
x
x
x 2 x
x 2


When the coefficients of the rotational vector
 u w 
2   

 z x 
 v u 
2    
 x y 
Are introduced, the following expression for the inertia forces along the OX axis results
 u   V 2 

      2  w   v  
 t x  2 

Similarly, it may be found that the inertia forces along the OY and OZ axes are
 v

 t
 w

 t


 V 2 

  2  u   w  
y  2 



 V 2 

  2  v   u  
z  2 

These three expressions may be written more concisely in vector notation as shown in Eq. 3-4.

V

t
Local
acceleration
grad
V2
2
Kinetic energy
term
 curl V x V 
(3.4)
Rotational
term
Convective acceleration
 grad
  V 2 
V2
 V 2 
  V 2 
  i 

j

k





2
z  2  
 x  2  y  2 
It has to be noticed that the convective inertia term is, in fact, the derivative with respect to space of the
kinetic energy,
Inertia Forces
 V 2 2 , of the particle.
Page
- 19