KINDS OF MOTION
by
Eka Oktariyanto Nugroho
Basic Equation 1
Eka O. N.
1.1.
KINDS OF MOTION
In mathematical terms, the motion of the fluid elements along their own paths is considered as the
superposition of different kinds of primary motions. The physical interpretation of these motions is
given first by considering the simple case of a two-dimensional fluid element, where all velocities
are parallel to the OX axis and depend only on y (like a laminar flow between two parallel planes).
Consider the infinitesimal square element ABCD of area
dx dy at time t and the same element at
time t  dt : A1B1C1D1 (Fig. 1.1).
The velocity of A and D is u , and the velocity of B and C is
u  u  u  u y dy since
AB  dy and u in this case is a function of y only.
It is possible to go from ABCD to A1B1C1D1 in three successive steps.
1. A translatory motion which gives A1B2C2D1; the speed of translation is
u
2. A rotational motion which turns the diagonals A1C2 and D1B2 to A1C3 and D1B3, respectively
3. A deformation which displaces C3 to C1 and B3 to B1.
Figure 1. 1 Elementary analysis of different kinds of motion of a fluid particle.
If in the limit dt tends to zero, C1C2 tends to zero. If this occurs the angle C2C1C3 will tend to 450
when dx  dy . Hence
C 2 C3 
C1C 2
2

u
y dydt
2
The rate of angular rotation is:
dr d  segment  d C2C3 d C2C3
, by using trigonometric rule’s
 


dt dt  radius  dt AC
dt 2dy
1 2
Introducing the value C2C3 previously given, it is found that the rate of angular rotation is:
dr 1 u

dt 2 y
Similarly, the rate of deformation would be found to be equal to:
  C3C1  1 u


t  A1C3  2 y
In the general case, there are three major constituents of particle motions and deformations. They
are:
1. The velocity components V
u, v, w : translation
2. The variation of velocity components in their own direction: dilatation
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Basic Equation 1
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3. The variation of velocity components with respect to a direction normal to their own direction:
rotation and angular deformation
These three constituents are successively discussed in the following sections.
1.2.
TRANSLATORY MOTION
Ax, y, z  at time t .The point is a corner of a small rectangular
element, the edges of which are parallel to the three axes OX , OY , OZ respectively (Fig.1.2).
Consider the particle at the point
When the particle moves so that the edges of the rectangular elements remain parallel to these
axes, and maintain a constant length, it is a translatory motion only. This implies no space
dependence of the velocity components. The translation can be along a straight line or a curved
line. If x, y, and z are the coordinates of A at time t , then x  x , y  y and z  z are the
coordinates at time t  t . The translatory motion is defined by the equations.
x  ut
dx  udt
y  vt or dy  vdt
z  wt
dz  wdt
The flow of particles along parallel and straight streamlines with a constant velocity (so-called
uniform flow) is a case of translatory motion only (Fig. 1.3). The translatory motion may be defined
more rigorously as the motion of the center of the rectangular element instead of the motion of the
corner of the element. However, this change complicates slightly the development of figures and
equations and gives, finally, the same result. Hence in the following discussion, translatory motion
will be defined as the motion of a corner. In the following, the physical meanings and the
corresponding mathematical expressions are studied in the case of a two-dimensional motion at
first, and then they are generalized for a three-dimensional motion.
Figure 1. 2 Translatory Motion.
Figure 1. 3 An example of translatory motion: uniform flow.
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Basic Equation 1
Eka O. N.
1.3.
DEFORMATION
It is easier to explain this kind of motion with the aid of an example. Two kinds of deformation have
to be distinguished: dilatational deformation and angular deformation.
1.3.1. Dilatational or Linear Deformation
In a converging flow, the velocity has a tendency to increase along the paths of particles.
Therefore, the velocities of the edges perpendicular to vector V (or to the streamlines) are not the
same (Fig. 1.4). The particle becomes longer and thinner. It is a case of dilatational or linear
deformation superimposed on a translation only provided the angles between the edges do not
change.
Figure 1. 4 Dilatational deformation of fluid particle in a convergent.
Now consider the two-dimensional particle ABCD of which the velocity in the


x direction of the
edge AB is u , and the velocity of the edge CD is u  du  u  u x dx , since AD  dx (Fig.
1.5). Similarly, the velocity of AD in the, y direction is v , and the velocity of BC
is: v  dv  v 
v y dy .
u with y or v with x are not
considered and the derivatives of velocity u x dx and v y dy , do not depend upon time.
The velocities of dilatational deformations are u x dx and v y dy . After a time dt and, BC
Note that here the derivatives of
becomes B'C', the length BB' being equal to the product of the change of velocity and the time, that
is, BB  v y dydt . [The velocity v y dy is negative in the case of Fig. 1.5] CD becomes



C'D' and similarly DD' is equal to:

DD  u xdxdt .
The velocities of dilatational deformation are per unit length:
u
x dx u v y dy v

,

dx
x
dy
y
The sum u x  v y is the total rate of dilatational deformation, i.e., the rate of change of area
per unit area. Areas BCEB' and D'C'ED must be equal in the case of an incompressible fluid. Their
difference gives the rate of expansion or compression in the case of a compressible fluid.
Figure 1. 5 Components of dilatational deformation.
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Basic Equation 1
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Figure 1. 6 Shear deformation in a bend.
1.3.2. Angular Deformation or Shear Strain
Angular deformation may be illustrated by the behavior of a fluid particle flowing without friction
around a bend. It is a matter of common observation that it is windier at a corner than it is in the
middle of a street. In the similar case of fluid flow around a bend, neglecting the effects of friction,
the velocity has a tendency to be greater on the inside than it is on the outside of the bend. The law
V x R = constant may be approximately applied where V is the velocity and R is the radius of
curvature of the paths. Hence, if the particle A is the corner of the rectangle ABCD, the edge AB of
the rectangle moves at a greater velocity than the edge CD and there is angular deformation (Fig.
1.6). This angular deformation is proportional to the difference of velocity between AB and CD.
Now considering, for example, the case presented in Fig. 1.7, in which the velocity of AB is u , and
the velocity of CD is u  du  u  u y dy , then the distance CC' (or DD') after a time dt is

u y dydt .The angular velocity is
u y dy  u
dy

y
Note that in contrast to the case of dilatational deformation. the derivatives of
u with y ,and v with
x are retained here. The derivative of the velocity u y dy does not depend upon time.
Similarly BB' (or DD") is equal to v x dxdt . When these two deformations exist at the same
time, the sum of the angular velocities u y  v x is the rare of angular deformation.
Note that u y was chosen equal to v x in Figure 1.7, and the bisectors of the angles made
by the edges of the square element tend to remain parallel to their initial positions during the
angular deformation. When the bisectors do not remain parallel to their initial positions, the motion
is said to be rotational.
Figure 1. 7 Angular or shear deformation.
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Basic Equation 1
Eka O. N.
1.4.
ROTATION
Although flow motions can be classified in various ways according to some of their typical
characteristics (such as laminar or turbulent, frictionless or viscous, with or without friction, steady,
or unsteady), one of the most important divisions in hydrodynamics consists of considering whether
a flow is rotational or irrotational. Hence, the abstract concept of irrotationality is fully developed in
the following sections.
For a two-dimensional motion, it has been shown that the angular velocities of deformation are
u y and v x . The rotation of a particle is proportional to the difference between these
components. Indeed, if u
y  v x , there is angular deformation without rotation and the
bisectors do not rotate (Fig. 1.7). But if u y  v x , the bisectors change their direction, and
there is either both rotation and angular deformation, or rotation only (Fig. 1.8). The difference
u y  v x defines the rate of rotation, and therefore, a two-dimensional irrotational motion is
u y  v x  0 . Angular deformation can be considered without
rotation when u y  v x  0 and u y  v x  0 , and theoretically, rotation can exist
without deformation when u y  v x  0 and u y  v x  0 . This case is rare in
defined mathematically by
practice, since rotation generally occurs with angular deformation in physical situations.
Figure 1. 8 Rotation and deformation.
1.5.
MATHEMATICAL EXPRESSIONS DEFINING THE MOTION OF A FLUID
PARTICLE
1.5.1. Two -Dimensional Motion
u and v are functions
of x and y such that du   u x  dx   u y  dy and dv   u x  dx   v y  dy . At time
t the space coordinates of A are x , y and of D are x  dx , y  dy . The coordinates of A and D at time
t  dt are qiven in Equation 1-1
Consider a fluid element ABCD at time t (Fig. 1.9). The velocity components
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Figure 1. 9 Two-dimensional coordinate system.
 x  udt
A'
 y  vdt
 x  dx   u  du  dt
D'
 y  dy   v  dv  dt
(1.1)
Or

 u
u 
 x  dx  udt   dx  dy  dt
y 

 x
D '
 y  dy  vdt   v dx  v dy  dt



y 
 x

1
1
Adding and subtracting
 v x  dydt to the x coordinate and  u y  dxdt to the
2
2
y coordinate leads to the form for the coordinates of D' expressed in Equation 1-2. The physical
meaning of the terms becomes apparent by reference to the previous paragraphs.

u
1  u v 
1  v u 
 x  dx  udt  dxdt     dydt     dydt
x
2  y x 
2  x y 


v
1  u v 
1  v u 
D '  y  dy  vdt  dydt     dxdt     dxdt
y
2  y x 
2  x y 
Translation
 Initial
coordinates

Dilatational
Rate of angular
Rate of
or linear
deformation
rotation

deformation

Angular or shear deformation
Rotation
Kinds of Motion
(1-2)
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Basic Equation 1
Eka O. N.
1.5.2. Three-Dimensional Motion: Definition of the Vorticity
Similar to the two-dimensional case, the coordinates of a point
three-dimensional fluid element after a time
D '  x  dx, y  dy, z  dz  of a
dt become Equation 1-3.
 u
u
u 
x  dx  udt   dx  dy  dz  dt
y
z 
 x
 v
v
v 
y  dy  vdt   dx  dy  dz  dt
y
z 
 x
(1-3)
 w
w
w 
z  dz  wdt  
dx 
dy 
dz  dt
y
z 
 x
Adding and subtracting
1 v
1 w
dydt and
dzdt to the first line
2 x
2 x
1 w
1 u
dzdt and
dxdt to the second line
2 y
2 y
1 u
1 v
dxdt and
dydt to the third line
2 z
2 z
Leads to equation 1-4
The following notations will be used: The coefficients of dilatational deformation will be defined as
a
u
x
b
v
y
c
w
z
x  dx  udt 
 1  v u 
u
1  u w 
1  u w 
1  v y  
dxdt      dy   
 dz   
 dz     dy  dt
x
2  z x 
2  z x 
2  x y  
 2  x y 
y  dy  vdt 
 1  w v 
v
1  v u 
1  v u 
1  w v  
dydt   
  dz     dx     dx  
  dz  dt
y
2  x y 
2  x y 
2  y z  
 2  y z 
z  dz  wdt 
 1  u w 
w
1  w v 
1  w v 
1  u w  
dzdt    
  dy  
  dy   
 dx  
 dx  dt
z
2  y z 
2  y z 
2  z x  
 2  z x 
(1-4)
The coefficients of shear deformation will be defined as
1  w v 
f  
 
2  y z 
1  u w 
g  

2  z x 
1  v u 
h   
2  x y 
The coefficients of rotation will be defined as
1  w
v 
   
2  y z 
1  u
w 
   
2  z x 
1  v
u 
    
2  x y 
The coordinates of the point D’ are now written as equation 1-5, in which 2 , 2 , and 2 are the
components of a vector which represents the vorticity of the fluid at a point. A three-dimensional
irrotational motion is defined by   0,   0, and   0 ; that is,
w v u w v u

,
 ,

y z z x x y
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Basic Equation 1
Eka O. N.
x  dx  udt
 adxdt   hdy  gdz  dt   dz   dy  dt
y  dy  vdt
 bdydt   fdz  hdx  dt   dx   dz  dt
z  dz  wdt  cdzdt   gdx  fdy  dt   dy   dx  dt
Initial
coordinates
Translational
Kinds of Motion
Dilatational
deformation
Angular
deformation
(1-5)
Rotation
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