Chapter II. Continued
Pathlines, Streamlines, Voriticity and Bernoulli
Some definitions:
Pathline: A pathline is simply the trajectory of a single fluid particle over a period of
time. We should think of it as a time-exposure on a slow film. In unsteady flows, the
pathlines of two particles starting at the same initial point may differ, depending on the
instance in time when they started. Think of cars travelling on a busy multi-lane highway.
Even if many cars are on the same lane one behind the other, their individual trajectories
will be different. Some of them zig-zag, others will travel straight, etc. Pathlines are like
the trajectories of the individual cars.
Streamline: A streamline is any curve in space, whose tangent at any point on the curve is
in the direction of the velocity vector. As in the case of pathlines, in unsteady flows,
streamlines may change in shape with time.
In steady flows, pathlines and streamlines are one and the same and become railroad
tracks fluid particles travel on, always on the same track. Whether we are tracking a group
of particles (multiple cars in a train) or a single particle (single car in a train) they will all
travel on the same path. At any instance in time, the velocity vector will be tangential to
the track.
In this course we are primarily concerned about steady flows. Therefore, we can use the
phrase "pathline" or "streamline" interchangeably.
Consider an infinitesimal segment of a streamline, bounded by two points in space,
separated by distances dx, dy and dz in along the three axes, respectively. Then, we can
assign a vector associated with this line segment:




ds  dxi  dyj  dzk


Since the velocity vector V is tangential to the streamline, the cross product of V and

ds will be zero.

 
i
j k
 
V  ds  u v w  0
dx dy dz
Or ,
udy  vdx  0
udz  wdx  0
vdz  wdy  0
(1)
The above vector equation (or the three components) holds at any points on the
streamline.
Vorticity: Vorticity is simply twice the angular velocity of a fluid particle. Since the
particle can rotate about any of the three axes x-, y- or z-, vorticity is prescribed by
specifying three components. In another words, vorticity is a vector, just as velocity is a

vector. It is given the symbol  in our text.
To define vorticity in mathematical terms, we need to think about how we will measure
the angular velocity of a fluid element. To simplify matters, let us assume that this
element has an ice-cube shape to start with. We will also initially worry only about its
rotation with respect to the z- axis.
Finding angular velocity of solid objects (e.g. a coin or a disk, or a piece of stamp) is
easy. We will draw a line on the body, and see how much the line has rotated over a small
period of time. For a rectangular element, we can track one of the edges AB and see how
much the edge rotates over a period of time. The angle of rotation divided by the time
elapsed is the angular velocity.
B
A
B
A
A solid body retains is shape.
We can determine how much it rotates by looking at a single edge,
or examining how much a line drawn on the the object rotates.
Fluid particles deform with time. A cube of fluid will not remain a cube for long, as it is
subjected to all kinds of forces (pressure, shear, etc.) Like a soft piece of cheese, it will
deform with time. We can not therefore look at just one edge of the fluid element
(initially a cube) and tell anything about how much the particle has rotated. We should
evaluate the angular velocity of at least two edges, preferably perpendicular to each other.
C
C'
B'
A
B
A'
The fluid element has deformed. The two edges have rotated by different amounts,
and even different directions. We need to average the rotational velocity of AB
and AC.
Let us consider the edge AB first. Let us assume that it is X long. The rotation is caused
by the fact that the point A and B travel at different rates along the y- direction over a
period of time. Any differences in the x- component of velocity between the end points A
and B will cause the line to stretch, and not rotate.
Let us assume that the v- component of velocity at point A is at the start is v A . Then the
v
v- component at point B is v A 
x . The second part simply says that vx A
component of the velocity is changing with x. This is what causes the edge to rotate.
Over a small time t, the point A will move to A'. The vertical displacement of point A
v
is v A t . The vertical displacement of point B is ( v A 
x ) times t. The relative
x A
vertical motion of B compared to A (which causes AB to rotate in the counterclockwise
v
direction in the figure above) is
xt .
x A
The angle by which edge AB rotates is estimated as the vertical displacement divided by
v
v
the length of segment AB. This equals
xt divided by x, or
t . The angular
x A
x A
v
velocity of the edge AB is this angle divided by the time elapsed, or
. This angular
x A
velocity will be in the counterclockwise direction.
By a similar logic, we can show that the vertical face AC will rotate by an angular
u
velocity
in the clockwise direction (Check. This may be a question in a forthcoming
y A
quiz or homework!) The average angular velocity of the two faces AB and AC is, then,
1  v u 
   . Notice the negative sign. This occurs because the edge AB is rotating in
2  x y  A
the counterclockwise direction (positive by common convention) while the edge AC is
rotating in the clockwise direction. The 1/2 factor simply says we have averaged over the
two edges.
In the above derivation, we considered a single face of a cube, and its rotation in a
plane containing this face. Extending this to three-dimensions, and multiplying the

angular velocity by 2, we get the vorticity 



i
j
k
  



   V 
x y z
u
v
w
(2)
In short, the vorticity vector is the curl of the velocity vector.
You should remember the definitions (in words and as equations 1 and 2) the definitions
of streamlines, pathlines and vorticity.
Bernoulli Eqution: The u-, v- and w- momentum equations are nonlinear PDEs and are
hard to solve. Fortunately, for inviscid flows without body forces, these equations yield a
simple algebraic equation linking pressure and velocity. We derive this simple equation
below.
We start with the u- momentum equation. It has many forms. But the form that we use is:
u
u
u
u 1 p
v
w 
0
x
y
z  x
(3)
Multiplying this equation by dx we get:
u
u
u
1 p
u dx  v dx  w dx 
dx  0
x
y
z
 x
Along a streamline, according to equation (1), udy=vdx, udz=wdx. The above equation
becomes:
u
u
u
u
1 p
dx  u dy  u dz 
dx  0
x
y
z
 x
(4)
Now,
u
u   u 2 
  
x x  2 
u   u 2 
u
  
y y  2 
u   u 2 
  
z z  2 
Equation (4) therefore becomes along a streamline:
u
  u2 
  u2 
  u2 
1 p
 dx   dx   dx 
dx  0
x  2 
x  2 
x  2 
 x
(5)
From Calculus, for any function f(x,y,z) the total differential df is given by:
df 
f
f
f
dx  dy  dz
x
y
z
(6)
Thus, equation (5) may be written as:
 u 2  1 p
d   
dx  0
 2   x
(7)
Thus far we worked with the u- momentum equation. We can similarly show that the vand w- momentum equations become:
 v 2  1 p
d   
dy  0
 2   y
(8)
2
 w  1 p
 
d 
dz   gdz
 2   z
(9)
Notice that we have added gravitational forces to the right side of equation (9). In
aerodynamic problems this term makes a negligible contribution. It has a negative sign
because gravitational forces are downward directed, whereas the z- axis points upwards.
In hydrodynamic and hydraulic applications, this term may make a big enough difference.
Add equations (7), (8) and (9). Use equation (6) to replace the partial derivatives
involving p with a total derivative. We get:
 u 2  v 2  w2  1
  dp  gdz  0
d 
2

 
Since
2
u 2  v 2  w2  V  V 2
V 2  1
  dp  gdz  0
d 
2

 
(10)
Here V is the velocity magnitude. Equation (10) may be integrated. Thus,
Along a streamline, for steady incompressible inviscid flow, pressure, height z, and
velocity are related by:
V2 p
  gz  Constant
2 
(11)
Along a streamline, the potential energy (due to gravity) plus the kinetic energy plus the
pressure energy is a constant.
Equation (11) is called the Bernoulli equation which links pressure to velocity, and it is
algebraic!
The differential form given in equation (10), neglecting gravity effects,
V 2  1
  dp  0
d 
 2  
Or
1
VdV  dp  0

(12)
is called the Euler relation. It holds for compressible and incompressible inviscid flows.
Equation (11) holds only for incompressible flows, since we had to assume density was
constant in order to integrate equation (10).