Faros University
ME 253
Fluid Mechanics II
Fluid Kinematics
Dr. A. Shibl
Text Book: Frank White
“ Fluid Mechanics”
Flow Classification
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Real (Viscous) or Ideal (Inviscid) Flow
Compressible or Incompressible Flow
Turbulent or Laminar Flow
Unsteady or Steady Flow
3, 2, or 1 dimensional Flow
Rotational or Iroratational Flow
Fluid Kinematics: Introduction
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Fluids subject to shear, flow
Fluids subject to pressure imbalance, flow
Kinematics : Description of Fluid motion.
Visualization.
Fluid Motion
• Two ways to describe
fluid motion
– Lagrangian
• Follow particles around
– Eularian
V
dx dy
dz
i
j k
dt
dt
dt
• Watch fluid pass by a
V  ui  vj  wk
point or an entire region
– Flow pattern
• Streamlines – velocity
is tangent to them
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STEADY AND UNSTEADY FLOW:
Steady flow: the flow in which conditions at any
point do not change with time is called steady
P
V

flow.
 0,
 0,
 0,
t
t
t
• Unsteady flow: the flow in which conditions at
any point change with time, is called unsteady
flow.
P
V

t
 0,
t
 0,
t
 0,
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UNIFORM AND NON-UNIFORM:
• The flow in which the conditions
at all points are the same at the
same instant is uniform flow.
P
V

 0,
 0,
 0,
s
s
s
• The flow in which the conditions
navy from point to point at the
same instant is non-uniform
flow.
P
V

 0,
 0,
 0,
s
s
s
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Velocity Field
Velocity, acceleration, and pressure of these particles at a given time are:
Velocity Field: Eulerian and Lagrangian
Eulerian: the fluid motion is given by completely describing properties as a
function of space and time. We obtain information about the flow by noting
what happens at fixed points.
Lagrangian: following individual fluid particles as they move about and
determining how the fluid properties of these particles change as a function of
time.
Measurement of Temperature
Eulerian
Lagrangian
Eulerian methods are
commonly used in fluid
experiments or analysis—a
probe placed in a flow.
Lagrangian methods can also
be used if we “tag” fluid
particles in a flow.
Velocity Field: 1D, 2D, and 3D Flows
Three-Dimensional Flow: All three velocity components are
important and of equal magnitude. Flow past a wing is 3D flow
Two-Dimensional Flow: If one of the velocity components is small relative to
the other two, thus it is reasonable to assume 2D flow.
One-Dimensional Flow: If two of the velocity components may be small
relative to the other one, thus it is reasonable to assume 1D flow.
Steady Flow: The velocity at a given point in space does not vary with time.
Velocity Field: Streamlines
Streamline: the line that is everywhere tangent to the velocity field.
Analytically, for 2D flows, integrate the equations defining lines tangent to the
velocity field:
ACCELERATION
• Acceleration = rate of change
of velocity
• Components:
– Normal – changing direction
– Tangential – changing speed


V  V ( s , t )et


det
 dV dV 
a

et  V
dt
dt
dt
dV
V V
V

dt
s t

det V 
 en
dt
r
V V  V 2 

a  (V

)et 
en
s t
r
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Acceleration Field
Lagrangian Frame:
Eulerian Frame: we describe the acceleration in terms of position and time
without following an individual particle.
A fluid particle can accelerate due to a change in velocity in time (“unsteady”)
or in space (moving to a place with a greater velocity).
Acceleration Field
time dependence
spatial dependence
We note:
Then, substituting:
The above is good for any fluid particle, so we drop “A”:
Acceleration Field
Writing out these terms in vector components:
x-direction:
y-direction:
z-direction:
Writing these results in “short-hand”:
where,
() 
 ˆ  ˆ  ˆ
i
j k ,
x
y
z
Acceleration Field: Unsteady Effects
Consider flow in a constant diameter pipe, where the flow is assumed to be
spatially uniform:
0
0
0
0 0
Acceleration Field: Convective Effects
The portion of the material derivative represented by the spatial derivatives is
termed the convective term or convective accleration:
It represents the fact the flow property associated with a fluid particle may
vary due to the motion of the particle from one point in space to another.
Convective effects may exist whether the flow is steady or unsteady.
Example 1:
Example 2:
Acceleration = Deceleration
ACCELERATION
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Cartesian coordinates
 


V  ui  vj  wk




a  ax i  a y j  az k
du u dx u dy u dz u u
u
u
u




. u  v w
dt x dt y dt z dt t
x
y
z
t
dv v dx v dy v dz v v
v
v
v
ay 



 . u  v w
dt x dt y dt z dt t
x
y
z
t
dw w dx w dy w dz w w
w
w
w
az 




. u
v
w
dt x dt y dt z dt t
x
y
z
t
ax 
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In steady flow ∂u/∂t = 0 , local acceleration is zero.
Convective
In unsteady flow ∂u/∂t ≠ 0 ; local acceleration
Occurs.
Other terms u ∂u/∂x, v ∂u/∂y,.. are called convective
accelerations. Convective acceleration Occurs when
the velocity varies with position.
Uniform flow: convective acceleration = 0
Non-uniform flow: convective acceleration ≠0
Local
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Example



2
Given : V  3ti  xzj  ty k

Find : Acceleration, a
u  3t ; v  xz; w  ty 2
u
u
u
u
u  v w
 0(3t )  0( xz )  0(ty 2 )  3  3
x
y
z
t
v
v
v
v
 u  v  w   z (3t )  0( xz )  x(ty 2 )  0  3 zt  xy 2t
x
y
z
t
w
w
w
w

u
v
w
 0(3t )  2ty( xz )  0(ty 2 )  y 2  2 xyzt  y 2
x
y
z
t
ax 
ay
az





2 
2 
a  a x i  a y j  a z k  3i  (3tz  txy ) j  (2 xyzt  y )k
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