Motion of Fluid Particles and Stream

1. FLUID MECHANICS :-
Motion of fluid particles and stream

2.  Fluid Flow :-
The study of fluid in motion is called
Fluid Kinematics .
 The motion of fluid is usually extremely complex.
 The study of fluid at rest was simplified by absence
of shear force, but when fluid flows over boundary
a velocity gradient created.
 the resulting change in velocity from one layer to
other parallel to boundary gives rise to shear stress.
 So, the study of fluid flow needs study of shear
stresses .

3.  The following terms are considered suitable to
describe the motion of the fluid:-
 Path line :- The path traced by a single fluid particle
over a period of time is called its path line .
 Streak line :-It is an instantaneous picture of the
position of all fluid particles in the flow which have
passed through a given fixed point.
 Streamline :-a streamline is an imaginary line drawn
through flow field such that the velocity vector of
the fluid flow at each pint is tangent to streamline.
 Stream tube :-it is defined as tubular space formed
by the collection of streamlines passing through
the perimeter of closed curve.

4. Stream tube

5.  Classification of fluid flow : -
 Steady flow :-it is defined as that type
of flow in which fluid characteristics like pressure,
density at a point do not change with time.
Unsteady flow :-it is defined as flow in which fluid
characteristics at a point change with time .
 Uniform flow :-it is defined as flow in which flow
parameters like pressure, velocity do not change
with respect to space(length of flow).

6.  Non uniform flow :-it is the flow in which flow
parameters like pressure , velocity at a given time
change with respect to space .
 Laminar Flow :-It is the flow in which fluid particles
flow in layers or lamina with one layer sliding over
the other .Fluid elements move in well defined
path and they retain the same relative position at
successive cross section of the fluid passage .
The laminar flow is also called the streamline or
viscous flow. This type of flow occurs when
the velocity of flow is low and liqiud having high
viscosity.

7.  Turbulent Flow :-It is that type of flow in which fluid
particles move in zig-zag way.
 All the fluid particles are disturbed and they mix
with each other.
 Thus there is continuous transfer of momentum
to adjacent layers.
 Due to movement of fluid particles in a zig-zag way,
the eddies formation takes place which are
responsible for high energy loss.
 At Reynold number less than 2000 the flow is
laminar and Re greater than 4000 the flow is
turbulent.

9.  Compressible flow :- When the volume and thereby
density of fluid changes appreciably during flow
the flow is said to be compressible flow
 Incompressible flow :-Flow is incompressible if the
volume and thereby the density of fluid changes
insignificantly in fluid flow. For all practical purposes
liquids can be considered as incompressible.
This means that pressure and temperature changes
have little effect on their volume .

10.  Real and Ideal fluids
 When a real fluid flows over a boundary the adjacent
layer of fluid in contact with the boundary will have
the same velocity as the boundary.
 The velocity of successive layer increases as we
move away from boundary
 So, it indicates that shear stresses are produced
between the layers of fluid moving with different
velocities as a result of viscosity and the interchange
of momentum due to turbulence causing particles
fluid move from one layer to another.

11.  An ideal fluid is defined as fluid in which there are
no viscosity and shear stresses .
 If the viscosity of fluid is less and velocity is high
the boundary layer is comparatively thin and
the assumption that a real fluid is treated as an
ideal fluid.

12.  Motion of fluid particle :-
 As fluid is composed of particles each one of them
has its own velocity and acceleration.
 Further these both quantities may change with respect
to time as well as with respect to position of
the particle in the flow passages .
 Its behaviour can be predicted from newton’s law
of motion when a force is applied.
 Newton’s law can be written ,
Force = mass×acceleration

13.  The relationship between acceleration and velocity
is as follows :-
V’ = V + at
S = Vt + ½ at^2
V’^2 = V^2 + 2aS
 Consider two points A and B δs apart so cahnge of
velocity which occurs when particle moves from
A to B in time δt .

14.  Difference in velocity between A and B at the given
time is as follows : -
difference in velocity = (әv/әs)(δs)
Also ,
Change of velocity at B in time t,
t = (әv/әt)δt .
Thus total change of velocity ,
dv = (әv/әs)(δs) + (әv/әt)δt

15.  Lagrangian frame of reference :-
 This approach refers to description of the behaviour
of individual fluid particles during their motion
through space .
 The observer travels with the particle beeing studied
The fluid velocity and acceleration are then determi-
ned as function of position and time .

16.  Let the initial coordinates of fluid particles are a,b,c
and final coordinates after time interval are x,y,z
 The kinetic flow pattern is described by following
equations of motion ,
x = x(a,b,c,t)
y = y(a,b,c,t)
z = z(a,b,c,t)
 These equations can be stated as “final position x of
a fluid particle is function of initial space coordinates
and time”
 The initial space coordinates a,b,c and time t are
known as the Lagrangian variables .

17.  The acceleration components are :
a = du/dt
 Here we get acceleration in all three direction or
in all three planes
 The motion of one individual fluid particle is not
sufficient to describe the entire flow field , motion
of all the fluid particles has to be considered
simultaneously .

18.  Eulerian frame of reference : -
 In this method our co-ordinates are fixed in space,
and we observe the fluid as it passes by as if we
had described a set of coordinate lines on a glass
window.
 The observer remains stationary and observe what
happens at some particular point.

19.  Let x,y,z be the space coordinates at time t.
 Then the component of velocity vector are functions
of these space coordinates and time .
u = u(x,y,z,t)
v = v(x,y,z,t)
w = w(x,y,z,t)
 Each component is represented as the rate of
change of displacement ,
u = dx/dt , v = dy/dt , w = dz/dt

20.  The acceleration in Eulerian frame of reference
is given by : -
1. a(x) = (әu/әt) + u(әu/әx) + v(әu/әy) + w(әu/әz)
2. a(y) = (әv/әt) + u(әv/әx) + v(әv/әy) + w(әv/әz)
3. a(z) = (әw/әt) + u(әw/әx) + v(әw/әy) + w(әw/әz)

21.  Discharge and mean velocity : -
 The discharge is defined as the total quantity of
fluid flowing per unit time at any particular cross
section of a stream .
 It is also called flow rate . It can be measured in
terms of mass, in which case it is referred as
the mass flow rate (kg/s) or in terms of volume,
when it is called as the volume flow rate .

22.  In an ideal fluid in which there is no fluid friction
the velocity of the fluid would be the same at
every point of cross section .
 The fluid stream would pass thee given cross
section per unit time
 Let the cross sectional area normal to direction of
flow is A
So, discharge Q = AV

23.  In real fluid the velocity of adjacent layer of fluid
to a solid boundary will be zero or equal to velocity
velocity of solid boundary in the flow direction,
a condition called ‘ no slip ’
 Let us say V is velocity at any radius r , the flow δQ
through an annular element of radius r and thickness
dr will be
δQ = 2πr dr × V
 Total discharge Q = 2πᶘ Vr dr

24.  Continuity of Flow :-
 According to the principle of mass conservation ,
matter can be neither created nor destroyed except
nuclear processes. This principle can be applied to a
flowing fluid.
 Considering any fixed region in the constituting a
control volume.
fig. (1)

25. Mass of fluid Mass of fluid Increase of mass of
entering per leaving per fluid in the control
unit time unit timr volume per unit time
     
           
          
The mass of fluid in the control volume remains constant
for steady flow and the relation is reduced to
Mass of fluid entering = Mass of fluid leaving per
per unit time unit time

26. Let = Average velocity of fluid at section -
1
= Area of stream tube at section-1
= Density of fluid at section-1
1V
1A
1

27. And , , are corresponding values at section-2
Then, mass flow rate at section-1=
mass flow rate at section-2=
According to law of conservation of mass,
mass flow rate at section-1= mass flow rate at section-2
=
2V 2A 2
1 1 1AV
2 2 2A V
2 2 2A V1 1 1AV

28. • If fluid is incompressible , and continuity
equation reduces to
Volume flow rate,
• The continuity equation can also be applied to
determine the relation between the flows into and out
of a junction.
fig.(2)
1 2 
1 1 2 2AV A V 
1 1 2 2Q AV A V 

29. From fig.(2), for steady conditions,
Total inflow to junction=Total outflow from junction
For an incompressible fluid,
1 1 2 2 3 3Q Q Q   
1 2 3   
1 2 3Q Q Q  
1 1 2 2 3 3AV A V AV  

30.  Continuity Equation for 3-D flow :-
Mass flux out of differential volu
Rate of change of mass in
differential volume
= Mass flux into differential volume
v
v y
y

 
 
 
y
y

 
 
 
 
x z 
x
y
z
y x z
t

  


v x z  

31.   



  v
v
y
y v
y
y
v
y y
y x z










F
HG I
KJ2
  

     

  v
v
y
y v
y
y x z v x z
t
y x z





F
HG I
KJ   


Out In
Rate of mass decrease
v
v
y y t
 

  
  
  
0
v
y t
  
 
 
One dimensional equation
Mass flux out of differential volume
Higher Order term

32.       0
u v w
t x y z
     
   
   
3-D Continuity Equation,

33. THANK YOU