Fluids dynamics

1. Fluid Dynamics
VIVEK KUMAR
21101158043
Dept. of Civil Engg.
GEC, LAKHISARAI
email: vivek.roy7320@gmail.com
Submitted to:- Randhir Sir

2. Introduction:
• In the kinematics of flow, we studied the velocity and acceleration at a point
in a fluid flow, without taking into consideration the forces causing the flow.
• Dynamics of fluid flow includes the study of forces causing fluid flow.
• It is the study of fluid motion with the forces causing flow.
• The dynamic behaviour of the fluid flow is analysed by the Newton's
second law of motion (F = ma), which relates the acceleration with the
forces.

3. Forces in a fluid
flow:
1. Pressure force ‘Fp’ is exerted on the fluid mass, if there exists a pressure gradient
between the 2 points in the direction of flow.
2. Gravity force ‘Fg’ is due to the weight of the fluid and it is equal to ‘mg’. The gravity
force for unit volume is equal to ‘ρg’
3. Viscous force ‘Fv’ is due to the viscosity of the flowing fluid and thus exists in the case
of all real fluid.
4. Turbulent force ‘Ft’ is due to the turbulence of the flow
5. Surface tension force ‘Fs’ is due to the cohesive property of the fluid mass.
6. Compressibility force ‘Fc’ is due to elastic property of fluid and it is important only for
compressible fluid flow

4. • If a certain mass of fluid in the motion is influenced by all the above
mentioned forces,
• Net force F = Fg+ Fp+ Fv+ Ft+ Fs+ Fc
Forces in a fluid
flow:

5. Equations of Motion
• There are three important equations of motions applicable to the
dynamics of fluid flow:
1. Reynold’s equation of motion
2. Navier-Stokes equation of motion
3. Euler’s equation of motion (Bernoulli’s equation)

6. Equations of Motion
(i)Reynolds’s equation of motion: While developing the equation of
motion, if the surface tension force Fs and the compressibility force Fc
are not significant and hence neglected, the net force on the fluid becomes
• F = Fg+ Fp+ Fv+ Ft
• The Reynolds’s equation of motion is useful in the analysis of the turbulent
flows.

7. (ii)Navier-Stokes equation of motion: Further, while developing
the equation of motion for laminar or viscous flows the turbulent force Ft
are less significant and hence they may be neglected. The net force on
the fluid becomes,
• F = Fg + Fp + Fv
• The developed equations above are known as Navier-stokes equations
which are useful in the analysis of viscous flow.
Equations of Motion

8. (iii)Euler’s equation of motion: Further, while developing the equation of motion, if
the viscous force Fv is also of less significance, then this force may also be neglected.
• The viscous force will become insignificant if the flowing fluid is an ideal fluid.
• However, in case of real fluids, the viscous forces may be considered insignificant if the
viscosity of flowing fluid is small.
• In such cases the net force on the fluid becomes,
• F = Fg + Fp
• These modified Navier-stokes equation by letting Fv = 0 are known as Euler’s equations.
Equations of Motion

9. Euler's equation of motion:
The assumptions made while developing this equation are:
• The fluid is ideal and incompressible.
• Flow is steady and continuous.
• Flow is along streamline and it is 1-D.
• The velocity is uniform across the section and is equal to the mean velocity.
• Flow is irrotational.
• The only forces acting on the fluid are gravity and the pressure forces.

10. Euler's equation of motion:

11. Euler's equation of motion:
From Newton’s second law,
Net force acting on the fluid element in the direction of
s
= mass of fluid element  acceleration in the direction s.
s
pdA  
p 
p
ds 
dA  g  dA ds  cos    dA
ds  a


s


where, as is the acceleration in the s direction
as = Local acceleration ( V/t) + Convective acceleration (VV/s)
a 
DV

V
V
V
s
Dt t s
where the velocity V is a function of s and t
But the flow is a steady flow. Therefore the local acceleration
V
 0
t

12. Euler's equation of motion:
s
a 
DV
 V
V
 V
dV
Dt s
ds
Since the flow is one dimensional (along direction s), the partial derivatives can be
expressed as a total derivatives
and
V

dV p

dp
s ds s ds
s
Substituting for a and simplifying, we
get

dp
dsdA  gdAds cos  dAds V
dV
ds ds
dp
dsdA  gdAds cos  dAds V
dV

0
ds ds
pdA  
p 
p
ds 
dA  g  dA ds 
cos


s
 
  dA ds 
as

13. Euler's equation of motion:
dp
 g cos   V
dV
 0
ds ds
dp
 g 
dz
  V
dV
 0 cos 
dz
ds ds ds ds
Multiplying through out by ds and dividing through out by
,
dp
+ gdz + VdV
= 0 ρ
This equation is known as Euler’s Equation of motion.

14. Bernoulli’s Equation from Euler’s equation
for motion:
dividing by ‘g’ we
get, p

V 2
 g 2g
 z  constant
This equation is known as Bernoulli’s Equation.

15. z1
z2
1
3
p1 , V1
p2 , V2
2
p3 , V3
Reference datum line
H
z3
= H
H
Bernoulli’s Equation

16. Bernoulli’s Equation
• Applies to all points on the streamline and thus provides a useful relationship
between pressure p, the magnitude of the velocity V, and the height z above
an arbitrary reference datum.
• The Bernoulli constant H is also termed the total head.

17. Bernoulli’s Equation
p

V 2
 g 2g
 z  constant
Potential head or datum head (m)
Kinetic head or velocity head(m)
Pressure head (m)
The three terms in the Bernoulli’s equation are identified as:

18. Bernoulli’s equation for real fluid:
• Bernoulli’s equation earlier derived was based on the assumption that fluid is non viscous
and therefore frictionless.
• Practically, all fluids are real (and not ideal) and therefore are viscous fluids.
• Due to the viscous effects, the layers of the fluid near to the surface of the pipe or
passage
of flow tend to stick to the solid surfaces (no slip condition).
• This causes frictional resistance to flow. Thus some energy of the fluid is utilized
in overcoming this frictional resistance.
• This loss in energy has to be taken into consideration in the application of
Bernoulli’s
equation which gets modified (between sections 1 and 2) for real fluids.

19. • If a fluid is flowing from section 1 to section 2, it implies that the total head
H1 at section 1 is greater than that (H2) at section 2.
• Thus the head lost hL must be added to H2 so as to make it equal to
H1.
(Note: For ideal fluid, H1 = H2)
Bernoulli’s equation for real fluid:
Section 1 Section 2
H1 H2
Flow Flow

20. THANK YOU