1. Mathematical Models for FLUID MECHANICS
P M V Subbarao
Associate Professor
Mechanical Engineering Department
IIT Delhi
Convert Ideas into A Precise Blue Print before
feeling the same....

4. A path line is the trace of the path followed by a selected fluid particle

5. Few things to know about streamlines
• At all points the direction of the streamline is the direction of the
fluid velocity: this is how they are defined.
• Close to the wall the velocity is parallel to the wall so the
streamline is also parallel to the wall.
• It is also important to recognize that the position of streamlines
can change with time - this is the case in unsteady flow.
• In steady flow, the position of streamlines does not change
• Because the fluid is moving in the same direction as the
streamlines, fluid can not cross a streamline.
• Streamlines can not cross each other.
• If they were to cross this would indicate two different velocities at
the same point.
• This is not physically possible.
• The above point implies that any particles of fluid starting on one
streamline will stay on that same streamline throughout the fluid.

6. A useful technique in fluid flow analysis is to consider only a part of the
total fluid in isolation from the rest. This can be done by imagining a
tubular surface formed by streamlines along which the fluid flows. This
tubular surface is known as a streamtube.
A Streamtube
A two dimensional version of the streamtube
The "walls" of a streamtube are made of streamlines.
As we have seen above, fluid cannot flow across a streamline, so
fluid cannot cross a streamtube wall. The streamtube can often be
viewed as a solid walled pipe.
A streamtube is not a pipe - it differs in unsteady flow as the walls
will move with time. And it differs because the "wall" is moving
with the fluid

7. Fluid Kinematics
• The acceleration of a fluid particle is the rate of change of its
velocity.
• In the Lagrangian approach the velocity of a fluid particle is a
function of time only since we have described its motion in terms of
its position vector.

8. In the Eulerian approach the velocity is a function of both
space and time; consequently,

V ˆ j ˆ
u ( x, y, z, t )i v( x, y, z, t ) ˆ w( x, y, z, t )k
   
Velocityco
mponents
x,y,z are f(t) since we must follow the total derivative approach
in evaluating du/dt.

10. Similarly for ay & az,
In vector notation this can be written concisely

13. x
Conservation laws can be applied to an infinitesimal element or
cube, or may be integrated over a large control volume.

14. Basic Control-Volume Approach

15. Control Volume
• In fluid mechanics we are usually interested in a region of space, i.e,
control volume and not particular systems.
• Therefore, we need to transform GDE’s from a system to a control
volume.
• This is accomplished through the use of Reynolds Transport
Theorem.
• Actually derived in thermodynamics for CV forms of continuity and
1st and 2nd laws.

16. Flowing Fluid Through A CV
• A typical control volume for flow
in an funnel-shaped pipe is
bounded by the pipe wall and the
broken lines.
• At time t0, all the fluid (control
mass) is inside the control
volume.

17. The fluid that was in the control volume at time t0 will be seen at
time t0 + t as: .

18. The control volume at time t0 + t .
The control mass at time t0 + t .
The differences between the fluid (control mass) and the control volume
at time t0 + t .

19. • Consider a system and a control volume (C.V.) as follows:
• the system occupies region I and C.V. (region II) at time t0.
• Fluid particles of region – I are trying to enter C.V. (II) at time t0.
III
II
I
• the same system occupies regions (II+III) at t0 + t
• Fluid particles of I will enter CV-II in a time t.
•Few more fluid particles which belong to CV – II at t0 will occupy
III at time t0 + t.

20. The control volume may move as time passes.
III has left CV at time t0+ t
III
II
I is trying to enter CV at time t0
II At time t0+ t

I VCV
At time t0

21. Reynolds' Transport Theorem
• Consider a fluid scalar property b which is the amount of this property
per unit mass of fluid.
• For example, b might be a thermodynamic property, such as the
internal energy unit mass, or the electric charge per unit mass of fluid.
• The laws of physics are expressed as applying to a fixed mass of
material.
• But most of the real devices are control volumes.
• The total amount of the property b inside the material volume M ,
designated by B, may be found by integrating the property per unit
volume, M ,over the material volume :

22. Conservation of B
• total rate of change of any extensive property B of a
system(C.M.) occupying a control volume C.V. at time t is
equal to the sum of
• a) the temporal rate of change of B within the C.V.
• b) the net flux of B through the control surface C.S. that
surrounds the C.V.
• The change of property B of system (C.M.) during Dt is
BCM Bt t
Bt
0 0
BCM BII t0 t
BIII t0 t
BI t0
BII t0
add and subtract B t t
0

23. BCM BII t0 t
BIII t0 t
BI t0
BII t0
BI t0 t
BI t0 t
BCM BI BII t0 t
BIII t0 t
BI BII t0
BI t0 t
BCM BCV t0 t
BCV t0
BIII t0 t
BI t0 t
The above mentioned change has occurred over a time t, therefore
Time averaged change in BCM is
BCM BCV t0 t
BCV t0
BIII t0 t
BI t0 t
t t t t

24. For and infinitesimal time duration
BCM BCV t0 t
BCV t0
BIII t0 t
BI t0 t
lim lim lim lim
t o t t o t t o t t o t
• The rate of change of property B of the system.
dBCM dBCV  
BIII BI
dt dt

25. Conservation of Mass
• Let b=1, the B = mass of the system, m.
dmCM dmCV

mout 
min
dt dt
The rate of change of mass in a control mass should be zero.
dmCV

mout 
min 0
dt

26. Conservation of Momentum
• Let b=V, the B = momentum of the system, mV.
 
d mV d mV  
CM CV

mV out

mV in
dt dt
The rate of change of momentum for a control mass should be equal
to resultant external force.

d mV   
CV

mV out

mV in F
dt

27. Conservation of Energy
• Let b=e, the B = Energy of the system, mV.
d me d me
CM CV

me out

me in
dt dt
The rate of change of energy of a control mass should be equal
to difference of work and heat transfers.
d me  
CV

me out

me in Q W
dt

35. Applications of Momentum Analysis
  

M out 
M in Vn A Vout Vn A Vin F
out in
This is a vector equation and will have three components in x, y and z
Directions.
X – component of momentum equation:
UA U out UA U in Fx
out in

36. X – component of momentum equation:
UA U out UA U in Fx
out in
Y – component of momentum equation:
VA Vout VA Vin Fy
out in
Z – component of momentum equation:
WA Wout WA Win Fz
out in
For a fluid, which is static or moving with uniform velocity, the
Resultant forces in all directions should be individually equal to zero.

37. X – component of momentum equation:
UA U out UA U in Fx
out in
Y – component of momentum equation:
VA Vout VA Vin Fy
out in
Z – component of momentum equation:
WA Wout WA Win Fz
out in
For a fluid, which is static or moving with uniform velocity, the
Resultant forces in all directions should be individually equal to zero.

38. X – component of momentum equation:
Max Fx FB, x FS , x
Y – component of momentum equation:
May Fy FB, y FS , y
Z – component of momentum equation:
Maz Fz FB, z FS , z
For a fluid, which is static or moving with uniform velocity, the
Resultant forces in all directions should be individually equal to zero.

39. Vector equation for momentum:
   
Ma F FB FS
Vector momentum equation per unit volume:
   
a f fB fS

f
Body force per unit volume:B

Gravitational force: f B oi 0 ˆ
ˆ j ˆ
gk

40. Electrostatic Precipitators
Electric body force: Lorentz force density
The total electrical force acting on a group of free charges (charged
ash particles) . Supporting an applied volumetric charge density.
   
fe fE Jf B
Where = Volumetric charge density
f

E = Local electric field
B = Local Magnetic flux density field

Jf = Current density

41. Electric Body Force
• This is also called electrical force density.
• This represents the body force density on a ponderable
medium.
• The Coulomb force on the ions becomes an electrical body
force on gaseous medium.
• This ion-drag effect on the fluid is called as
electrohydrodynamic body force.

42. 0
Ideal Fluids….

43. Pressure Variation in Flowing Fluids
• For fluids in motion, the pressure variation is no longer hydrostatic and is
determined from and is determined from application of Newton’s 2nd Law
to a fluid element.

44. Various Forces in A Flow field
• For fluids in motion, various forces are important:
 
• Inertia Force per unit volume : finertia a

• Body Force: f body ˆ
gk

• Hydrostatic Surface Force: f surface, pressure p
 

• Viscous Surface Force: f surface ,viscous .
• Relative magnitudes of Inertial Forces and Viscous Surface
Force are very important in design of basic fluid devices.

45. Comparison of Magnitudes of Inertia Force and Viscous Force
• Internal vs. External Flows
• Internal flows = completely wall bounded;
• Both viscous and Inertial Forces are important.
• External flows = unbounded; i.e., at some distance from body
or wall flow is uniform.
• External Flow exhibits flow-field regions such that both
inviscid and viscous analysis can be used depending on the
body shape.

46. Ideal or Inviscid Flows
Euler’s Momentum Equation
X – Momentum Equation:

48. Euler’s Equation for One Dimensional Flow
Define an exclusive direction along the
axis of the pipe and corresponding unit
ˆ
direction vector el
Along a path of zero acceleration the
pressure variation is hydrostatic

49. Pressure Variation Due to Acceleration
V V p z
V
t l l
For steady flow along l – direction (stream line)
V P z
V
l l l
Integration of above equation yields

50. Momentum Transfer in A Pump
• Shaft power Disc Power Fluid Power.
2 TN
P Td Td m vdp or m pdv
60
• Flow Machines & Non Flow Machines.
• Compressible fluids & Incompressible Fluids.
• Rotary Machines & Reciprocating Machines.

51. Pump
• Rotate a cylinder containing
fluid at constant speed.
• Supply continuously fluid
from bottom.
• See What happens?
Flow in
•Any More Ideas?

55. Momentum Principle
P M V Subbarao
Associate Professor
Mechanical Engineering Department
IIT Delhi
A primary basis for the design of flow devices ..

56. Momentum Equation

57. Applications of of the Momentum Equation
Initial Setup and Signs
• 1. Jet deflected by a plate or a vane
• 2. Flow through a nozzle
• 3. Forces on bends
• 4. Problems involving non-uniform velocity distribution
• 5. Motion of a rocket
• 6. Force on rectangular sluice gate
• 7. Water hammer

58. Navier-Stokes Equations
Differential form of momentum equation
X-component:
2 2 2
u u u u (p z) u u u
u v w
t x y z x x2 y2 z2
Y-component:
2 2 2
v v v v (p z) v v v
u v w
t x y z y x2 y2 z2

59. z-component:
2 2 2
w w w w (p z) w w w
u v w
t x y z z x2 y2 z2

60. Applications of Momentum Equation

61. Generation of Motive Power Through
Newton’s Second Law

62. Jet Deflected by a Plate or Blade
Consider a jet of gas/steam/water turned through an angle
CV and CS are
for jet so that Fx
and Fy are blade
reactions forces on
fluid.
2 2 2
u u u u (p z) u u u
u v w
t x y z x x2 y2 z2

63. Steady 2 Dimensional Flow
X-component:
2 2
u u (p z) u u
u v
x y x x2 y2
Y-component:
2 2
v v (p z) v v
u v
x y y x2 y2
Continuity equation:
u v u v
0 0
x y x y

64. Steady 2 Dimensional Invisicid Flow
X-component:
u u p
u v
x y x
Y-component:
v v p
u v
x y y
Continuity equation: u v
0
x y
Inlet conditions : u = U & v = 0

65. Pure Impulse Blade
Pressure remains constant along the entire jet.
u u
X-component: u v o
x y
Y-component: v v
u v 0
x y
Continuity equation: u v u v
0
x y x y