This document discusses forces on bodies immersed in fluids, specifically drag and lift forces. It defines drag as the force component in the direction of fluid flow and lift as the perpendicular component. Drag and lift depend on factors like fluid density, velocity, and body size/shape. Dimensionless coefficients are used to characterize drag and lift. Specific examples like Stokes' law for drag on a sphere in low Reynolds number flow are provided. Applications like calculating terminal velocity and fluid viscosity are also mentioned.

1. Forces on Immersed Bodies

2. • Up till now fluid flow within pipes have been
studies (internal flow).
• Next, flow of fluids over bodies that are
immersed in a fluid, called external flow will
be studied.
• emphasis on the resulting lift and drag forces

3. • When a body is immersed in a real fluid, which is
flowing at a uniform velocity U, the fluid will exert
a force on the body.
• The total force (FR) can be resolved in two
components:
– 1. Drag (FD): Component of the total force in the
direction of motion of fluid.
– 2. Lift (FL): Component of the total force in the
perpendicular direction of the motion of fluid. It
occurs only when the axis of the body is inclined to
the direction of fluid flow. If the axis of the body is
parallel to the fluid flow, lift force will be zero.

5. • Examples include drag force acting on
automobiles, power lines, trees, and
underwater pipelines;
• Lift developed by airplane wings; upward draft
of rain, snow, hail, and dust particles in high
winds; the transportation of red blood cells by
blood flow;
• the vibration and noise generated by bodies
moving in a fluid; and the power generated by
wind turbines.

6. • It is a common experience that a body meets
some resistance when it is forced to move
through a fluid, especially a liquid.
• It is very difficult to walk in water because of
the much greater resistance it offers to motion
compared to air.
• High winds knocking down trees, power lines,
and even trailers and felt the strong “push”
the wind exerts on your body

8. • Same feeling is observed when you extend your
arm out of the window of a moving car.
• A fluid may exert forces and moments on a body
in and about various directions.
• The force a flowing fluid exerts on a body in the
flow direction is called drag.
• The drag force can be measured directly by
simply attaching the body subjected to fluid flow
to a calibrated spring and measuring the
displacement in the flow direction

9. • Bodies subjected to fluid flow are classified as being
streamlined or blunt, depending on their overall shape.
• A body is said to be streamlined if a conscious effort is
made to align its shape with the anticipated
streamlines in the flow.
• Streamlined bodies such as race cars and airplanes
appear to be contoured and sleek. Otherwise, a body
(such as a building) tends to block the flow and is said
to be bluff or blunt.
• Usually it is much easier to force a streamlined body
through a fluid, and thus streamlining has been of
great importance in the design of vehicles and
airplanes

11. • Drag is usually an undesirable effect, like
friction, and we do our best to minimize it.
• Reduction of drag is closely associated with
the reduction of fuel consumption in
automobiles, submarines, and aircraft;
improved safety and durability of structures
subjected to high winds; and reduction of
noise

12. • A stationary fluid exerts only normal pressure forces on
the surface of a body immersed in it.
• A moving fluid, however, also exerts tangential shear
forces on the surface because of the no-slip condition
caused by viscous effects.
• Both of these forces, in general, have components in
the direction of flow, and thus the drag force is due to
the combined effects of pressure and wall shear forces
in the flow direction.
• The components of the pressure and wall shear forces
in the direction normal to the flow tend to move the
body in that direction, and their sum is called lift.

13. • The pressure and shear
forces acting on a
differential area dA on the
surface are PdA and τw
dA, respectively.
• Drag force and the lift
force acting on dA in two-
dimensional flow are

14. • where θ is the angle that the outer normal of
dA makes with the positive flow direction.
• The total drag and lift forces acting on the
body are determined by

15. • In the special case of a thin flat plate aligned
parallel to the flow direction, the drag force
depends on the wall shear only and is
independent of pressure since θ=90°.

16. • When the flat plate is placed normal to the flow
direction, however, the drag force depends on
the pressure only and is independent of wall
shear since the shear stress in this case acts in
the direction normal to flow and θ=0°. If the flat
plate is tilted at an angle relative to the flow
direction, then the drag force depends on both
the pressure and the shear stress

17. • The drag and lift forces depend on the density r
of the fluid, the upstream velocity V, and the size,
shape, and orientation of the body, among other
things, and it is not practical to list these forces
for a variety of situations.
• Instead, it is found convenient to work with
appropriate dimensionless numbers that
represent the drag and lift characteristics of the
body.
• These numbers are the drag coefficient CD, and
the lift coefficient CL, and they are defined as

18. • where A is ordinarily the frontal area (the area
projected on a plane normal to the direction of flow) of
the body. In other words, A is the area that would be
seen by a person looking at the body from the
direction of the approaching fluid.
• The frontal area of a cylinder of diameter D and length
L, for example, is A = LD. In lift calculations of some
thin bodies, such as airfoils, A is taken to be the
planform area, which is the area seen by a person
looking at the body from above in a direction normal to
the body.

19. Drag on a sphere (stokes law)
• For case of an ideal fluid flowing past a sphere, there is no
drag
• Consider the case of flow of a real fluid past a sphere
• Let D be the diameter, V be the velocity of flow of fluid with
density ρ and viscosity μ
• If the velocity of flow is very small or the fluid is very viscous
such that Reynold’s number is quite small (Re<=2)
• Viscous forces are much more predominant than inertial
forces

20. • George Gabriel Stokes analysed the flow around a sphere
under very low velocities and found that total drag force is
given by
• Two third of the drag force is contributed by skin friction and
one third by pressure difference
(skin friction drag)
DV
FD 
3

DV
DV
FD 
 2
3
3
2
3
2



21. (pressure drag)
The total drag is given by
Where A is the projected area=πD2/4
DV
DV
FD 
 
 3
3
1
3
1
A
V
C
F D
D
2
2



22. By equating the two equations for FD
Which is called Stokes Law.
2
2
4
2
3 D
V
C
DV D


 

e
D
R
VD
C
24
24





23. Terminal velocity of a body
• Maximum velocity attained by a falling body is known as the
terminal velocity
• If a body is allowed to fall from rest, its velocity increases due
to gravitational acceleration
• The increase in velocity also increases the drag force opposing
the motion of the body
• when drag force = weight of body, acceleration ceases, net
external force acting on the body becomes zero making the
body to move at constant speed called terminal velocity

24. W=FD+B
Or
FD=W-B
  g
d
g
d
W s
s 


 3
3
6
2
/
3
4


  g
d
g
d
B f
f 


 3
3
6
2
/
3
4



25. Drag force will be given by
 g
d
B
W
F f
s
D 





 3
6

27. Application of stokes law
• To calculate the terminal velocity of a falling
sphere and hence the viscosity of the fluid
• Desilting the river flow
• Sanitary engineering-treatment of raw water
and sewerage water