Forces Acting on a Control Volume
Body forces: Act through the entire body
of the control volume:
gravity, electric, and magnetic forces.
Surface forces: Act on the control
surface: Pressure and viscous
forces at points of contact
Most common body force is that of gravity, which exerts a
downward force on every differential element of the control
volume. Gravitational force acting on a fluid element:
d F gravity   gdV '
Total body force acting on control volume:
F
body

  gdV '  m
CV
CV
g
Forces Acting on a Control Volume
• Surface forces are not as simple to
analyze as they have both normal and
tangential components.
• While the physical force acting on the
surface is independent of orientation of
the coordinate axes, the description of
the force in terms of its coordinate
changes with orientation,
• Rarely control surface aligns with
coordinate axes.
• Second order tensor, stress tensor,
Stress
• Diagonal components of the
stress tensor are called normal
stresses
• Off-diagonal components are
called shear stress
• Since pressure acts only
normal to surface, shear
stresses are composed mainly
of viscous stress
 xx ,  yy ,  zz
 xy ,  zx ,  yz
Surface Force
• Surface force acting on a
differential surface element
d F surface   ij ndA
• Total surface force acting on
control surface:
F
surface


ij
ndA
CS
• Total force:
F
total force
  F gravity   F pressure   F viscous   F other
Newton’s Second Law- Linear Momentum
Equation
Momentum = mass x velocity
Time rate of change of the linear momentum of the system =
sum of external forces acting on the system
D
Dt
 V  dV '   F
sys
sys
For a system and a coincidental non-deforming fixed control
D

volume:
V  dV ' 
V dV ' V V ndA

Dt sys
t CV

CS
For a fixed and nondeforming control volume

V dV '  V V ndA =  F

CV
CS
t
Linear momentum equation
From RTT
Example 1
• A horizontal jet of water exits a nozzle with a uniform speed
of V1=10 ft/s, strikes a vane and is turned through an angle
θ. Determine the anchoring forces need to hold the vane
stationary.
Example 2
• Water flows through a horizontal 1800 pipe end. The
cross sectional area of flow is constant at 0.1 ft2. The
flow velocity everywhere in the bend is axial and 50 ft/s.
The absolute pressures at the entrance and exit are 30
psia and 24 psia, respectively. Calculate the horizontal
components of the anchoring force required to hold the
bend in place.
Example 3
Assuming uniform velocity distribution, determine the
frictional force exerted by the pipe wall on the air flow
between sections (1) and (2) of 4-in inside diameter.
Example 4
A static thrust stand is to be designed for testing a jet
engine for the following conditions: intake air velocity =200
m/s, exhaust gas velocity=500 m/s, intake cross-sectional
area =1 m2,intake static pressure, temperature =78.5 kPa
(abs), 268 K Exhaust static pressure =101 kPa. Estimate
the nominal design thrust