Forces Acting on a Control VolumeBody forces: Act through the entire bodyof the control volume:gravity, electric, and magnetic forces.

Surface forces: Act on the controlsurface: Pressure and viscousforces 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:

Total body force acting on control volume:

'gravitydF gdV����������������������������

'body CV

CV

F gdV m 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

• Total surface force acting on

control surface:

• Total force:

surface ijdF ndA

��������������

surface ij

CS

F ndA ��������������

total force gravity pressure viscous otherF F F F F ����������������������������������������������������������������������

Newton’s Second Law- Linear MomentumEquation

Momentum = mass x velocity

sys

' sys

DV dV F

Dt

Time rate of change of the linear momentum of the system = sum of external forces acting on the system

For a system and a coincidental non-deforming fixed control volume:

For a fixed and nondeforming control volumesys

' ' CV CS

DV dV V dV V V ndA

Dt t

' = FCV CSV dV V V ndA

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 cros

s 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