Page 101 - 48Fundamentals of Compressible Fluid Mechanics

4.5. THE IMPULSE FUNCTION 63

The net force is denoted here as . The mass conservation also can be applied
to our control volume
(4.90)

Combining equations (4.89) with equation (4.90) and utilizing the identity in equa-

tion (??) results in

(4.91)

results in

Rearranging equation (4.91) and dividing it by
(4.92)

Examining equation (4.92) shows

that right hand side is only function of x-direction

Mach number and specific heat ratio,

. Hence, if the right hand side is only
function of the Mach number and than
the left hand side must be function of
and .
Defining a function that depends only Fig. 4.9: Schematic to explain the signifi-
only the same parameters,
on the Mach number creates the con- cances of the Impulse function
venience for calculation the net forces
action on any device. Thus, defining the Impulse function as
(4.93)
In the Impulse function when ) is denoted as

(4.94)
(

The ratio of the impulse function is defined as

see function (4.92)
(4.95)

This ratio is different only in a coefficient from the ratio defined in equation (??)

which make the ratio function of and the Mach number. Hence, the net force is

(4.96)