Page 83 - 48Fundamentals of Compressible Fluid Mechanics

4.2. ISENTROPIC CONVERGING-DIVERGING FLOW IN CROSS SECTION 45

from the mass balance equation (4.26) and
utilizing it in equation (4.28) reads
Obtaining an expression for

(4.30)

, can be replaced by the static

yields
Rearranging equation (4.30) so that the density,

pressure,

(4.31)

and substitute the speed of sound into equation (4.31)

to obtain

Recalling that
(4.32)

&

Or in a dimensionless form (
(4.33)

Equation (4.33) is a differential equation for the pressure as a function of the cross

section area. It is convenient to rearrange equation (4.33) to obtain a variables

separation form of
(4.34)

4.2.1.1 The pressure relationship to Mach number
Before going further in the mathematical derivation it is worth in looking at the
is always positive (because

can
and Mach number. The meaning of

physical meaning equation (4.34). The term

the sign change for the pressure differential is that the pressure can increase or
all the three terms can be only positive). Now, it can be observed that
be positive or negative deepening on the
decrease. It can be observed that the critical Mach number is one. If the Mach
. If Mach number is

has opposite sign of
number is larger than one than