Decreasing Area of Axial Compressor

in axial compressor, why the area is decreasing until it reaches the combustor?

and if the answer because of the pressure is inversely proportional to the volume. that's happened when the temperature is constant, and the temperature in not constant in axial compressor!

and if the answer because of Bernoulli statement, the area going decreasing and then the velocity will increase and the pressure will decrease!

Without entering into the details about how an axial compressor works, let's consider the following plot about how the characteristics of the airflow change in a typical turbojet engine:

We see that in the axial compressor:

1. pressure steadily increases from some 15 to 140 psia (i.e. from 10⁶ to 10⁷ Pa);
2. temperature rises from some 20 to 300 °C (i.e. 290 to 570 K);
3. velocity slightly decreases but we just consider it constant for simplicity.

Considering the simple ideal gas law (which retains its validity here) we calculate the density change between the beginning and the end of the compressor:

$$P=\rho RT \Rightarrow \rho=\frac{P}{RT} \Rightarrow \frac{\rho_{end}}{\rho_{begin}}=\frac{P_{end}T_{begin}}{P_{begin}T_{end}} = \frac{10⁷ \times 290}{10⁶ \times 570} = 5$$

That means that the air has decreased its volume by a factor 5: the section of the compressor must simply keep up with this reduction.

The following is a schematic (source this NASA report) of a J57-P-1 turbojet engine which, by a technological point of view, resembles the previous engine:

As visible in the table, the section area changes from 952 (section 3) to 5800 ft² i.e. of a factor 6, very close to our calculation. Actually if we scale this value taking into account the different pressure ratio of the two engines (11.5 for the J57 and 9.5 for the turbojet in the first picture) we get a practically equal density change $$6 \times \frac{9.5}{11.5}=5$$.