How are temperature differences handled in a jet engine?
1. The combustion chamber

Temperature here is usually controlled via design of the chamber shape, in particular of the location where combustion occurs vs the airflow.



Image source

In the image above you can see that the fuel line (2) terminates in a nozzle. The nozzle will turn the fuel into tiny droplets to increase the surface area and hasten the combustion process.

Indicated with (6) are instead the main air holes: the cool air coming from the compressor (on the left side) enters the chamber with high pressure from these holes. Note that the holes are distributed in an annular pattern: high pressure air will be entering from all sides of the chamber.


The result is that the flame will not directly touch the walls of the
chamber, we then got rid of conductive heat transmission.
Convective is a non-issue: cool air keeps coming and hot air goes
into the turbine.
Radiative heat transmission might be an issue, but the new air
incoming will cool the walls via conduction.


The engine can thus run without the combustion chamber melting. Note that the use of specific materials does greatly improve the conditions, as higher temperatures can be tolerated, and thus better efficiencies achieved.



2. The turbine cooling air

The air is usually bled from a stage of the compressor, usually a high pressure stage, in a similar manner as done for the air used for the passenger compartment.



3. Thermal stresses

a. Short term

Over the duration of a flight the blades will warm up when the engine is turned on and cool down at the end of the flight. Intermediate load variation do not have a significant impact.

Even the initial warming and the final cooling do not pose a significant problem. The only effect to be taken into account is that the rotating hot blades will become slightly longer: a special material is placed on the opposing stationary side so that the gap is minimal during operation and the losses due to the air going around the blades (instead of between them) are reduced.

b. Prolonged

The most problematic aspect here is the so called "creep": the blades, after prolonged use, will tend to change shape, because they tend to be more malleable at high temperatures, exactly when they are under high tensile load. This becomes a problem over the lifetime of an engine, not during a single flight.

To mitigate this problem the blades are nowadays manufactured as single crystals: each blade is a single solid block of metal and not made up af many tiny crystals as the metal tools we use everyday. This is because under tensile load each of the crystals can slide over its neighbors (and high temperatures facilitate this) changing the shape of the blade. To visualize it, think of a rubber band that does not go back to its original form.

Having a single crystal prevents this sliding, giving more strength to the blade.