How to phrase omissions correctly in a thesis?
You could explain one case in detail, then cover the other cases by listing what terms to substitute for the other cases. Especially if you're dealing with applying the same equation with different terminology, you could provide a table of which variable is named what in each case.
E.g.
Because the principles only differ in terminology, only the solid to liquid transition is covered in detail. The equivalent terminology for each transition is in table 3.
Find a good source of the appropriate discussion and include a reference. "This is discussed in detail in Smith, Jones and Brown, 2005."
The mathematicians have a good way of doing it: declare it trivial.
In your heater example, let's say that you demonstrated some results in a liquid. Then you start your section with:
In this section, we present results obtained from heating a liquid sample with a Widget2000. These results can be trivially extended to samples in a gas or solid phase. Since the liquid case is of particular interest for this thesis, other types of sample are not described in detail.
As long as your readers know very well that your statement is correct and intuitively agree that the other cases are trivial, without having to think about why they are equivalent, this is a good way to do it. If you have some doubt - e.g. because somebody has to have a certain level of background in your field to know that it is indeed trivial to demonstrate the equivalency - it is also advisable to add a citation of a source which discusses the equivalency.
Alternatively, if you think that your readers will disagree with the assessment of "trivial", but also don't have sources which show proof, you can write instead:
In this section, we present results obtained from heating a liquid sample with a Widget2000. Solid and gaseous samples show analogous behavior, but demonstrating a proof for that is outside of the scope of this thesis.
Or, if you want to play it safe
In this section, we present results obtained from heating a liquid sample with a Widget2000. We also assume that solid and gaseous samples show analogous behavior, which is a standard assumption in the field of heating samples by widgets [here you cite 2-3 popular papers which do the same handwaving as you do].
I know that this might sound strange, since there "trivial" also has a different, negative meaning in everyday language. In a scientific text, it means "it can be proven with somewhat tedious, but entirely straightforward work, without any gotchas". There is no negative valence attached to it in this sense, and it is used for exactly the purpose you want: to signify that you are streamlining your text and omitting the true but boring parts.