The question has had an answer two hours ago, 12/31/2024 1056 CDT.
And it's quite a good answer, albeit misdirected for the purpose of this thread. Najak asked that forum a different question than he asked this one. He asked that forum for a clarification of steady-state operation only, and got one—along with the standard simplifications. He asked this forum to explain a transient observation, which naturally requires us to consider transient phenomena that the simplifications don't cover. So any dreams he might have had of cribbing that answer for the question here falls into the same pitfall as demanding that simplified, straightforward rules explain the uncommon.
At least in that forum, where he's not overtly peddling conspiracy theories, he's willing to confess his ignorance. And there, as here, people first have to disabuse him of the wacky notions he's concocted from his desire to build his own Chartres from children's wooden blocks. Here he stomped and whined until I reminded him of the list of factors we can consider for LM liftoff performance. As usual he's latched onto a new one, conjured up a new straw man, and Gish-galloped it down a completely new rabbit hole that we're going to have to undo when we get there.
The endorsed answer accurately depicts nominal steady-state thrust in three escalating degrees of complexity. The first, 𝐹 = 𝑐
𝑓𝑃
𝑐𝐴
𝑡 is, as stated, the first-order approximation: throat area times chamber pressure, scaled by a coefficient. As explained—and as is the case any time "coeffiicient of ____" appears in engineering, the coefficient is the first-order approximation for a number of factors that under other circumstances may behave in complex ways, but under the conditions appropriate to the use of this simplification they may be approximated by a single number. One of the conditions given as being hidden behind this approximation is unideal expansion. That's exactly part of the transient phenomena we're considering in this thread, which is not about normal, steady-state operation but rather something that occurs under very rare conditions where fluid expansion can produce a number of uncommon effects. Rocketry has traditionally assiduously avoided those conditions, which is why they're not part of the standard canon.
The second form, 𝐹 = 𝑚˙𝑣
𝑒+(𝑃
𝑒−𝑃
𝑎)𝐴
𝑒 (where
m-dot doesn't seem to want to paste accurately) is a second-order approximation that allows two important terms to vary separately. The author was kind enough to provide a bolded clarification that disabuses Najak of his conflation between thrust chamber pressure and pressure thrust (as considered collateral with momentum thrust). This equation is especially important where 𝑃
𝑎 varies significantly. It's the classic form because it gives the most accurate answer for the least math.
The third form, ∫𝑃
𝑥 𝑑𝐴, is the ultimately correct answer in the sense that it correctly and completely expresses the actual physics. Calculus is entitled to make people's eyes glaze over, but this is just a mathsy way of saying "The vector sum of all pressure parallel to the direction of travel." 99 times out of 100 that's overkill, and one of the approximations would be simpler and safer. LM liftoff is that hundredth case, and why we have to belabor the solution. The author of the answer basically gives away the method we're reaching for here; we'll see if Najak can think outside the box. That's why we're working with the underlying thermodynamics: because the unconventional ways in which the pressures act require us to start with the basic elements of pressure-volume work. And that's why we started with understanding the gap through which exhaust gases will escape while the LM is just lifting off. ∫𝑃
𝑥 𝑑𝐴 is the solution we need, and as the author warned, it's a "nasty integral." It's about to get monumentally more nasty, which is why we're approaching with caution and ensuring that we have solid footing in every step of that approach.
LM stability is only tangentially part of this thread, and it's indeed gratifying and amusing that someone linked to my explanation (which seems to be missing a key diagram). I understand why that forum downvoted it in their context, but the endorsed answer hints at the solution I invited the reader to consider by looking at the front view of the LM with its principal mass elements collocated. The general form of the solution to finding center of mass, center of gravity (which in terms of orbits can be different), center of pressure, and moment of inertia are "nasty integrals." But there are some coarser techniques that apply such as considering the moments provided by only the high-order masses. It's still math, but less nasty.