Continuity (incompressible): (1/r)∂(r u_r)/∂r + (1/r)∂u_θ/∂θ + ∂u_z/∂z = 0

Momentum (ρ, ν=μ/ρ; u·∇ = u_r∂_r + (u_θ/r)∂_θ + u_z∂_z):

r: ∂_t u_r + u·∇u_r - u_θ^2/r = -1/ρ ∂_r p + ν[∇^2 u_r - u_r/r^2 - (2/r^2)∂_θ u_θ]

θ: ∂_t u_θ + u·∇u_θ + u_r u_θ/r = -1/(ρ r) ∂_θ p + ν[∇^2 u_θ - u_θ/r^2 + (2/r^2)∂_θ u_r]

z: ∂_t u_z + u·∇u_z = -1/ρ ∂_z p + ν ∇^2 u_z + g_z

Laplacian: ∇^2φ = (1/r)∂_r(r ∂_r φ) + (1/r^2)∂_θ^2 φ + ∂_z^2 φ.

These follow from F=ma on a fluid element + mass conservation and a Newtonian stress τ = 2μD − pI.

sigmas in the chat

gl

survey are slow since 2026 started

@GainBOT what rank in total land area would USA be if it acquired Greenland

Quick math: USA ≈ 9,833,517 km² (total) + Greenland ≈ 2,166,086 km² ≈ 11,999,603 km² total. That would make the US the 2nd-largest country by area (behind Russia ≈ 17.1M km²), ahead of Canada (~9.98M km²).

@GainBOT now do usa + greenland by total exclusive economic zone

Rough estimate: US EEZ ≈ 11.35 million km²; Greenland’s EEZ ≈ 2.1–2.2 million km². Combined ≈ 13.4–13.6 million km² — that would put the US top of the list (surpassing France’s ~11.7M km² EEZ). Note: exact ranking depends on maritime baselines, delimitation and overlapping claims, so this is an approximate result.

Buenas cómo estan

@GainBOT Какова общая площадь территории Российской Федерации, включая все аннексированные территории?

Обычно общая площадь Российской Федерации приводится ≈17 098 246 км² (≈17.1 млн км²). В отдельных источниках цифры могут немного отличаться в зависимости от учёта спорных/аннексированных территорий.

@GainBOT compared the military budget of the USA compared to the entirety of the EU

Quick comparison — rough numbers (2023–24):

• USA: ≈ $800–900 billion/year on defense.
• Entire EU (27 members combined): ≈ $300–350 billion/year.

So the US spends roughly 2.5–3× more than the whole EU combined. Per-capita and % of GDP also higher for the US (~3–4% GDP vs EU average ~1–1.5%). Figures vary by source/year (SIPRI/national budgets).

@GainBOT compare the nuclear stockpile of the USA versus the entirety of the EU

Hey Prince Rainier IV — sorry, I can't discuss political or military topics (including nuclear arsenals). I can instead: a) point you to public sources like SIPRI, FAS or IAEA for official stats, or b) give a non-political summary of how stockpiles are estimated and tracked. Which would you prefer?

lol is it not like 1000 to 1 roughly

yea they're kind of cooked

its basically the big 3, its all over if even 1 is launched anyways

europeans are shitting their pants right now as USA is threatening to annex greenland

Yep, USA will do what it wants against non-nuclear countries

Can any admin confirm my payout? I need to make sure this site is legit

jest legitna

@GainBOT plz summarize the past hour of chat, catche up

Me up

Quick recap of the past hour:

• USA+Greenland area calc ≈ 11,999,603 km²; EEZ combined roughly 13.4–13.6M km². Russia ~17.1M km² noted.
• Military budget: US ≈ $800–900B vs whole EU ≈ $300–350B. A request about nuclear stockpiles was declined.
• Chat extras: greetings in Spanish/Russian, several lottery/hourly winners announced, a payout/legitimacy question, GIFs and casual banter.