How Does the Poluly Equation Explain Speed and Pressure at a Microscopic Level?

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The discussion centers on understanding the Poluly equation, specifically its relationship between speed and pressure at a microscopic level, without delving into atomic reactions. The equation is viewed as elegant and simple, prompting curiosity about its underlying principles. Participants express uncertainty about whether the equation relates to the Navier-Stokes (N-S) equations, suggesting that the explanation should not be overly complex. The desire for a straightforward yet profound understanding of the equation is emphasized. Overall, the conversation seeks clarity on how the Poluly equation operates within a simplified model of particles.
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So many years.
I always confused in some principle(i mean sharp pointed)questiones.
first one is the biggest one.

How to explain the Poluly equation(i mean the relationship between the speed and pressure) in the level of microcosmical. (spoted! i mean the level of microcosmical but not consider the atomic reactions,we just consider it's a lot of small ball,as small as possball)

I think this equation is a very simple very beautiful equation.So there must be a very beautiful and very wonderful and very depth and very interesting reasones.
 
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Somebody says it's N-S equation.
I'm not sure.
The question can't be so complex as N-S question.
If it's,must be very simple,but comefrom N-S question.
 
good
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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