Preliminary: the partitioning into dependent and independent units, a = a (a1 , a2 , ..., an ); ∃{a1 , a2 , ..., ak
(1.1)
the pi theorem, Π≡
a p 1
a ...ak
r
=
a k1′′ k 2′′ 1 2
a a
k k ′′ k
...a
=
a
∏
k ′= k k ′=1
ak ′
kk′′′
=
a (a1 , a2 , ..., an )
∏
k′=k k ′ =1
ak ′
kk′′′
k′
k ′
pk +1
=
a (a1 , ..., ak ; Π1 a1
r
p
r
...ak k +1 ; ...; Πn − k a1 n ...ak n )
∏
k ′=k k ′ =1
ak ′
k k ′′′
(1.2)
k ′
≡ F (a1 , ..., ak ; Π1 , ..., Π n − k ) → a = a (a1 , a2 , ..., an ) = a1 1 a2 2 ...ak k ⋅ F ( Π1 , Π 2 , ..., Π n− k ) 1
Example: strong explosion goes off . How does blast front move in time, given the explosion is strong? Step 1: quantify the thing you’re computing, the blast-radius vs. time, R = R(t ) , where R(0) = 0 (ah, already, this 2 captures some physics). Step 2: What are parameters at work? (1) Energy of the bomb, (2) density of the air (3) time (obviously, as the blast-front evolves [quickly] in time). Indicate all dimensions, [ parameters ] = { R, E0 , ρ 0 , t } ; R = R (t ) = [blast radius]; (1.3) E0 = [energy released in explosion] ρ 0 = [density of air]; t = [time];
Step 3: separate things into independent and dependent parameters; i.e., effect the partitioning (1.1), {dependent} = { E0 , ρ 0}; {independent} = { M , t, L} = {mass, ti time, le length};
(1.4)
Step 4: write how the dependent parameters depend on the independent ones, − − [ E0 ] = [ E0 ]( R, t , ρ0 ) = [M ][L ]2 [t ] 2 ; [ρ 0 ] = [ρ 0 ](R , t ) = [M ][L ] 3 ;
(1.5)
Step 5: assert that a “power law ansatz” that has the correct dimensions will capture the thing of interest to 3 within an order of magnitude, but fail to capture any characteristic scales , 1 2 a −3 b c a b c a b c a b c a +b R ≡ R (t ) ~ R ( E0 , ρ0 ) ⋅ t ≡ E0 ρ 0 t = E 0 ρ 0 ⋅ F ( ρ 0 / t ); [ R ] ≡ [ L ] ≡ [ E0 ] [ ρ 0 ] [t ] = [ M ] [ L] [t ]c − 2 a ;
1 = 2a − 3b 2 −3 0 a 1 a 15 E 1 0 b = 0 ↔ b = −51 → R = R (t ) ~ E01/ 5 ρ 0 −1/ 5t 2/ 5 = 5 0 t 2 ; → 0 = a+b ↔ 1 ρ 0 0 = c − 2a −2 0 1 c 0 c 25 Plot, and see if you’ve found all qualitative behavior: (1) t = 0 behaviour, (2) the expanding front, etc.;
1
Taken directly from research on the theoretical yield of an atomic bomb. since explosion is modeled as spherical front of hot gas effusing (quickly) into cold gas, you have hot temperature and low temperature, forming a pressure gradient. Temperature inside (hot) should be function of E0, so possibly not independent. Temperature outside is independent parameter (in which mass-density of air, specific heat, etc., will be subsumed) 3 Recall: power-laws don’t have any characteristic scales. See MM 07 - 007 – functions. 2
(1.6)
R = R Ht L
3.0
2.5
E0 = r0 = 1
2.0
E0 = 100
Hhi -EL
r0 = 100
Hthick air L
1.5
1.0
0.5
t 0.5
1.0
1.5
2.0
basically, units says “you can only combine all parameters in one way; mathematically speaking, you can find two solutions: (1) expanding front (2) spherical blast-front (manifest in positing a blast-radius), (3) scale not captured due to power law, (4) to within an order of magnitude, the muffling-effect of thick/dense air has the 4 same effect as cranking up the energy of the blast . Source of the problem: essentially, this is nonlinear fluid dynamics. This is a hard problem (nonlinear fluid dynamics equation, etc.). Before this method, nobody looked at strong explosions, since that was considered a difficult problem. This particular solution was most welcome.
4
However: if an explosion happens in vacuum, it’s likely th at the blast won’t propagate infinitely-fast—some lost physics.
