020 - pr 05 - nonlinear diffusion: In non-linear diffusion; the diffusion coefficient, D, depends on the number 1 a density of diffusing particles : D = D(n) = D (n( x, t )) . Suppose that: D = D0 n . −∇ • j and Fick's (a) write down the diffusion equation in this case starting from the continuity equation, nɺ = −∇ law j = − D ⋅ ∇n . a a nɺ = −∇ • (− D ⋅ ∇n) = ∇ • (ψ a) = a • ∇ψ + ψ ∇ • a = ∇ n • ∇ ( D0 n ) + D0n ⋅ ∇ 2n
(
= D0 an
a −1
a
2
)
⋅ ∇n • ∇n + n ⋅ ∇ n =
(∇n) 2 D0 n a n a
+∇
2
(∇n)2 n = D (n ) a n
2
+∇ n
(1.1)
For a = 0 , we obviously have Fick’s law, nɺ = D (n) ⋅ ∇ 2 n = D0 ⋅ ∇ 2 n . b) using dimensional analysis, determine the dependence of the “characteristic front” of the density profile on t. 2
Parameters: the a-dependence comes in through D0. For diffusion in D-dimensions , [nɺ ] 2 −1 2+ aD −1 a − aD [ D] = 2 = L T = [ D0 ][ n ] = [ D0 ][ L ] → [ D0 ] = L T ; [∇ ][ n]
(1.2)
3
An invocation of the Buckingham pi theorem then requires us to solve a 2x2 system, and this appears as, {parameters} = { D0 , t, R}; R = R( t ); ); [ R] = L1 = [ D0 ]α [t ]β = L2α + aDα T −α T β →
2α + aDα = 1
α −β
=0
→
2α + aDα = 1
β
= α = (2 + aD)
−1
→ R = R (t ) ~
2 + aD
D0t = 4 + 2 aD D0 2t 2
(1.3)
Example: Suppose we wanted to model an explosion in three dimensions as nonlinear diffusion. Then, we would have D = 3 and thus 4 + 2a ⋅ 3 = 5 ↔ a = 16 , and thus D0 2 = E 0 / ρ 0 , whose dimensions are verified to
work out as
D02 ρ 0 E 0
4+ 2( 2 ( 3/ 6 )
= L
T − 2 ML−3 −2 ML T 2
4 +1− 3+ 2
= L
0
=L
.
Example: For a = 0 , we have the familiar R(t ) ~
Dt , as can be seen directly from (1.3).
lly, for a ≠ 0 , this “front” happen happenss to be a real one: one: there is a sharp sharp boundary boundary between between Afterword: Accidenta Afterword: Accidentally, the regions of finite and zero densities. c) Analyze the behavior of the front for all possible a (both positive and negative). Some cases: a = ±1, ±2, ±3; D = 1, 1, 2, 2, 3 . See if the switch from D = 1 to D = 2 to D = 3 produces any interesting results,
1
A more realistic and quite common situation is diffusion of heat: temperature T(r, t) plays the role of the number density. An “oops” for using the symbol D to indicate both dimensions and diffusion coefficient; however, contexts are obvious. 3 Those of you who need to use Mathematica at this point should be ashamed of your math skills. 2
a 1ê6 Hexplosion L =
a 1 =
a 2 =
2.0
a 3 =
a
=-
a
=-
1
a
=-
2 3
a 0 Hlinear diffusion L
o i s u1.5 f f i d r a e n i l n o n , 31.0
=
=
D , t R L H
0.5
0.0 0
1
2
3
4
5
t
(1.4) 2.5
a 1ê6 Hexplosion L =
a 1 =
a 2 =
a 3 =
2.0
o i s u f f i d r a 1.5 e n i l n o n , 2
a
=-
1
a
=-
a
=-
2 3
a 0 Hlinear diffusion L =
=1.0
D , t R L H
0.5
0.0 1
2
3
4
5
t
(1.5)
3.5
a a a a a a a a
3.0
o i s 2.5 u f f i d r a e n 2.0 i l n o n , 11.5
=
=
=
=
1ê6 Hexplosion L 1 2 3 1 2 3 0 Hlinear diffusion L
=-
=-
=-
=
=
D , t 1.0 R L H
.5
.0 1
2
3
4
5
t
(1.6) Discuss the physical origin of all divergences or irregularities you may encounter. For 4 + 2aD = 0 ↔ a = − 2 / D , the R(t) diverges; e.g., D = (1,2,3) → a ≠ (−2, −1, −32 ) are the forbidden values of
a for nonlinear diffusion. Recall that (1.2) says [ D0 ] = L2 + aDT −1 ; notice these coincide with solutions to 2 + aD = 0 . Recall, also, that any physical/dimensionful quantity α is written as, α = [dimensions]× [scaling function of dimensionless argument]= α 0 × f (α ′ )
(1.7)
To set 2 + aD = 0 would be to remove dimensions of length from the prefactor D0 (recall: Eq. (1.1) used a a −a D = D0 n = D0 ℓ ( n / a ) ), and a characteristic length
ℓ
would not exist. Naturally, a divergence would occur
since length-dimensions would be meaningless. This tells us our scaling function f (α ′ ) must have some sort of dependence such that the divergence is cancelled, for physical meaningfulness. (That is to say: the values D = (1, 2,3) → a ≠ (−2, −1, −32 ) really aren’t forbidden at all; we could have such a physical process, but it would 4
be poorly modeled by dimensional analysis alone ).
Appendix I – dimensional analysis for blast-front of strong explosion
(1) Energy of the bomb, (2) density5 of the air (3) time (obviously, as the blast-front evolves [quickly] in time). Indicate all dimensions,
4
See http://xkcd.com/687/ Explosion is considered spherical front of hot gas effusing (quickly) into cold gas, you have hot temperature and low temperature, forming a pressure gradient. 5
[ parameters ] = { R, E0 , ρ 0 , t};
R = R( t ) = [blast radius];
E0 = [energy released in explosion] ρ 0 = [density of air]; t = [time];
(1.8)
Assert that a “power law ansatz” that has the correct dimensions will capture the thing of interest to within an order of magnitude, but fail to capture any characteristic scales6, R ≡ R (t ) ~ R ( E0 , ρ 0 ) ⋅ t c ≡ E0 a ρ 0b t c = E0a ρ 0b ⋅ F ( ρ 0 / t c ); [ R] ≡ [ L]1 ≡ [ E0 ]a [ ρ 0 ]b [ t]c = [ M ]a +b [ L]2a −3b [ t ]c − 2a ;
1 = 2 a − 3b 2 → 0 = a+b ↔ 1 0 = c − 2a −2
6
−3
0 a
1
0
0
b 1 c
1 = 0 0
a ↔ b c
15 E 2 1/5 −1/5 2/5 t = 5 0 t ; = −51 → R = R (t ) ~ E0 ρ 0 ρ 0 25
Recall: power-laws don’t have any characteristic scales. See MM 07 - 007 – functions.
(1.9)
