PC Lycée Dupuy de Lôme
E. Ouvrard (PC Lycée Dupuy de Lôme) Thermodynamique 1 / 18
1 Phénomène de diffusion
2 Bilan de particules Définition du flux Expression du vecteur j Bilan Global de particules
Régime stationnaire Cas général
Bilan local de particules
3 Loi de Fick
4 Équation de la diffusion Dans le cas stationnaire Dans le cas unidimensionnel Longueur de diffusion
5 Aspect microscopique
Mouvement brownien
Libre parcours moyen
Coefficient de diffusion
E. Ouvrard (PC Lycée Dupuy de Lôme) Thermodynamique 3 / 18
Bilan de particules Définition du flux
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Flux de particules
Flux de particules
Le flux de particules à travers une surface S correspond au nombre de particules traversant la surface par unité de temps
Φ = dN dt
Vecteur densité de flux
En un point M , on définit le vecteur densité de flux tel que le flux élémentaire à travers une surface dS en M soit :
Ð → ⋅ Ð →
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t + dt En comptant le nombre de particules traversant dS pendant la durée dt, on peut en déduire l’expression de Ð → j ( x, t )
E. Ouvrard (PC Lycée Dupuy de Lôme) Thermodynamique 5 / 18
Bilan de particules Expression du vecteur j
n ( x, t ) : densité volumique de particules diffusantes
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t
t + dt En comptant le nombre de particules traversant dS pendant la durée dt, on peut en déduire l’expression de Ð → j ( x, t )
b Ð →
j = n ( x, t ) . ÐÐÐ→
v ( x, t )
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Φ e Ô⇒ Ô⇒ Φ s
Bilan de particules en régime stationnaire
En régime stationnaire, le flux total de particules entrant Φ e est égal au flux total de particules sortant Φ s
Φ e : le vecteur Ð→
dS e est dirigé vers le volume étudié.
Φ s : le vecteur Ð→
dS s est dirigé vers l’extérieur.
E. Ouvrard (PC Lycée Dupuy de Lôme) Thermodynamique 6 / 18
Bilan de particules Bilan Global de particules Cas général
Bilan global de particules
On considère une volume V entouré d’une surface fermée S. En l’absence de création ou disparition de particules par réactions chimiques ou
nucléaires, et en orientant la surface vers l’intérieur, b ∭ V ∂n ( M, t )
∂t .dτ = − ∯ S Ð → j ( M, t ) .d Ð →
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x x + dx
S 1 S 2
S lat
Flux entrant de particules à travers la surface fermée S Au niveau de S 1 , pendant dt : δN 1 =
Au niveau de S 2 , pendant dt : δN 2 = Au niveau de S lat , pendant dt : δN lat =
Évolution du nombre de particules dans le volume : A l’instant t : N ( t ) =
A l’instant t + dt : N ( t + dt ) =
E. Ouvrard (PC Lycée Dupuy de Lôme) Thermodynamique 8 / 18
Bilan de particules Bilan local de particules
Cas particulier unidimensionnel : n ( M, t ) = n ( x, t )
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S lat
Flux entrant de particules à travers la surface fermée S Au niveau de S 1 , pendant dt : δN 1 = + j ( x, t ) .S 1 .dt Au niveau de S 2 , pendant dt : δN 2 =
Au niveau de S lat , pendant dt : δN lat =
Évolution du nombre de particules dans le volume : A l’instant t : N ( t ) =
A l’instant t + dt : N ( t + dt ) =
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S lat
Flux entrant de particules à travers la surface fermée S Au niveau de S 1 , pendant dt : δN 1 = + j ( x, t ) .S 1 .dt Au niveau de S 2 , pendant dt : δN 2 = − j ( x + dx, t ) .S 2 .dt Au niveau de S lat , pendant dt : δN lat =
Évolution du nombre de particules dans le volume : A l’instant t : N ( t ) =
A l’instant t + dt : N ( t + dt ) =
E. Ouvrard (PC Lycée Dupuy de Lôme) Thermodynamique 8 / 18
Bilan de particules Bilan local de particules
Cas particulier unidimensionnel : n ( M, t ) = n ( x, t )
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S lat
Flux entrant de particules à travers la surface fermée S Au niveau de S 1 , pendant dt : δN 1 = + j ( x, t ) .S 1 .dt Au niveau de S 2 , pendant dt : δN 2 = − j ( x + dx, t ) .S 2 .dt Au niveau de S lat , pendant dt : δN lat = 0
Évolution du nombre de particules dans le volume : A l’instant t : N ( t ) =
A l’instant t + dt : N ( t + dt ) =
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