Description des écoulements
v(r 0, t 0, t) r 0, t 0
Description lagrangienne
Trajectoire des particules de fluide
Description des écoulements
v(r 0, t 0, t) r 0, t 0
u(r , t 0 ) u(r 0, t 0 )
u(r , t) u(r 0, t)
Description lagrangienne
Description eulerienne
Trajectoire des particules de fluide
Lignes de courant à t 0
Lignes de courant à t
Un exemple de champ de vitesse eulérienne
Vecteurs vitesse
Lignes de courant
Conservation de la masse dans un écoulement
Conservation de la masse dans un écoulement
Ω
dS n
S = ∂Ω
u
u
n
n
u.n
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Volume entrant par unité de surface
par unité de temps
n
u dt
u
u.n dS dS
Conservation de la masse dans un écoulement
Ω
dS n
S = ∂Ω
u
u
n
n
Z
@ ⌦
⇢u.n dS + Q
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
u.n
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Volume entrant par unité de surface
par unité de temps
Conservation de la masse dans un écoulement
Ω
dS n
S = ∂Ω
u
u
Z
⌦
@⇢
@ t dV =
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
n
n
Z
@ ⌦
⇢u.n dS + Q
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
u.n
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Volume entrant par unité de surface
par unité de temps
Z
⌦
@⇢
@ t dV =
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Z
@ ⌦
⇢u.n dS + Q
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Z
⌦
@⇢
@ t dV =
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Z
@ ⌦
⇢u.n dS + Q
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Z
⌦
@⇢
@ t dV + Z
⌦ r .(⇢u)dV = Q
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
r
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>. ⌘
divergence
Z
⌦
@⇢
@ t dV =
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Z
@ ⌦
⇢u.n dS + Q
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Z
⌦
@⇢
@ t dV + Z
⌦ r .(⇢u)dV = Q
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
@⇢
@ t + ⇢ r .u + u. r ⇢ = 0
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
En l’absence de source de masse r<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> . ⌘
divergence
gradient
<latexit sha1_base64="5z2iFyo9csDsWOJJwK4wPDs17DQ=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0WPRi8cKthaaUDbbSbt0s0l3N4VS+ju8eFDEqz/Gm//GbZuDtj4YeLw3w8y8MBVcG9f9dgpr6xubW8Xt0s7u3v5B+fCoqZNMMWywRCSqFVKNgktsGG4EtlKFNA4FPoWDu5n/NEKleSIfzTjFIKY9ySPOqLFS4EsaCkp8HGZ81ClX3Ko7B1klXk4qkKPeKX/53YRlMUrDBNW67bmpCSZUGc4ETkt+pjGlbEB72LZU0hh1MJkfPSVnVumSKFG2pCFz9ffEhMZaj+PQdsbU9PWyNxP/89qZiW6CCZdpZlCyxaIoE8QkZJYA6XKFzIixJZQpbm8lrE8VZcbmVLIheMsvr5LmRdW7qroPl5XabR5HEU7gFM7Bg2uowT3UoQEMhvAMr/DmjJwX5935WLQWnHzmGP7A+fwBnnmSAA==</latexit>
r ⌘
Z
⌦
@⇢
@ t dV =
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Z
@ ⌦
⇢u.n dS + Q
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Z
⌦
@⇢
@ t dV + Z
⌦ r .(⇢u)dV = Q
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
@⇢
@ t + ⇢ r .u + u. r ⇢ = 0
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
En l’absence de source de masse
Si le fluide est « incompressible », masse volumique constante
r
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>.u = 0
r
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>. ⌘
divergence
gradient
<latexit sha1_base64="5z2iFyo9csDsWOJJwK4wPDs17DQ=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0WPRi8cKthaaUDbbSbt0s0l3N4VS+ju8eFDEqz/Gm//GbZuDtj4YeLw3w8y8MBVcG9f9dgpr6xubW8Xt0s7u3v5B+fCoqZNMMWywRCSqFVKNgktsGG4EtlKFNA4FPoWDu5n/NEKleSIfzTjFIKY9ySPOqLFS4EsaCkp8HGZ81ClX3Ko7B1klXk4qkKPeKX/53YRlMUrDBNW67bmpCSZUGc4ETkt+pjGlbEB72LZU0hh1MJkfPSVnVumSKFG2pCFz9ffEhMZaj+PQdsbU9PWyNxP/89qZiW6CCZdpZlCyxaIoE8QkZJYA6XKFzIixJZQpbm8lrE8VZcbmVLIheMsvr5LmRdW7qroPl5XabR5HEU7gFM7Bg2uowT3UoQEMhvAMr/DmjJwX5935WLQWnHzmGP7A+fwBnnmSAA==</latexit>
r ⌘
Fluide comme “incompressible” ?
Fluide comme “incompressible” ?
⇢
⇢ = 1
⇢
@⇢
@ p p
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Fluide comme “incompressible” ?
⇢
⇢ = 1
⇢
@⇢
@ p p
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
⇢
⇢ = 1
⇢c 2 p
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
: vitesse du son
<latexit sha1_base64="IAXJ5b2I81udfgHAcvJ7DyY45fo=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0ZMUvHhswX5AG8pmO2nXbjZhdyOU0F/gxYMiXv1J3vw3btsctPXBwOO9GWbmBYng2rjut1NYW9/Y3Cpul3Z29/YPyodHLR2nimGTxSJWnYBqFFxi03AjsJMopFEgsB2M72Z++wmV5rF8MJME/YgOJQ85o8ZKDdYvV9yqOwdZJV5OKpCj3i9/9QYxSyOUhgmqdddzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1IQ3fsZlkhqUbLEoTAUxMZl9TQZcITNiYgllittbCRtRRZmx2ZRsCN7yy6ukdVH1rqpu47JSu83jKMIJnMI5eHANNbiHOjSBAcIzvMKb8+i8OO/Ox6K14OQzx/AHzucPxoOM5w==</latexit>
c
Fluide comme “incompressible” ?
⇢
⇢ = 1
⇢
@⇢
@ p p
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
⇢
⇢ = 1
⇢c 2 p
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
En écoulement dominé par l’inertie du fluide:
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>p ⇠ ⇢u 2
⇢
⇢ ⇠ u 2
c 2 = M 2
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
: Nombre de Mach : vitesse du son
<latexit sha1_base64="IAXJ5b2I81udfgHAcvJ7DyY45fo=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0ZMUvHhswX5AG8pmO2nXbjZhdyOU0F/gxYMiXv1J3vw3btsctPXBwOO9GWbmBYng2rjut1NYW9/Y3Cpul3Z29/YPyodHLR2nimGTxSJWnYBqFFxi03AjsJMopFEgsB2M72Z++wmV5rF8MJME/YgOJQ85o8ZKDdYvV9yqOwdZJV5OKpCj3i9/9QYxSyOUhgmqdddzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1IQ3fsZlkhqUbLEoTAUxMZl9TQZcITNiYgllittbCRtRRZmx2ZRsCN7yy6ukdVH1rqpu47JSu83jKMIJnMI5eHANNbiHOjSBAcIzvMKb8+i8OO/Ox6K14OQzx/AHzucPxoOM5w==</latexit>
c
<latexit sha1_base64="qiSGP287cgZFWaq8XXo6g1RMCMk=">AAAB6HicdVDJSgNBEK2JW4xb1KOXxiB4CjOi6EkCXrwICZgFkiH0dGqSNj09Q3ePEIZ8gRcPinj1k7z5N3YWIW4PCh7vVVFVL0gE18Z1P5zc0vLK6lp+vbCxubW9U9zda+g4VQzrLBaxagVUo+AS64Ybga1EIY0Cgc1geDXxm/eoNI/lrRkl6Ee0L3nIGTVWqt10iyWv7E5B3F/kyyrBHNVu8b3Ti1kaoTRMUK3bnpsYP6PKcCZwXOikGhPKhrSPbUsljVD72fTQMTmySo+EsbIlDZmqixMZjbQeRYHtjKgZ6J/eRPzLa6cmvPAzLpPUoGSzRWEqiInJ5GvS4wqZESNLKFPc3krYgCrKjM2msBjC/6RxUvbOym7ttFS5nMeRhwM4hGPw4BwqcA1VqAMDhAd4gmfnznl0XpzXWWvOmc/swzc4b5+mq4zS</latexit>
M
Fluide comme “incompressible” ?
⇢
⇢ = 1
⇢
@⇢
@ p p
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
⇢
⇢ = 1
⇢c 2 p
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
En écoulement dominé par l’inertie du fluide:
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>p ⇠ ⇢u 2
⇢
⇢ ⇠ u 2
c 2 = M 2
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
: Nombre de Mach : vitesse du son
Fluide quasi incompressible
M
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>⌧ 1
<latexit sha1_base64="IAXJ5b2I81udfgHAcvJ7DyY45fo=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0ZMUvHhswX5AG8pmO2nXbjZhdyOU0F/gxYMiXv1J3vw3btsctPXBwOO9GWbmBYng2rjut1NYW9/Y3Cpul3Z29/YPyodHLR2nimGTxSJWnYBqFFxi03AjsJMopFEgsB2M72Z++wmV5rF8MJME/YgOJQ85o8ZKDdYvV9yqOwdZJV5OKpCj3i9/9QYxSyOUhgmqdddzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1IQ3fsZlkhqUbLEoTAUxMZl9TQZcITNiYgllittbCRtRRZmx2ZRsCN7yy6ukdVH1rqpu47JSu83jKMIJnMI5eHANNbiHOjSBAcIzvMKb8+i8OO/Ox6K14OQzx/AHzucPxoOM5w==</latexit>c
<latexit sha1_base64="qiSGP287cgZFWaq8XXo6g1RMCMk=">AAAB6HicdVDJSgNBEK2JW4xb1KOXxiB4CjOi6EkCXrwICZgFkiH0dGqSNj09Q3ePEIZ8gRcPinj1k7z5N3YWIW4PCh7vVVFVL0gE18Z1P5zc0vLK6lp+vbCxubW9U9zda+g4VQzrLBaxagVUo+AS64Ybga1EIY0Cgc1geDXxm/eoNI/lrRkl6Ee0L3nIGTVWqt10iyWv7E5B3F/kyyrBHNVu8b3Ti1kaoTRMUK3bnpsYP6PKcCZwXOikGhPKhrSPbUsljVD72fTQMTmySo+EsbIlDZmqixMZjbQeRYHtjKgZ6J/eRPzLa6cmvPAzLpPUoGSzRWEqiInJ5GvS4wqZESNLKFPc3krYgCrKjM2msBjC/6RxUvbOym7ttFS5nMeRhwM4hGPw4BwqcA1VqAMDhAd4gmfnznl0XpzXWWvOmc/swzc4b5+mq4zS</latexit>
M
M
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>⇠ 1 M >
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>1
Ondes de choc
ou
dS n
Ω
S = ∂Ω
n
u
u n
Conservation de la quantité de mouvement
Z
⌦
@⇢u
@ t dV =
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Z
⌦
⇢f dV
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
+ Z
@ ⌦
.n dS
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Z
@ ⌦
⇢u u.n dS
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
scalaire
vecteur contraintes
i
j k T(n)
T(n) : vecteur contrainte totale sur la face de normale n
n
δA -i
δA i
-j
δA j
-k δA k
T(n) A
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Forces de surface
T(
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>i) A i T( j) A j
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
T( k) A k
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
T(
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>i) = T(i) A i = A i.n
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
i
k T(n)
n
δA -i
δA i
-j
δA j
-k δA k
T(n) A
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Forces de surface
T(i) i.n A
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
T(j) j.n A
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
T(k) k.n A
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
F surface = P
T A
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
pfd sur l’élément de volume :
⇢ V a = ⇢ f V + X
T A
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
⇢ a = ⇢ f + X
T A V
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
A
V ⇠ 1
L ! 1
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
L
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>! 0 X T = 0
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
T(n) =
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>T(i) i.n + T(j) j.n + T(k) k.n
T(n) = T(i) n x + T(j) n y + T(k) n z
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
T x (n) = xx n x + xy n y + xz n z
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
=
0
@ xx yx xy yy xz yz
zx zy zz
1 A
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Tenseur des contraintes
contraintes normales
contraintes tangentielles
T x (n) = T x (i) n x + T x (j) n y + T x (k) n z
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Composante x
T
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>= .n 0
@ T x T y T z
1
A =
0
@ xx yx xy yy xz yz
zx zy zz
1 A
0
@ n x n y n z
1 A
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Vecteur contrainte (force/unité de
surface)
Tenseur des contraintes
Normale à la
surface considérée
Z
⌦
@⇢u
@ t dV =
Z
@ ⌦
⇢u u.n dS + Z
⌦
⇢f dV + Z
@ ⌦
.n dS
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Z
⌦
@⇢u
@ t dV + Z
⌦ r .⇢uu dV = Z
⌦
⇢f dV + Z
⌦ r . dV
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Théorème de la divergence:
Conservation de la quantité de mouvement
: Flux de quantité de mouvement,
<latexit sha1_base64="8AJo72ZtQEcBA0U2dutJ5sIFyn4=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiSi6LLoxmUF+4AmlMl00g6dTMI8CiX0T9y4UMStf+LOv3HSZqHVAwOHc+7lnjlRxpnSnvflVNbWNza3qtu1nd29/QP38KijUiMJbZOUp7IXYUU5E7Stmea0l0mKk4jTbjS5K/zulErFUvGoZxkNEzwSLGYEaysNXDeQ4xQFCdbjKM6NmQ/cutfwFkB/iV+SOpRoDdzPYJgSk1ChCcdK9X0v02GOpWaE03ktMIpmmEzwiPYtFTihKswXyefozCpDFKfSPqHRQv25keNEqVkS2ckiolr1CvE/r290fBPmTGRGU0GWh2LDkU5RUQMaMkmJ5jNLMJHMZkVkjCUm2pZVsyX4q1/+SzoXDf+q4T1c1pu3ZR1VOIFTOAcfrqEJ99CCNhCYwhO8wKuTO8/Om/O+HK045c4x/ILz8Q3LlJPE</latexit>
⇢uu
⇢u i u j
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
tenseur de composantes produit tensoriel
@
@ x j (⇢u i u j ) = u i @⇢u j
@ x j + ⇢u j @ u i
@ x j
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Sa divergence :
(somme sur j)
r . = P
j
@ ij
@ x j
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Divergence des contraintes
Conservation de la quantité de mouvement
⇢ @ u
@ t + u
✓ @⇢
@ t + r .⇢u
◆
+ ⇢u. r u = ⇢f + r .
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
⇢ @ u
@ t + u @⇢
@ t + u r .⇢u + ⇢u. r u = ⇢f + r .
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Pour un volume élémentaire de fluide :
⇢
✓ @ u
@ t + u. r u
◆
= ⇢f + r .
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
ma = F
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
= 0
Conservation de la masse
Accélération d’un élément de fluide
⇢
✓ @ u
@ t + u. r u
◆
= ⇢f + r .
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
ma = F
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
a = @ u
@ t + u. r u
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
instationnarité accélération convective
un écoulement stationnaire où u.<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> r u 6 = 0
Écriture de l’accélération convective
Variation spatiale de u x projetée sur la vitesse (u x , u y , u z ) Pour la composante i : u j @ u i
@ x j = X
j =1,3
u j @ u i
@ x j
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Pour la composante x : u x @ u x
@ x + u y @ u x
@ y + u z @ u x
@ z
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Les contraintes dans un fluide
Dans un fluide à l’équilibre, sans écoulement macroscopique
n La résultante des interactions entre atomes ou molécules est une pression isotrope
=
0
@ p 0 0
0 p 0
0 0 p
1 A
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
ij = p ij
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
r . = r p =
0 B @
@ p
@ x
@ p
@ y
@ p
@ z
1 C A
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
⇢
✓ @ u
@ t + u. r u
◆
= ⇢f + r .
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Dans un fluide à l’équilibre, sans écoulement macroscopique
0 =
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>⇢g r p p = p 0 ⇢gz
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
