Cinématique des uides Les points du cours à connaître

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d~v
dt ~v ·−−→
grad~v
−−→
grad v2
2−→
rot ~v ∧~v
div~v = 0
−→
rot ~v =~ 0
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Dg
Dt =∂ ∂t +−→
v .−−→ grad.g
~v
−→
d` ~v ∧−→ d` =~ 0−→ d` =k.~v
(X(t), Y (t), Z(t)) ~v
dX(t)
dt =vx(X(t), Y (t), Z(t), t)
div (~v) = 1
µ1.µ2.µ3∂(µ2.µ3.v1) ∂s1
+∂(µ3.µ1.v2)
∂s2
+∂(µ1.µ2.v3)
∂s3
ZZ
~v.−−→ d2S=ZZZ div (~v).d3τ
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−→
rot (~v) = 
1
µ2.µ3h∂(µ3.v3) ∂s2−∂(µ2.v2) ∂s3i
1
µ3.µ1h∂(µ1.v1) ∂s3−∂(µ3.v3) ∂s1i
1
µ1.µ2h∂(µ2.v2) ∂s1−∂(µ1.v1) ∂s2i
I~v.−→
d` =ZZ −→ rot (~v).−−→ d2S
φ ~v =−−→ grad (φ)
−−→
grad (φ) = 
1
µ1.∂φ ∂s1
1
µ2.∂φ ∂s2
1
µ3.∂φ ∂s3
Zb
a
~v−→ d` =Zb
a
−−→
grad (φ)−→ d` =φ(b)−φ(a)
Dm=RRSµ.−→ v .−−→ d2S Dv=RRS−→ v .−−→ d2S
•t
•t+dt
DM
Dt = 0
•t
•t+dt
D~ P
Dt = Σ ~
Fext
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•t
•t+dt
DEc
Dt = ΣPext +
ΣPint = ΣPext
x < 0y > 0
~v(~r, t) = −k.x.~ux+k.y.~uy
~a =k2.~r
~v(~r, t) = +k.x.~ux+k.y.~uy
~a =k2.~r
X=X0.(1 + b.t)
Y=Y0
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