td mécanique fluide

D
D
h2h2
v2
A
2+pA ρ=v2
B
2+pB ρvA= 0 pA=
patm +ρg (z−h1)pB=patm vB=p2g(z−h1)
d=vBs=sp2g(z−h1)
•sp2g(h3−h1)< D
•sp2g(h2−h1)> D zmin sp2g(zmin −h1) = D
P2−P1

patm
ρ=p1 ρ+v2
avant
2
patm
ρ+v2
2
2=p2 ρ+v2
apres
2
vavant =vapres P2−P1=1
2ρv2
2
t−−−−→ pouvert(t) + Savantvavantdtρvavant
−→
ux
−−−−→
pouvert(t+dt)−Sapresvapresdtρvapres
−→
ux
Sapres =Savant vapres =vavant dp
dt = 0
Fhelice−jet =−Fpression = (P2−P1)S=1
2Sρv2
2
1
2Sρvavantv2
2=1
2Dmv2
0
e×l
v0
−→
ux
M
e1×l e2×l v1v2
e e1e2v0v1v2
e e1e2
e1e2
−→
F
−→
uz
d
α
lev0le1v1+le2v2ev0=e1v1+e2v2
p=p0v2
0
2+p0 ρ=v2
1
2+p0 ρ=v2
2
2+p0 ρ
v0=v1=v2e=e1+e2
t−−−−→ pouvert(t) + lev0dtρv0
−→
ux
−−−−→
pouvert(t+dt)−le1v1dtρv1
−→
n0+le2v2dtρv2
−→
n0
−−−−→
pouvert =−−→ cste d−−−−→
pf erme
dt =−le1v1ρv1
−→
n0+le2v2ρv2
−→
n0−lev0ρv0
−→
ux
•
lLp0
−→
n pressiontube ferme −pressioncontact plaque = 0 −pressioncontact plaque
∗ −F−→ n

−le1v1ρv1
−→
n0+le2v2ρv2
−→
n0−lev0ρv0
−→
ux=lLp0
−→
n−F−→ n−→ ux=cosα−→ n−sinα−→ n0
v0=v1=v2
(−e1+e2+esinα = 0
−leρv2
0cosα =lLp0−F⇔(e1/2=1
2e(1 ±sinα)
F=lLp0+leρv2
0cosα
Mpoids =−mgdsinα−→ uzMF=F l =l2Lp0+l2eρv2
0cosα
l2Lp0l2Lp0+l2eρv2
0cosα =Mgdsinα tanα 'l2eρv2
0
Mgd
R L
v∞
−→
ux
A
−→
v−→ v=−−→ gradφ
φ.
φ
φ
φ(−→ r) = Arcosθ +Bcosθ
r2+C r+D
−→
rot−→ v=−→ 0⇒−→ v=−−→ gradφ div−→ v= 0 ⇔∆φ= 0
div−→ E=−∆V= 0
V=q
4π0
1
r
−→
E=q
4π0
1
r2
−→
urdr
rdθ ∧Er 0= 0 ⇔
rErdθ = 0 ⇔θ=cste
V=p
4π0
cosθ
r2
−→
E=p
4π0
2cosθ/r3
sinθ/r3
dr
rdθ ∧
Er
Eθ= 0
dr
r= 2cosθdθ sinθ = 2d(sinθ) sinθ
⇒r=r0sin2θ
r cosθ =x

φ(−→ r) = Arcosθ +Bcosθ
r2+C r+D
−→
v=
Acosθ −2Bcosθ
r3−C r2
−Asinθ −Bsinθ
r3
0
=
v∞cosθ 1−R3
r3
−v∞sinθ 1 + R3
r3
0
−→
v(r=∞) = −→ v∞A=v∞
vr(r=R)=0∀θ2B=R2A C = 0
v2
2+p ρ=cste =v2
∞
2+p∞ ρp(R, θ) = p∞+ρ 2v2
∞1−4sin2θ=p∞+ρ
2v2
∞−2ρv2 ∞sin2θ
θ=π p(R, θ) = pApext(R, θ) = pA−2ρv2
∞sin2θ
LRdθ pALRdθ−→ ur−pA−2ρv2
∞sin2θLRdθ−→
ur
−→
uz
2ρv2
∞sin2θLRdθ (sinθ)
2ρv2
∞R´π
0sin2θsinθdθ =−2ρv2
∞R´π
01−cos2θd(cosθ)=2ρv2
∞Rh−cosθ +cos3θ
3iπ
0=8
3ρv2
∞R