Formule MFII vjezbe 01 - FSB Online · PDF fileFORMULE: 1. VJEŽBE MEHANIKA FLUIDA II 1 / 3...
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FORMULE: 1. VJEŽBE MEHANIKA FLUIDA II 1 / 3 Izvor: A. Werner: Odabrana poglavlja iz mehanike fluida-zbirka zadataka, FSB Zagreb OSNOVNI OPERATORI U KARTEZIJEVIM, CILINDARSKIM I SFERNIM KOORDINATAMA ⎪ ⎭ ⎪ ⎬ ⎫ = = = z z sin r y cos r x ϑ ϑ 2 2 r x y y arctg x z z ϑ ⎫ =+ + ⎪ ⎪ = ⎬ ⎪ = ⎪ ⎭ ⎪ ⎭ ⎪ ⎬ ⎫ = = = ψ ϑ ψ ϑ ψ cos R z sin sin R y cos sin R x ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎬ ⎫ = + = + + + = x y arctg z y x arctg z y x R ϑ ψ 2 2 2 2 2 Cilindarske koordinate Sferne koordinate TABLICA 1.1 Osnovne operacije deriviranja skalarnih ( s ) , vektorskih ( w , v G G ) i tenzorskih ( T ) polja u kartezijevim koordinatama k j i G G G G G G G z y x z z y y x x v v v e v e v e v v + + ≡ + + = k j i G G G G G G G z y x z z y y x x w w w e w e w e w w + + ≡ + + = ( ) ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = zz zy zx yz yy yx xz xy xx T T T T T T T T T T z v y v x v v z y x ∂ ∂ + ∂ ∂ + ∂ ∂ = ⋅ ∇ G 2 2 2 2 2 2 2 z s y s x s s ∂ ∂ + ∂ ∂ + ∂ ∂ = ∇ [ ] x s s x ∂ ∂ = ∇ [ ] z v y v v y z x ∂ ∂ − ∂ ∂ = × ∇ G [ ] z T y T x T zx yx xx x ∂ ∂ + ∂ ∂ + ∂ ∂ = T Div [ ] y s s y ∂ ∂ = ∇ [ ] x v z v v z x y ∂ ∂ − ∂ ∂ = × ∇ G [ ] z T y T x T zy yy xy y ∂ ∂ + ∂ ∂ + ∂ ∂ = T Div [ ] z s s z ∂ ∂ = ∇ [ ] y v x v v x y z ∂ ∂ − ∂ ∂ = × ∇ G [ ] z T y T x T zz yz xz z ∂ ∂ + ∂ ∂ + ∂ ∂ = T Div [ ] z w v y w v x w v w v x z x y x x x ∂ ∂ + ∂ ∂ + ∂ ∂ = ∇ ⋅ G G [ ] z w v y w v x w v w v y z y y y x y ∂ ∂ + ∂ ∂ + ∂ ∂ = ∇ ⋅ G G [ ] z w v y w v x w v w v z z z y z x z ∂ ∂ + ∂ ∂ + ∂ ∂ = ∇ ⋅ G G ϑ x y z (x,y,z)ili(r,ϑ,z) x y z (x,y,z)ili(R,ψ,ϑ) R ϑ x y z Ψ x y z r Alternativne oznake i i e x s s s G ∂ ∂ = = ∇ grad i i x v v v ∂ ∂ = = ⋅ ∇ G G div rot k ijk i j v v v e x ε ∂ ∇× = = ∂ G G G 2 Δ ∇ = ∇ ⋅ ∇ =
Transcript of Formule MFII vjezbe 01 - FSB Online · PDF fileFORMULE: 1. VJEŽBE MEHANIKA FLUIDA II 1 / 3...
FORMULE: 1. VJEŽBE MEHANIKA FLUIDA II 1 / 3
Izvor: A. Werner: Odabrana poglavlja iz mehanike fluida-zbirka zadataka, FSB Zagreb
OSNOVNI OPERATORI U KARTEZIJEVIM, CILINDARSKIM I SFERNIM KOORDINATAMA
⎪⎭
⎪⎬
⎫
===
zzsinrycosrx
ϑϑ
2 2r x yyarctgx
z z
ϑ
⎫= + +⎪⎪= ⎬⎪
= ⎪⎭
⎪⎭
⎪⎬
⎫
===
ψϑψϑψ
cosRzsinsinRycossinRx
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
=
+=
+++=
xy
arctg
zyx
arctg
zyxR
ϑ
ψ22
222
Cilindarske koordinate Sferne koordinate TABLICA 1.1 Osnovne operacije deriviranja skalarnih ( s ) , vektorskih ( w,v ) i tenzorskih (T ) polja u kartezijevim koordinatama
kji zyxzzyyxx vvvevevevv ++≡++=
kji zyxzzyyxx wwweweweww ++≡++= ( )⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
=
zzzyzx
yzyyyx
xzxyxx
TTTTTTTTT
T
zv
yv
xv
v zyx
∂∂
+∂
∂+
∂∂
=⋅∇
2
2
2
2
2
22
zs
ys
xss
∂
∂+
∂
∂+
∂
∂=∇
[ ]xss x ∂
∂=∇ [ ]
zv
yvv yz
x ∂
∂−
∂∂
=×∇ [ ]z
Ty
Tx
T zxyxxxx ∂
∂+
∂
∂+
∂∂
=TDiv
[ ]yss y ∂
∂=∇ [ ]
xv
zvv zx
y ∂∂
−∂∂
=×∇ [ ]z
Ty
Tx
T zyyyxyy ∂
∂+
∂
∂+
∂
∂=TDiv
[ ]zss z ∂
∂=∇ [ ]
yv
xv
v xyz ∂
∂−
∂
∂=×∇ [ ]
zT
yT
xT zzyzxz
z ∂∂
+∂
∂+
∂∂
=TDiv
[ ]z
wv
yw
vx
wvwv x
zx
yx
xx ∂∂
+∂
∂+
∂∂
=∇⋅
[ ]z
wv
yw
vx
wvwv y
zy
yy
xy ∂
∂+
∂
∂+
∂
∂=∇⋅
[ ]z
wv
yw
vx
wvwv z
zz
yz
xz ∂∂
+∂
∂+
∂∂
=∇⋅
ϑ
x y
z
(x,y,z)ili(r,ϑ,z)
xy
z
(x,y,z)ili(R,ψ,ϑ)
R
ϑ
x y
z
Ψ
xy
z
r
Alternativne oznake
ii
exs
ss∂∂
==∇ grad
i
i
xv
vv∂∂
==⋅∇ div
rot kijk i
j
vv v ex
ε ∂∇× = =
∂
2Δ ∇=∇⋅∇=
FORMULE: 1. VJEŽBE MEHANIKA FLUIDA II 2 / 3
Izvor: A. Werner: Odabrana poglavlja iz mehanike fluida-zbirka zadataka, FSB Zagreb
TABLICA 1.2 Osnovne operacije deriviranja skalarnih ( s ) , vektorskih ( w,v ) i tenzorskih (T ) polja u cilindarskim koordinatama
zzrr evevevv ++= ϑϑ ϑϑ sinvcosvv yxr +=
zzrr eweweww ++= ϑϑ ϑϑϑ cosvsinvv yx +−= zz vv =
( )⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
zzzzr
zr
rzrrr
TTTTTTTTT
ϑ
ϑϑϑϑ
ϑ
T
zvv
rrv
rrv z
r ∂∂
+∂∂
+∂∂
=⋅∇ϑϑ1)(1)(
2
2
2
2
22 1)(1)(
zss
rrsr
rrs
∂
∂+
∂
∂+
∂∂
∂∂
=∇ϑ
[ ]rss r ∂
∂=∇ [ ]
zvv
rv z
r ∂∂
−∂∂
=×∇ ϑ
ϑ1
[ ]ϑϑ ∂
∂=∇
sr
s 1 [ ]r
vz
vv zr
∂∂
−∂
∂=×∇ ϑ
[ ]zss z ∂
∂=∇ [ ]
ϑϑ ∂∂
−∂∂
=×∇ rz
vr
)rv(rr
v 11
[ ]r
TT
zT
rrT
rr zrrrrrϑϑ
ϑϑ−
∂∂
+∂∂
+∂∂
=1)(1DivT
[ ]r
TTT
zT
rTr
rrrr
zrϑϑ
ϑϑϑϑϑ ϑ−
+∂∂
+∂∂
+∂∂
=1)(1Div 2
2T
[ ] zzzrzz Tz
Tr
rTrr ∂
∂+
∂∂
+∂∂
= ϑϑ1)(1DivT
[ ] )()1()(z
wv
rww
rv
rw
vwv rz
rrrr ∂
∂+−
∂∂
+∂
∂=∇⋅ ϑ
ϑ ϑ
[ ] )()1()(z
wv
rww
rv
rw
vwv zr
r ∂∂
++∂
∂+
∂∂
=∇⋅ ϑϑϑ
ϑϑ ϑ
[ ] )()1()(z
wv
wr
vr
wvwv z
zzz
rz ∂∂
+∂∂
+∂
∂=∇⋅
ϑϑ
xy
z
reϑe
ze
ϑ
r
xyz
FORMULE: 1. VJEŽBE MEHANIKA FLUIDA II 3 / 3
Izvor: A. Werner: Odabrana poglavlja iz mehanike fluida-zbirka zadataka, FSB Zagreb
TABLICA 1.3 Osnovne operacije deriviranja skalarnih ( s ) , vektorskih ( w,v ) i tenzorskih (T ) polja u sfernim koordinatama
ϑϑψψ evevevv RR ++= ψϑψϑψ cosvsinsinvcossinvv zyxR ++=
ϑϑψψ eweweww RR ++= ψϑψϑψψ sinvsincosvcoscosvv zyx −+=
ϑϑϑ cosvsinvv yx +−=
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
=
ϑϑϑψϑ
ψϑψψψ
ϑψ
TTTTTTTTT
R
R
RRRR
T
ϑψψ
ψψϑ
ψ ∂∂
+∂∂
+∂∂
=⋅∇v
sinRsinv
sinRvR
RRv R
1)(1)(1)( 22
2
2
2222
22 1)(1)(1)(
ϑψψψ
ψψ ∂
∂+
∂∂
∂∂
+∂∂
∂∂
=∇s
sinRssin
sinRRsR
RRs
[ ]Rss R ∂
∂=∇ [ ]
ϑψψ
ψψψ
ϑ ∂
∂−
∂∂
=×∇v
sinRsinv
sinRv R
1)(1
[ ]ψψ ∂
∂=∇
sR
s 1 [ ] )(11ϑψ ϑψ
RvRR
vsinR
v R
∂∂
−∂∂
=×∇
[ ]ϑψϑ ∂
∂=∇
ssinR
s 1 [ ]ψψϑ ∂
∂−
∂∂
=×∇ RvR
RvRR
v 1)(1
[ ]R
TTT
sinRsinT
sinRTR
RRRRRRR
ϑϑψψϑψ ϑψ
ψψψ
+−
∂∂
+∂∂
+∂∂
=1)(1)(1Div 2
2T
[ ]R
cotTTTT
sinRsinT
sinRTR
RRRR
Rψ
ϑψψ
ψψϑϑψψ
ϑψψψψψ−−
+∂∂
+∂∂
+∂∂
=)(1)(1)(1Div 3
3T
[ ]R
cotTTTT
sinRsinT
sinRTR
RRRR
Rψ
ϑψψ
ψψϑψϑϑ
ϑϑϑψϑϑ+−
+∂∂
+∂∂
+∂∂
=)(1)(1)(1Div 3
3T
[ ] ⎟⎟⎠
⎞⎜⎜⎝
⎛−
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛−
∂∂
+⎟⎠
⎞⎜⎝
⎛∂
∂=∇⋅
Rww
Rv
Rww
Rv
Rw
vwv RRRRR
ϑϑ
ψψ ϑψψ sin
11
[ ] ⎟⎟⎠
⎞⎜⎜⎝
⎛−
∂
∂+⎟
⎟⎠
⎞⎜⎜⎝
⎛+
∂
∂+⎟
⎟⎠
⎞⎜⎜⎝
⎛
∂
∂=∇⋅ ψ
ϑψψϑψ
ϑψ
ψψ
ψ cotR
wwsinR
vR
wwR
vR
wvwv R
R11
[ ] ⎟⎟⎠
⎞⎜⎜⎝
⎛++
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂=∇⋅ ψ
ϑψψψϑ
ϑϑ
ψϑ
ϑ cotsin11
Rw
Rww
Rv
wR
vR
wvwv R
R
y
z
x
ψ
ϑ
Re
y
ϑe
ψeR z
x
