NAVIER STOKES EQ -...
1
Navier - Stokes equation: We consider an incompressible , isothermal Newtonian flow (density ρ =const, viscosity μ =const), with a velocity field )) ( ) ( ) ( ( x,y,z , w x,y,z , v x,y,z u V = r Incompressible continuity equation: 0 = ∂ ∂ + ∂ ∂ + ∂ ∂ z w y v x u eq1. Navier - Stokes equation: vector form: V g P Dt V D r r r 2 ∇ + + −∇ = μ ρ ρ x component: ) ( 2 2 2 2 2 2 z u y u x u g x P z u w y u v x u u t u x ∂ ∂ + ∂ ∂ + ∂ ∂ + + ∂ ∂ − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ μ ρ ρ eq2. y component: ) ( 2 2 2 2 2 2 z v y v x v g y P z v w y v v x v u t v y ∂ ∂ + ∂ ∂ + ∂ ∂ + + ∂ ∂ − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ μ ρ ρ eq3. z component: ) ( 2 2 2 2 2 2 z w y w x w g z P z w w y w v x w u t w z ∂ ∂ + ∂ ∂ + ∂ ∂ + + ∂ ∂ − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ μ ρ ρ eq4. Cylindrical coordinates ) , , ( z r θ : We consider an incompressible , isothermal Newtonian flow (density ρ =const, viscosity μ =const), with a velocity field . u u u V z r ) , , ( θ = r Incompressible continuity equation: 0 ) ( 1 ) ( 1 = ∂ ∂ + ∂ ∂ + ∂ ∂ z u u r r ru r z r θ θ eq a) r-component: ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ + ∂ ∂ − ∂ ∂ + − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + + ∂ ∂ − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + − ∂ ∂ + ∂ ∂ + ∂ ∂ 2 2 2 2 2 2 2 2 2 1 1 z u u r u r r u r u r r r g r P z u u r u u r u r u u t u r r r r r r z r r r r θ θ μ ρ θ ρ θ θ θ eq b) θ -component: ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ + ∂ ∂ + ∂ ∂ + − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + + ∂ ∂ − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + + ∂ ∂ + ∂ ∂ + ∂ ∂ 2 2 2 2 2 2 2 2 1 1 1 z u u r u r r u r u r r r g P r z u u r u u u r u r u u t u r z r r θ θ θ θ θ θ θ θ θ θ θ θ θ μ ρ θ θ ρ eq c) z-component: ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ + ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + + ∂ ∂ − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ 2 2 2 2 2 1 1 z u u r r u r r r g z P z u u u r u r u u t u z z z z z z z z r z θ μ ρ θ ρ θ eq d)
Transcript of NAVIER STOKES EQ -...
Navier - Stokes equation:
We consider an incompressible , isothermal Newtonian flow (density ρ =const, viscosity μ =const), with a velocity field ))()()(( x,y,z, w x,y,z, vx,y,zuV =
r
Incompressible continuity equation:
0=∂∂
+∂∂
+∂∂
zw
yv
xu eq1.
Navier - Stokes equation:
vector form: VgPDt
VD rrr
2∇++−∇= μρρ
x component:
)( 2
2
2
2
2
2
zu
yu
xug
xP
zuw
yuv
xuu
tu
x ∂∂
+∂∂
+∂∂
++∂∂
−=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
+∂∂ μρρ eq2.
y component:
)( 2
2
2
2
2
2
zv
yv
xvg
yP
zvw
yvv
xvu
tv
y ∂∂
+∂∂
+∂∂
++∂∂
−=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
+∂∂ μρρ eq3.
z component:
)( 2
2
2
2
2
2
zw
yw
xwg
zP
zww
ywv
xwu
tw
z ∂∂
+∂∂
+∂∂
++∂∂
−=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
+∂∂ μρρ eq4.
Cylindrical coordinates ),,( zr θ : We consider an incompressible , isothermal Newtonian flow (density ρ =const, viscosity μ =const), with a velocity field .uuuV zr ),,( θ=
r
Incompressible continuity equation:
0)(1)(1
=∂∂
+∂
∂+
∂∂
zuu
rrru
rzr
θθ eq a)
r-component:
⎥⎦
⎤⎢⎣
⎡∂∂
+∂∂
−∂∂
+−⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
++∂∂
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+−∂∂
+∂∂
+∂∂
2
2
22
2
22
2
211zuu
ru
rru
rur
rrg
rP
zuu
ruu
ru
ruu
tu
rrrrr
rz
rrr
r
θθμρ
θρ
θ
θθ
eq b)
θ -component:
⎥⎦
⎤⎢⎣
⎡∂∂
+∂∂
+∂∂
+−⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
++∂∂
−=
⎟⎠⎞
⎜⎝⎛
∂∂
++∂∂
+∂∂
+∂∂
2
2
22
2
22
2111zuu
ru
rru
ru
rrr
gPr
zu
uruuu
ru
ru
ut
u
r
zr
r
θθθθθ
θθθθθθ
θθμρ
θ
θρ
eq c)
z-component:
⎥⎦
⎤⎢⎣
⎡∂∂
+∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
++∂∂
−=
⎟⎠⎞
⎜⎝⎛
∂∂
+∂∂
+∂∂
+∂∂
2
2
2
2
2
11zuu
rrur
rrg
zP
zuuu
ru
ruu
tu
zzzz
zz
zzr
z
θμρ
θρ θ
eq d)
