Chee2940 Lecture 7 Part B - Navier-stokes Eqs (addedum)
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Table 7.1 Continuity and Navier-Stokes equations in the Cartesian coordinate system (x, y, z)
Continuity:
∂Wx ∂W y ∂Wz + + =0 ∂x ∂y ∂z
x–component:
∂Wx ∂Wx ∂Wx ∂Wx 1 ∂P µ + Wx + Wy + Wz =− + ∆W x ∂t ∂x ∂y ∂z δ ∂y δ
y–component:
z–component:
∂W y ∂t
+ Wx
∂W y ∂x
+ Wy
∂W y ∂y
+ Wz
∂W y ∂z
=−
1 ∂P µ + ∆W y δ ∂y δ
∂Wz ∂Wz ∂Wz ∂Wz 1 ∂P µ + Wx + Wy + Wz =− + ∆Wz ∂t ∂x ∂y ∂z δ ∂y δ
Note: The pressure in these equations also includes the hydrostatic pressure term.
where the Laplace operator is described by ∆≡
∂2 ∂2 ∂2 + + ∂x 2 ∂y 2 ∂z 2
Table 7.2 Continuity and Navier-Stokes equations in the spherical coordinate system (r, ϕ, φ)
∂ 2 r ∂ r ∂Wφ r Wr + Wϕ sin ϕ + =0 ∂r sin ϕ ∂ϕ sin ϕ ∂φ
(
Continuity:
)
(
)
r–component: Wφ ∂Wr Wϕ2 + Wφ2 ∂Wr ∂Wr Wϕ ∂Wr + Wr + + − = ∂t ∂r r ∂ϕ r sin ϕ ∂φ r −
1 ∂P µ 2 ∂Wϕ 2 ∂Wφ 2Wr 2Wϕ cot ϕ + ∆Wr − 2 − 2 − 2 − δ ∂r δ r ∂ϕ r sin ϕ ∂φ r r2
ϕ–component: ∂Wϕ ∂t
+ Wr
∂Wϕ ∂r
+
Wϕ ∂Wϕ r
∂ϕ
+ −
Wφ
∂Wϕ
r sin ϕ ∂φ
+
WrWϕ − Wφ2 cot ϕ r
=
Wϕ 1 ∂P µ 2 ∂Wr 2cos ϕ ∂Wφ + ∆Wϕ + 2 − 2 2 − 2 2 δ r ∂ϕ δ r ∂ϕ r sin ϕ ∂φ r sin ϕ
φ–component: ∂Wφ ∂t
+ Wr
∂Wφ ∂r
+
Wϕ ∂Wφ r
∂ϕ
+ −
Wφ
∂Wφ
r sin ϕ ∂φ
+
WφWr + WφWϕ cot ϕ r
=
2Wφ 1 ∂P µ 2 ∂Wr 2cos φ ∂Wϕ + ∆Wφ + 2 + 2 2 − 2 2 δ r sin ϕ ∂φ δ r sin ϕ ∂φ r sin φ ∂φ r sin ϕ
Laplace operator: ∆≡
1 ∂ 2 ∂ 1 ∂ ∂ 1 ∂2 ϕ r + sin + ∂r r 2 sin 2 ϕ ∂φ 2 r 2 ∂r ∂r r 2 sin ϕ ∂ϕ
Table 7.3 Continuity and Navier-Stokes equations in the cylindrical coordinate system (ϕ, r, z)
∂Wr 1 ∂Wϕ ∂Wz Wr + + + =0 r ∂ϕ r ∂r ∂z
Continuity:
ϕ−component:
∂Wϕ ∂t
+ Wr
∂Wϕ ∂r
+ −
r−component:
Wϕ ∂Wϕ r
∂ϕ
+ Wz
∂Wϕ ∂z
+
WrWϕ r
=
1 ∂P µ 2 ∂Wr Wϕ + ∆Wϕ + 2 − 2 δ r ∂ϕ δ r ∂ϕ r
2 ∂Wr ∂Wr Wϕ ∂Wr ∂Wr Wϕ + Wr + + Wz − = ∂t ∂r ∂z r ∂ϕ r
−
1 ∂P µ 2 ∂Wϕ Wr + ∆Wr − 2 − 2 δ ∂r δ r ∂ϕ r
z−component:
∂Wz ∂Wz Wϕ ∂Wz ∂Wz 1 ∂P µ + Wr + + Wz =− + ∆Wz ∂t ∂r ∂z δ ∂z δ r ∂ϕ
Laplace operator:
∆≡
∂2 1 ∂2 ∂2 1 ∂ + + + ∂r 2 r 2 ∂ϕ 2 ∂z 2 r ∂r
Continuity:
∂Wx ∂W y ∂Wz + + =0 ∂x ∂y ∂z
x–component:
∂Wx ∂Wx ∂Wx ∂Wx 1 ∂P µ + Wx + Wy + Wz =− + ∆W x ∂t ∂x ∂y ∂z δ ∂y δ
y–component:
z–component:
∂W y ∂t
+ Wx
∂W y ∂x
+ Wy
∂W y ∂y
+ Wz
∂W y ∂z
=−
1 ∂P µ + ∆W y δ ∂y δ
∂Wz ∂Wz ∂Wz ∂Wz 1 ∂P µ + Wx + Wy + Wz =− + ∆Wz ∂t ∂x ∂y ∂z δ ∂y δ
Note: The pressure in these equations also includes the hydrostatic pressure term.
where the Laplace operator is described by ∆≡
∂2 ∂2 ∂2 + + ∂x 2 ∂y 2 ∂z 2
Table 7.2 Continuity and Navier-Stokes equations in the spherical coordinate system (r, ϕ, φ)
∂ 2 r ∂ r ∂Wφ r Wr + Wϕ sin ϕ + =0 ∂r sin ϕ ∂ϕ sin ϕ ∂φ
(
Continuity:
)
(
)
r–component: Wφ ∂Wr Wϕ2 + Wφ2 ∂Wr ∂Wr Wϕ ∂Wr + Wr + + − = ∂t ∂r r ∂ϕ r sin ϕ ∂φ r −
1 ∂P µ 2 ∂Wϕ 2 ∂Wφ 2Wr 2Wϕ cot ϕ + ∆Wr − 2 − 2 − 2 − δ ∂r δ r ∂ϕ r sin ϕ ∂φ r r2
ϕ–component: ∂Wϕ ∂t
+ Wr
∂Wϕ ∂r
+
Wϕ ∂Wϕ r
∂ϕ
+ −
Wφ
∂Wϕ
r sin ϕ ∂φ
+
WrWϕ − Wφ2 cot ϕ r
=
Wϕ 1 ∂P µ 2 ∂Wr 2cos ϕ ∂Wφ + ∆Wϕ + 2 − 2 2 − 2 2 δ r ∂ϕ δ r ∂ϕ r sin ϕ ∂φ r sin ϕ
φ–component: ∂Wφ ∂t
+ Wr
∂Wφ ∂r
+
Wϕ ∂Wφ r
∂ϕ
+ −
Wφ
∂Wφ
r sin ϕ ∂φ
+
WφWr + WφWϕ cot ϕ r
=
2Wφ 1 ∂P µ 2 ∂Wr 2cos φ ∂Wϕ + ∆Wφ + 2 + 2 2 − 2 2 δ r sin ϕ ∂φ δ r sin ϕ ∂φ r sin φ ∂φ r sin ϕ
Laplace operator: ∆≡
1 ∂ 2 ∂ 1 ∂ ∂ 1 ∂2 ϕ r + sin + ∂r r 2 sin 2 ϕ ∂φ 2 r 2 ∂r ∂r r 2 sin ϕ ∂ϕ
Table 7.3 Continuity and Navier-Stokes equations in the cylindrical coordinate system (ϕ, r, z)
∂Wr 1 ∂Wϕ ∂Wz Wr + + + =0 r ∂ϕ r ∂r ∂z
Continuity:
ϕ−component:
∂Wϕ ∂t
+ Wr
∂Wϕ ∂r
+ −
r−component:
Wϕ ∂Wϕ r
∂ϕ
+ Wz
∂Wϕ ∂z
+
WrWϕ r
=
1 ∂P µ 2 ∂Wr Wϕ + ∆Wϕ + 2 − 2 δ r ∂ϕ δ r ∂ϕ r
2 ∂Wr ∂Wr Wϕ ∂Wr ∂Wr Wϕ + Wr + + Wz − = ∂t ∂r ∂z r ∂ϕ r
−
1 ∂P µ 2 ∂Wϕ Wr + ∆Wr − 2 − 2 δ ∂r δ r ∂ϕ r
z−component:
∂Wz ∂Wz Wϕ ∂Wz ∂Wz 1 ∂P µ + Wr + + Wz =− + ∆Wz ∂t ∂r ∂z δ ∂z δ r ∂ϕ
Laplace operator:
∆≡
∂2 1 ∂2 ∂2 1 ∂ + + + ∂r 2 r 2 ∂ϕ 2 ∂z 2 r ∂r
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