Navier-Stokes Equations - Information Management …mefm/me19b/handouts/Equations.pdfIn spherical...
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Navier-Stokes Equations In cylindrical coordinates, (r, θ, z ), the continuity equation for an incompressible fluid is 1 r ∂ ∂r (ru r )+ 1 r ∂ ∂θ (u θ )+ ∂u z ∂z =0 In cylindrical coordinates, (r, θ, z ), the Navier-Stokes equations of motion for an incompress- ible fluid of constant dynamic viscosity, μ, and density, ρ, are ρ Du r Dt - u 2 θ r = - ∂p ∂r + f r + μ 5 2 u r - u r r 2 - 2 r 2 ∂u θ ∂θ ρ Du θ Dt + u θ u r r = - 1 r ∂p ∂θ + f θ + μ 5 2 u θ - u θ r 2 + 2 r 2 ∂u r ∂θ ρ Du z Dt = - ∂p ∂z + f z + μ 5 2 u z where u r ,u θ ,u z are the velocities in the r, θ, z cylindrical coordinate directions, p is the pressure, f r ,f θ ,f z are the body force components in the r, θ, z directions and the operators D/Dt and 5 2 are D Dt = ∂ ∂t + u r ∂ ∂r + u θ r ∂ ∂θ + u z ∂ ∂z 5 2 = ∂ 2 ∂r 2 + 1 r ∂ ∂r + 1 r 2 ∂ 2 ∂θ 2 + ∂ 2 ∂z 2 and for an incompressible, Newtonian fluid σ rr = -p +2μ ∂u r ∂r ; σ θθ = -p +2μ 1 r ∂u θ ∂θ + u r r σ zz = -p +2μ ∂u z ∂z σ rθ = μ 1 r ∂u r ∂θ + ∂u θ ∂r - u θ r ; σ rz = μ ∂u r ∂z + ∂u z ∂r σ θz = μ 1 r ∂u z ∂θ + ∂u θ ∂z **************************************************** 1
Transcript of Navier-Stokes Equations - Information Management …mefm/me19b/handouts/Equations.pdfIn spherical...
Navier-Stokes Equations
In cylindrical coordinates, (r, θ, z), the continuity equation for an incompressible fluid is
1
r
∂
∂r(rur) +
1
r
∂
∂θ(uθ) +
∂uz∂z
= 0
In cylindrical coordinates, (r, θ, z), the Navier-Stokes equations of motion for an incompress-ible fluid of constant dynamic viscosity, µ, and density, ρ, are
ρ
[DurDt− u2
θ
r
]= −∂p
∂r+ fr + µ
[52ur −
urr2− 2
r2
∂uθ∂θ
]
ρ
[DuθDt
+uθurr
]= −1
r
∂p
∂θ+ fθ + µ
[52uθ −
uθr2
+2
r2
∂ur∂θ
]
ρDuzDt
= −∂p∂z
+ fz + µ52 uz
where ur, uθ, uz are the velocities in the r, θ, z cylindrical coordinate directions, p is thepressure, fr, fθ, fz are the body force components in the r, θ, z directions and the operatorsD/Dt and 52 are
D
Dt=
∂
∂t+ ur
∂
∂r+uθr
∂
∂θ+ uz
∂
∂z
52 =∂2
∂r2+
1
r
∂
∂r+
1
r2
∂2
∂θ2+
∂2
∂z2
and for an incompressible, Newtonian fluid
σrr = −p+ 2µ∂ur∂r
; σθθ = −p+ 2µ
(1
r
∂uθ∂θ
+urr
)σzz = −p+ 2µ
∂uz∂z
σrθ = µ
(1
r
∂ur∂θ
+∂uθ∂r− uθ
r
); σrz = µ
(∂ur∂z
+∂uz∂r
)σθz = µ
(1
r
∂uz∂θ
+∂uθ∂z
)
****************************************************
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In spherical coordinates, (r, θ, φ), the continuity equation for an incompressible fluid is :
1
r2
∂
∂r
(r2ur
)+
1
r sin θ
∂
∂θ(uθ sin θ) +
1
r sin θ
∂uφ∂φ
= 0
In spherical coordinates, (r, θ, φ), the Navier-Stokes equations of motion for an incompressiblefluid with uniform viscosity are:
ρ
[DurDt−u2θ + u2
φ
r
]= −∂p
∂r+ fr + µ
[52ur −
2urr2− 2
r2
∂uθ∂θ− 2uθ cot θ
r2− 2
r2 sin θ
∂uφ∂φ
]
ρ
[DuθDt
+uθurr−u2φ cot θ
r
]= −1
r
∂p
∂θ+ fθ +µ
[52uθ +
2
r2
∂ur∂θ− uθr2 sin θ sin θ
− 2 cot θ
r2 sin θ
∂uφ∂φ
]
ρ
[DuφDt
+uφurr
+uθuφ cot θ
r
]= − 1
r sin θ
∂p
∂φ+fφ+µ
[52uφ +
2
r2 sin θ
∂ur∂φ− uφr2 sin θ sin θ
+2 cot θ
r2 sin θ
∂uθ∂φ
]where ur, uθ, uφ are the velocities in the r, θ, φ directions, p is the pressure, ρ is the fluiddensity and fr, fθ, fφ are the body force components. The Lagrangian or material derivativeis
D
Dt=
∂
∂t+ ur
∂
∂r+uθr
∂
∂θ+
uφr sin θ
∂
∂φ
and the Laplacian operator is
52 =1
r2
∂
∂r
(r2 ∂
∂r
)+
1
r2 sin θ
∂
∂θ
(sin θ
∂
∂θ
)+
1
r2 sin θ sin θ
∂2
∂2φ
and for an incompressible, Newtonian fluid
σrr = −p+ 2µ∂ur∂r
; σθθ = −p+ 2µ
(1
r
∂uθ∂θ
+urr
)
σφφ = −p+ 2µ
(1
r sin θ
∂uφ∂φ
+urr
+uθ cot θ
r
)σrθ = µ
(r∂
∂r
(uθr
)+
1
r
∂ur∂θ
); σrφ = µ
(1
r sin θ
∂ur∂φ
+ r∂
∂r
(uφr
))σθφ = µ
(1
r sin θ
∂uθ∂φ
+sin θ
r
∂
∂θ
( uφsin θ
))
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