basic seminar ii a · basic seminar ii a -+reviews of chap. 2-5+-sokendai, m1 hideaki takemura...
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![Page 1: Basic seminar II A · Basic seminar II A -+Reviews of Chap. 2-5+-SOKENDAI, M1 Hideaki Takemura 2018/10/05 The physics of fluids and plasmas -An introduction for astrophysicists-](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f1047517e708231d4485203/html5/thumbnails/1.jpg)
Basic seminar II A -+Reviews of Chap. 2-5+-
SOKENDAI, M1 Hideaki Takemura
2018/10/05
The physics of fluids and plasmas -An introduction for astrophysicists-
![Page 2: Basic seminar II A · Basic seminar II A -+Reviews of Chap. 2-5+-SOKENDAI, M1 Hideaki Takemura 2018/10/05 The physics of fluids and plasmas -An introduction for astrophysicists-](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f1047517e708231d4485203/html5/thumbnails/2.jpg)
The full set of hydrodynamic equation for neutral fluids- The continuity equation (conservation of mass)
- Incompressible Navier-Stokes equation (conservation of momentum)
- Conservation of energy
∂ρ∂t
+ ∇ ⋅ (ρv) = 0
∂v∂t
+ (v ⋅ ∇)v = −1ρ
∇p + F + ν∇2v
ρ( ∂ϵ∂t
+ v ⋅ ∇ϵ) − ∇ ⋅ (K∇T) + p∇v = 0
![Page 3: Basic seminar II A · Basic seminar II A -+Reviews of Chap. 2-5+-SOKENDAI, M1 Hideaki Takemura 2018/10/05 The physics of fluids and plasmas -An introduction for astrophysicists-](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f1047517e708231d4485203/html5/thumbnails/3.jpg)
The continuity equation (conservation of mass)∂ρ∂t
+ ∇ ⋅ (ρv) = 0
vn
SdS
V
total mass
increase of mass per unit time
the mass flows out from the region V
conservation of the mass
So that this makes ends meet for any arbitrary regions V
∫ ∫ ∫VρdV
∂∂t ∫ ∫ ∫V
ρdV = ∫ ∫ ∫V
∂ρ∂t
dV
∫ ∫Sρv ⋅ n dS = ∫ ∫ ∫V
∇ ⋅ (ρv) dV
∫ ∫ ∫V
∂ρ∂t
dV = − ∫ ∫ ∫V∇ ⋅ (ρv) dV
∫ ∫ ∫V (∂ρ∂t
+ ∇ ⋅ (ρv)) dV = 0
∂ρ∂t
+ ∇ ⋅ (ρv) = 0
![Page 4: Basic seminar II A · Basic seminar II A -+Reviews of Chap. 2-5+-SOKENDAI, M1 Hideaki Takemura 2018/10/05 The physics of fluids and plasmas -An introduction for astrophysicists-](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f1047517e708231d4485203/html5/thumbnails/4.jpg)
The full set of hydrodynamic equation for neutral fluids- The continuity equation (conservation of mass)
- Incompressible Navier-Stokes equation (conservation of momentum)
- Conservation of energy
∂ρ∂t
+ ∇ ⋅ (ρv) = 0
∂v∂t
+ (v ⋅ ∇)v = −1ρ
∇p + F + ν∇2v
ρ( ∂ϵ∂t
+ v ⋅ ∇ϵ) − ∇ ⋅ (K∇T) + p∇v = 0
![Page 5: Basic seminar II A · Basic seminar II A -+Reviews of Chap. 2-5+-SOKENDAI, M1 Hideaki Takemura 2018/10/05 The physics of fluids and plasmas -An introduction for astrophysicists-](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f1047517e708231d4485203/html5/thumbnails/5.jpg)
Incompressible Navier-Stokes equation (conservation of momentum)
∂v∂t
+ (v ⋅ ∇)v = −1ρ
∇p + F + ν∇2v
∇ ⋅ v = 0∂ρ∂t
+ ∇ ⋅ (ρv) = 0The continuity equation
∂ρ∂t
= 0 , ρ = const .
variation
convection
diffusion
internal source external source
dvdt
=Fm
(force per unit mass)
equation of motion
ν =μρ
kinematic viscosity : : viscosity coefficient
![Page 6: Basic seminar II A · Basic seminar II A -+Reviews of Chap. 2-5+-SOKENDAI, M1 Hideaki Takemura 2018/10/05 The physics of fluids and plasmas -An introduction for astrophysicists-](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f1047517e708231d4485203/html5/thumbnails/6.jpg)
The full set of hydrodynamic equation for neutral fluids- The continuity equation (conservation of mass)
- Incompressible Navier-Stokes equation (conservation of momentum)
- Conservation of energy
∂ρ∂t
+ ∇ ⋅ (ρv) = 0
∂v∂t
+ (v ⋅ ∇)v = −1ρ
∇p + F + ν∇2v
ρ( ∂ϵ∂t
+ v ⋅ ∇ϵ) − ∇ ⋅ (K∇T) + p∇v = 0
![Page 7: Basic seminar II A · Basic seminar II A -+Reviews of Chap. 2-5+-SOKENDAI, M1 Hideaki Takemura 2018/10/05 The physics of fluids and plasmas -An introduction for astrophysicists-](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f1047517e708231d4485203/html5/thumbnails/7.jpg)
conservation of energy
ρ( ∂ϵ∂t
+ v ⋅ ∇ϵ) − ∇ ⋅ (K∇T) + p∇v = 0
going in and out of energy with fluids
going in and out of energy due to the heat flow
going in and out of energy due to the work
![Page 8: Basic seminar II A · Basic seminar II A -+Reviews of Chap. 2-5+-SOKENDAI, M1 Hideaki Takemura 2018/10/05 The physics of fluids and plasmas -An introduction for astrophysicists-](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f1047517e708231d4485203/html5/thumbnails/8.jpg)
The full set of hydrodynamic equation for neutral fluids- The continuity equation (conservation of mass)
- Incompressible Navier-Stokes equation (conservation of momentum)
- Conservation of energy
∂ρ∂t
+ ∇ ⋅ (ρv) = 0
∂v∂t
+ (v ⋅ ∇)v = −1ρ
∇p + F + ν∇2v
ρ( ∂ϵ∂t
+ v ⋅ ∇ϵ) − ∇ ⋅ (K∇T) + p∇v = 0
![Page 9: Basic seminar II A · Basic seminar II A -+Reviews of Chap. 2-5+-SOKENDAI, M1 Hideaki Takemura 2018/10/05 The physics of fluids and plasmas -An introduction for astrophysicists-](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f1047517e708231d4485203/html5/thumbnails/9.jpg)
Assumptions
∂ρ∂t
+ ∇ ⋅ (ρv) = 0
∂v∂t
+ (v ⋅ ∇)v = −1ρ
∇p + F + ν∇2v
ρ( ∂ϵ∂t
+ v ⋅ ∇ϵ) − ∇ ⋅ (K∇T) + p∇v = 0
- The continuity equation
- Incompressible Navier-Stokes equation
- Conservation of energy
1. Spatial variation of viscosity μ is neglected.
2. Heat production due to the viscous damping of motion is neglected.
3. Neutral incompressible fluids ∇ ⋅ v = 0
No radiation No chemical evolution
We will consider the solutions of the equations under different circumstance.
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Ideal fluids
Viscous flows
Chapter 4, no viscosity
Chapter 5
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Euler equation
If ν = 0 (μ = 0),
∂v∂t
+ (v ⋅ ∇)v = −1ρ
∇p + F + ν∇2vIncompressible Navier-Stokes equation
∂v∂t
+ (v ⋅ ∇)v = −1ρ
∇p + F Euler equation
&(v ⋅ ∇) v =
12
∇(v ⋅ v) − v × (∇ × v)
∂v∂t
+ (v ⋅ ∇) v +12
∇(v ⋅ v) − v × (∇ × v) = −1ρ
∇p + F
・F is conservative force ・vorticity ・incompressible fluids
ω = ∇ × vtake a curl
ρ = const .∂ω∂t
= ∇ × (v × ω)
∇ ⋅ v = 0
the complete dynamical theory of incompressible fluids
no viscosity
![Page 12: Basic seminar II A · Basic seminar II A -+Reviews of Chap. 2-5+-SOKENDAI, M1 Hideaki Takemura 2018/10/05 The physics of fluids and plasmas -An introduction for astrophysicists-](https://reader033.vdocuments.mx/reader033/viewer/2022060410/5f1047517e708231d4485203/html5/thumbnails/12.jpg)
Newtonian fluidShear force is proportional to the gradient of velocity
Force { acts in a vertical direction on the surface of fluids
acts in a horizontal direction on the surface of fluids pressure
shear force
Tensor Pij = p δij + πij
πij = a ( ∂vi
∂xj+
∂vj
∂xi ) + b δij∇ ⋅ v
The equation of motion &
∂v∂t
+ (v ⋅ ∇)v = −1ρ
∇p + F + ν∇2v Navier-Stokes equation
take a curl ∂ω∂t
= ∇ × (v × ω) + ν∇2ω
ρdvi
dt= ρFi −
∂Pij
∂xj
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Reynolds numberFluid flows around geometrically similar object of different sizes
let L, V are the typical length of the object and fluid velocity
x = x′�L, v = v′�V, t = t′�LV
, ω = ω′�VL
x’, v’, t’ and ω’ are the values of length, velocity, time and vorticity measured in these scaled units
∂ω∂t
= ∇ × (v × ω) + ν∇2ω∂ω′�∂t′�
= ∇ × (v′� × ω′�) +1ℛ
∇′�2ω′�
ℛ =LVν
Reynolds numberthe equation is in the scaled variables