Post on 28-Jun-2020
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-
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
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
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
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
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
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
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
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.
Ideal fluids
Viscous flows
Chapter 4, no viscosity
Chapter 5
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
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
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