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