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Ideal Fluids
PHY 688: Astrophysical Fluids and Plasmas
Navier-Stokes Equatons
● We have
PHY 688: Astrophysical Fluids and Plasmas
Euler Equatons
● We have
– Note: the last equaton is internal energy evoluton. Many tmes we want total energy:
PHY 688: Astrophysical Fluids and Plasmas
Eulerian vs. Lagrangian Descripton
● We can imagine diferent reference frames when thinking about fluids
– Eulerian: watch a partcular region and look at how the fluid changes
● Advecton of new fluid and local sources will both contributon to the change at that locaton
– Lagrangian: move along with the flow, tracking a partcular fluid element
● Propertes in that fluid element change due to local sources, or compression of the fluid element
PHY 688: Astrophysical Fluids and Plasmas
Eulerian vs. Lagrangian Descripton
● Take a property g(x(t), t)
● Eulerian descripton:
– We are looking at a fied i, so we take
● Lagrangian descripton:
– We move with the fluid:
● We’ll use D/Dt to denote the Lagrangian derivatve:
PHY 688: Astrophysical Fluids and Plasmas
Macroscopic Derivaton
● Mass conservaton
M
Ω is an arbitrary volume over which mass M is considered conserved.
The only changes to M will be due to the fluies across the boundary
PHY 688: Astrophysical Fluids and Plasmas
Macroscopic Derivaton
● In Lagrangian form:
– This tells us that density of a fluid element changes locally only due to compressive or eipansive flow
● Note that if the density of a flid element is constant, then
PHY 688: Astrophysical Fluids and Plasmas
Macroscopic Derivaton
● We can get the momentum and energy equatons from similar macroscopic ideas:
– Momentum: Newton’s second law
– Internal energy: First law of thermodynamics
● Blackboard derivaton...
PHY 688: Astrophysical Fluids and Plasmas
Other “Energy” Equatons
● We’ve seen equatons for specifc internal energy and total energy
● Sometmes we want evoluton for other thermodynamic variables that can take the place of energy, e.g.,
– Enthalpy
– Pressure
– Temperature
– Entropy● This follows from the frst law of thermodynamics:
you’ll derive these on your homework
In the absence of heat sources
PHY 688: Astrophysical Fluids and Plasmas
Ideal Fluid
● The Navier-Stokes equatons (with simplifed transport efects) read:
– An ideal fluid ignores these transport efects
PHY 688: Astrophysical Fluids and Plasmas
Reynolds Number
● Importance of viscosity:
– Where we assumed μ is constant, and defned ν = μ/ρ
● Now, dimensionless quanttes:
– then
● where we have defned the Reynolds number,
PHY 688: Astrophysical Fluids and Plasmas
Reynolds Number
● Dimensionless numbers provide convenient ways to compare vastly diferent problems.
– Idea: systems with the same dimensionless numbers (and confguratons) should behave the same.
● Reynolds number measures the relatve importance of the advecton to viscous dissipaton.
– Large Re: viscosity is unimportant (advecton term dominates flow). These flows are turbulent.
– Smal Re: viscosity is important. The flow remains laminar.
● Eiperiments show that the transiton to turbulence occurs at Re ~ 2000, although the shape of the object influences this.
PHY 688: Astrophysical Fluids and Plasmas
Reynolds Number
● Reynolds number scaling plays a big role in aircraf design.
Here a scale model of the aircraf is placed in a windtunnel, and the flow is adjusted (in speed and temperature) to match the Reynolds number eipected during flight.
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PHY 688: Astrophysical Fluids and Plasmas
Reynolds Number
● Many astrophysical environments have large Re, and we will ignore viscosity
● Some Reynolds numbers of everyday objects (thanks Wikipedia!)
– Ciliate ~ 1 × 10−1
– Smallest fsh ~ 1
– Blood flow in brain ~ 1 × 102
– Blood flow in aorta ~ 1 × 103
– Onset of turbulent flow ~ 2.3 × 103 to 5.0 × 104 for pipe flow to 106 for boundary layers
– Typical pitch in Major League Baseball ~ 2 × 105
– Person swimming ~ 4 × 106
– Fastest fsh ~ 1 × 108
– Blue whale ~ 4 × 108
– A large ship (RMS Queen Elizabeth 2) ~ 5 × 109
PHY 688: Astrophysical Fluids and Plasmas
Thermal Difusion
● Another process we neglect in the Euler equatons is thermal difusion
● In astrophysics, advecton of heat lslally dominates over heat conducton
– Here, Pe is called the Peclet number
– We introduced the thermal difusivity, χ
PHY 688: Astrophysical Fluids and Plasmas
Thermal Difusion
One situaton where thermal difusion is important is flame propagaton:
fuel ash
PHY 688: Astrophysical Fluids and Plasmas
Incompressible and Baratropic
● In your homework, you’ll show that:
– Allowing us to rewrite the momentum equaton as:
● Taking the curl of this (blackboard), we get:
– Where vortcity is defned as:
PHY 688: Astrophysical Fluids and Plasmas
Incompressible and Baratropic
● If the pressure and density gradients are aligned, then the last term vanishes, and we have
● Together with the relaton between vortcity and velocity, we this completely specifes the flow
● This approiimaton arises when:
– Density is constant (note Choudhuri says this is incompressible, but incompressible is more general and can let variable density flows)
– Baratropic equaton of state, p = p(ρ)
● Notce that the energy equaton is not needed
PHY 688: Astrophysical Fluids and Plasmas
Conservatve Form
● When we do numerical methods, we’ll want to consider these equatons in conservatve form:
– With
● Introducing a momentum tensor:
PHY 688: Astrophysical Fluids and Plasmas
Applicaton: Solar Corona(Choudhuri Ch. 4)
● Solar corona: hot tenuous gas eitending beyond the solar surface.
– Parker (1958) constructed a simple spherical model.
● Corona is much hoter than the surface of the Sun.
– Assume heated at lower layers
– Boundary conditon: T = T0 at r = r0
● Assume that the corona is statc and of negligible mass:
PHY 688: Astrophysical Fluids and Plasmas
Applicaton: Solar Corona(Choudhuri Ch. 4)
● Assume an ideal gas:
– Into HSE:
● Conductvity from kinetc theory:
PHY 688: Astrophysical Fluids and Plasmas
Applicaton: Solar Corona(Choudhuri Ch. 4)
● Soluton:
– We see that T → 0 as r → ∞
● Using this with HSE:
– So, with p = p0 at r = r0,
● Pressure is non-zero as r → ∞
● This suggests that it is not in equilibrium… solar wind
PHY 688: Astrophysical Fluids and Plasmas
Bernoulli’s Principle
● Previous eiample used hydrostatcs (v = 0)
● Now let’s consider a steady flow (no tme dependence)
● Useful construct: streamline—curve tangent to v at every point
– For steady flows, streamlines don’t change in tme, so streamlines are the path a fluid element takes
● Back to our momentum equaton, assuming conservatve force:
● Assuming a baratropic EOS, we can introduce:
This requires a bit more discussion… blackboard...
PHY 688: Astrophysical Fluids and Plasmas
Bernoulli’s Principle
● Defning:
● We have:
● Finally, dotng with v,
● So B must be constant on steamlines
PHY 688: Astrophysical Fluids and Plasmas
Bernoulli's Theorem(Shu Ch. 6, Shore Ch. 1)
● Consider an incompressible flow, φ = 0
– Now consider flow through a pipe with a constricton
● Mass conservaton:
– At smaller diameter, v increases, so p must decrease (by Bernoulli’s theorem)
PHY 688: Astrophysical Fluids and Plasmas
Bernoulli's Theorem(Shu Ch. 6, Shore Ch. 1)
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PHY 688: Astrophysical Fluids and Plasmas
Bernoulli's Theorem(Shu Ch. 6, Shore Ch. 1)
● Consider the flow of water from a faucet:
– Fluid is incompressible, and Bernoulli's theorem tells us:
– Pressure of the fluid is in equilibrium with the atmosphere, and then we see that the fluid speeds up as it drops.
PHY 688: Astrophysical Fluids and Plasmas
Bernoulli's Theorem(Shu Ch. 6, Shore Ch. 1)
● Contnuity equaton tells us that
● ρ is constant, v is increasing, so cross-sectonal area must be decreasing.
● Eventually, the stream of water will narrow enough that surface tension becomes important.
● Also, the shear on the air-water interface is unstable, causing oscillatons in the surface.
PHY 688: Astrophysical Fluids and Plasmas
Bernoulli's Theorem(Shu Ch. 6, Shore Ch. 1)
● Consider water coming out of a spout
– Incompressible limit:
– Pressure at the top and eiit are atmospheric pressure
● Other eiamples: shower curtain blowing in, blowing between two sheets of paper
water level
PHY 688: Astrophysical Fluids and Plasmas
Kelvin Circulaton Theorem(Shu Ch. 6)
● Kelvin's circulaton theorem:
– Number of vortei lines passing through a surface element remains unchanged for inviscid barotropic flow.
● Defne the circulaton as
● Note that Γ = 0 if circulatory component of the fluid velocity averages to zero in the circuit C
⊙
PHY 688: Astrophysical Fluids and Plasmas
Kelvin Circulaton Theorem(Shu Ch. 6)
● Rewrite using Stokes's theorem
– Circulaton enclosed by C is simply the number of vortei lines that thread A
● Evoluton equaton:
PHY 688: Astrophysical Fluids and Plasmas
Kelvin Circulaton Theorem(Shu Ch. 6)
● Therefore, we have
– With triple product identty
Each line element contributes |v × dl| to the rate of change of area. The cross product also gives the proper sign to the area contributon.
PHY 688: Astrophysical Fluids and Plasmas
Kelvin Circulaton Theorem(Shu Ch. 6)
● Finally, we see
● and therefore
PHY 688: Astrophysical Fluids and Plasmas
Kelvin Circulaton Theorem(Shu Ch. 6)
● Kelvin circulaton theorem implies:
– If a fluid has no circulaton (or vortcity) at tme zero, then no vortcity will be generated at subsequent tmes.
● Note, this assumes that the viscosity is negligible.
– If there are some vortces in a region of fluid, they will stck with that fluid element as it moves (think whirlpools in a river)
– No mater what disturbances the fluid is subjected to (as long as it is not dissipatve), you will not add net angular momentum.
● Eiample: moving an oar through water.
– Counterrotatng vortces are generated on either side, but there is no net spin up of the fluid.
PHY 688: Astrophysical Fluids and Plasmas
Potental Flows
● A flow that is initally irrotatonal ( ▽ × v = 0) remains so
– Consequence of Kelvin circulaton theorem
– This means we can write:
● If we are incompressible, then further:
– This is a scalar equaton (Laplace equaton)
● Consider flow past a cylinder. Boundary conditons:
– Normal velocity goes to zero at surface:
– Velocity tends to uniform velocity at infnity
PHY 688: Astrophysical Fluids and Plasmas
Potental Flows
● Asymptotc velocity
– Take:
– Then:
● General soluton (we’ll skip the details):
PHY 688: Astrophysical Fluids and Plasmas
Potental Flows
● Streamline plot
PHY 688: Astrophysical Fluids and Plasmas
Reynolds Number Efects
Taneda 1956, from htps://authors.library.caltech.edu/25017/4/chap5.htm
PHY 688: Astrophysical Fluids and Plasmas
Limitatons
● The real world has viscosity
● Choudhuri shows that if we move the cylinder though the fluid at a constant velocity, there is no force actng on the cylinder
– This is against our intuiton
– Known as d’Alembert’s paradoi● If there were a drag force, then the kinetc energy removed from
the cylinder needs to go somewhere—heat to the fluid● But without viscosity, there is no mechanism to convert KE into
heat● Ideal fluids, therefore cannot eiert drag
PHY 688: Astrophysical Fluids and Plasmas
Viscosity
● Shu has a nice discussion of the efects of viscosity on this problem:
– Notce that the Laplace equaton does not allow us to additonally supply a boundary conditon on the tangental velocity (no slip conditon)
– Tangental velocity should go to zero● Real world: boundary layer (thin) forms● Viscous efects dominate in the boundary layer (Kelvin’s theorem
no longer applies, vortces form and get swept downstream)● Potental flow descripton applies well outside of the boundary
layer