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Fluids dynamics and plasma P 1 dynamics and plasma physics P 2

ASTROPHYSICAL DYNAMICS DYNAMICS OF PARTICLES, FLUIDS AND PLASMAS Astrophysical processes frequently involve fluids and plasmas in motion. We will need to study fluid dynamics to understand the following. We will need to study: • The internal structure of the major planets, including the Earth. a) systems of particles moving under ; • b) incompressible and compressible fluids with ; The structure and evolution of the and other . • c) behaviour of plasmas; d) . The formation and dynamics of the solar and its interaction with • the planets. The following approaches all provide insight, depending on the • circumstances: The properties and dynamics of the various components of the • - The single particle approach This gives useful hints if interactions 6 are few. It can be extended by to the - The cold (100 K, 10 particles m−3) neutral . N -particle case, where it provides powerful constraints on - Bright nebulae ionised by hot, young stars. theoretical approximations. - Dark, dense clouds of molecular hydrogen and dust (10 K, - Statistical approach When there are very many particles, we can 1012 particles m−3). keep a track on the number of particles in any given “energy level” Accretion discs and jets in protostars and exotic binary systems. (the particle velocity distribution function) and work out how it varies • as a function of space and time. Some details of the individual The dynamics of supernovae and their remnants. • dynamics will be lost as a result of the approximation. The overall dynamics of the in spiral . - Fluid approach When there are even more particles we may not • care at all about microscopic properties and just use the average or The properties of hot gas in clusters of galaxies (108 K, • macroscopic properties: density; velocity; and 103 particles m−3) . The dynamics of the jets in radio galaxies and . • Fluids dynamics and plasma physics P 3 Fluids dynamics and plasma physics P 4

STATISTICAL TREATMENT — THE DISTRIBUTION FUNCTION REVISION: THE PERFECT GAS

Distribution function f(x, v, t) The interiors of stars, the interstellar medium and cluster gas are all • • describes the density of particles in collisional, since the mean free paths of particles are much smaller 6-dimensional space (x, v). than the overall scale of interest. We can usually treat the as an inviscid perfect gas. We now recall some results from previous years. The probability of finding a particle • A perfect gas obeys Sir Edward Boyle’s lawa. The V , pressure (assumed all of equal mass m) • in a volume element is P and temperature T of one mole of a perfect gas has the equation of f(x, v, t)d3xd3v. state P V = RT , where the gas constant is −1 R = NAk = 8.31 J K . The normalisation is such that d3v f(x, v, t) = n(x, t), where • The First Law of n(x, t) is the particle density. Z • dU = T dS P dV − Kinetic theory (statistical treatment) tries to predict the evolution of the • internal energy work distribution function. Specific • There is a flux of particles (fv, fa) through any region of phase At constant volume, no work done: CVdT = T dS = dU • space, where a is the acceleration acting on the particles. At constant pressure: CPdT = T dS = dU + P dV = dU + RdT C C = R (γ 1)C Conservation of particles and use of the 6-dimensional divergence ⇒ P V V • − ≡ − theorem leads to the governing equation, the Boltzmann equation; Important property of gas: ratio of specific heats γ = CP/CV df(x, v, t) ∂f ∂f ∂f ∂f • = + v + a = dt ∂t ·∂x ·∂v ∂t RT P coll Can rearrange: U = CVT = = V   • γ 1 γ 1 The term on the RHS will often be ignored. − − • P The Boltzmann equation is a perfectly horrible approximation to the Specific internal energy u: U = dV u u = • • ⇒ γ 1 physics of an N -body system, but still provides very useful insights Z − (and is still very difficult to solve!). a“The greater the external pressure, the greater the volume of hot air.” Fluids dynamics and plasma physics P 5 Fluids dynamics and plasma physics P 6

PRESSURE AND HYDROSTATIC EQUILIBRIUM

ADIABATIC COMPRESSION OF PERFECT GAS Apparently, Archimedes knew something • about pressure and hydrostatic equilibrium. Adiabatic compression No heat — no change in entropy (dS = 0). • The barometer was invented by • dU = P dV Evangelista Torricelli in 1643. − P V 1 dU = d = (P dV + V dP ) Atmospheric pressure can balance a γ 1 γ 1 •  −  − column of mercury of height 760 mm. ∼ dP dV −γ Mean atmospheric pressure at sea level = γ P V • ⇒ P − V ⇒ ∝ on Earth is about 1.0 105 Pa. × For an adiabatic fluid P ργ . • ∝ The weight of the above us per unit area is • This is true as long as the flow is adiabatic. However, even if the equal to this pressure, so the mass of air • 4 −2 thermal conductivity is negligible, the entropy can increase we support is 1.0 10 kg m . ≈ × catastrophically at shock fronts.

γP 1/2 γkT 1/2 The speed of in a gas is v = = The pressure needed to support s ρ m • • the weight ρg dz per unit area (m is the mass of an individual molecule).    is dP = ρg dz (g negative). [Air: γ = 1.4; non-relativistic monatomic gas: γ = 5/3; relativistic gas: γ = 4/3] Generalising, differential equation of hydrostatic equilibrium is •

∇P = ρg Fluids dynamics and plasma physics P 7 Fluids dynamics and plasma physics P 8

HOT GAS IN CLUSTERS OF GALAXIES

When the mean free path λ of particles in a gas or plasma is small • compared to the scales of interest, we can treat the system as a hydrodynamical fluid.

We imagine the fluid composed of fluid elements, which are regions • bigger than λ, large enough to have well-defined values of macroscopic properties; density ρ(x, t), velocity v(x, t); pressure P (x, t); u(x, t) and anything else that is relevant, e.g. magnetic field.

A fluid element is acted on by gravity (and other body forces), by • pressure forces and other stresses on the surface.

These fluid elements stay more or less well-defined (on scales λ), •  but the surface of an element is moving at the local fluid velocity, so elements distort as they move around. The distortions can become very large — in the presence of turbulent motions the fluid elements can be dispersed

There are still some microscopic physical processes occurring on Abell 2029 is a dynamically relaxed cluster of galaxies at z = 0.0779 • • scales of λ that determine important properties of a fluid: heat (300 Mpc). It contains some active galaxies, but none that disturb the conduction; viscosity. These are transport processes resulting from hot cluster gas very much. It is seen here in X-rays emitted by gas that interpenetration of the of the velocity distributions from nearby points. fell into the potential well. The spatial extent is about 300 kpc The macroscopic properties can also change discontinuously on scale The temperature of the gas drops from 9 keV (almost exactly 108 K) in • so λ at shock fronts. the outer regions to 4 107 K in the centre, where there is a massive × We generalise the equation of : 0 = ∇P + ρg to elliptical that is eating everything that comes close. • − include the acceleration of a fluid element: 4 3 The density of the gas is about 10 particles m− and Dv • ρ = ∇P + ρg shows that the gas near the centre has been enriched in metals Dt − (metals are anything with Z > 2) due to supernovae. Fluids dynamics and plasma physics P 9 Fluids dynamics and plasma physics P 10

LAGRANGE OR EULER? THE CONVECTIVE DERIVATIVE

In order to work out the accelerations, • we need to keep track of movement of the fluid elements.

A material element moves with the Here we have a choice: we can watch • • velocity of the fluid, which is a the fluid flowing through space function of space and time: v(x, t). (Eulerian viewpoint), so that we work directly with the ρ(x, t), v(x, t) and P (x, t) For any change of dt, dx the change in velocity is (in components) • variables and work how things are changing ∂v ∂v ∂v at points fixed in space. dv = dt + dx ∇v = dt + dxi ∂t · ∂t ∂xi Alternately, (Lagrangian formulation) we assign time-independent • An element of fluid moves with velocity v, so in time dt, the position labels ξ to the fluid elements and track their positions x(ξ, t). • change is dx = vdt. In the Lagrangian view the mass of fluid in an element is fixed so • 3 Hence for changes at position of fluid elements: d ξ ρ(ξ), is not a function of time. • The dynamics is easier in ’s view: ∂v ∇ dv = dt + v v • ∂x Dv ∂v ∂t · the velocity is v(ξ, t) = and = .   ∂t Dt ∂t  ξ  ξ Dv dv From Lagrange’s viewpoint we keep track of the whole history of the Hence define convective derivative = • • Dt dt flow, but the surfaces of constant ξ can become horribly scrambled as moving with fluid the flow progresses.

v v D ∂ ∇ The Langrangian viewpoint is preferable in principle, but it is really only = + v v • Dt ∂t · possible for one-dimensional flows, but this case includes some very important applications. Fluids dynamics and plasma physics P 11 Fluids dynamics and plasma physics P 12

THE EQUATIONS OF HYDRODYNAMICS

BERNOULLI EQUATION FOR COMPRESSIBLE FLOW The equation of continuity is only needed in the Eulerian formulation: • ∂ρ There is an important theorem which is often useful for cases involving + ∇ (ρv) = 0 • ∂t · steady flow, expressing the as it transported along a streamline: the Bernoulli equation. The Lagrangian force equation (can divide by ρ(x)): • For steady flow, consider streamlines connecting areas A1 and A2: • Dv ∂v 1 = = ∇P (x) + g(x) Dt ∂t −ρ(x)  ξ

The Eulerian force equation:

Dv ∂v ρ = ρ + ρv ∇v = ∇P + ρg 1 2 Energy flowing in: A1v1(ρφ1 + ρ1v + u1) Dt ∂t · − • 2 1 (u is internal energy per unit volume). We also need an energy equation: for Lagrangian hydrodynamics we • Pressure does work: A1v1P1. can use dU = T dS P dV . For adiabatic flow we can often get • − away with using P ργ . The energy equation in Eulerian form is ∝ Similarly at area A2, but mass flow is the same: A1v1ρ1 = A2v2ρ2. complicated by the fact that the volume of fluid elements is changing. • Bernoulli equation: along a streamline the quantity The following equation is only given for completeness ( is the heat • input per unit volume): u + P + 1 v2 + φ = constant Du ρ 2 =  + (T s u P )∇ v Dt − − · γ For a perfect gas specific enthalpy To make a realistic model of any fluid we also need to consider the = u + P = P • • γ 1 effects of viscosity, and thermal conduction, and the any anisotropic − For incompressible flow γ = set u = 0. component to the pressure of a plasma arising from any magnetic field • ∞ present. Fluids dynamics and plasma physics P 13 Fluids dynamics and plasma physics P 14

HYDROSTATIC EQUILIBRIUM OF THE SUN

APPROXIMATIONS TO HYDRODYNAMICS The equation of hydrostatic equilibrium for a spherically-symmetric • body of radius R and mass M : The equations of compressible hydrodynamics (already a massive • dP (r) Gm(r)ρ(r) simplification of the microscopic picture) are clearly very complicated. = dr − r2 They are also notoriously difficult to integrate numerically — in 3-D where m(r) is the mass interior to r. • computing time scales like (grid size)4. Accuracy also a problem due to Multiply by 4πr3 and integrate: advection through grid, which will blur density contrasts. • R R 3 R 2 2 Gm(r) Very useful approximations that yield insight: 4πr P (r) 3 dr 4πr P (r) = dr 4πr ρ(r) 0 − − r • Z0 Z0 ∇ The integrated part is zero if the pressure falls to zero at r = R, and - Incompressible flow: v = 0. Clearly a good model for , • · the integral on the LHS is 3 times the integral of the pressure over but also a surprisingly good approximation for provided the − flow is subsonic. the volume. The RHS is the stored gravitational , i.e. the total energy needed to disassemble the sphere. - Irrotational flow: this is the case when there is no Egrav ∇×v = 0. The vorticity field Ω ∇×v moves with the fluid, and The volume-averaged pressure is therefore P = ≡ • h i − 3V vorticity is usually generated at boundaries, so often the bulk of a The gravitational stored energy depends on the density distribution flow is irrotational. • GM 2 Egrav = f , where f is of order unity. For a uniform sphere If ∇×v = 0 the velocity field can be generated from a scalar potential − R f = 3/5, but it would be greater if the density increases towards the v = ∇Φ (sorry, but fluid dynamics has a plus sign here. . . ). If the centre. fluid is both incompressible and irrotational, the velocity potential 30 8 satisfies Laplace’s equation ∇2Φ = 0. For M = 2 10 kg and R = 7 10 m, we find a stored energy • × × of 4 1041 J and an average pressure of P = 9 1013 Pa In this important case a we can use potential theory to find v and × h i × • Bernoulli’s theorem to find the pressure. Assuming that the Sun is made of hydrogen, we calculate that it • contains 1057 and the same number of . Using a ≈ Feynman calls it the “flow of dry water” (Lectures: Volume 2) P V = NkT we estimate the temperature as 5 106 K, three ≈ × orders of magnitude greater than its surface temperature (6000 K) Fluids dynamics and plasma physics P 15 Fluids dynamics and plasma physics P 16

BERNOULLI EQUATION FOR INCOMPRESSIBLE FLOW THE SUN AND BERNOULLI’S THEOREM Flow from water tank or bathtub. • The pressure is the same at the top of We have concluded that the Sun’s surface is very cool compared to its • • the tank at point A where v = 0 and at interior (the central temperature is 1.56 107 K). × point B where the water escapes. We can get many insights into by applying Bernoulli’s • Use Bernoulli: theorem to (hypothetical) streamlines connecting the Sun’s surface to • P + 1 v2 + gh = constant. infinity. 2 u + P This says the outflow velocity is 2gh. In Bernoulli’s theorem + 1 v2 + φ = constant set • • ρ 2 The actual draining rate might notpbe equal to this velocity times the v∞ = 0, P∞ = u∞ = 0, so the constant is zero. Then at the • area of the hole, since the flow can still be converging as the water surface: leaves the tank. - if the material is cold (P = u = 0) we conclude the velocity −1 Efflux coefficient (effective area of hole divided by geometric area) (inwards or outwards) is 600 km s . • varies between 0.5 to 1. For a simple hole in the side of a - alternatively, if the velocity is low, the temperature must be tank it is 0.62. (approximately)

P kT γ 1 GM Cylindrical jet hitting a wall = = − T = 5 106 K • ρ m γ R ⇒ × Ignore gravity for this problem. • In fact, both alternatives happen: in quiescent conditions there is a hot • The stagnation point at A must have solar corona, which produces a slow 400 km s−1 by heat • 1 2 pressure P0 + ρv0. conduction to radius of about 2 R . 2 The pressure at B (large radius In solar flares and coronal ejections, hot material is thrown upwards, • • compared to jet radius) must be where it escapes as a faster solar wind at 700 km s−1. ≈ P0 again, so Bernoulli says that the velocity is v0 there. Fluids dynamics and plasma physics P 17 Fluids dynamics and plasma physics P 18

VISCOSITY IN GASES COLLISIONAL PROCESSES: VISCOSITY AND THERMAL CONDUCTIVITY

Viscous stresses arise from transport Q = ρvi and we • 1 2 will expect a diffusive term ρλvT∇ v to appear in the force equation. Viscosity and thermal conductivity in fluids and plasmas are 3 • determined by the extent to which the particle distributions at different spatial position can interpenetrate; i.e. it depends on the mean free path λ.

For collisional fluids this is related to the thermal velocity vT and • vT collision frequency νc by λ . ≈ νc Suppose the velocity varies as in a shear layer: v y. The • x ∝ Consider the transport of a quantity Q tangential (force per unit area) on the surface at A is due to the • dvx which varies macroscopically with material coming from B where the flow is faster by λ . dy position. 1 In a time λ/vT a volume λ is transported per unit area, with • 3 Q might be the specific energy u dvx • momentum density ρλ . (thermal conductivity) or a component dy of the momentum ρv (viscosity). The tangential viscous stress is the momentum per unit area per unit • dvx 1 1 dvx dvx time τ = ρλ λ/(λ/vT) = ρλvT = η , where Transport of a Q then occurs by exchange yx dy × 3 3 dy dy • is the viscosity. of “blobs” of amount ∆Q, travelling a distance λ at velocity vT . η v=V This random walk leads to a diffusion equation for property Q, in the In a uniform shear layer the t • • yx B frame moving with the fluid element: viscous shear stress y d (force per unit area) is t 1 2 DQ yx λv ∇ Q = given by τyx = ηV/d. t 3 T Dt yx A τyx = ηV/d is the x-component The factor of 1 in the diffusion coefficient accounts for the fact that the x • 3 of the stress transmitted normal to y-direction. blob is free to move in any of the three . The direction of τ indicates the direction of the stress acting on the • yx inside of the plates at A and B. Fluids dynamics and plasma physics P 19 Fluids dynamics and plasma physics P 20

VISCOSITY IN GASES II

HYDRODYNAMICAL EQUATIONS WITH VISCOSITY

A proper derivation yields a viscous force term η(∇2v + ∇∇ v) • · The mean free path λ is related to the number density of particles n which is the gradient of a symmetric stress tensor: η ( jvi + ivj). • ∇ ∇ and the collision cross-section σ by nσλ = 1. Compressible fluids also have a resistance to compression — a bulk • The cross-section is related to the size of the and various other viscosity η0(∇∇ v). • · assumptions about the , e.g. hard spheres: σ = 4πa2. The force equation now becomes • 1 1 1 3kT The viscosity is given by η = ρλvT mn , where Dv • 3 ∼ 3 × nσ m ρ = ∇P + ρg + η∇2v + (η + η0)∇∇ v m is the particle mass. r Dt − · There are two interesting limits depending on the ratio of the viscous Gas viscosity is independent of density, and proportional to the square • mkT/3 and inertial stresses: root of temperature: η . (Analysis of the hard sphere ≈ p σ - if the viscosity is large we get laminar flow (streamlines are smooth). model suggests a numerical factor of 5√π/16 (4% different. . . .) - if the inertial stresses ρvivj are greater than the viscous stress the Viscosity of air at STP is 1.8 10−5 Pa s, suggesting • × flow may become unstable and turbulence will result. σ = 8 10−19 m2 and a = 2 10−10 m. The mean free path × × implied is 4 microns. We will now do some examples on fluid flow to help you with Problem • Sheet 2. For atomic hydrogen in the interstellar medium, we can scale the • measurements from He4 (46 µPa s at 100 K), to get an estimate of - Sources and sinks. Illustrating potential flow and Bernoulli’s −19 2 6 −3 σ 10 m . At interstellar densities of 10 particles m this equation. ≈ gives λ 1013 m, which is larger than the , but small on ≈ - Potential flow past sphere. Illustrates Bernoulli and problems with interstellar scales. “dry water” approach. The solar wind has a similar density 106 particles m−3, but is • ∼ - Viscous flow. ionised and its mean free path will be smaller, due to plasma processes and magnetic fields. Fluids dynamics and plasma physics P 21 Fluids dynamics and plasma physics P 22

POTENTIAL FLOW — SOURCES AND SINKS II

POTENTIAL FLOW — SOURCES AND SINKS

Sources/sinks: analogue of charges in electrostatics (∇ v = 0). • · 6 Flow rate Q from an isotropic • has velocity potential and flow Q Φ = ; −4πR Q v = uˆ 4πR2 R At the plate the velocity potential is • Q 2 Φ = −4π (d2 + r2)1/2 Use the same methods to solve problems as • where r = (x2 + y2)1/2 is the cylindrical radial coordinate. in electrostatics. Q r The radial velocity at the plate is Source at (0, 0, d), at (0, 0, d). 2 2 3/2 • − • 2π (d + r ) Potential at point P at x, y, z is Q2ρ r2 • Apply Bernoulli to streamline AB: Q 1 1 P = P0 2 2 2 3 Φ = + . • − 8π (d + r ) −4π R1 R2 | | | | Integrate the pressure defect δP = P P0 to get the total force The velocity field is then given by v = ∇Φ. • − • ∞ Q2ρ F = 2πrdr δP = We then use Bernoulli to find the pressure on the plate. −16πd2 • Z0 T. wo sources (or two sinks) distance D apart are attracted to each • Q2ρ other by force (c.f. electrostatics). A source and a sink repel 4πD2 each other (net momentum transfer). Fluids dynamics and plasma physics P 23 Fluids dynamics and plasma physics P 24

POTENTIAL FLOW PAST SPHERE

Steady flow past sphere of radius a. POTENTIAL FLOW PAST SPHERE •

Velocity v = ∇Φ: • a3 At large distances flow is uniform vR = V0 cos θ 1 ; − R3 •   velocity in z direction a3 v = V0 sin θ 1 + Velocity potential θ − 2R3 ⇒   Φ = V0z = V0r cos θ 3 At R = a the velocity v = V0 sin θ. • θ 2 Boundary condition: vr = 0 at r = a. • Apply Bernoulli to the streamline AB: pressure on sphere 2 • Cylindrically-symmetric solutions of Φ = 0 which tend to zero at • ∇ 1 2 9 2 2 r = are P (θ) = P0 + 2 ρV0 8 ρV0 sin θ ∞ − A B C 1 2 High pressure at stagnation points at front and rear of sphere. Φ = + 2 cos θ + 3 2 (3 cos θ 1) . . . • r r r − Pressure symmetric at front and rear in potential flow approximation B ⇒ Present case needs dipole at centre: Φ = V0r cos θ + cos θ no force. • r2 ∂Φ 2B Real flow: fluid cannot slip past surface boundary shear layer, Radial velocity v = = cos θ V0 . • ⇒ • r ∂r − r3 which detaches from surface, causing a slow-moving wake.   1 3 Boundary conditions gives B = V0a . Momentum defect of wake causes drag force (depends on value of • 2 • viscosity via dimensionless Reynolds number (NR = (2a)vρ/η). Velocity potential for flow past sphere • Potential flow solution quite good for rest of the system. • a3 Φ = V cos θ r + 0 2r2   Fluids dynamics and plasma physics P 25 Fluids dynamics and plasma physics P 26

VISCOUS SHEAR LAYER

Dv Equation of motion ρ = ∇P + ρg + η∇2v BOUNDARY LAYERS • Dt − Simple case — shear flow between two flat plates at y = 0 and y = d. v • v=V Boundary condition for surface • t of body: no radial or tangential y yx velocity (no slip). d d g Boundarylayer t Flow past a solid body must develop yx t • v=0 yx a boundary layer with a velocity x gradient. This is a region with vorticity (∇×v = 0), which can then 6 v=0 enter the fluid flow. Steady flow: 0 = ∇P + ρg + η∇2v • − The ratio of the inertial stress ρv2 to the viscous stress ηv/L is an • The equation of motion is a vector equation: important dimensionless quantity known as the Reynolds number • dP ρvL Vertically = ρg N = . dy − R η

2 If the inertial stress is too high in a flow random transverse motions will d vx • Horizontally η = 0 cause turbulent mixing, which in turn increases the effective “eddy dy2 η viscosity” veddyLeddy. Vertical forces imply P (y) = P (0) ρgy (as expected). ρ ∼ • −  effective Horizontal forces give: v = A + By, where A and B are constants. Turbulence transports energy to smaller scales (and to larger scales) • x • V y by splitting up of eddies on a timescale Leddy/veddy. Boundary condition : no slip at either plate v = • x d Energy is rapidly transported to the smallest scales, where viscous In steady flow the horizontal forces on a fluid element are zero, so the • • dissipation can occur. stresses on either side of the element must balance: dvx τ = η = ηV/d. yx dy The stress = ηV/d is constant in y and is transmitted to both plates. • Fluids dynamics and plasma physics P 27 Fluids dynamics and plasma physics P 28

POISEUILLE FLOW IN PIPES — TURBULENCE

Flow in pipe of circular cross-section POISEUILLE FLOW — DRAINING PLATE • with longitudinal pressure gradient.

Turn plate to vertical with free edge Balance of forces on element • • dP dv (keeping x longitudinal and y transverse) 2πrdr = d 2πrη z dz dr (Situation models oil draining from plate.).   dvz 1 2 dP rη = 2 r Pressure is equal to P0 everywhere dr dz • (constant of integration is zero since no stress at r = 0). (ignoring density of air).

Integrating and setting vz = 0 at r = a we get The net viscous force on an element • 1 dP πa4 dP • v = a2 r2 and flow rate Q = must now balance gravity: z 4η dz − 8η dz 2 dτyx = ρgdy, so that dP ρv − 2 Friction factor f: f dτyx d vx • dz ≡ 2a = η 2 = ρg dy dy − Lamina flow: f = 32/NR • where Reynolds number dvx Solution: η = A ρgy, • dy − NR = ρ(2a)v/η. where A is a constant. Flow becomes turbulent at • Boundary condition: the stress must vanish at y = d A = ρgd Reynolds number NR 2000. • ⇒ ≈ After that for turbulent flow. Integrating and applying the boundary condition vx = 0 at y = 0 we f 0.03 • gρ • ≈ get v = yd 1 y2 . In turbulent flow the velocity x η 2 • − profile across bulk of pipe  Characteristic parabolic (Poiseuille) flow profile. The surface flow rate is irregular, but of approximately • gρd2 is constant mean velocity. 2η d gρd3 dP The total volume flow rate per unit area is dy v = Q = . Laminar flow: Q . • x 3η • ∝ dz Z0 dP 1/2 Turbulent flow: Q . • ∝ dz   Fluids dynamics and plasma physics P 29 Fluids dynamics and plasma physics P 30

FLOW PAST SMOOTH SPHERES AND GOLF BALLS

EXPERIMENTAL DRAG COEFFICIENT FOR SPHERE

Boundary layer swept round to low-pressure • region at θ = π/2.

Thereafter pressure predicted by potential • solution increases (adverse pressure gradient).

2 2 Method of dimensions: drag force on sphere:F = ρv d f(NR) Boundary layer detaches and eddies form • • (d is diameter, NR = ρvd/η, f is drag coefficient). behind sphere, decreasing the pressure and giving a drag force. Low Reynolds number: ignore ρ F ηdv (f 1/NR). • ⇒ ∝ ∝ Laminar boundary layer around sphere Drag force on sphere in laminar flow is 6πηav (Solved by Stokes). • • is more likely to detach than a turbulent one. High Reynolds number (> 106) ignore η F ρv2d2 • ⇒ ∝ For smooth sphere the drag coefficient suddenly decreases by factor of (f constant). • 6 two at NR 2 10 due to turbulence. ≈ × Golf balls exploit this by having dimples that cause the transition to • 6 occur earlier at NR 10 . ≈ Fluids dynamics and plasma physics P 31 Fluids dynamics and plasma physics P 32

SUBSONIC TURBULENT JETS

Entrainment of external material slows the jet as it expands. • WAVES IN COMPRESSIBLE FLUIDS

γP Sound waves in a gas travel at vs = . • s ρ

The sound waves carry “information” around the fluid about moving Assumption: turbulent velocity w is proportional to average central • • objects, etc. velocity v: w = αv. If a fluid is moving subsonically Dr • Turbulent motions entrain external fluid into element dz: = αv past a source of waves • Dt (e.g. a body), the waves Element dz of jet shares its forward momentum with entrained fluid, so • are carried slowly downstream, that πr2 dzρv is constant along the jet. but can still propagate upstream. Although the centre of the jet is moving faster than the edges, the • Subsonic movement of fluid is shape of the velocity profile remains the same at all distances, so we • qualitatively different from supersonic flow. set dz = vdt. In terms of the fluid element injected per unit time, we have πr2 ρv2 is constant along the jet.

dr dz Waves get carried away quickly = w = αv = α r = α(z z0) • • dt dt ⇒ − downstream, leaving an The jet expands sideways linearly with distance travelled. The constant undisturbed flow of fluid upstream • α 0.08 for turbulent jets in external medium of equal density. The of the caustic at the Mach angle ≈ 1/2 jet slows down v 1/r. In time: v 1/t . sin α = v /v. ∝ ∝ s High density jets can have lower spreading ratio. Have also assumed • The Mach number M = v/vs where vs is the upstream sound speed. jet was fully turbulent from the outset. • The average velocity profile v(r) is approximately Gaussian. • Fluids dynamics and plasma physics P 33 Fluids dynamics and plasma physics P 34

SHOCK FRONTS II

SHOCK FRONTS Shock fronts are regions dominated by microscopic physics: effects of • viscosity; thermal conduction; generation of turbulence; plasma wave; Supersonic flows are characterised by the presence of shock fronts, magnetic viscosity. • with a sudden discontinuity of density, pressure and velocity. This is Shocks are contained to a length scale of the mean free path (though accompanied by dissipation leading to an increase of entropy. • shocks often determine the mean free path).

Conservation laws of the fluxes • of mass, momentum and energy imply continuity across a normal shock front of

In a one-dimensional shock tube a piston moves into a fluid at a speed Mass flux: ρ1v1 = ρ2v2 • greater than the initial sound speed of the gas in the tube. 2 2 vp vs Momentum flux: P1 + ρ1v1 = P2 + ρ2v2 γP1v1 1 3 γP2v2 1 3 A shock advances into the colder, stationary gas, and hot, shocked gas Energy flux: + 2 ρ1v1 = + 2 ρ2v2 • γ 1 γ 1 then moves with the velocity of the piston. − − Lead to the Rankine-Hugoniot shock conditions. • v2 ρ1 γ 1 2 = = − 1 + 2 v1 ρ2 γ + 1 (γ 1)M  −  Look at the process in the frame moving with the shock. Upstream of • 2 the shock, the particles of the fluid have an ordered, supersonic 2ρ1v1 1 P2 = P1 + 1 γ + 1 − M 2 motion. The downstream particles have a much more disordered   motion (hotter), and a lower bulk velocity. v1 γP1 where M is the upstream Mach number and vs = . vs s ρ1 Collisional gases/plasmas: thickness λ collisional mean free path. (These formulae are not for examination, but very much easier to prove • ∼ than appears at first sight. . . ) Collisionless plasma: thickness v/λDi due to plasma waves. • ∼ Density compression of 4 for the usual case of a strong shock in a • 5 monatomic non-relativistic gas (γ = 3 ). Fluids dynamics and plasma physics P 35 Fluids dynamics and plasma physics P 36

ASTROPHYSICAL PLASMA PHYSICS ASTROPHYSICAL PLASMAS

Plasma is a state of containing significant ionisation. Plasma is • very different from other states of matter (solid, , gas). Why?

A plasma is a soup of oppositely charged particles: electrons/ (or • e−/e+). These particles interact via strong, long-range electromagnetic forces. In gases the interparticle forces are short range and only important in collisions.

Plasmas show a wide range of new, collective phenomena (plasma • waves). These can interact with each other and the particles, producing plasma turbulence and particle acceleration.

2 1/2 nee Most important is the plasma frequency ωpe = . The • 0me whole plasma undergoes longitudinal oscillations at this frequency .

The extent to which particle behave individually or collectively is • determined by the — the distance a thermal particle

travels in time 1/ωpe. Electric fields are screened on scales larger 1/2 0kTe than λ = v /ω = . De T p n e2  e  Influence of magnetic field is hugely important: • eB - Particles circulate the field lines at the gyro frequency ωge = me This limits the mobility of individual particles to the Larmor radius, Almost all material (normal, star stuff ) in the is ionised which is tiny on astrophysical scales. Particles can still stream • (plasma). A tiny fraction of the universe is made up of ordinary , along field lines, but will scatter off magnetic irregularities. liquids and gases (0.1% of the solar system). - the anisotropic can be very important There is a very great range of densities and of plasma dynamically, particularly near compact objects (stars/neutron stars). • found in . Fluids dynamics and plasma physics P 37 Fluids dynamics and plasma physics P 38

DISPERSION CURVES FOR PLASMA WAVES INDIVIDUAL PARTICLES IN PLASMA

Motion of particles in electromagnetic fields:

Electric field gives uniform acceleration, leading to currents opposing • E. This means that plasmas are highly conducting so there are no large-scale E fields.

Plasmas are usually electrically neutral (on scales greater than the • Debye length).

Magnetic field Determines the basic behaviour of particles in • plasma: motion is helix of constant pitch angle.

Velocity parallel to field v is constant (no force). • k Moves in circle perpendicular to field. • mv2 Balance of forces: ⊥ = ev B gives gyro (Larmor) frequency: • r ⊥ t = transverse l = longitudinal eB mv w = s = -sound ω = ; r = ⊥ ⇒ g m eB a = alfven` m = magnetosound ms = magnetised sound Fluids dynamics and plasma physics P 39 Fluids dynamics and plasma physics P 40

PLASMA FREQUENCY AND LONGITUDINAL WAVES COLLECTIVE BEHAVIOUR — DEBYE LENGTH

Consider a plasma of fixed ions and warm electrons of temperature T Electrons in cold plasma can oscillate at the plasma frequency. • e • and average density n0e. Consider a slab of plasma. The ions are • The density depends on the electrostatic potential Φ(r) via too heavy to move far in the of a • Boltzmann factor: , so consider them as a eΦ(r) ne(r) = n0e exp − kTe fixed, neutralising background.   for small eΦ/kTe we approximate exp(x) 1 + x so that the net • 2 ≈ Suppose the electrons are displaced from n0ee • charge is ρe = Φ (the ions cancel out the average charge). the ions by a distance ξ. − kTe 2 σ neeξ 2 ρe n0ee This generates an electric field E = = Equation for electrostatic potential: ∇ Φ = = Φ • 0 0 • − 0 0kTe

Field tends to restore electrons to original position: 1 0kTe The potential satisfies ∇2 , where 2 is the • Φ 2 Φ = 0 λD = 2 • − λ n0ee 2 D ¨ nee ¨ 2 Debye length. meξ = eE = ξ ξ + ωp ξ = 0 − 0 ⇒ Charges move to neutralise the 2 • nee 2 applied field on scales > λD. where ωp = . me0 Consider plasma between charged plates. Simple harmonic motion at the electron plasma frequency • 2 • d Φ 1 In one 2 φ 2 Φ = 0 dx − λD 2 nee has solutions Φ exp( x/λD) ωp = ∝  sme0 Excess charge confined to sheath of size λD near plates. • e−r/λD Fundamental frequency of plasma oscillations. Spherically symmetric field: Φ(r) . • • ∝ r Determines electromagnetic properties; interaction of particles with Screened Coulomb potential (Yukawa). • plasma waves; acceleration of particles. Every ion is surrounded by shielding a “coat” of excess electrons • Fluids dynamics and plasma physics P 41 Fluids dynamics and plasma physics P 42

COLLISIONS IN PLASMA

Mean free paths in plasma are determined by long-range interactions • between ions and electrons. The collision process is almost exactly the same as we have seen for gravitational interactions in a galaxy.

Electron-ion collisions. Consider an electron moving at v going past DEBYE NUMBER • a stationary ion with impact parameter b.

We assume that the interaction is so weak that we can approximate the Consider a sphere of radius the Debye length λD. • • 4 3 trajectory as a straight line, moving at constant velocity. It contains ND πλ ne electrons: the Debye number. ≡ 3 D The Debye number is the number of electrons in the “coat” shielding • any ion in the plasma.

The Debye number is a measure of the importance of collective effects • in the plasma. Ze2 2b Maximum lateral force applied for a time 2 • 4π0b ≈ v If ND < 1 there are no collective effects. The “plasma” is merely a • 2Ze2 collection of individual particles. Momentum transfer per “collision” ∆p = (exact result). • 4π0bv If ND > 1 it is a true plasma and cooperative effects are important. • The lateral momentum changes have 4 32 • Usually ND 1, with ND ranging from 10 (laboratory) to 10 random directions, but the effect •  (cluster of galaxies). on ∆p2 is cumulative: 2 2 ∆p (t) = ∆p Ncoll. ×

Number of collisions with • impact parameter b b + db → per unit time is 2πb db ni v,

where ni = ne/Z. Fluids dynamics and plasma physics P 43 Fluids dynamics and plasma physics P 44

MEAN FREE PATHS IN COLLISIONLESS SYSTEMS PLASMAS AS FLUIDS

A plasma is a highly conducting fluid. In a stationary medium we have The rate of increase of ∆p2(t) per unit time is • • the constitutive relation (Ohm’s law) j = σE. 2 2 4 2Ze Ze In a (non-relativistic) medium this becomes j = σ(E + v×B). If the db 2πb niv = ne 2 log(bmax/bmin) • ≈ 4π0bv 2π0v medium is highly conducting and we let σ , so E = v×B. Z   → ∞ − The cumulative effects of the ∂B • Taken together with the ∇×E = , this gives small collisions build up and • − ∂t are significant when the overall 2 2 2 ∂B ∆p m v = ∇× (v×B) ≈ ∂t

Setting v = v kTe/me and expressing everything in terms of ne One. can show that this means that the magnetic field lines are “frozen • ≈ • and Te we get a meanp free path λei = τcollv = v/νei in” and move with the fluid velocity.

2 4 2 2 2 2 m v 0 k Te 0 NDλD In the force equation of fluid dynamics we have to add a term j×B. λei 4 4 = • ≈ neZe log Λe ≈ Znee log Λe Z log Λe For fluid motions we ignore the displacement current and set

∇×B = µ0J to get where log Λe = log(bmax/bmin). Dv 1 ρ = ∇P + ρg (B× (∇×B)) The largest impact parameter is bmax λD since the ions are • ≈ Dt − − µ0 screened beyond this distance. 1 mag We can write (B× (∇×B)) = ∇ P where • µ j ij  0 i The largest amount of momentum that can be transferred is B2δ B B • mag ij i j ∇ Ze2m Ze2m Z Pij = (used B = 0). e e 2µ0 − µ0 · ∆p mev, so bmin = 2 = 2 .   ≈ 4π0v ≈ 0kT neλ D There is an additional anisotropic magnetic pressure, which has a • 2 The logarithm log Λe is therefore simply log(ND/Z), showing how B • magnitude . Magnetic field provides a pressure force all aspects of the collision process involve collective effects. 2µ0 perpendicular to the field, but a tension in the direction of the field lines. 3/2 1/2 νii Te me νee νei, but ions are heavier so The tension leads to a new form of magnetohydrodynamic wave — • ≈ νee ≈ Ti mi •     Alfven waves, which are like waves on magnetic strings. Fluids dynamics and plasma physics P 45 Fluids dynamics and plasma physics P 46

PROPERTIES OF MAGNETIC PRESSURE TYPES OF MAGNETOHYDRODYNAMIC WAVE

2 mag B δij BiBj Magnetic pressure: Pij = . (1) k parallel to B • 2µ0 − µ0 •   (A) v parallel to k (B) v perpendicular to k Take B = (0, 0, B), so that • 2 B 2µ0 0 0 2 B Pij =  0 0  2 2µ0 ω γsP B 2 B = vs = va =  0 0  k s ρ sµ0ρ  − 2µ0   

Magnetic pressure is positive (i.e. pushes) • perpendicular to B but has tension parallel to B. (2) k perpendicular to B • Magnetic pressure produces waves, just like (A) v parallel to k (B) v perpendicular to k • ω γP sound waves: = (γ is adiabatic index). k s ρ

Parallel to B: (vertical stretch) • B stays the same. P B2 V 0 ∝ ∝ γ = 0. ⇒ k

Perpendicular to B: (horizontal stretch) 2 2 • vm = vs + va B V −1. ∝ P B2 V −2 For arbitrary pdirections of k and v the magnetohydrodynamic modes ∝ ∝ • γ = 2. are mixtures of the above. ⇒ ⊥ Fluids dynamics and plasma physics P 47 Fluids dynamics and plasma physics P 48

GENERAL FEATURES OF TURBULENCE

The spatial power spectrum E(k) of the velocity distribution in • SUMMARY OF IMPORTANT hydrodynamical turbulence shows the general features of all types of turbulence. 2 1/2 nee Plasma frequency ωpe (and ωpi) = Fundamental −5/3 •  m E(k) k  0 e  ∝ frequency of plasma oscillations. Determines electromagnetic properties; interaction of particles with plasma waves; acceleration of particles.

1/2 0kTe Debye length λDe (and λDi) λD = vT/ωp = • n e2  e  Debye number N = 4 πn λ3 1. Energy generation at some spatial scale (k-number). De 3 e De Determine importance of collective effects; screening; Landau 2. Energy flow from this region due to nonlinear effects, through a damping of plasma waves. region with a stationary spectrum (E(k) k−5/3). ∝ Debye number is the figure of merit for a plasma. 3. A region where energy is dissipated in the form of heat. ωpe Collision frequencies (νee, νei, νii) νee = Λe In hydrodynamical turbulence energy is generated on large scales and • NDe • 3/2 1/2 is dissipated by viscosity at the scale of the mean free path k 1/λ. νii Te me ∼ Ions are heavier so νee ≈ Ti mi Determines conductivities; viscosity;  damping of non-resonant waves;

thermal equalisation time. (Λe log(NDe)) ≈ eB Gyro frequency ωge (and ωgi) ωge = • me Usually ωpe. Determines polarisation of electromagnetic waves;  ; mobility of particles in collisionless plasma.

In collisionless plasma turbulence the eventual dissipation will be at low • k-number (again at k 1/λ), so the energy flows the other way. ∼