Fluid Dynamics: Conservation, Flow and Stability: /20 Summary

Context and scope

the supplied physics reference, Applications of Classical Physics — Chapters 13–19 Foundations · Vorticity · Turbulence · Waves · Compressible & Supersonic Flow · Convection · Magnetohydrodynamics

/20 Summary

• A fluid is a continuum that cannot support shear stress at rest ($\mu_{\text{shear}} = 0$). It flows under arbitrarily small applied shear.
• The fluid equations are the conservation laws of Part I, with constitutive relations for stress and energy flux:
  • Mass: $\partial_t \rho + \nabla\cdot(\rho\mathbf{v}) = 0$
  • Momentum (Euler / Navier-Stokes): $\rho\,D\mathbf{v}/Dt = -\nabla P + \mu\nabla^2\mathbf{v} + \rho\mathbf{g}$
  • Energy / entropy: transport with possible heating, viscous dissipation, radiation losses
• One dimensionless group dominates each regime:
  • Reynolds $\text{Re} = vL/\nu$: viscous vs inertial — onset of turbulence
  • Mach $M = v/c_s$: compressibility — subsonic / supersonic / shocks
  • Froude $\text{Fr} = v/\sqrt{gL}$: gravity vs inertia — open-channel / atmospheric
  • Rossby $\text{Ro} = v/(fL)$: rotation vs inertia — geophysical flows
  • Rayleigh $\text{Ra}$: buoyancy vs diffusion — convection onset
  • Magnetic Reynolds $\text{Rm} = vL/\eta_m$: advection vs diffusion of $\mathbf{B}$ — frozen-in vs reconnection
• Vorticity $\boldsymbol\omega = \nabla\times\mathbf{v}$ is the protagonist of incompressible flow. Kelvin's theorem makes it a near-Lagrangian invariant in ideal flow.
• Turbulence is the unsolved problem — but the Kolmogorov $k^{-5/3}$ spectrum in the inertial range is universal and predictive.
• Magnetohydrodynamics = fluid + $\mathbf{B}$. Frozen-in field, Alfvén waves, reconnection, dynamos. Foundation of solar/stellar physics, fusion, magnetosphere.
• Practical reach: weather, oceans, blood, jet engines, plasma reactors, stellar interiors, accretion disks, supernova explosions.

Master Map

mindmap
  root((Fluid Dyn))
    Ch.13 Foundations
      Continuum
      ρ v T fields
      Material deriv D/Dt
      Continuity
      Euler eq
      Navier-Stokes
      Bernoulli
      Hydrostatics
      Surface tension
      Re number
      Stokes flow
      Poiseuille
    Ch.14 Vorticity
      ω = ∇×v
      Kelvin circulation
      Helmholtz theorems
      Vortex tubes
      Rotating frames
      Coriolis
      Taylor-Proudman
      Ekman layer
      Geostrophic
      Rossby
    Ch.15 Turbulence
      Reynolds decomp
      Reynolds stress
      K41 theory
      Cascade
      E(k) ∝ k^-5/3
      Kolmogorov scales
      Intermittency
      Boundary layer
      Log law
      Closure problem
    Ch.16 Waves
      Sound
      Gravity (surface)
      Internal
      Capillary
      Shallow / deep water
      Dispersion ω(k)
      Phase / group v
      Tsunami
      Rossby waves
      WKB
    Ch.17 Compressible
      Mach M
      Sound c_s
      Riemann invariants
      Shocks
      Rankine-Hugoniot
      Oblique shocks
      Prandtl-Meyer
      Nozzles
      Choked
      Sedov-Taylor
    Ch.18 Convection
      Buoyancy
      Boussinesq
      Schwarzschild
      Rayleigh-Bénard
      Ra, Nu, Pr
      Pattern formation
      MLT stellar
    Ch.19 MHD
      Frozen-in
      Magnetic Reynolds Rm
      Alfvén waves
      Fast / slow MS
      Reconnection
      Sweet-Parker
      Dynamos
      MHD instabilities

Foundations of Fluid Dynamics

Continuum Hypothesis & Fluid Variables

Continuum assumption: scale of interest $L$ ≫ mean free path $\ell$ ≫ molecular scale. Define fluid parcels large compared to molecules, small compared to flow.

Equation of state closes the system: $P = P(\rho, T)$ or $P = P(\rho, s)$.

Lagrangian vs Eulerian Descriptions

Material (convective/substantial) derivative:

$$\boxed{\frac{D}{Dt} \equiv \frac{\partial}{\partial t} + \mathbf{v}\cdot\nabla}$$

— rate of change as seen by a fluid element. Bridges Eulerian fields to Lagrangian particle dynamics.

Conservation Laws

Mass — Continuity Equation

$$\boxed{\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v}) = 0 \quad\Leftrightarrow\quad \frac{D\rho}{Dt} = -\rho\nabla\cdot\mathbf{v}}$$

Incompressible flow: $D\rho/Dt = 0 \Rightarrow \nabla\cdot\mathbf{v} = 0$. Valid when $M^2 \ll 1$ AND density variations from heating/stratification are small.

Momentum — Euler & Navier-Stokes

Stress tensor for fluid:

$$T_{ij} = P\,\delta_{ij} - 2\mu\,\sigma_{ij} - \zeta\,(\nabla\cdot\mathbf{v})\,\delta_{ij}$$

with dynamic shear viscosity $\mu$, bulk viscosity $\zeta$, and rate-of-strain $\sigma_{ij} = \tfrac{1}{2}(\partial_i v_j + \partial_j v_i) - \tfrac{1}{3}(\nabla\cdot\mathbf{v})\delta_{ij}$ (deviatoric).

Euler equation (ideal, $\mu = \zeta = 0$):

$$\boxed{\rho\frac{D\mathbf{v}}{Dt} = -\nabla P + \rho\,\mathbf{g}}$$

Navier-Stokes (Newtonian viscous fluid):

$$\boxed{\rho\frac{D\mathbf{v}}{Dt} = -\nabla P + \mu\nabla^2\mathbf{v} + (\zeta + \tfrac{\mu}{3})\nabla(\nabla\cdot\mathbf{v}) + \rho\mathbf{g}}$$

Kinematic viscosity: $\nu = \mu/\rho$ (units m²/s — diffusivity of momentum).

Energy

Internal-energy form: $$\rho T \frac{Ds}{Dt} = \Phi - \nabla\cdot\mathbf{q} + Q$$

with viscous dissipation rate $\Phi = 2\mu\sigma_{ij}\sigma_{ij} + \zeta(\nabla\cdot\mathbf{v})^2 \ge 0$, heat flux $\mathbf{q}$ (Fourier: $-\kappa\nabla T$), volumetric heat source $Q$.

Important: entropy can only increase via $\Phi$ (viscous) and $-\nabla\cdot\mathbf{q}$ (heat conduction). Ideal flow ($\mu=\kappa=0$) is adiabatic and isentropic: $Ds/Dt = 0$.

Hydrostatics

Velocity $= 0$ ⇒ Euler reduces to $\nabla P = \rho\mathbf{g}$.

Archimedes' principle: buoyant force on submerged body = weight of displaced fluid.

Standard atmosphere scale height: $H \approx 8.5$ km on Earth.

Bernoulli's Equation

For steady, ideal, barotropic ($\rho = \rho(P)$) flow with conservative body force $\mathbf{g} = -\nabla\Phi$:

$$\boxed{\frac{v^2}{2} + \int\!\frac{dP}{\rho} + \Phi = \text{const along streamlines}}$$

Incompressible form: $$P + \tfrac{1}{2}\rho v^2 + \rho g z = \text{const}$$

Compressible (adiabatic, ideal gas): $$\tfrac{1}{2}v^2 + \frac{\gamma}{\gamma-1}\frac{P}{\rho} + gz = \text{const}$$

— equivalent to $\tfrac{1}{2}v^2 + c_p T + gz = $ const.

Applications: Venturi tube, Pitot probe, lift on airfoils, fountains, draining tanks (Torricelli $v = \sqrt{2gh}$).

Beware: Bernoulli requires streamline analysis; constants differ on different streamlines (unless flow is also irrotational).

Reynolds Number — The Master Dimensionless Group

$$\text{Re} = \frac{\rho v L}{\mu} = \frac{v L}{\nu} = \frac{\text{inertial}}{\text{viscous}}$$

Typical values: bacteria swimming $\sim 10^{-4}$; insect flight $\sim 10^2$; car at highway $\sim 10^6$; jet aircraft $\sim 10^8$; ocean basin circulation $\sim 10^{12}$.

Low-Reynolds (Stokes) Flow

Drop inertial term:

$$\nabla P = \mu\nabla^2\mathbf{v}, \quad \nabla\cdot\mathbf{v} = 0$$

Linear — solutions superpose.

Stokes drag on sphere (radius $a$, velocity $U$):

$$\boxed{F_{\text{drag}} = 6\pi\mu a U}$$

— foundation of Brownian-motion calibration (Part II), settling velocities, microswimmer physics.

Reversibility: Stokes-flow equations are time-symmetric. Consequence — scallop theorem: a swimmer with only one reciprocal degree of freedom (like a scallop opening/closing) makes no net progress at low Re.

Classic Laminar Flows

Poiseuille (Pipe Flow)

Steady, axisymmetric flow in pipe radius $a$, pressure gradient $-dP/dz$:

$$v_z(r) = \frac{(-dP/dz)}{4\mu}(a^2 - r^2)$$

Volume flux: $Q = \pi a^4(-dP/dz)/(8\mu)$ — the famous $a^4$ dependence (blood vessels constricting by 20% halve their flow).

Couette Flow

Two parallel plates separated by $h$, top moves at $U$, bottom stationary:

$$v(y) = U y/h, \quad \tau = \mu U/h$$

Linear velocity profile. Measurement principle for $\mu$ (rotating cylinder rheometer).

Surface Tension & Capillarity

Surface tension $\gamma$ [N/m]: free energy per area of interface.

Young-Laplace equation (pressure jump across curved interface):

$$\boxed{\Delta P = \gamma(1/R_1 + 1/R_2) = 2\gamma/R \;\text{(sphere)}}$$

Capillary waves (next chapter): $\omega^2 = (\gamma/\rho)k^3$ at small wavelengths (high $k$).

Vorticity

The Vorticity Field

$$\boxed{\boldsymbol\omega \equiv \nabla\times\mathbf{v}}$$

— a pseudovector, twice the local angular velocity of an infinitesimal fluid element.

Decomposition of velocity gradient: $$\partial_j v_i = \sigma_{ij} + \tfrac{1}{3}(\nabla\cdot\mathbf{v})\delta_{ij} - \tfrac{1}{2}\epsilon_{ijk}\omega_k$$

(deviatoric strain + isotropic expansion + rotation).

Irrotational = $\boldsymbol\omega = 0$ ⇒ $\mathbf{v} = \nabla\phi$ (velocity potential). Combined with incompressibility ⇒ $\nabla^2\phi = 0$ (Laplace) — potential flow theory.

The Vorticity Equation

Take curl of Navier-Stokes:

$$\boxed{\frac{D\boldsymbol\omega}{Dt} = (\boldsymbol\omega\cdot\nabla)\mathbf{v} - \boldsymbol\omega(\nabla\cdot\mathbf{v}) + \frac{\nabla\rho\times\nabla P}{\rho^2} + \nu\nabla^2\boldsymbol\omega}$$

Crucial 3-D effect: vortex stretching only exists in 3-D — it has no 2-D analog. This makes turbulence in 3-D qualitatively different (and energy cascades to small scales).

Kelvin's Circulation Theorem

Circulation: $\Gamma = \oint_C \mathbf{v}\cdot d\boldsymbol\ell = \int_S \boldsymbol\omega\cdot d\mathbf{S}$ (Stokes).

For ideal, barotropic flow with conservative forces:

$$\boxed{\frac{D\Gamma}{Dt} = 0 \quad \text{(material loop)}}$$

— circulation around any closed material loop is conserved.

Consequences (Helmholtz vortex theorems):

1. Vortex lines move with the fluid (Lagrangian invariant).
2. Vortex tubes have constant strength along their length.
3. Vortex tubes can't end in the fluid — must close on themselves or terminate at boundaries.

Caveats: broken by viscosity (re-organizes $\omega$), baroclinicity (creates $\omega$ from misaligned gradients), and external curl forces (e.g. magnetic in MHD).

Rotating Reference Frames

Transform to a frame rotating at angular velocity $\boldsymbol\Omega$ (e.g., Earth-frame for geophysics):

$$\rho\frac{D\mathbf{v}}{Dt} = -\nabla P - 2\rho\boldsymbol\Omega\times\mathbf{v} - \rho\boldsymbol\Omega\times(\boldsymbol\Omega\times\mathbf{r}) + \rho\mathbf{g}$$

Coriolis parameter: $f = 2\Omega\sin\lambda$ (latitude $\lambda$).

Geophysical Flow Regimes

Rossby number $\text{Ro} = U/(fL)$: inertia / Coriolis.

• $\text{Ro} \ll 1$: rotation dominates ⇒ geostrophic balance $-\nabla P/\rho = 2\boldsymbol\Omega\times\mathbf{v}$. Flow follows isobars (low to right of velocity, N hemisphere).
• $\text{Ro} \sim 1$: mesoscale weather, large-scale ocean
• $\text{Ro} \gg 1$: small-scale, rotation negligible

Taylor-Proudman theorem: for $\text{Ro}\to 0$ flow, slow steady incompressible motion is 2-D — velocity has no variation along $\boldsymbol\Omega$:

$$(\boldsymbol\Omega\cdot\nabla)\mathbf{v} = 0$$

Spectacular demo: stir tank gently in rotating frame — fluid forms "Taylor columns" parallel to axis, can hide obstacle by acting as if it spanned the entire vertical extent.

Ekman layer: thin boundary layer where viscosity restores no-slip on a rotating surface. Thickness $\delta_E = \sqrt{2\nu/f}$. Generates secondary flow ("Ekman pumping") — drives ocean circulation.

Rossby Waves

• Restoring force: gradient of Coriolis parameter $\partial f/\partial y = \beta$.
• Dispersion: $\omega = -\beta k_x/(k^2 + \kappa^2)$, $\kappa = $ deformation wavenumber.
• Propagate westward (always).
• Govern weather patterns; jet stream undulations are Rossby waves.

Turbulence

The Transition to Turbulence

Above critical Re, laminar flow becomes unstable → cascade through wavenumbers → chaotic 3-D flow.

Critical Re for various configurations:

Transition is sensitive to perturbations — Reynolds's own experiments showed subcritical transition with disturbances and could maintain laminar flow well above nominal Re$_c$ with controlled inlet.

Reynolds Decomposition & Reynolds Stress

Decompose $\mathbf{v} = \bar{\mathbf{v}} + \mathbf{v}'$ (mean + fluctuation, $\langle\mathbf{v}'\rangle = 0$).

Substituting into Navier-Stokes and averaging:

$$\rho\frac{D\bar{v}_i}{Dt} = -\partial_i\bar P + \mu\nabla^2\bar v_i - \partial_j(\rho\overline{v'_i v'_j})$$

— the last term is the Reynolds stress:

$$\boxed{\tau^R_{ij} = -\rho\overline{v'_i v'_j}}$$

Acts on mean flow exactly like a stress — but it's quadratic in fluctuations, hence the closure problem: equations for $\bar v$ involve $\overline{v'v'}$; equations for $\overline{v'v'}$ involve $\overline{v'v'v'}$; ... infinite hierarchy. Must be closed by modeling (mixing length, $k$-$\epsilon$, RANS, LES, DNS).

Kolmogorov (K41) Theory

Premises (1941):

1. Turbulence is statistically homogeneous and isotropic at small scales, independent of large-scale geometry.
2. Energy injected at large scale $L$ cascades to smaller scales, dissipated by viscosity at the Kolmogorov scale $\eta$.
3. In the intermediate "inertial range" $\eta \ll \ell \ll L$, the only relevant parameter is the energy dissipation rate $\epsilon$ [m²/s³].

Kolmogorov Scales (from dimensional analysis with $\nu, \epsilon$)

$$\boxed{\eta = (\nu^3/\epsilon)^{1/4}, \quad v_\eta = (\nu\epsilon)^{1/4}, \quad \tau_\eta = (\nu/\epsilon)^{1/2}}$$

Re at Kolmogorov scale = 1 by construction (viscous and inertial in balance).

Range of scales: $L/\eta \sim \text{Re}^{3/4}$. So $\text{Re} = 10^4 \Rightarrow L/\eta \sim 10^3$; $\text{Re} = 10^8 \Rightarrow L/\eta \sim 10^6$. Why DNS is impossible at high Re.

Energy Spectrum

In the inertial range:

$$\boxed{E(k) = C_K\, \epsilon^{2/3}\, k^{-5/3}}$$

with Kolmogorov constant $C_K \approx 1.5$. Universally observed in atmospheric, oceanic, lab, astrophysical turbulence.

Velocity structure functions: $$\langle (\delta v_\|)^n\rangle = C_n (\epsilon\, \ell)^{n/3} \quad \text{(K41)}$$

Famous 4/5 law (exact, no model assumption beyond K41): $$\langle (\delta v_\|)^3\rangle = -\tfrac{4}{5}\epsilon\, \ell$$

Intermittency Corrections

K41 predicts $\zeta_n = n/3$ (scaling exponents). Reality: $\zeta_n < n/3$ at high $n$ — intermittency. Models (She-Lévêque 1994): $\zeta_n = n/9 + 2[1 - (2/3)^{n/3}]$.

Boundary Layer Turbulence — The Log Law

Near a wall, momentum is carried by turbulent eddies. Self-similar matching argument:

Friction velocity: $u_\tau = \sqrt{\tau_w/\rho}$ where $\tau_w$ is wall shear stress. Wall units: $y^+ = y u_\tau/\nu$, $U^+ = U/u_\tau$.

with von Kármán constant $\kappa \approx 0.41$, $B \approx 5.0$.

The log law is a robust prediction; observed in pipes, channels, atmospheric boundary layer, astrophysical settings.

Free Turbulence — Jets, Wakes, Shear Layers

Coherent structures — large eddies (Kelvin-Helmholtz rolls, hairpin vortices) persist within seemingly chaotic background. Modern view: turbulence = coherent structures + cascade.

Waves

Linearized Wave Equations

Perturb about a uniform background: $\rho = \rho_0 + \rho'$, $\mathbf{v} = \mathbf{v}'$, $P = P_0 + P'$ (primes small).

Linearize continuity and Euler:

$$\partial_t\rho' + \rho_0\nabla\cdot\mathbf{v}' = 0$$ $$\rho_0\partial_t\mathbf{v}' = -\nabla P'$$

Plus equation of state $P' = c_s^2\rho'$.

⇒ Sound wave equation:

$$\boxed{\partial_t^2\rho' = c_s^2\nabla^2\rho', \quad c_s = \sqrt{(\partial P/\partial\rho)_S}}$$

Sound Waves

Plane-wave solution: $\rho' = A\,e^{i(\mathbf{k}\cdot\mathbf{x} - \omega t)}$, $\omega = c_s|\mathbf{k}|$.

Non-dispersive: $v_{\text{phase}} = v_{\text{group}} = c_s$ — sound packets keep shape.

Acoustic impedance $Z = \rho c_s$; reflection coefficient at interface $(Z_2 - Z_1)/(Z_2 + Z_1)$.

Gravity Waves on a Free Surface

Liquid layer of depth $h$, perturbation $\eta(x,t)$. Solving the Laplace equation for the velocity potential + linearized BC at surface:

Full dispersion relation:

$$\boxed{\omega^2 = \left(gk + \frac{\gamma}{\rho}k^3\right)\tanh(kh)}$$

Minimum phase speed of water surface wave: $v_{\min} = (4g\gamma/\rho)^{1/4} \approx 23$ cm/s at $\lambda \approx 1.7$ cm — what your finger creates pushing through water.

Tsunamis

Ocean depth ~ 4 km; $\lambda \sim 100$–$300$ km ⇒ deep ocean is shallow water for tsunamis. Speed:

$$v = \sqrt{gh} \approx 200\,\text{m/s} \approx 720\,\text{km/h}$$

Approaching shore (smaller $h$): slower BUT $\eta$ amplifies as $h^{-1/4}$ (Green's law) → enormous on landfall.

Internal Gravity Waves

In a stratified medium (density $\rho_0(z)$), restoring force = buoyancy. The Brunt-Väisälä frequency:

$$N^2 = -\frac{g}{\rho_0}\frac{d\rho_0}{dz} \;\text{(adiabatic)}$$

(Modify with adiabatic gradient for compressible.) $N^2 > 0$ ⇒ stable stratification.

Dispersion: $\omega = N\cos\theta$ where $\theta$ is angle of $\mathbf{k}$ from horizontal.

Bizarre property: group velocity perpendicular to phase velocity, energy propagates perpendicular to wavefronts. Allow "St. Andrew's Cross" patterns from oscillating sources.

Group vs Phase Velocity

$$v_{\text{phase}} = \omega/k, \quad v_g = d\omega/dk$$

Energy travels at $v_g$ (for narrow-band wavepacket).

Compressible & Supersonic Flow

Mach Number & Compressibility

$$M = v/c_s$$

• $M < 0.3$: density variations $\lesssim 5\%$, incompressible approximation OK.
• $M < 1$: subsonic; disturbances propagate upstream
• $M = 1$: sonic
• $M > 1$: supersonic; upstream cannot "know" about downstream → shocks possible
• $M \gg 1$: hypersonic

Steady, Quasi-1-D Flow

Incompressible + Bernoulli gives $P + \rho v^2/2 = $ const. Compressible version:

$$\tfrac{1}{2}v^2 + \frac{\gamma}{\gamma-1}\frac{P}{\rho} = \text{const} = \tfrac{1}{2}v_{\max}^2$$

Stagnation properties:

$$\frac{T_0}{T} = 1 + \frac{\gamma-1}{2}M^2$$ $$\frac{P_0}{P} = \left[1 + \tfrac{\gamma-1}{2}M^2\right]^{\gamma/(\gamma-1)}$$ $$\frac{\rho_0}{\rho} = \left[1 + \tfrac{\gamma-1}{2}M^2\right]^{1/(\gamma-1)}$$

Subscript 0 = adiabatic stagnation (brought to rest reversibly).

Riemann Invariants (1-D Unsteady)

For 1-D isentropic flow, define $J_\pm = v \pm \int dP/(\rho c_s) = v \pm 2c_s/(\gamma-1)$. Then:

$$\left(\partial_t + (v\pm c_s)\partial_x\right) J_\pm = 0$$

— $J_+$ propagates rightward at $v+c_s$, $J_-$ leftward at $v-c_s$. Characteristic curves of the hyperbolic system.

Shock Waves — Rankine-Hugoniot Jump Conditions

When characteristics converge, smooth flow develops discontinuity. Conservation across shock (frame in which shock is stationary):

For ideal gas, normal shock, define $M_1 = v_1/c_{s,1}$ (upstream Mach):

$$\frac{P_2}{P_1} = 1 + \frac{2\gamma}{\gamma+1}(M_1^2 - 1)$$

$$\frac{\rho_2}{\rho_1} = \frac{v_1}{v_2} = \frac{(\gamma+1)M_1^2}{(\gamma-1)M_1^2 + 2}$$

$$M_2^2 = \frac{(\gamma-1)M_1^2 + 2}{2\gamma M_1^2 - (\gamma-1)}$$

Properties:

• $M_1 > 1 \Rightarrow M_2 < 1$ (shock decelerates flow to subsonic).
• Entropy increases across shock (2nd law).
• For strong shocks ($M_1 \to \infty$): $\rho_2/\rho_1 \to (\gamma+1)/(\gamma-1) = 4$ for $\gamma = 5/3$.
• For weak shock: thickness $\sim$ several mean free paths.

Oblique Shocks & Prandtl-Meyer Expansion

Oblique shock at angle $\beta$ relative to upstream flow, with flow deflection $\theta$:

$$\tan\theta = 2\cot\beta\,\frac{M_1^2\sin^2\beta - 1}{M_1^2(\gamma + \cos 2\beta) + 2}$$

For given $M_1$, maximum deflection $\theta_{\max}$ (beyond which shock detaches).

Prandtl-Meyer expansion fan: smooth supersonic turn through convex corner. Defines:

$$\nu(M) = \sqrt{\tfrac{\gamma+1}{\gamma-1}}\arctan\sqrt{\tfrac{(\gamma-1)(M^2-1)}{\gamma+1}} - \arctan\sqrt{M^2-1}$$

Flow turning angle = $\nu(M_2) - \nu(M_1)$.

De Laval Nozzle (Converging-Diverging)

Steady 1-D flow with area $A(x)$:

$$\frac{dA}{A} = (M^2 - 1)\frac{dv}{v}$$

→ to accelerate gas from subsonic to supersonic, need converging-diverging nozzle. Sonic point $M=1$ at throat (smallest $A$).

Choked flow: at $M = 1$ at throat, mass flux is maximal for given stagnation properties:

$$\dot m_{\max} = A_t P_0\sqrt{\frac{\gamma}{RT_0}}\left(\frac{2}{\gamma+1}\right)^{(\gamma+1)/[2(\gamma-1)]}$$

Foundation of rocket engines, jet engines, gas-turbine cycles.

Sedov-Taylor Blast Wave

Strong point explosion releases energy $E$ in uniform medium of density $\rho_0$. Self-similar solution: shock radius

$$\boxed{R_s(t) = \left(\frac{\xi E t^2}{\rho_0}\right)^{1/5}}$$

with dimensionless $\xi \approx 1$. Famous use: G. I. Taylor declassified the Trinity test yield from photographs of the fireball expansion.

Astrophysical analog: supernova remnants in their Sedov-Taylor phase (after free expansion, before snowplow).

Convection

Stability of Stratification — Schwarzschild Criterion

A fluid parcel displaced upward adiabatically retains its entropy. If its new density is less than ambient → buoyant → unstable.

Schwarzschild criterion for stability:

$$\boxed{\frac{dT}{dz} > \frac{dT}{dz}\bigg|_{\text{ad}} = -\frac{g}{c_p}}$$

(temperature decreases slower than adiabatic ⇒ stable).

In atmospheric science: lapse rate $\Gamma = -dT/dz$.

• $\Gamma < \Gamma_{\text{ad}} \approx 9.8$ K/km: stable.
• $\Gamma > \Gamma_{\text{ad}}$: convectively unstable.

In stellar physics: convective vs radiative zones determined by $\nabla = d\ln T/d\ln P$ vs $\nabla_{\text{ad}}$.

Boussinesq Approximation

When density variations are small ($\Delta\rho/\rho \ll 1$) but driving buoyancy: keep $\rho$ constant everywhere except the buoyancy term:

$$\rho_0\frac{D\mathbf{v}}{Dt} = -\nabla P' + \rho'\mathbf{g} + \mu\nabla^2\mathbf{v}, \quad \rho' = -\rho_0\alpha(T - T_0)$$

with $\alpha = -\rho_0^{-1}(\partial\rho/\partial T)_P$ (thermal expansion).

Excellent approximation for: oceans, atmosphere (locally), Earth's mantle, lab thermal convection.

Rayleigh-Bénard Convection

Fluid between two horizontal plates, bottom hot, top cool. Three key dimensionless groups:

Critical Rayleigh (for plate-to-plate, rigid free boundaries; Rayleigh 1916; Pellew & Southwell 1940):

• Rigid–rigid: $\text{Ra}_c = 1707.76$
• Free–free: $\text{Ra}_c = 657.51$
• Rigid–free: $\text{Ra}_c \approx 1100.65$

At onset, critical wavenumber $k_c d \approx 3.117$ ⇒ horizontal wavelength ≈ 2$d$.

Beyond onset:

• $\text{Ra} \sim 10^3$–$10^4$: hexagonal cells, rolls
• $\text{Ra} \sim 10^4$–$10^5$: oscillatory time-dependence
• $\text{Ra} \gg 10^6$: turbulent convection
• $\text{Nu} \sim \text{Ra}^{1/3}$ (Malkus); $\text{Ra}^{2/7}$ in some experiments

Mixing Length Theory (Stellar Convection)

Phenomenological model: convective "blobs" travel a mixing length $\ell_m \sim H_p$ (pressure scale height) before dissolving.

Convective velocity: $$v_{\text{conv}} \sim \sqrt{g\beta\Delta\nabla\,\ell_m}$$

with $\Delta\nabla = \nabla - \nabla_{\text{ad}}$ (super-adiabatic excess).

Convective heat flux: $$F_{\text{conv}} \sim \rho c_p v_{\text{conv}} T(\Delta\nabla)$$

In stellar interiors, $\Delta\nabla$ is tiny ($\sim 10^{-7}$), so convection is enormously efficient. In stellar atmospheres, $\Delta\nabla$ is order unity — radiative losses matter — granulation we see on the Sun.

Pattern Formation

Above onset, convection picks definite spatial patterns: rolls, hexagons, squares, spirals (in rotation), traveling waves. Amplitude (Landau-Ginzburg) equation captures the weakly nonlinear regime, akin to phase transitions in Part II.

Magnetohydrodynamics (MHD)

The MHD Equations

Combine fluid dynamics with Maxwell's equations under the assumption that the fluid is highly conducting and slowly varying compared to $\omega_{\text{plasma}}$.

MHD assumptions:

1. Non-relativistic ($v \ll c$); drop displacement current.
2. Charge neutrality (quasineutral plasma).
3. Ideal Ohm's law in fluid frame: $\mathbf{E}' = 0 \Rightarrow \mathbf{E} = -\mathbf{v}\times\mathbf{B}$ (resistive: $\mathbf{E} = -\mathbf{v}\times\mathbf{B} + \mathbf{J}/\sigma$).

Full ideal MHD system:

$$\partial_t\rho + \nabla\cdot(\rho\mathbf{v}) = 0$$

$$\rho\frac{D\mathbf{v}}{Dt} = -\nabla P + \frac{1}{\mu_0}(\nabla\times\mathbf{B})\times\mathbf{B} + \rho\mathbf{g}$$

$$\partial_t\mathbf{B} = \nabla\times(\mathbf{v}\times\mathbf{B}) + \eta_m\nabla^2\mathbf{B}$$

$$\nabla\cdot\mathbf{B} = 0$$

with magnetic diffusivity $\eta_m = 1/(\mu_0\sigma)$.

Lorentz force decomposed:

$$\frac{(\nabla\times\mathbf{B})\times\mathbf{B}}{\mu_0} = -\nabla\!\left(\frac{B^2}{2\mu_0}\right) + \frac{(\mathbf{B}\cdot\nabla)\mathbf{B}}{\mu_0}$$

= magnetic pressure $-\nabla(B^2/2\mu_0)$ + magnetic tension along field lines.

Frozen-In Field Theorem (Alfvén's Theorem)

For ideal MHD ($\eta_m \to 0$): magnetic flux through any material loop is conserved.

$$\frac{d\Phi}{dt} = 0$$

Equivalent: field lines move with the fluid. The induction equation reduces to $\partial_t\mathbf{B} = \nabla\times(\mathbf{v}\times\mathbf{B})$ which is structurally identical to Kelvin's vorticity theorem.

Astrophysical consequence: stretching matter stretches frozen-in fields, amplifying $|B|$ — basis of dynamos. Compressing reduces flux per area but increases field strength.

Magnetic Reynolds Number

$$\text{Rm} = vL/\eta_m$$

• $\text{Rm} \gg 1$: ideal MHD, frozen-in dominates (most astro contexts).
• $\text{Rm} \ll 1$: diffusion dominates (e.g., liquid metals in lab).
• $\text{Rm} \sim 1$: reconnection / dynamo / instability regime.

Typical Rm: | System | Rm | |---|---| | Earth's outer core | $10^3$ | | Sun's interior | $10^{10}$ | | Interstellar medium | $10^{15}$+ | | Lab plasma | $10^2$–$10^5$ |

Alfvén Waves

Linearize MHD about uniform $\mathbf{B}_0$. Three wave modes:

with $\theta$ = angle between $\mathbf{k}$ and $\mathbf{B}_0$.

Alfvén speed:

$$\boxed{v_A = \frac{B}{\sqrt{\mu_0\rho}}}$$

Properties of shear Alfvén wave:

• Propagates along $\mathbf{B}_0$ ($\theta = 0$): $\omega = v_A k_\|$.
• Polarization: $\mathbf{v}_1, \mathbf{B}_1 \perp \mathbf{B}_0$ and to each other.
• Incompressible ($\nabla\cdot\mathbf{v}_1 = 0$).

Typical $v_A$: | System | $v_A$ | |---|---| | Solar corona | $\sim 1000$ km/s | | Solar wind | $\sim 50$ km/s | | Earth's magnetosphere | $\sim 100$ km/s | | Tokamak | $\sim 10^6$ m/s | | ISM (warm phase) | $\sim 5$ km/s |

Magnetic Reconnection

When opposite-polarity field lines come together, frozen-in flux locally breaks down (large gradients → large diffusion). Topology changes; magnetic energy converted to kinetic + thermal.

Sweet-Parker Reconnection

Thin current sheet of length $L$, thickness $\delta$:

• Mass conservation: $v_{\text{in}} L = v_A \delta$
• Resistive diffusion at sheet: $v_{\text{in}} \delta = \eta_m$
• ⇒ $\delta = L/\sqrt{\text{S}}$, reconnection rate $v_{\text{in}}/v_A = 1/\sqrt{\text{S}}$

where Lundquist number $\text{S} = L v_A/\eta_m$.

Problem: for solar values $\text{S} \sim 10^{12}$, predicted reconnection time scale far too slow to explain observations.