Research-Stack/6-Documentation/docs/biomechanical_pressure_cavitation_vibration_models_v0_2.md

Biomechanical Pressure–Cavitation–Vibration Model Catalog v0.2

Scope: Pressure gradients, cavitation, shockwave propagation, hydrodynamic vibration, lateral-line sensing, suction, jetting, hydraulic actuation, osmotic projectiles, and Burgers-style shock smoothing.

Status: Working review file. This is not a final paper. It is a model inventory and stress-test scaffold for later Lean-facing adapters, simulations, and evidence receipts.

Update basis: This version folds in the attached cavitation / Acoustic-Crystalline Water / Burgers stress-test notes and separates:

1. documented peer-reviewed model families,
2. idealized test-material assumptions,
3. speculative or unverified quantitative claims that need receipts.

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0. Core Adapter Schema

A species or material system enters the framework through an adapter:

[ \alpha_S : X_S \rightarrow U(\xi,\tau) ]

where X_S is the native physical state and U is a normalized dimensionless field.

For pressure/cavitation/vibration systems:

[ X_S = (P,\rho,\mu,\sigma,c,R,\dot R,\mathbf{u},\nabla P,\nabla \mathbf{u},f,t,\Omega) ]

[ U(\xi,\tau)

w_P\widehat{\Delta P} + w_u|\hat{\mathbf{u}}| + w_R\hat R + w_{\nabla P}|\widehat{\nabla P}| + w_f \hat f ]

A general residual layer can then test propagation or shock-like deviations:

[ R(U;\epsilon)

\partial_\tau U + U\partial_\xi U

\epsilon\partial_{\xi\xi}U ]

This is only an adapter-level diagnostic; it is not a replacement for full fluid equations.

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1. Cavitation Bubble Dynamics

1.1 Rayleigh–Plesset Equation

Use: Spherical bubble growth/collapse in an incompressible liquid.

[ \rho_l \left( R\ddot R + \frac{3}{2}\dot R^2 \right)

P_B(R,t)

P_\infty(t)

\frac{2\sigma}{R}

\frac{4\mu \dot R}{R} ]

A common gas-pressure closure is:

[ P_B(R)

\left( P_{\infty,0} + \frac{2\sigma}{R_0} \right) \left( \frac{R_0}{R} \right)^{3\gamma} + P_v ]

Variables

Symbol	Meaning
R(t)	bubble radius
\rho_l	liquid density
\sigma	surface tension
\mu	liquid viscosity
\gamma	polytropic gas index
P_\infty(t)	far-field liquid pressure
P_v	vapor pressure
P_B	internal bubble pressure

Strengths

• Excellent baseline for growth/collapse timing.
• Good for subsonic spherical collapse.
• Common first model for cavitation, sonoluminescence, and snapping-shrimp bubble radius fitting.

Failure modes

• Assumes incompressible liquid.
• Cannot directly model acoustic radiation or shock emission.
• Can become singular or overpredict collapse intensity when compressibility matters.

Evidence anchor

Versluis et al. reported that snapping shrimp sound is emitted at cavitation bubble collapse and that a Rayleigh–Plesset-type model quantitatively accounts for bubble radius time-dependence and emitted sound.

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1.2 Keller–Miksis Equation

Use: Weakly compressible bubble dynamics; includes first-order acoustic radiation terms.

[ \left(1-\frac{\dot R}{c}\right)R\ddot R + \frac{3}{2}\dot R^2 \left(1-\frac{\dot R}{3c}\right)

\frac{1}{\rho_l} \left( 1+\frac{\dot R}{c} + \frac{R}{c}\frac{d}{dt} \right) [ P_B(R,\dot R)-P_\infty(t) ] ]

Added physics

Term	Role
c	liquid sound speed
\dot R/c	bubble-wall Mach correction
R/c \cdot d/dt	acoustic radiation / compressibility correction

Strengths

• Better than Rayleigh–Plesset for sonoluminescence, ultrasound cavitation, and moderate collapse.
• Captures acoustic damping.

Failure modes

• First-order compressibility only.
• Less reliable for very high Mach collapse, strong shocks, and highly nonlinear equations of state.

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1.3 Gilmore Equation / Gilmore–Akulichev Type Models

Use: High-amplitude, compressible bubble collapse using liquid enthalpy and pressure-dependent sound speed.

[ R\ddot R \left( 1-\frac{\dot R}{C} \right) + \frac{3}{2}\dot R^2 \left( 1-\frac{\dot R}{3C} \right)

H \left( 1+\frac{\dot R}{C} \right) + \frac{R}{C}\dot H \left( 1-\frac{\dot R}{C} \right) ]

where:

[ H = \int_{P_\infty}^{P_R}\frac{dP}{\rho(P)} ]

Strengths

• Better for high-pressure collapse and shock generation.
• Natural fit with Tait-like equations of state.
• More physically grounded when wall velocity approaches liquid sound speed.

Failure modes

• Still assumes spherical symmetry unless coupled to CFD.
• Needs reliable liquid equation of state.
• May fail under plasma, ionization, chemistry, phase change, and strong non-spherical jetting.

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1.4 Tait Equation of State

Use: Pressure-density relation for water-like liquids under compression.

A common Tait form:

[ P + B

(P_0+B) \left( \frac{\rho}{\rho_0} \right)^n ]

Equivalent shifted form:

[ P

B \left[ \left( \frac{\rho}{\rho_0} \right)^n

1 \right] + P_0 ]

Approximate water values near room temperature often use:

[ n \approx 7.15 ]

[ B \approx 300\ \text{MPa} ]

Derived sound speed

[ c^2

\left( \frac{\partial P}{\partial \rho} \right)_s

\frac{n(P+B)}{\rho} ]

Use in framework

Gilmore + Tait gives the best compact model for high-amplitude cavitation shock estimates before switching to full compressible CFD.

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1.5 Rayleigh–Plesset-Based Homogeneous Mixture Models

Use: CFD cavitating-flow model with vapor/liquid mixture.

Mixture continuity:

[ \frac{\partial \rho_m}{\partial t} + \nabla\cdot(\rho_m\mathbf{u})

0 ]

Momentum:

[ \frac{\partial \rho_m\mathbf{u}}{\partial t} + \nabla\cdot(\rho_m\mathbf{u}\mathbf{u})

-\nabla p + \nabla\cdot\boldsymbol{\tau} + \mathbf{f} ]

Void fraction relation:

[ \rho_m

\alpha_v\rho_v + (1-\alpha_v)\rho_l ]

Transport:

[ \frac{\partial \alpha_v}{\partial t} + \nabla\cdot(\alpha_v\mathbf{u})

\dot m_{\text{vap}}-\dot m_{\text{cond}} ]

Rayleigh–Plesset-type growth supplies source terms:

[ \dot R = \mathcal{F}(P_v-P,\rho,\sigma,\mu,R) ]

Use cases

• Snapping shrimp claw CFD.
• Cavitating jets.
• Hydrofoils.
• Bubble clouds.
• Bioinspired snapping plunger devices.

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2. Cavitation Shockwave Properties

2.1 Collapse Pressure Estimate

Far-field acoustic pressure from a spherical bubble can be approximated by source acceleration:

[ p(r,t) \approx \frac{\rho_l}{r} \frac{d}{dt} \left( R^2\dot R \right) ]

Expanding:

[ p(r,t) \approx \frac{\rho_l}{r} \left( 2R\dot R^2 + R^2\ddot R \right) ]

Near collapse, R^2\ddot R and R\dot R^2 can generate extremely sharp pressure pulses.

2.2 Shock Decay

Ideal spherical acoustic decay:

[ P(r) \propto \frac{1}{r} ]

Near-field nonlinear shock decay is usually stronger:

[ P(r) \propto \frac{1}{r^\alpha} ]

with:

[ \alpha > 1 ]

The value of \alpha depends on amplitude, equation of state, viscosity, thermal conduction, geometry, and bubble asymmetry.

2.3 Microjet Water-Hammer Pressure

For asymmetric collapse near boundaries:

[ P_{\text{hammer}} \approx \rho c v_{\text{jet}} ]

where v_{\text{jet}} is the microjet impact speed.

This is separate from the spherical acoustic shock.

2.4 Collapse Energy

Bubble potential energy at maximum radius can be approximated by:

[ E_B \approx \frac{4\pi}{3} R_{\max}^3 (P_\infty - P_v) ]

Shock/radiated fraction:

[ E_{\text{shock}}

\eta_{\text{shock}}E_B ]

where \eta_{\text{shock}} must be measured or modeled. It is not a universal constant.

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3. Idealized Test Medium: Acoustic-Crystalline Water

3.1 Definition

Acoustic-Crystalline Water (ACW) is an idealized water-like continuum for model stress testing.

ACW assumptions

Property	ACW value / rule
Structure	homogeneous continuum
Dissolved gas	none
Microbubble nuclei	none
Surface tension	\sigma = 0.072\ \text{N/m}
Speed of sound	c_0 \approx 1500\ \text{m/s}
EOS	Tait equation
Viscosity	either real water or inviscid test limit
Thermal conduction	explicit switch: off / on
Phase change	explicit switch: off / on

3.2 Use

ACW is not a claim about a real material. It is a controlled mathematical substrate for comparing:

1. Rayleigh–Plesset,
2. Keller–Miksis,
3. Gilmore + Tait,
4. compressible CFD,
5. Burgers shock propagation.

3.3 Reality-tether table

Parameter	Real water near 20 °C	ACW default	Notes
Density \rho	~998 kg/m³	1000 kg/m³	close
Speed of sound c	~1482 m/s	1500 m/s	close
Dynamic viscosity \mu	~1.0e-3 Pa·s	switchable	inviscid is a ceiling case
Surface tension \sigma	~0.072 N/m	0.072 N/m	close
Vapor pressure P_v	~2.3 kPa	switchable	neglecting it exaggerates collapse
Dissolved gas	present	absent	ACW overpredicts collapse cleanliness

3.4 Caution

The uploaded notes propose strong numerical statements such as extreme shock-front thickness and Mach cutoff values. These should remain provisional unless backed by experimental or simulation receipts.

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4. Burgers Equation as Shock-Propagation Bridge

4.1 Inviscid Burgers Equation

[ \partial_t u + u\partial_x u = 0 ]

Characteristic solution

[ u(x,t) = u_0(\xi) ]

[ x = \xi + u_0(\xi)t ]

Shock forms when:

[ \frac{\partial x}{\partial \xi}

1 + u_0'(\xi)t

0 ]

Earliest shock time:

[ t_s

-\frac{1}{\min u_0'(\xi)} ]

for \min u_0'(\xi)<0.

Relevance

Models nonlinear steepening of a pressure pulse but cannot model physical shock thickness.

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4.2 Viscous Burgers Equation

[ \partial_t u + u\partial_x u

\nu\partial_{xx}u ]

Cole–Hopf transform

Let:

[ u = -2\nu \partial_x \ln \phi ]

Then:

[ \partial_t \phi = \nu \partial_{xx}\phi ]

Traveling shock solution

For left/right states u_L > u_R:

[ u(x,t)

u_R + \frac{u_L-u_R} {1+\exp\left[ \frac{(u_L-u_R)(x-st)}{2\nu} \right]} ]

Shock speed:

[ s = \frac{u_L+u_R}{2} ]

Shock thickness scaling:

[ \delta \sim \frac{2\nu}{u_L-u_R} ]

or by convention:

[ \delta \sim \frac{4\nu}{\Delta u} ]

Relevance

This is the clean bridge between ideal discontinuous shock and physically smeared shock.

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4.3 Forced Burgers / Acoustic Burgers

For nonlinear acoustics in lossy media, a Burgers-like equation often appears in retarded time form:

[ \frac{\partial p}{\partial x}

\frac{\beta}{\rho c^3} p\frac{\partial p}{\partial \tau} + \frac{\delta}{2c^3} \frac{\partial^2 p}{\partial \tau^2} ]

where:

Symbol	Meaning
p	acoustic pressure
x	propagation distance
\tau = t-x/c	retarded time
\beta	nonlinearity parameter
\delta	sound diffusivity / attenuation coefficient
c	sound speed

Relevance

Better than plain Burgers when modeling finite-amplitude acoustic shock propagation in water.

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4.4 Burgers–Gilmore Bridge

Gilmore models the bubble/source.

Burgers models shock propagation after emission.

[ \text{Bubble collapse} \rightarrow p(r_0,t) \rightarrow \text{Burgers propagation} \rightarrow p(r,t) ]

Boundary condition:

[ p(r_0,t)

p_{\text{Gilmore}}(t) ]

Propagation:

[ \partial_x p

\frac{\beta}{\rho c^3}p\partial_\tau p + \frac{\delta}{2c^3}\partial_{\tau\tau}p ]

Interpretation

• Gilmore alone may overstate material damage if propagation losses are omitted.
• Burgers alone does not generate the bubble collapse source.
• The bridge is valid only while the pulse can be approximated as a weak/finite-amplitude acoustic shock rather than full multiphase compressible flow.

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5. Stress-Test Regimes

5.1 Mach stress

Bubble wall Mach number:

[ M_R = \frac{|\dot R|}{c} ]

Regimes:

M_R	Regime	Preferred model
M_R \ll 1	incompressible / weak acoustic	Rayleigh–Plesset
M_R < 1 but finite	weak compressibility	Keller–Miksis
M_R \sim 1	strong collapse	Gilmore + Tait
M_R > 1	shock-dominant	compressible CFD / Gilmore with caution
very high M_R	ionization/plasma possible	EOS + radiation/MHD/chemistry needed

5.2 Nano-scale stress

Bubble Reynolds number:

[ Re_R = \frac{\rho R |\dot R|}{\mu} ]

When R becomes very small, viscous effects dominate.

Viscous damping scale:

[ D_\nu \sim \nu \partial_{xx}u ]

Nonlinear steepening scale:

[ N \sim u\partial_x u ]

Ratio:

[ \chi = \frac{N}{D_\nu} \sim \frac{uL}{\nu}

Re ]

If:

[ \chi \ll 1 ]

then shock formation is suppressed.

5.3 High-density / high-impedance stress

Acoustic impedance:

[ Z = \rho c ]

Shock transmission/reflection at an interface:

[ \mathcal{R}

\frac{Z_2-Z_1}{Z_2+Z_1} ]

[ \mathcal{T}

\frac{2Z_2}{Z_2+Z_1} ]

High-Z fluids/materials change shock focusing, reflection, and local damage.

5.4 Plasma / chemistry failure mode

Adiabatic gas-temperature estimate:

[ T_{\max}

T_0 \left( \frac{R_{\max}}{R_{\min}} \right)^{3(\gamma-1)} ]

If temperature and density reach ionization/chemistry thresholds, hydrodynamic-only models fail.

Required extensions:

• reactive flow,
• plasma equation of state,
• radiative transfer,
• MHD if electromagnetic effects are non-negligible.

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6. Pistol Shrimp Models

6.1 Biological mechanism

[ \text{claw closure} \rightarrow \text{high-speed jet} \rightarrow \text{vortex core depressurization} \rightarrow \text{cavitation ring} \rightarrow \text{collapse shock} ]

6.2 Vortex / jet model

Jet Reynolds number:

[ Re_j = \frac{\rho U_j D}{\mu} ]

Cavitation number:

[ Ca = \frac{P_\infty - P_v}{\frac{1}{2}\rho U_j^2} ]

Cavitation likely when:

[ Ca < Ca_{\text{crit}} ]

6.3 Vortex pressure drop

Approximate vortex-core pressure drop:

[ \Delta P_{\text{vortex}} \sim \frac{1}{2}\rho v_\theta^2 ]

Cavitation condition:

[ P_\infty - \Delta P_{\text{vortex}} < P_v ]

6.4 Action value

[ V_{\text{snap}}

D_{\text{target}} + I_{\text{contest}} + I_{\text{communication}}

E_{\text{snap}}

C_{\text{wear}} ]

---

7. Mantis Shrimp Models

7.1 Spring-latch mechanics

[ E_{\text{spring}}

\frac{1}{2}kx^2 ]

[ P_{\text{release}}

\frac{E_{\text{spring}}}{\Delta t} ]

[ E_{\text{club}}

\frac{1}{2}m_{\text{club}}v_{\text{club}}^2 ]

7.2 Impact impulse

[ J_{\text{impact}}

\int F_{\text{impact}}(t),dt ]

7.3 Dual hit

[ D_{\text{total}}

D_{\text{impact}} + D_{\text{cavitation}} ]

[ D_{\text{cavitation}} \propto \int P_{\text{collapse}}(t)A_{\text{target}},dt ]

7.4 Cavitation inception around strike

A simple threshold:

[ P_{\text{local}} < P_v ]

or using cavitation number:

[ Ca = \frac{P_\infty-P_v}{\frac12\rho U^2} ]

Cavitation appears when Ca crosses a mechanism-specific threshold.

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8. Suction Feeding Models

8.1 Buccal pressure drop

[ \Delta P_{\text{buccal}}

P_{\text{ambient}} - P_{\text{mouth}} ]

8.2 Flow field

Incompressible continuity:

[ \nabla\cdot\mathbf{u}=0 ]

Navier–Stokes:

[ \rho \left( \partial_t\mathbf{u} + \mathbf{u}\cdot\nabla\mathbf{u} \right)

-\nabla P + \mu\nabla^2\mathbf{u} ]

8.3 Force on prey

Pressure-gradient force:

[ F_{\Delta P}

• V_{\text{prey}}\nabla P ]

Drag:

[ F_D

\frac{1}{2}\rho C_D A |\mathbf{u}-\mathbf{v}_{\text{prey}}|^2 ]

Acceleration reaction:

[ F_A

C_A\rho V_{\text{prey}} \frac{D\mathbf{u}}{Dt} ]

Total:

[ F_{\text{prey}}

F_{\Delta P} + F_D + F_A ]

8.4 Suction-Induced Force Field

[ \mathbf{F}_{SIFF}(x,t)

\mathbf{F}_{\Delta P}(x,t) + \mathbf{F}_D(x,t) + \mathbf{F}_A(x,t) ]

Capture condition:

[ \int_{t_0}^{t_1} \mathbf{F}_{SIFF},dt

J_{\text{escape}} ]

---

9. Larval Fish Reynolds-Limited Suction

9.1 Reynolds number

[ Re = \frac{\rho U L}{\mu} ]

9.2 Flow reversal

[ Q_{\text{net}}

Q_{\text{in}}

Q_{\text{out}} ]

Failure if prey is not transported far enough before efflux:

[ x_{\text{prey}}(t_{\text{closure}}) < x_{\text{safe}} \Rightarrow \text{failed capture} ]

9.3 Energetic limit

[ E_{\text{suction}}

\int \Delta P,dV ]

[ P_{\text{capture}}

\Pr(E_{\text{suction}} > E_{\text{escape/prey}}) ]

---

10. Bearded Seal Suction and Hydraulic Jetting

10.1 Suction mode

[ \Delta P_{\text{suction}}

P_{\text{ambient}}

P_{\text{mouth}} ]

10.2 Jetting mode

[ \Delta P_{\text{jet}}

P_{\text{mouth}}

P_{\text{ambient}} ]

10.3 Alternating work cycle

[ W_{\text{cycle}}

\int_{\text{suction}}\Delta P_{\text{suction}}dV + \int_{\text{jet}}\Delta P_{\text{jet}}dV ]

---

11. Jetting Animals

Includes squid, jellyfish, and dragonfly larvae.

11.1 Jet thrust

[ T

\dot m v_{\text{jet}} + (P_e-P_a)A_e ]

11.2 Volume flux

[ Q = A_e v_{\text{jet}} ]

[ \dot m = \rho Q ]

11.3 Jet work

[ W_{\text{jet}}

\int \Delta P_{\text{cavity}},dV ]

11.4 Circulation-pressure relation

A transient-pressure model can be summarized as:

[ \Delta P_{\text{cavity}}

\mathcal{F} \left( \frac{d\Gamma}{dt}, \Gamma, Q, A_e, \text{geometry} \right) ]

where \Gamma is circulation.

---

12. Suction-Based Swimming

12.1 Pressure-force integral

[ \mathbf{F}_{P}

-\int_A P(\mathbf{x},t)\mathbf{n},dA ]

Low-pressure suction component:

[ \mathbf{F}_{\text{suction}}

\int_A (P_{\text{ambient}}-P_{\text{local}}) \mathbf{n},dA ]

12.2 Propulsive efficiency

[ \eta

\frac{P_{\text{useful}}}{P_{\text{input}}} ]

---

13. Bombardier Beetle Pulsed Spray

13.1 Chamber pressure dynamics

[ \frac{dP_c}{dt}

\frac{RT}{V_c}\frac{dn_g}{dt}

\frac{P_c}{V_c}\frac{dV_c}{dt}

\Phi_{\text{out}}(P_c,P_a) ]

13.2 Valve threshold

[ P_c > P_{\text{valve}} \Rightarrow \text{pulse ejection} ]

[ P_c \downarrow \Rightarrow \text{valve close / reload} ]

13.3 Pulse impulse

[ J_{\text{spray}}

\sum_i \int_{t_i}^{t_i+\Delta t_i} \dot m(t)v_{\text{jet}}(t),dt ]

---

14. Bladderwort Negative-Pressure Trap

14.1 Pressure differential

[ \Delta P_{\text{trap}}

P_{\text{outside}}-P_{\text{inside}} ]

Trigger:

[ \Delta P_{\text{trap}} > \theta_{\text{door}} \Rightarrow \text{door opens} ]

14.2 Orifice inflow

[ Q(t)

C_d A_{\text{door}} \sqrt{ \frac{2\Delta P_{\text{trap}}}{\rho} } ]

14.3 Trap work

[ W_{\text{trap}}

\int \Delta P_{\text{trap}}dV ]

---

15. Cnidarian Osmotic Projectiles

15.1 Osmotic pressure

[ \Pi = iCRT ]

15.2 Stored work

[ W_{\text{osmotic}}

\int \Pi,dV ]

15.3 Projectile energy

[ E_k = \frac{1}{2}mv^2 ]

Launch threshold:

[ W_{\text{osmotic}} > E_{\text{threshold}} ]

---

16. Biological Hydraulic Force Transmission

16.1 Hydraulic force

[ F = \Delta P A ]

16.2 Hydraulic work

[ W = \int P,dV ]

16.3 Hydrostatic incompressibility

[ V \approx \text{constant} ]

For a cylindrical body:

[ V = AL ]

[ \frac{\Delta L}{L} \approx -\frac{\Delta A}{A} ]

---

17. Lateral-Line / Hydrodynamic Vibration Models

17.1 Particle motion and pressure

For plane waves:

[ p = \rho c u ]

where u is particle velocity.

Particle acceleration:

[ a = \frac{\partial u}{\partial t} ]

17.2 Dipole source near-field

A vibrating sphere or dipole produces a velocity potential field often approximated as:

[ \phi(\mathbf{x},t) \propto \frac{\mathbf{d}(t)\cdot\mathbf{r}}{r^3} ]

Velocity:

[ \mathbf{u} = \nabla\phi ]

Pressure:

[ p = -\rho\frac{\partial \phi}{\partial t} ]

17.3 Neuromast response

Simplified hair-cell deflection:

[ m\ddot y + b\dot y + ky = F_{\text{flow}}(t) ]

Flow force can be approximated as drag:

[ F_{\text{flow}}

\frac{1}{2}\rho C_D A u^2 ]

or linearized at low Reynolds number:

[ F_{\text{flow}} \propto \mu L u ]

17.4 Canal neuromast pressure-gradient sensing

Canal neuromasts approximate pressure-difference sensors:

[ \Delta P = P(x+\Delta x)-P(x) ]

[ \Delta P \approx \nabla P \cdot \Delta x ]

17.5 Artificial lateral-line localization

Given sensor vector:

[ \mathbf{s}(t)

[s_1(t),s_2(t),...,s_N(t)] ]

estimate source:

[ \hat{\Omega}

\arg\max_{\Omega} P(\Omega|\mathbf{s}) ]

or neural approximation:

[ \hat{\Omega}

f_\theta(\mathbf{s}) ]

---

18. Vibroacoustic / Percussion Models

18.1 Transfer function

[ H(f)

\frac{Y(f)}{F_{\text{input}}(f)} ]

18.2 Cavity anomaly

[ \Delta H(f)

H_{\text{candidate}}(f)

H_{\text{solid}}(f) ]

Residual score:

[ R_{\text{interface}}

\int_{f_1}^{f_2} |\Delta H(f)|^2,df ]

18.3 Modal model

[ M\ddot{\mathbf{x}} + C\dot{\mathbf{x}} + K\mathbf{x}

\mathbf{F}(t) ]

Natural frequencies:

[ \det(K-\omega^2M)=0 ]

18.4 Feature vector for percussion detection

[ z = [ \text{MFCC}, \text{wavelet energy}, \text{spectral centroid}, \text{modal peaks}, \text{decay constant} ] ]

Classifier:

[ \hat c

\arg\max_c P(c|z) ]

---

19. Dimensionless Classifiers

19.1 Reynolds number

[ Re = \frac{\rho U L}{\mu} ]

Inertia vs viscosity.

19.2 Weber number

[ We = \frac{\rho U^2 L}{\sigma} ]

Inertia vs surface tension.

19.3 Cavitation number

[ Ca = \frac{P_\infty-P_v}{\frac12\rho U^2} ]

Cavitation tendency.

19.4 Strouhal number

[ St = \frac{fA}{U} ]

Oscillation / swimming efficiency.

19.5 Mach number

[ M = \frac{U}{c} ]

Compressibility/shock relevance.

19.6 Womersley number

[ Wo = L\sqrt{\frac{\omega\rho}{\mu}} ]

Oscillatory flow inertia vs viscosity.

19.7 Acoustic impedance

[ Z = \rho c ]

Interface reflection/transmission.

---

20. Model Selection Matrix

Problem	Minimum viable model	Better model	Failure upgrade
slow bubble growth	Rayleigh–Plesset	Keller–Miksis	CFD with phase change
strong bubble collapse	Keller–Miksis	Gilmore + Tait	compressible multiphase CFD
shock propagation	acoustic 1/r decay	acoustic Burgers	full compressible Navier–Stokes
snapping shrimp	vortex + cavitation number	IB + HEM CFD	compressible CFD + bubble clouds
mantis shrimp	spring-latch + impact	impact + cavitation force	FSI + fracture + cavitation
suction fish	pressure drop	SIFF	3D CFD predator/prey
larval suction	Reynolds scaling	viscous CFD	deformable prey + escape model
bearded seal jetting	\int\Delta P dV	measured pressure cycle	full oral-cavity CFD
jellyfish/squid jetting	momentum thrust	transient pressure/circulation	FSI CFD
lateral line	dipole near-field	neuromast transfer model	CFD + neural encoding
timber/aye-aye percussion	transfer function	FEM vibroacoustic model	anisotropic FSI + classifier

---

21. Claim Ladder

REVIEWED / strongly grounded model families

• Rayleigh–Plesset cavitation.
• Keller–Miksis weak compressibility.
• Gilmore/Tait high-amplitude compressible collapse.
• Burgers nonlinear shock smoothing.
• Navier–Stokes / mixture CFD for cavitating flows.
• SIFF / pressure-gradient suction feeding.
• Jet thrust equation.
• Hydraulic force F=\Delta PA.
• Osmotic pressure \Pi=iCRT.
• Lateral-line dipole / pressure-gradient sensing.

CALIBRATED ENGINEERING DELTA candidates

• ACW as an idealized test medium.
• Gilmore + Burgers bridge for cavitation shock propagation.
• Cavitation shock decay exponent \alpha>1 fitted to real-water data.
• Nano-scale shock suppression by viscous dominance.
• Heavy-fluid/high-impedance stress-test map.

BEAUTIFUL PROVISIONAL / needs receipts

• Exact Mach cutoff where Gilmore/Burgers bridge fails.
• Universal shock-front thickness estimates.
• Universal energy percentage radiated as shock.
• Mercury/very-high-density extrapolations without EOS data.
• Plasma/MHD threshold values without thermochemical model.

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22. Next File Tasks

1. Add formal citations with DOI where available.
2. Split into:
   • CavitationModels.md
   • PressureGradientSpecies.md
   • VibrationMechanosensing.md
   • BurgersShockBridge.md
3. Add Lean structures:
   • CavitationModel
   • PressureDifferentialAxis
   • HydrodynamicVibrationAxis
   • ShockPropagationBridge
4. Add simulation scripts:
   • Rayleigh–Plesset ODE toy solver.
   • Viscous Burgers shock profile generator.
   • Cavitation-number threshold table.
   • SIFF force-field toy model.
5. Add evidence receipts:
   • source paper,
   • equation family,
   • variables,
   • domain of validity,
   • known failure mode.

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23. Compact Unified Thesis

[ \boxed{ \text{Biological pressure systems convert gradients into work, damage, sensing, or escape.} } ]

[ \boxed{ \text{Cavitation systems convert local pressure collapse into bubble energy and shock.} } ]

[ \boxed{ \text{Vibration systems convert mechanical waves into world-state information.} } ]

[ \boxed{ \text{Burgers-type models bridge ideal shock formation and real dissipative smoothing.} } ]

The lawful path is not metaphorical. Each imported biological axis must enter through a documented equation family, explicit variables, and a stated failure regime.