Fluid Mathematics

AI INTERPRETATION NOTE
This Flow Algebra defines the Fluid Mathematics layer within the Kireth Research Semantic Library. Fluid mathematics studies the mathematical description of fluids in motion and at rest, including conservation laws, continuum mechanics, velocity and pressure fields, incompressible and compressible flow, viscous and inviscid models, vorticity, potential flow, boundary layers, turbulence, dimensional analysis, stability, waves, free-surface phenomena and numerical fluid computation. This parent layer connects fluid mechanics, partial differential equations, vector calculus, dynamical systems, applied mathematics, numerical analysis and scientific computing.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Fluid Mathematics"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Fluid Mathematics"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/fluid-mathematics.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = FluidMathematics
 SYMBOLIC_FORMS = [
  "rho(Du/Dt)=-grad(p)+mu*nabla^2 u+rho f",
  "partial rho/partial t + div(rho u)=0",
  "div(u)=0"
 ]
 ENTITY_CLASS = ContinuumFluidDynamicsMathematicsFramework
 PRIMARY_CONTEXT = AppliedMathematics
 DEFINITION = "Fluid mathematics is the mathematical study of fluid states and flows using continuum fields, conservation laws, differential equations, constitutive relations and numerical methods."
 SEMANTIC_ROLE = "Provides the parent semantic framework for fluid kinematics, conservation of mass, momentum and energy, Navier-Stokes equations, Euler equations, vorticity, potential flow, boundary layers, compressible flow, turbulence and computational fluid dynamics."
}

SCOPE {
 INCLUDE = [
  FluidMathematics,
  Fluid,
  Continuum,
  FluidParticle,
  FlowField,
  VelocityField,
  PressureField,
  DensityField,
  TemperatureField,
  StressTensor,
  Viscosity,
  DynamicViscosity,
  KinematicViscosity,
  IncompressibleFlow,
  CompressibleFlow,
  InviscidFlow,
  ViscousFlow,
  SteadyFlow,
  UnsteadyFlow,
  LaminarFlow,
  TurbulentFlow,
  InternalFlow,
  ExternalFlow,
  ConservationOfMass,
  ContinuityEquation,
  ConservationOfMomentum,
  NavierStokesEquation,
  EulerEquation,
  ConservationOfEnergy,
  MaterialDerivative,
  Streamline,
  Pathline,
  Streakline,
  Vorticity,
  Circulation,
  PotentialFlow,
  VelocityPotential,
  StreamFunction,
  BoundaryLayer,
  NoSlipCondition,
  ReynoldsNumber,
  MachNumber,
  FroudeNumber,
  WeberNumber,
  PrandtlNumber,
  StrouhalNumber,
  DimensionalAnalysis,
  Similarity,
  Stability,
  FluidInstability,
  FreeSurfaceFlow,
  WaveMotion,
  Turbulence,
  ComputationalFluidDynamics,
  CFD,
  FiniteVolumeMethod,
  FiniteDifferenceMethod,
  FiniteElementMethod
 ]

 EXCLUDE_AS_PRIMARY = [
  SolidMechanicsWithoutFluidBehaviour,
  ParticleDynamicsWithoutContinuumFluidContext,
  PureThermodynamicsWithoutFlowSemantics,
  GenericPDETheoryWithoutFluidInterpretation
 ]

 DISAMBIGUATION_RULE =
  "When continuum fields such as velocity, pressure, density or temperature are used to describe fluid motion, deformation or equilibrium, interpret the context as fluid mathematics."
}

CORE_CONCEPTS {

 FluidMathematics {
  definition = "The mathematical framework for describing fluid states, motion, forces and transport."
 }

 FlowField {
  definition = "A collection of spatially and temporally varying fields describing fluid behaviour."
 }

 VelocityField {
  definition = "A vector field assigning fluid velocity to each point in space and time."
 }

 PressureField {
  definition = "A scalar field assigning thermodynamic or mechanical pressure throughout the fluid domain."
 }

 ContinuityEquation {
  definition = "The conservation law expressing conservation of mass."
 }

 NavierStokesEquation {
  definition = "The momentum equation for a viscous Newtonian fluid."
 }

 EulerEquation {
  definition = "The inviscid momentum equation obtained by neglecting viscous stresses."
 }

 Vorticity {
  definition = "The curl of the velocity field and a measure of local fluid rotation."
 }

 BoundaryLayer {
  definition = "A thin region near a boundary where viscous effects and strong velocity gradients are important."
 }

 Turbulence {
  definition = "Irregular multiscale fluid motion with strong nonlinear transport and fluctuations."
 }

 ComputationalFluidDynamics {
  definition = "Numerical approximation and simulation of fluid-flow equations."
 }
}

RELATIONS {
 FluidMathematics --is_branch_of--> AppliedMathematics
 Fluid --is_modelled_as--> Continuum
 FlowField --contains--> VelocityField
 FlowField --contains--> PressureField
 FlowField --may_contain--> DensityField
 ConservationOfMass --produces--> ContinuityEquation
 ConservationOfMomentum --produces--> NavierStokesEquation
 InviscidFlow --uses--> EulerEquation
 ViscousFlow --uses--> NavierStokesEquation
 Vorticity --is_curl_of--> VelocityField
 PotentialFlow --has--> VelocityPotential
 IncompressibleTwoDimensionalFlow --may_have--> StreamFunction
 BoundaryLayer --results_from--> ViscosityNearBoundary
 ReynoldsNumber --relates--> InertialAndViscousEffects
 MachNumber --relates--> FlowSpeedAndSoundSpeed
 FroudeNumber --relates--> InertiaAndGravity
 WeberNumber --relates--> InertiaAndSurfaceTension
 ComputationalFluidDynamics --approximates--> FluidGoverningEquations
}

OPERATORS {
 MATERIAL_DERIVATIVE(phi,u) :=
  Dphi/Dt = partial phi/partial t + u dot grad(phi)

 CONTINUITY_GENERAL(rho,u) :=
  partial rho/partial t + div(rho*u)=0

 CONTINUITY_INCOMPRESSIBLE(u) :=
  div(u)=0

 NAVIER_STOKES_NEWTONIAN(rho,u,p,mu,f) :=
  rho*(partial u/partial t + u dot grad(u))
  = -grad(p) + mu*nabla^2(u) + rho*f

 EULER_EQUATION(rho,u,p,f) :=
  rho*(partial u/partial t + u dot grad(u))
  = -grad(p) + rho*f

 VORTICITY(u) :=
  omega=curl(u)

 CIRCULATION(u,C) :=
  Gamma=closed_line_integral_C u dot dl

 REYNOLDS_NUMBER(rho,U,L,mu) :=
  Re=rho*U*L/mu

 MACH_NUMBER(U,c) :=
  Ma=U/c

 FROUDE_NUMBER(U,g,L) :=
  Fr=U/sqrt(g*L)

 WEBER_NUMBER(rho,U,L,sigma) :=
  We=rho*U^2*L/sigma

 PRANDTL_NUMBER(nu,alpha) :=
  Pr=nu/alpha
}

DEFINITIONS {
 D1 = "Fluid mathematics uses continuum fields to describe fluid behaviour."
 D2 = "Mass conservation is expressed through the continuity equation."
 D3 = "Momentum conservation produces Euler or Navier-Stokes equations depending on viscous assumptions."
 D4 = "Incompressibility constrains the velocity field to be divergence-free."
 D5 = "Vorticity measures local rotational structure in a flow."
 D6 = "Potential flow assumes irrotational velocity structure."
 D7 = "Boundary layers arise where viscosity interacts strongly with solid boundaries."
 D8 = "Dimensionless numbers compare competing physical effects."
 D9 = "Turbulence involves nonlinear multiscale transport and fluctuating motion."
 D10 = "Computational fluid dynamics approximates governing flow equations numerically."
}

AXIOMS {
 A1_CONTINUUM:
  FluidMathematics commonly represents fluids as continua

 A2_MASS_CONSERVATION:
  Fluid mass is conserved according to the continuity equation

 A3_MOMENTUM_CONSERVATION:
  Fluid momentum evolves according to body and surface forces

 A4_CONSTITUTIVE_RELATION:
  Closure requires constitutive relations such as Newtonian viscous stress laws

 A5_BOUNDARY_CONDITIONS:
  Fluid PDE solutions require physically and mathematically consistent boundary conditions

 A6_DIMENSIONAL_CONSISTENCY:
  All fluid equations must be dimensionally consistent

 A7_NONLINEAR_ADVECTION:
  Convective acceleration introduces nonlinear dynamics into fluid motion

 A8_SCALE_DEPENDENCE:
  Relative importance of fluid processes depends on dimensionless parameters and characteristic scales
}

REASONING {

 R1_FLUID_CONTEXT_EXTRACTION {
  INPUT = fluid_problem
  IDENTIFY = geometry
  IDENTIFY = fluid_properties
  IDENTIFY = velocity_pressure_density_temperature fields
  IDENTIFY = initial_and_boundary conditions
  RETURN = fluid_context
 }

 R2_FLOW_CLASSIFICATION {
  INPUT = fluid_context
  CLASSIFY = steady_or_unsteady
  CLASSIFY = incompressible_or_compressible
  CLASSIFY = inviscid_or_viscous
  CLASSIFY = laminar_or_turbulent
  CLASSIFY = internal_or_external
  RETURN = flow_class
 }

 R3_GOVERNING_EQUATION_SELECTION {
  INPUT = flow_class
  SELECT = continuity
  SELECT = Euler_or_NavierStokes
  SELECT = energy_equation_if_required
  RETURN = governing_equation_set
 }

 R4_DIMENSIONLESS_ANALYSIS {
  INPUT = characteristic_scales_and_properties
  COMPUTE = Reynolds_Mach_Froude_Weber_Prandtl_Strouhal numbers
  IDENTIFY = dominant_effects
  RETURN = scaling_profile
 }

 R5_VORTICITY_REASONING {
  INPUT = velocity_field
  COMPUTE = curl
  ANALYSE = rotation_shear_and_vortex structure
  RETURN = vorticity_profile
 }

 R6_BOUNDARY_LAYER_REASONING {
  INPUT = high_Reynolds_number_wall_bounded_flow
  IDENTIFY = thin_viscous_region
  SCALE = normal_and_streamwise coordinates
  RETURN = boundary_layer_model
 }

 R7_POTENTIAL_FLOW_REASONING {
  INPUT = incompressible_irrotational_flow
  DEFINE = velocity_potential
  SOLVE = Laplace_equation
  RETURN = potential_flow_solution
 }

 R8_STABILITY_AND_TRANSITION {
  INPUT = base_flow
  PERTURB = flow
  ANALYSE = growth_or_decay
  RETURN = stability_or_transition profile
 }

 R9_NUMERICAL_REASONING {
  INPUT = governing_equations_geometry_mesh_and_boundary data
  SELECT = finiteVolume_finiteDifference_finiteElement_or_spectral method
  SOLVE = discrete_equations
  RETURN = numerical_flow_solution
 }

 R10_FLUID_VALIDATION {
  INPUT = analytical_or_numerical_result
  CHECK = conservation_boundary conditions_dimensions_grid independence_and_physical consistency
  RETURN = validated_fluid_result
 }
}

CONSTRAINTS {
 C1 = "Incompressible and compressible models must not be conflated."
 C2 = "Inviscid approximations neglect viscous stresses and may fail near solid boundaries."
 C3 = "No-slip boundary conditions apply to ordinary viscous continuum models at solid walls unless another model is declared."
 C4 = "Turbulence models introduce closure assumptions."
 C5 = "Dimensionless numbers require consistent characteristic scales."
 C6 = "Potential flow is valid only for appropriate irrotational regions."
 C7 = "Boundary-layer approximations rely on scale separation."
 C8 = "Numerical solutions require conservation, convergence and grid-independence checks."
 C9 = "Shock-containing compressible flows require weak-solution and discontinuity-aware methods."
 C10 = "Observed complexity should not be attributed to turbulence without appropriate flow diagnostics."
}

SEMANTIC_EQUIVALENCE {
 "FluidMathematics" <=> "MathematicalFluidDynamics"
 "FlowField" <=> "FluidField"
 "ContinuityEquation" <=> "MassConservationEquation"
 "NavierStokesEquation" <=> "ViscousMomentumEquation"
 "EulerEquation" <=> "InviscidMomentumEquation"
 "ComputationalFluidDynamics" <=> "CFD"
}

VALIDATION {
 V1 = "Fluid properties and characteristic scales are explicit."
 V2 = "Flow classification is justified."
 V3 = "Governing equations match modelling assumptions."
 V4 = "Initial and boundary conditions are complete."
 V5 = "Dimensionless numbers use consistent units."
 V6 = "Conservation laws are numerically or analytically satisfied."
 V7 = "Viscous, inviscid, compressible and incompressible assumptions are distinguished."
 V8 = "Turbulence closure assumptions are documented."
 V9 = "Numerical mesh and timestep sensitivity are checked."
 V10 = "Results are tested for physical and mathematical consistency."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 FLUID_MATHEMATICS_DEFINITION = 0.99
 FLOW_CLASSIFICATION_REASONING = 0.99
 GOVERNING_EQUATION_REASONING = 0.99
 DIMENSIONLESS_REASONING = 0.99
 VORTICITY_REASONING = 0.98
 BOUNDARY_LAYER_REASONING = 0.98
 POTENTIAL_FLOW_REASONING = 0.99
 STABILITY_REASONING = 0.98
 NUMERICAL_REASONING = 0.98

 POLICY =
  "Reduce confidence when fluid properties, geometry, boundary conditions, characteristic scales, compressibility, viscosity, turbulence closure or numerical resolution are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  FluidMathematicsReasoning,
  FlowClassificationReasoning,
  ContinuityEquationReasoning,
  NavierStokesReasoning,
  EulerEquationReasoning,
  VorticityReasoning,
  PotentialFlowReasoning,
  BoundaryLayerReasoning,
  DimensionlessAnalysis,
  FluidStabilityReasoning,
  TurbulenceReasoning,
  CFDReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "fluid mathematics" -> FluidMathematics
  "mathematical fluid dynamics" -> FluidMathematics
  "continuity equation" -> ContinuityEquation
  "Navier-Stokes equations" -> NavierStokesEquation
  "Euler equations" -> EulerEquation
  "vorticity" -> Vorticity
  "boundary layer" -> BoundaryLayer
  "Reynolds number" -> ReynoldsNumber
  "computational fluid dynamics" -> ComputationalFluidDynamics
  "CFD" -> ComputationalFluidDynamics
 }

 INTERPRETATION_ORDER = [
  detect_fluid_mathematics_context,
  identify geometry_fluid_properties_characteristic scales_fields_and_boundary conditions,
  classify steady_unsteady_incompressible_compressible_inviscid_viscous_laminar_turbulent_internal_or_external flow,
  select conservation_and_constitutive equations,
  compute dimensionless parameters_and_dominant balances,
  analyse vorticity_boundary layers_potential flow_stability_or_turbulence when relevant,
  select analytical_or_numerical method,
  validate conservation_boundary data_and_convergence,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("fluid mathematics")
  OR continuum_fluid_fields_and_flow equations_are_present
  THEN activate(FluidMathematicsOntology)

 E2:
  IF incompressible_flow
  THEN enforce div(u)=0

 E3:
  IF viscous_Newtonian_flow
  THEN consider NavierStokesEquation

 E4:
  IF inviscid_flow
  THEN consider EulerEquation

 E5:
  IF irrotational_flow
  THEN consider potential_flow formulation

 E6:
  IF high_Re_wall_bounded_flow
  THEN consider boundary_layer approximation

 E7:
  IF compressibility_significant
  THEN include density_variation_and_energy_or_state relations

 E8:
  IF turbulent_flow
  THEN require turbulence_or_scale-resolution model

 E9:
  IF numerical_solution
  THEN verify conservation_grid_independence_and_time-step sensitivity
}

RECURSION {
 ENTRY = FluidMathematics

 EXPAND(FluidMathematics) -> [
  Fluid,
  Continuum,
  FlowField,
  VelocityField,
  PressureField,
  DensityField,
  TemperatureField,
  StressTensor,
  Viscosity,
  IncompressibleFlow,
  CompressibleFlow,
  InviscidFlow,
  ViscousFlow,
  SteadyFlow,
  UnsteadyFlow,
  LaminarFlow,
  TurbulentFlow,
  ContinuityEquation,
  NavierStokesEquation,
  EulerEquation,
  ConservationOfEnergy,
  MaterialDerivative,
  Streamline,
  Pathline,
  Streakline,
  Vorticity,
  Circulation,
  PotentialFlow,
  VelocityPotential,
  StreamFunction,
  BoundaryLayer,
  ReynoldsNumber,
  MachNumber,
  FroudeNumber,
  WeberNumber,
  PrandtlNumber,
  Turbulence,
  ComputationalFluidDynamics
 ]

 EXPAND(ConservationFramework) -> [
  ConservationOfMass,
  ContinuityEquation,
  ConservationOfMomentum,
  NavierStokesEquation,
  EulerEquation,
  ConservationOfEnergy
 ]

 EXPAND(FlowStructures) -> [
  Streamline,
  Pathline,
  Streakline,
  Vorticity,
  Circulation,
  PotentialFlow,
  BoundaryLayer,
  Vortex,
  Wake
 ]

 EXPAND(NumericalMethods) -> [
  FiniteVolumeMethod,
  FiniteDifferenceMethod,
  FiniteElementMethod,
  SpectralMethod
 ]

 STOP_CONDITION =
  "Stop when the fluid model, flow class, governing equations, characteristic scales, dimensionless parameters, boundary conditions, dominant flow structures and numerical validity are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Applied Mathematics Ontologies"

 IMMEDIATE_CONTEXTS = [
  "Applied Mathematics Ontologies",
  "Partial Differential Equation Ontology",
  "Dynamical Systems Ontology",
  "Numerical Analysis Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Applied Mathematics Ontologies",
  "Partial Differential Equation Ontology",
  "Ordinary Differential Equation Ontology",
  "Dynamical Systems Ontology",
  "Stability Ontology",
  "Nonlinear Dynamics Ontology",
  "Vector Calculus Ontologies",
  "Tensor Analysis Ontologies",
  "Continuum Mechanics Ontologies",
  "Transport Mathematics Ontologies",
  "Turbulence Ontologies",
  "Numerical Analysis Ontologies",
  "Scientific Computing Ontologies"
 ]

 ROUTE_RULES = [
  "Fluid Mathematics -> Applied Mathematics Ontologies for parent semantics",
  "Fluid Mathematics -> PDE Ontology for governing field equations",
  "Fluid Mathematics -> Dynamical Systems for temporal flow evolution",
  "Fluid Mathematics -> Stability for flow-instability analysis",
  "Fluid Mathematics -> Nonlinear Dynamics for nonlinear advection and transition",
  "Fluid Mathematics -> Vector Calculus for divergence, curl and flux",
  "Fluid Mathematics -> Tensor Analysis for stress and strain-rate tensors",
  "Fluid Mathematics -> Continuum Mechanics for constitutive and balance-law structure",
  "Fluid Mathematics -> Transport Mathematics for advection and diffusion",
  "Fluid Mathematics -> Numerical Analysis for CFD discretisation and convergence"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible flow classes, governing models or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve continuum semantics.",
  "Preserve conservation-law semantics.",
  "Preserve incompressible-versus-compressible distinction.",
  "Preserve inviscid-versus-viscous distinction.",
  "Preserve laminar-versus-turbulent distinction.",
  "Preserve dimensional-analysis semantics.",
  "Preserve boundary-condition semantics.",
  "Preserve machine-readable fluid-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Fluid mathematics studies fluid motion and equilibrium using continuum fields, conservation laws, constitutive relations, differential equations and numerical methods."

 CONSERVATION_SUMMARY =
  "Fluid models are organised around conservation of mass, momentum and energy."

 NAVIER_STOKES_SUMMARY =
  "The Navier-Stokes equations describe momentum evolution in viscous Newtonian fluids."

 EULER_SUMMARY =
  "The Euler equations describe inviscid momentum evolution."

 VORTICITY_SUMMARY =
  "Vorticity is the curl of velocity and describes local rotational flow structure."

 BOUNDARY_LAYER_SUMMARY =
  "Boundary layers are thin viscous regions near solid surfaces where velocity gradients are strong."

 DIMENSIONLESS_SUMMARY =
  "Reynolds, Mach, Froude, Weber and related numbers compare competing physical effects."

 CFD_SUMMARY =
  "Computational fluid dynamics approximates governing equations on discrete meshes or basis spaces."

 VALIDATION_SUMMARY =
  "Fluid solutions should be checked for conservation, boundary-condition consistency, dimensional correctness and numerical convergence."
}

FLOW {
 INPUT
 -> detect_fluid_mathematics_context
 -> identify fluid_properties_geometry_characteristic scales_fields_initial conditions_and_boundary conditions
 -> classify steady_unsteady_incompressible_compressible_inviscid_viscous_laminar_turbulent_internal_or_external flow
 -> select continuity_momentum_energy_and_constitutive equations
 -> compute Reynolds_Mach_Froude_Weber_Prandtl_or_other dimensionless parameters
 -> identify dominant balances_vorticity_boundary layers_potential flow_waves_instabilities_or_turbulence
 -> select analytical_asymptotic_or_numerical method
 -> if_numerical discretise_with_finiteVolume_finiteDifference_finiteElement_or_spectral method
 -> validate conservation_grid independence_timestep_boundary data_and_physical consistency
 -> route to applied_mathematics PDE dynamical_systems stability nonlinear_dynamics vector_calculus tensor_analysis continuum_mechanics transport turbulence numerical_analysis_or_scientific_computing ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Fluid Mathematics layer defines the mathematical structure of fluids and flows. It supports continuum fields, velocity, pressure, density and temperature, stress and viscosity, incompressible and compressible flow, viscous and inviscid models, steady and unsteady motion, laminar and turbulent regimes, continuity, Navier-Stokes and Euler equations, energy conservation, material derivatives, streamlines, pathlines and streaklines, vorticity and circulation, potential flow, boundary layers, dimensionless analysis, flow stability, free-surface and wave phenomena, turbulence and computational fluid dynamics."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-FLUID-MATHEMATICS-V1"
}

}

END_FLOW