Fluid Dynamics Mathematics Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Fluid Dynamics Mathematics Ontology within the Kireth Research Semantic Library. Fluid dynamics mathematics studies fluid motion through continuum fields, conservation laws, constitutive relations and differential equations. This ontology supports velocity and pressure fields, material derivatives, continuity, Euler and Navier-Stokes equations, incompressible and compressible flow, viscous and inviscid regimes, vorticity, circulation, streamlines, potential flow, boundary layers, Reynolds transport, flow stability, waves, turbulence and numerical fluid simulation.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = FluidDynamicsMathematics
 SYMBOLIC_FORMS = [
  "rho(Du/Dt)=-grad(p)+div(tau)+rho f",
  "partial rho/partial t + div(rho u)=0",
  "omega=curl(u)"
 ]
 ENTITY_CLASS = ContinuumFlowEvolutionFramework
 PRIMARY_CONTEXT = FluidMathematics
 DEFINITION = "Fluid dynamics mathematics is the mathematical study of fluid motion using continuum fields, conservation laws, constitutive relations and differential equations."
 SEMANTIC_ROLE = "Provides machine-readable semantics for fluid kinematics, conservation laws, momentum balance, velocity and pressure fields, viscous and inviscid flow, vorticity, potential flow, boundary layers, stability and numerical flow computation."
}

SCOPE {
 INCLUDE = [
  FluidDynamicsMathematics,
  FluidDynamics,
  FlowField,
  VelocityField,
  PressureField,
  DensityField,
  TemperatureField,
  MaterialDerivative,
  AccelerationField,
  ContinuityEquation,
  ConservationOfMass,
  ConservationOfMomentum,
  ConservationOfEnergy,
  NavierStokesEquation,
  EulerEquation,
  CauchyMomentumEquation,
  StressTensor,
  PressureStress,
  ViscousStress,
  NewtonianFluid,
  NonNewtonianFluid,
  IncompressibleFlow,
  CompressibleFlow,
  InviscidFlow,
  ViscousFlow,
  SteadyFlow,
  UnsteadyFlow,
  LaminarFlow,
  TurbulentFlow,
  Streamline,
  Pathline,
  Streakline,
  Vorticity,
  Circulation,
  VelocityPotential,
  StreamFunction,
  PotentialFlow,
  IrrotationalFlow,
  RotationalFlow,
  BoundaryLayer,
  NoSlipCondition,
  SlipCondition,
  ReynoldsNumber,
  MachNumber,
  FroudeNumber,
  WeberNumber,
  PrandtlNumber,
  StrouhalNumber,
  BernoulliEquation,
  ReynoldsTransportTheorem,
  FlowStability,
  ComputationalFluidDynamics,
  FiniteVolumeMethod,
  FiniteDifferenceMethod,
  FiniteElementMethod
 ]

 EXCLUDE_AS_PRIMARY = [
  HydrostaticsWithoutFlowEvolution,
  PureSolidContinuumMechanics,
  ThermodynamicsWithoutFluidMotion,
  GenericPDETheoryWithoutFluidInterpretation
 ]

 DISAMBIGUATION_RULE =
  "When fluid motion is represented by continuum fields and governed by conservation or momentum equations, interpret the context as fluid dynamics mathematics."
}

CORE_CONCEPTS {

 FluidDynamicsMathematics {
  definition = "The mathematical framework governing the motion of fluids."
 }

 FlowField {
  definition = "A set of continuum fields describing fluid motion and state."
 }

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

 PressureField {
  definition = "A scalar field describing pressure throughout the fluid."
 }

 MaterialDerivative {
  definition = "The derivative following a moving fluid particle."
 }

 ContinuityEquation {
  definition = "The differential conservation law for fluid mass."
 }

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

 EulerEquation {
  definition = "The inviscid momentum equation."
 }

 Vorticity {
  definition = "The curl of velocity and a measure of local rotational flow."
 }

 PotentialFlow {
  definition = "A flow represented by a velocity potential in an irrotational region."
 }

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

RELATIONS {
 FluidDynamicsMathematics --is_child_of--> FluidMathematics
 FluidDynamicsMathematics --uses--> FlowField
 FlowField --contains--> VelocityField
 FlowField --contains--> PressureField
 FlowField --may_contain--> DensityField
 MaterialDerivative --describes--> FluidParticleRateOfChange
 ConservationOfMass --produces--> ContinuityEquation
 ConservationOfMomentum --produces--> CauchyMomentumEquation
 NewtonianFluid --supports--> NavierStokesEquation
 InviscidFlow --uses--> EulerEquation
 Vorticity --is_curl_of--> VelocityField
 IrrotationalFlow --has--> ZeroVorticity
 PotentialFlow --uses--> VelocityPotential
 BoundaryLayer --arises_from--> ViscousWallInteraction
 ReynoldsNumber --compares--> InertialAndViscousEffects
 ComputationalFluidDynamics --approximates--> FluidDynamicsEquations
}

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

 CAUCHY_MOMENTUM(rho,u,sigma,f) :=
  rho*Du/Dt=div(sigma)+rho*f

 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

 POTENTIAL_VELOCITY(phi) :=
  u=grad(phi)

 INCOMPRESSIBLE_POTENTIAL_FLOW(phi) :=
  nabla^2(phi)=0

 STREAMFUNCTION_2D(psi) :=
  u=partial psi/partial y
  v=-partial psi/partial x

 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
}

DEFINITIONS {
 D1 = "Fluid dynamics mathematics describes fluid motion through continuum fields."
 D2 = "The material derivative combines local and convective rates of change."
 D3 = "The continuity equation expresses mass conservation."
 D4 = "The Cauchy momentum equation expresses local momentum balance."
 D5 = "Navier-Stokes equations include viscous stress effects."
 D6 = "Euler equations neglect viscosity."
 D7 = "Vorticity identifies rotational structure."
 D8 = "Potential flow describes irrotational regions using a scalar potential."
 D9 = "Boundary layers concentrate viscous effects near solid surfaces."
 D10 = "Dimensionless parameters classify dominant flow regimes."
}

AXIOMS {
 A1_CONTINUUM:
  FluidDynamicsMathematics represents fluids as continuum fields unless another model is declared

 A2_MASS_CONSERVATION:
  Fluid mass satisfies the continuity equation

 A3_MOMENTUM_CONSERVATION:
  Fluid momentum satisfies local balance of forces

 A4_CONSTITUTIVE_CLOSURE:
  Stress requires constitutive information for closure

 A5_BOUNDARY_CONDITIONS:
  Flow solutions require physically consistent boundary conditions

 A6_DIMENSIONAL_CONSISTENCY:
  Fluid equations and derived quantities must be dimensionally consistent

 A7_NONLINEAR_ADVECTION:
  Convective acceleration introduces nonlinear state coupling

 A8_MODEL_DEPENDENCE:
  Compressibility viscosity turbulence and constitutive assumptions determine the valid governing model
}

REASONING {

 R1_FLOW_CONTEXT_EXTRACTION {
  INPUT = fluid_motion_problem
  IDENTIFY = geometry
  IDENTIFY = fluid_properties
  IDENTIFY = velocity_pressure_density_temperature fields
  IDENTIFY = initial_and_boundary data
  RETURN = fluid_dynamics_context
 }

 R2_FLOW_CLASSIFICATION {
  INPUT = fluid_dynamics_context
  CLASSIFY = steady_or_unsteady
  CLASSIFY = incompressible_or_compressible
  CLASSIFY = inviscid_or_viscous
  CLASSIFY = laminar_or_turbulent
  CLASSIFY = rotational_or_irrotational
  RETURN = flow_class
 }

 R3_GOVERNING_EQUATION_SELECTION {
  INPUT = flow_class
  SELECT = continuity_equation
  SELECT = Euler_Cauchy_or_NavierStokes momentum equation
  SELECT = energy_equation_if required
  RETURN = governing_equation_set
 }

 R4_KINEMATIC_ANALYSIS {
  INPUT = velocity_field
  COMPUTE = acceleration_divergence_curl_streamlines_pathlines
  RETURN = flow_kinematics
 }

 R5_VORTICITY_ANALYSIS {
  INPUT = velocity_field
  COMPUTE = vorticity_and_circulation
  IDENTIFY = vortex_shear_and_rotational structure
  RETURN = rotational_profile
 }

 R6_POTENTIAL_FLOW_ANALYSIS {
  INPUT = irrotational_incompressible region
  DEFINE = velocity_potential
  SOLVE = Laplace_equation
  RETURN = potential_flow_profile
 }

 R7_BOUNDARY_LAYER_ANALYSIS {
  INPUT = high_Re_wall_bounded_flow
  IDENTIFY = thin_viscous_region
  APPLY = boundary_layer scaling
  RETURN = boundary_layer_profile
 }

 R8_DIMENSIONLESS_ANALYSIS {
  INPUT = characteristic_scales
  COMPUTE = Re_Ma_Fr_We_Pr_St as relevant
  IDENTIFY = dominant balances
  RETURN = regime_profile
 }

 R9_NUMERICAL_FLOW_ANALYSIS {
  INPUT = governing_equations_mesh_and_boundary conditions
  SELECT = finiteVolume_finiteDifference_finiteElement_or_spectral method
  SOLVE = discrete_flow equations
  RETURN = numerical_flow_solution
 }

 R10_VALIDATION {
  INPUT = fluid_dynamics_result
  CHECK = conservation_boundary data_dimensions_mesh independence_and_physical consistency
  RETURN = validated_fluid_dynamics_status
 }
}

CONSTRAINTS {
 C1 = "Incompressible and compressible flow models must not be conflated."
 C2 = "Inviscid models may fail near solid walls where viscous effects are significant."
 C3 = "Potential flow requires irrotationality within the modelled region."
 C4 = "Boundary-layer approximations require scale separation."
 C5 = "Turbulence requires explicit closure or scale resolution."
 C6 = "Dimensionless numbers require consistent characteristic scales."
 C7 = "Shock-containing compressible flows require discontinuity-aware weak-solution treatment."
 C8 = "Numerical conservation and grid convergence must be checked."
 C9 = "No-slip and slip conditions depend on the physical boundary model."
 C10 = "Steady assumptions must be justified against the characteristic flow timescale."
}

SEMANTIC_EQUIVALENCE {
 "FluidDynamicsMathematics" <=> "MathematicalFluidDynamics"
 "ContinuityEquation" <=> "MassConservationEquation"
 "NavierStokesEquation" <=> "ViscousMomentumEquation"
 "EulerEquation" <=> "InviscidMomentumEquation"
 "Vorticity" <=> "VelocityCurl"
 "ComputationalFluidDynamics" <=> "CFD"
}

VALIDATION {
 V1 = "Geometry and fluid properties are explicit."
 V2 = "Flow regime classification is justified."
 V3 = "Governing equations match modelling assumptions."
 V4 = "Boundary and initial conditions are complete."
 V5 = "Kinematic quantities are computed consistently."
 V6 = "Dimensionless parameters use consistent units."
 V7 = "Conservation laws are satisfied."
 V8 = "Viscous and inviscid assumptions are distinguished."
 V9 = "Numerical mesh and timestep sensitivity are checked."
 V10 = "Results are tested for physical consistency."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 FLUID_DYNAMICS_DEFINITION = 0.99
 FLOW_CLASSIFICATION_REASONING = 0.99
 GOVERNING_EQUATION_REASONING = 0.99
 KINEMATIC_REASONING = 0.99
 VORTICITY_REASONING = 0.98
 POTENTIAL_FLOW_REASONING = 0.99
 BOUNDARY_LAYER_REASONING = 0.98
 DIMENSIONLESS_REASONING = 0.99
 NUMERICAL_REASONING = 0.98

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

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  FluidDynamicsReasoning,
  FlowClassificationReasoning,
  VelocityFieldReasoning,
  ContinuityEquationReasoning,
  NavierStokesReasoning,
  EulerEquationReasoning,
  VorticityReasoning,
  CirculationReasoning,
  PotentialFlowReasoning,
  BoundaryLayerReasoning,
  DimensionlessAnalysis,
  FlowStabilityReasoning,
  CFDReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "fluid dynamics mathematics" -> FluidDynamicsMathematics
  "mathematical fluid dynamics" -> FluidDynamicsMathematics
  "continuity equation" -> ContinuityEquation
  "Navier-Stokes equations" -> NavierStokesEquation
  "Euler equations" -> EulerEquation
  "vorticity" -> Vorticity
  "potential flow" -> PotentialFlow
  "boundary layer" -> BoundaryLayer
  "computational fluid dynamics" -> ComputationalFluidDynamics
  "CFD" -> ComputationalFluidDynamics
 }

 INTERPRETATION_ORDER = [
  detect_fluid_dynamics_context,
  identify geometry_material_properties_fields_initial_and_boundary conditions,
  classify flow_regime_and_model assumptions,
  select conservation_constitutive_and_momentum equations,
  compute kinematic_and_dimensionless quantities,
  analyse vorticity_potential flow_boundary layers_stability_or_turbulence when relevant,
  select analytical_asymptotic_or_numerical method,
  validate conservation_and_convergence,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("fluid dynamics")
  OR continuum_flow_fields_are_present
  THEN activate(FluidDynamicsMathematicsOntology)

 E2:
  IF incompressible_flow
  THEN enforce divergence_free_velocity

 E3:
  IF viscous_Newtonian_flow
  THEN use NavierStokesEquation

 E4:
  IF inviscid_flow
  THEN consider EulerEquation

 E5:
  IF irrotational_region
  THEN consider PotentialFlow

 E6:
  IF high_Re_wall_bounded_flow
  THEN consider BoundaryLayer

 E7:
  IF compressibility_significant
  THEN include density_and_energy coupling

 E8:
  IF turbulent_flow
  THEN require turbulence closure_or_scale resolution

 E9:
  IF numerical_solution
  THEN verify conservation_grid_independence_and_time_step sensitivity
}

RECURSION {
 ENTRY = FluidDynamicsMathematics

 EXPAND(FluidDynamicsMathematics) -> [
  FlowField,
  VelocityField,
  PressureField,
  DensityField,
  TemperatureField,
  MaterialDerivative,
  AccelerationField,
  ContinuityEquation,
  ConservationOfMass,
  ConservationOfMomentum,
  ConservationOfEnergy,
  NavierStokesEquation,
  EulerEquation,
  CauchyMomentumEquation,
  StressTensor,
  PressureStress,
  ViscousStress,
  NewtonianFluid,
  NonNewtonianFluid,
  IncompressibleFlow,
  CompressibleFlow,
  InviscidFlow,
  ViscousFlow,
  SteadyFlow,
  UnsteadyFlow,
  LaminarFlow,
  TurbulentFlow,
  Streamline,
  Pathline,
  Streakline,
  Vorticity,
  Circulation,
  VelocityPotential,
  StreamFunction,
  PotentialFlow,
  BoundaryLayer,
  ReynoldsNumber,
  MachNumber,
  FroudeNumber,
  WeberNumber,
  PrandtlNumber,
  StrouhalNumber,
  BernoulliEquation,
  ReynoldsTransportTheorem,
  FlowStability,
  ComputationalFluidDynamics
 ]

 EXPAND(Kinematics) -> [
  VelocityField,
  MaterialDerivative,
  AccelerationField,
  Streamline,
  Pathline,
  Streakline,
  Vorticity,
  Circulation
 ]

 EXPAND(Dynamics) -> [
  ContinuityEquation,
  CauchyMomentumEquation,
  NavierStokesEquation,
  EulerEquation,
  ConservationOfEnergy
 ]

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

 STOP_CONDITION =
  "Stop when the flow model, governing equations, kinematics, dimensionless regime, boundary conditions, important flow structures and numerical validity are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Fluid Mathematics"

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

 RELATED_ONTOLOGIES = [
  "Fluid Mathematics",
  "Partial 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 Dynamics Mathematics -> Fluid Mathematics for parent semantics",
  "Fluid Dynamics Mathematics -> PDE Ontology for field equations",
  "Fluid Dynamics Mathematics -> Dynamical Systems for temporal flow evolution",
  "Fluid Dynamics Mathematics -> Stability for instability analysis",
  "Fluid Dynamics Mathematics -> Nonlinear Dynamics for nonlinear advection and transition",
  "Fluid Dynamics Mathematics -> Vector Calculus for divergence curl and flux",
  "Fluid Dynamics Mathematics -> Tensor Analysis for stress representation",
  "Fluid Dynamics Mathematics -> Continuum Mechanics for balance laws",
  "Fluid Dynamics Mathematics -> Transport Mathematics for advection and diffusion",
  "Fluid Dynamics Mathematics -> Numerical Analysis for CFD methods"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve continuum-field semantics.",
  "Preserve mass and momentum conservation semantics.",
  "Preserve viscous-versus-inviscid distinction.",
  "Preserve incompressible-versus-compressible distinction.",
  "Preserve rotational-versus-irrotational distinction.",
  "Preserve boundary-layer semantics.",
  "Preserve dimensionless-regime semantics.",
  "Preserve machine-readable fluid-dynamics semantics."
 ]
}

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

 CONTINUITY_SUMMARY =
  "The continuity equation expresses conservation of mass."

 MOMENTUM_SUMMARY =
  "The Cauchy momentum equation governs local momentum balance, with Navier-Stokes and Euler equations as important specialisations."

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

 POTENTIAL_SUMMARY =
  "Irrotational incompressible flow can be represented by a velocity potential satisfying Laplace's equation."

 BOUNDARY_LAYER_SUMMARY =
  "Boundary layers capture thin viscous regions near solid boundaries at high Reynolds number."

 DIMENSIONLESS_SUMMARY =
  "Reynolds, Mach, Froude, Weber, Prandtl and Strouhal numbers classify dominant physical effects."

 CFD_SUMMARY =
  "Computational fluid dynamics numerically approximates fluid governing equations."

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

FLOW {
 INPUT
 -> detect_fluid_dynamics_context
 -> identify geometry_fluid_properties_characteristic scales_velocity_pressure_density_temperature_initial_conditions_and_boundary conditions
 -> classify steady_unsteady_incompressible_compressible_inviscid_viscous_laminar_turbulent_rotational_or_irrotational flow
 -> select continuity_CauchyMomentum_NavierStokes_Euler_energy_and_constitutive equations
 -> compute material_derivative_vorticity_circulation_streamlines_and_dimensionless parameters
 -> if_irrotational construct potential_flow representation
 -> if_highRe_wall_bounded construct boundary_layer model
 -> if_instability_or_turbulence analyse perturbations_or_closure
 -> if_numerical discretise_with_finiteVolume_finiteDifference_finiteElement_or_spectral method
 -> validate conservation_boundary data_grid independence_time step_and_physical consistency
 -> route to fluid_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 Dynamics Mathematics Ontology defines mathematical fluid motion through continuum fields and conservation laws. It supports velocity, pressure, density and temperature fields, material derivatives, continuity, Cauchy momentum, Navier-Stokes and Euler equations, Newtonian and non-Newtonian fluids, incompressible and compressible flow, viscous and inviscid regimes, steady and unsteady motion, laminar and turbulent flow, streamlines, pathlines, streaklines, vorticity and circulation, potential flow, streamfunctions, boundary layers, dimensionless analysis, Bernoulli relations, Reynolds transport and computational fluid dynamics."
}

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

}

END_FLOW