Continuity Equation Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Continuity Equation Ontology within the Kireth Research Semantic Library. The continuity equation expresses conservation of mass in continuum systems by relating temporal density change to divergence of mass flux. This ontology supports differential and integral conservation forms, incompressible and compressible flow, steady and unsteady mass balance, control-volume reasoning, material derivatives, fluxes, source terms, coordinate transformations, numerical conservation and relationships with fluid dynamics mathematics, Navier–Stokes equations, transport equations, partial differential equations and computational fluid dynamics.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = ContinuityEquation
 SYMBOLIC_FORMS = [
  "partial rho/partial t + div(rho u)=0",
  "div(u)=0",
  "d/dt integral_V rho dV + surface_integral rho u dot n dA = 0"
 ]
 ENTITY_CLASS = ContinuumMassConservationFramework
 PRIMARY_CONTEXT = FluidMathematics
 DEFINITION = "The continuity equation is the local or integral conservation law expressing conservation of mass in a continuum."
 SEMANTIC_ROLE = "Provides machine-readable semantics for density evolution, mass flux, divergence, control volumes, incompressibility, compressibility, sources, sinks and conservative numerical formulations."
}

SCOPE {
 INCLUDE = [
  ContinuityEquation,
  ConservationOfMass,
  MassBalance,
  LocalConservationLaw,
  IntegralConservationLaw,
  DifferentialForm,
  IntegralForm,
  DensityField,
  VelocityField,
  MassFlux,
  VolumetricFlux,
  Divergence,
  MaterialDerivative,
  ControlVolume,
  ControlSurface,
  SurfaceFlux,
  SourceTerm,
  SinkTerm,
  CompressibleFlow,
  IncompressibleFlow,
  ConstantDensityFlow,
  VariableDensityFlow,
  SteadyFlow,
  UnsteadyFlow,
  EulerianDescription,
  LagrangianDescription,
  ReynoldsTransportTheorem,
  JacobianDeterminant,
  CoordinateTransformation,
  CylindricalCoordinates,
  SphericalCoordinates,
  ConservativeForm,
  NonconservativeForm,
  FiniteVolumeMethod,
  FluxBalance,
  NumericalConservation,
  Residual,
  BoundaryFlux,
  InitialCondition,
  BoundaryCondition
 ]

 EXCLUDE_AS_PRIMARY = [
  MomentumConservationWithoutMassBalanceContext,
  EnergyConservationWithoutMassSemantics,
  ProbabilityContinuityEquationWithoutFluidOrContinuumContext,
  ElectricalContinuityEquationWithoutMassTransportSemantics
 ]

 DISAMBIGUATION_RULE =
  "When a continuum model balances density change against divergence of mass flux, interpret the context as the mass continuity equation."
}

CORE_CONCEPTS {

 ContinuityEquation {
  definition = "The conservation law governing mass density and mass flux in a continuum."
 }

 DensityField {
  definition = "A scalar field representing mass per unit volume."
 }

 MassFlux {
  definition = "Mass crossing unit area per unit time, commonly rho*u."
 }

 ControlVolume {
  definition = "A selected region of space used for integral conservation analysis."
 }

 ControlSurface {
  definition = "The boundary of a control volume through which mass flux is evaluated."
 }

 IncompressibleFlow {
  definition = "Flow satisfying zero velocity divergence when density is constant along trajectories."
 }

 CompressibleFlow {
  definition = "Flow in which density variation is dynamically significant."
 }

 ConservativeForm {
  definition = "A formulation written explicitly as accumulation plus divergence of flux."
 }

 MaterialDerivative {
  definition = "The derivative following a moving material element."
 }

 ReynoldsTransportTheorem {
  definition = "A theorem relating system conservation to control-volume balance."
 }
}

RELATIONS {
 ContinuityEquation --is_child_of--> FluidMathematics
 ContinuityEquation --expresses--> ConservationOfMass
 DensityField --evolves_by--> ContinuityEquation
 MassFlux --equals--> DensityTimesVelocity
 ControlVolume --bounded_by--> ControlSurface
 SurfaceFlux --crosses--> ControlSurface
 ReynoldsTransportTheorem --derives--> IntegralConservationLaw
 IntegralConservationLaw --localises_to--> DifferentialForm
 CompressibleFlow --requires--> DensityEvolution
 ConstantDensityFlow --may_reduce_to--> DivergenceFreeVelocity
 ConservativeForm --supports--> NumericalConservation
 FiniteVolumeMethod --enforces--> FluxBalance
 BoundaryFlux --affects--> MassBalance
 SourceTerm --modifies--> ContinuityEquation
}

OPERATORS {
 LOCAL_CONTINUITY(rho,u) :=
  partial rho/partial t + div(rho*u)=0

 CONTINUITY_WITH_SOURCE(rho,u,s) :=
  partial rho/partial t + div(rho*u)=s

 INCOMPRESSIBLE_CONTINUITY(u) :=
  div(u)=0

 MATERIAL_FORM(rho,u) :=
  Drho/Dt + rho*div(u)=0

 MATERIAL_DERIVATIVE(rho,u) :=
  Drho/Dt=partial rho/partial t + u dot grad(rho)

 INTEGRAL_MASS_BALANCE(rho,u,V) :=
  d/dt integral_V rho dV
  + surface_integral_boundaryV rho*u dot n dA
  = 0

 STEADY_CONTINUITY(rho,u) :=
  div(rho*u)=0

 ONE_DIMENSIONAL_STEADY_FLOW(rho,A,U) :=
  rho*A*U=constant

 MASS_FLOW_RATE(rho,u,A) :=
  m_dot=surface_integral_A rho*u dot n dA

 CYLINDRICAL_INCOMPRESSIBLE(u_r,u_theta,u_z,r) :=
  (1/r)*partial(r*u_r)/partial r
  +(1/r)*partial u_theta/partial theta
  +partial u_z/partial z
  =0
}

DEFINITIONS {
 D1 = "The continuity equation is a conservation statement for mass."
 D2 = "The differential form balances local accumulation against flux divergence."
 D3 = "The integral form balances mass change inside a control volume against boundary flux."
 D4 = "For constant-density incompressible flow, continuity reduces to zero velocity divergence."
 D5 = "Compressible flow requires explicit density evolution."
 D6 = "Mass flux is density multiplied by velocity."
 D7 = "Steady flow removes the explicit local time derivative but not spatial flux variation."
 D8 = "Source terms represent mass creation or removal only when the physical model permits them."
 D9 = "Finite-volume discretisations naturally mirror integral conservation."
 D10 = "Coordinate-specific forms must preserve the underlying divergence operator."
}

AXIOMS {
 A1_MASS_CONSERVATION:
  Mass is conserved unless explicit source or sink terms are declared

 A2_LOCAL_BALANCE:
  Local density accumulation plus mass-flux divergence equals the declared source term

 A3_CONTROL_VOLUME_BALANCE:
  Control-volume mass change equals net inward mass flux plus sources

 A4_INCOMPRESSIBLE_REDUCTION:
  Constant-density material conservation implies divergence-free velocity

 A5_COORDINATE_INVARIANCE:
  The physical conservation law is coordinate independent even though its differential expression changes

 A6_CONSERVATIVE_FORM:
  Conservative form preserves flux-balance semantics

 A7_BOUNDARY_DEPENDENCE:
  Boundary fluxes determine net mass exchange with the domain

 A8_NUMERICAL_CONSERVATION:
  A conservative discretisation should balance discrete accumulation and discrete fluxes
}

REASONING {

 R1_CONTINUITY_CONTEXT {
  INPUT = continuum_mass_problem
  IDENTIFY = density_field
  IDENTIFY = velocity_field
  IDENTIFY = domain_and_control_volume
  IDENTIFY = source_or_sink terms
  RETURN = continuity_context
 }

 R2_FORM_SELECTION {
  INPUT = continuity_context
  IF local_field analysis
   SELECT = differential_form
  IF control_volume analysis
   SELECT = integral_form
  RETURN = selected_mass_balance
 }

 R3_COMPRESSIBILITY_REASONING {
  INPUT = density_behaviour
  IF density_constant
   REDUCE = material_form_to_divergence_constraint
  ELSE
   RETAIN = full_density_evolution
  RETURN = compressibility_profile
 }

 R4_STEADY_FLOW_REASONING {
  INPUT = steady_flow_assumption
  REMOVE = explicit_time_derivative
  SOLVE = div(rho*u)=source
  RETURN = steady_mass_balance
 }

 R5_CONTROL_VOLUME_REASONING {
  INPUT = control_volume_and_boundary fluxes
  COMPUTE = accumulation_and_net_surface_flux
  CHECK = source_terms
  RETURN = mass_balance_residual
 }

 R6_COORDINATE_REASONING {
  INPUT = chosen_coordinate_system
  EXPAND = divergence_operator_correctly
  RETURN = coordinate_specific_continuity equation
 }

 R7_BOUNDARY_FLUX_REASONING {
  INPUT = inflow_outflow_wall boundaries
  COMPUTE = signed_mass_fluxes
  RETURN = net_boundary_mass_flux
 }

 R8_NUMERICAL_CONSERVATION {
  INPUT = discrete_cells_faces_and_fluxes
  SUM = face_fluxes
  UPDATE = cell_mass
  RETURN = discrete_conservation_status
 }

 R9_RESIDUAL_ANALYSIS {
  INPUT = candidate_velocity_density fields
  COMPUTE = continuity_residual
  RETURN = local_and_global_mass_error
 }

 R10_VALIDATION {
  INPUT = continuity_solution_or_discretisation
  CHECK = units_boundary fluxes_global mass balance_local residual_and_source consistency
  RETURN = validated_continuity_status
 }
}

CONSTRAINTS {
 C1 = "Incompressibility and constant density are related but not identical concepts in every context."
 C2 = "Zero divergence alone does not generally imply constant density for arbitrary compressible models."
 C3 = "Source terms must be physically justified."
 C4 = "Steady flow does not imply uniform velocity or density."
 C5 = "Coordinate expressions must include the correct metric factors."
 C6 = "Numerical mass conservation requires consistent face fluxes."
 C7 = "Boundary conditions can introduce or remove mass from an open domain."
 C8 = "Residual tolerance should be interpreted relative to scale and discretisation."
 C9 = "Nonconservative algebraic rearrangements can be unsuitable across discontinuities."
 C10 = "Compressible shock problems should use conservative formulations."
}

SEMANTIC_EQUIVALENCE {
 "ContinuityEquation" <=> "MassConservationEquation"
 "MassBalance" <=> "ConservationOfMass"
 "MassFlux" <=> "DensityVelocityFlux"
 "ConservativeForm" <=> "FluxConservativeForm"
 "ControlVolume" <=> "FiniteControlRegion"
 "IncompressibleContinuity" <=> "DivergenceFreeVelocityConstraint"
}

VALIDATION {
 V1 = "Density and velocity variables are explicit."
 V2 = "Units are dimensionally consistent."
 V3 = "Compressible or incompressible assumptions are explicit."
 V4 = "Source and sink terms are documented."
 V5 = "Boundary flux signs are consistent."
 V6 = "Differential and integral forms agree."
 V7 = "Coordinate-system metric factors are correct."
 V8 = "Global mass balance closes within tolerance."
 V9 = "Local residuals are assessed where relevant."
 V10 = "Numerical fluxes are conservative across shared faces."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 CONTINUITY_DEFINITION = 0.99
 FORM_SELECTION_REASONING = 0.99
 COMPRESSIBILITY_REASONING = 0.99
 CONTROL_VOLUME_REASONING = 0.99
 COORDINATE_REASONING = 0.98
 BOUNDARY_FLUX_REASONING = 0.99
 NUMERICAL_CONSERVATION_REASONING = 0.99
 RESIDUAL_REASONING = 0.99
 VALIDATION_REASONING = 0.99

 POLICY =
  "Reduce confidence when density behaviour, boundary fluxes, source terms, coordinate system, discretisation or conservation tolerances are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  ContinuityEquationReasoning,
  ConservationOfMassReasoning,
  DensityEvolutionReasoning,
  MassFluxReasoning,
  IncompressibleFlowReasoning,
  CompressibleFlowReasoning,
  ControlVolumeReasoning,
  ReynoldsTransportReasoning,
  CoordinateFormReasoning,
  FiniteVolumeReasoning,
  NumericalConservationReasoning,
  ResidualReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "continuity equation" -> ContinuityEquation
  "mass conservation equation" -> ContinuityEquation
  "mass balance" -> ConservationOfMass
  "mass flux" -> MassFlux
  "control volume" -> ControlVolume
  "Reynolds transport theorem" -> ReynoldsTransportTheorem
  "divergence-free velocity" -> IncompressibleContinuityConstraint
 }

 INTERPRETATION_ORDER = [
  detect_mass_conservation_context,
  identify density_velocity_domain_sources_and_boundary fluxes,
  select differential_or_integral form,
  determine compressible_or_incompressible assumptions,
  compute local_or_control-volume balance,
  expand divergence in the declared coordinates,
  evaluate boundary_fluxes_and_residuals,
  if_numerical verify conservative discrete flux balance,
  validate global_and_local mass conservation,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("continuity equation")
  OR mass_conservation_in_continuum_is_present
  THEN activate(ContinuityEquationOntology)

 E2:
  IF constant_density_and_materially conserved
  THEN reduce_to_divergence_free_velocity

 E3:
  IF variable_density
  THEN retain full continuity equation

 E4:
  IF steady_flow
  THEN remove explicit time_accumulation term

 E5:
  IF control_volume formulation
  THEN use integral mass balance

 E6:
  IF cylindrical_or_spherical coordinates
  THEN use coordinate_correct divergence

 E7:
  IF source_or_sink present
  THEN include source_term explicitly

 E8:
  IF finite_volume discretisation
  THEN require equal_and_opposite shared_face fluxes

 E9:
  IF shock_or_discontinuity present
  THEN preserve conservative_form
}

RECURSION {
 ENTRY = ContinuityEquation

 EXPAND(ContinuityEquation) -> [
  ConservationOfMass,
  MassBalance,
  DifferentialForm,
  IntegralForm,
  DensityField,
  VelocityField,
  MassFlux,
  Divergence,
  MaterialDerivative,
  ControlVolume,
  ControlSurface,
  SurfaceFlux,
  SourceTerm,
  SinkTerm,
  CompressibleFlow,
  IncompressibleFlow,
  ConstantDensityFlow,
  VariableDensityFlow,
  SteadyFlow,
  UnsteadyFlow,
  ReynoldsTransportTheorem,
  CoordinateTransformation,
  CylindricalCoordinates,
  SphericalCoordinates,
  ConservativeForm,
  NonconservativeForm,
  FiniteVolumeMethod,
  FluxBalance,
  NumericalConservation,
  Residual,
  BoundaryFlux
 ]

 EXPAND(Forms) -> [
  DifferentialForm,
  IntegralForm,
  MaterialForm,
  ConservativeForm,
  CoordinateSpecificForm
 ]

 EXPAND(NumericalStructure) -> [
  FiniteVolumeMethod,
  CellMass,
  FaceFlux,
  BoundaryFlux,
  Residual,
  GlobalMassError
 ]

 STOP_CONDITION =
  "Stop when density evolution, flux form, control-volume balance, compressibility assumptions, coordinate representation, boundary exchange and numerical conservation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Fluid Mathematics"

 IMMEDIATE_CONTEXTS = [
  "Fluid Dynamics Mathematics Ontology",
  "Navier–Stokes Ontology",
  "Partial Differential Equation Ontology",
  "Transport Mathematics Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Fluid Mathematics",
  "Fluid Dynamics Mathematics Ontology",
  "Navier–Stokes Ontology",
  "Partial Differential Equation Ontology",
  "Vector Calculus Ontologies",
  "Conservation Law Ontologies",
  "Transport Mathematics Ontologies",
  "Compressible Flow Ontologies",
  "Incompressible Flow Ontologies",
  "Finite Volume Ontologies",
  "Computational Fluid Dynamics Ontologies",
  "Scientific Computing Ontologies"
 ]

 ROUTE_RULES = [
  "Continuity Equation -> Fluid Mathematics for parent semantics",
  "Continuity Equation -> Fluid Dynamics Mathematics for flow interpretation",
  "Continuity Equation -> Navier–Stokes for coupled momentum equations",
  "Continuity Equation -> PDE Ontology for conservation-law structure",
  "Continuity Equation -> Vector Calculus for divergence and flux",
  "Continuity Equation -> Transport Mathematics for advective mass transport",
  "Continuity Equation -> Compressible Flow for density evolution",
  "Continuity Equation -> Incompressible Flow for divergence-free constraints",
  "Continuity Equation -> Finite Volume for conservative discretisation",
  "Continuity Equation -> CFD for numerical mass-balance validation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible conservation forms, coordinate representations or numerical pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve mass-conservation semantics.",
  "Preserve differential-versus-integral form equivalence.",
  "Preserve compressible-versus-incompressible distinction.",
  "Preserve control-volume flux semantics.",
  "Preserve coordinate-invariant physical meaning.",
  "Preserve source-term semantics.",
  "Preserve conservative-discretisation semantics.",
  "Preserve machine-readable continuity-equation semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "The continuity equation expresses conservation of mass by balancing local density change against divergence of mass flux."

 DIFFERENTIAL_SUMMARY =
  "The local conservative form is partial rho/partial t plus div(rho u) equal to the declared source term."

 INTEGRAL_SUMMARY =
  "The integral form balances control-volume mass accumulation against net boundary mass flux."

 INCOMPRESSIBLE_SUMMARY =
  "For constant-density incompressible flow, continuity reduces to div(u)=0."

 COMPRESSIBLE_SUMMARY =
  "Compressible flow retains density evolution and mass-flux divergence."

 NUMERICAL_SUMMARY =
  "Finite-volume methods naturally preserve mass through face-by-face flux balance."

 VALIDATION_SUMMARY =
  "Continuity solutions should be checked using both local residuals and global mass balance."
}

FLOW {
 INPUT
 -> detect_continuity_equation_context
 -> identify density_velocity_domain_control_volume_sources_and_boundary conditions
 -> choose differential_integral_material_or_coordinate form
 -> classify compressible_incompressible_steady_or_unsteady flow
 -> compute local accumulation_mass flux_divergence_and_boundary exchange
 -> if_constant_density reduce_to_divergence_free constraint
 -> if_coordinate_specific expand divergence_with_correct metric factors
 -> if_numerical enforce conservative_cell_face flux balance
 -> evaluate local_residual_and_global_mass_error
 -> route to fluid_mathematics fluid_dynamics NavierStokes PDE vector_calculus conservation_laws transport compressible_flow incompressible_flow finite_volume_or_CFD ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Continuity Equation Ontology defines mass conservation in continuum systems. It supports differential, integral and material forms, density and velocity fields, mass flux, divergence, control volumes and surfaces, source and sink terms, compressible and incompressible flow, steady and unsteady flow, Reynolds transport, coordinate-specific formulations, conservative forms, finite-volume flux balance, numerical conservation, local residuals and global mass-balance validation."
}

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

}

END_FLOW