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Showing 181–210 of 1709 reference entries

InformationFinite graphs and trees

Graphs represent pairwise connections; trees impose acyclicity.

DefinitionInformationData-structure interfaces and operation costs

An interface defines operations; an implementation determines costs.

MethodInformationLoop invariants and total correctness

An invariant establishes correctness only with its proof obligations.

MethodInformationComparison sorting lower bound

Distinct-key comparisons need enough branches to distinguish orders.

Mathematical statementInformationBreadth-first and depth-first traversal

Graph searches visit reachable vertices using different orders.

MethodInformationShortest-path algorithm conditions

Edge weights determine which shortest-path method applies.

MethodInformationTuring machines and effective computation

Finite instructions can act on an unbounded writable tape.

DefinitionInformationRelations and tabular data

A relation is a set of tuples from declared domains.

DefinitionPhysicsCentre-of-mass dynamics

External force controls the mass-weighted centre's acceleration.

ModelPhysicsRigid-body inertia tensor

A tensor relates rotation to angular momentum.

ModelPhysicsEuler equations for a rigid body

Body-axis angular acceleration includes rotation of the axes.

ModelPhysics · Mathematics & logicEuler–Lagrange equations

Stationary action supplies coordinate equations of motion.

Mathematical statementPhysics · Mathematics & logicHamilton's canonical equations

Canonical coordinates and momenta evolve from the Hamiltonian.

Mathematical statementPhysicsFluid mass continuity

Density change balances mass flux through a region.

ModelPhysicsBernoulli's streamline relation

An ideal steady flow balances pressure, motion and height.

ModelPhysicsIncompressible Navier–Stokes equation

Fluid inertia balances pressure, viscosity and gravity.

ModelPhysicsReynolds number

Compare inertial and viscous scales of a flow.

ModelPhysicsDriven damped oscillator

A linear restoring force, damping and periodic drive determine response.

ModelPhysicsIdeal-string wave equation

A local differential equation determines small string disturbances.

ModelPhysics · Mathematics & logicWave superposition and interference

Linear disturbances add; their relative phase determines reinforcement.

ModelPhysicsFixed-end string modes

Boundary conditions select standing-wave frequencies.

ModelPhysicsSnell's refraction relation

Angles at an interface depend on refractive indices.

ModelPhysicsThin-lens image equation

A paraxial lens relates object and image distances.

ModelPhysicsSingle-slit diffraction minima

An aperture's finite width selects dark far-field directions.

ModelPhysicsLinear polarization and Malus's law

An ideal analyzer transmits the field component along its axis.

ModelPhysicsElectromagnetic potentials

Scalar and vector potentials encode electric and magnetic fields.

ModelPhysicsCapacitance

A linear two-conductor arrangement stores separated charge.

DefinitionPhysicsSelf-inductance

Changing current changes linked magnetic flux and induced emf.

ModelPhysicsElectromagnetic energy balance

The rate of field-energy change balances outward energy flow and power transferred to charges.

ModelPhysicsKirchhoff circuit balances

Lumped circuit models balance node currents and loop voltages.

Model
ModelSource checked

Fluid mass continuity

Physics

Density change balances mass flux through a region.

Key knowledge

Classical mass conservation without sources gives ∂ρ/∂t + ∇·(ρv) = 0. Constant positive density reduces this to ∇·v = 0; steady constant-density tube flow gives A₁v₁ = A₂v₂.

Values and equations

∂ρ/∂t + ∇·(ρv) = 0; for constant ρ > 0: ∇·v = 0; steady tube: A₁v₁ = A₂v₂

ρ
Fluid mass density kg m⁻³
v
Fluid velocity field m s⁻¹
t
Time s
A₁, A₂
Tube cross-sectional areas m²
v₁, v₂
Cross-section mean speeds m s⁻¹
∇
Spatial derivative operator m⁻¹

Conditions and scope

  • Continuum density and velocity fields are smooth enough for the displayed derivatives; no mass production or removal term.
  • The tube relation uses cross-section mean speeds and excludes leakage or side inflow.
Sources · 3

References

Sources retain their recorded scope. A source link alone does not establish verification of every statement.

  1. Massachusetts Institute of Technology2 sources
    1. Marine Hydrodynamics 13.021, Lecture 3: mass conservation and continuityKarl P. Burr / MIT Ocean EngineeringCourse notesweb.mit.eduRead source
    2. Classical Mechanics 8.01SC, Navier–Stokes chapter (2022 file)MIT OpenCourseWareCourse notes · PDFocw.mit.eduRead source
  2. University Physics 1: Fluid DynamicsSamuel J. Ling, Jeff Sanny and William Moebs / OpenStax, Rice UniversityTextbook chapteropenstax.orgRead source

Evidence and review

MIT Marine Hydrodynamics sections 1.3.1 and 1.5: material-volume conservation and general differential balance; OpenStax and MIT 29.4.14: specialization.

2026-10-10 · General smooth classical mass balance and stated specialization; source terms and discontinuous shock/jump formulations excluded.

Related reference entries

Explore relationships within this area of study and its connections to other areas.

Showing 1–3 of 3 relationships

Scientific relationships · 3 / 3

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

  1. Related to
    Relationship sources · 3

    Linked statements

    • Classical mass conservation without sources gives ∂ρ/∂t + ∇·(ρv) = 0. Constant positive density reduces this to ∇·v = 0; steady constant-density tube flow gives A₁v₁ = A₂v₂.

    A smooth classical no-source mass balance, rather than a claim that every quantity is conserved.

  2. Uses definition
    Relationship sources · 1

    Linked statements

    • A constant-density Newtonian fluid with constant viscosity satisfies ρ(∂v/∂t + (v·∇)v) = −∇p + μ∇²v + ρg, together with ∇·v = 0.

    This constant-density momentum model is closed with incompressible mass continuity.

  3. Uses definition
    Relationship sources · 1

    Linked statements

    • For a smooth classical fluid with mass continuity, material acceleration is D_tv=∂_tv+(v·∇)v and inviscid momentum obeys ρD_tv=−∇P+ρg. In conservation form, ∂_t(ρv)+∇·(ρv⊗v+P𝟙)=ρg; pressure, density and thermodynamic closure must be supplied with initial/boundary data.

    Mass continuity is required to equate material and conservative momentum forms.

Entry detailsPermanent link to this entry ↗
Record identity
physics-continuity-equation
Revision
university-reference.1
Structured reference record
University reference · University core expansion