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mathematics, an incompressible surface is a surface properly embedded in a 3-manifold, which, in intuitive terms, is a "nontrivial" surface that cannot be
Incompressible_surface
Topics referred to by the same term
mechanics incompressible vector field, in mathematics Incompressible surface, in mathematics Incompressible string, in computing This disambiguation page lists
Incompressible
In low-dimensional topology, a boundary-incompressible surface is a two-dimensional surface within a three-dimensional manifold whose topology cannot
Boundary-incompressible surface
Boundary-incompressible_surface
Equations of motion for viscous fluids
u = 0 {\textstyle \nabla \cdot \mathbf {u} =0} for an incompressible fluid. Incompressibility rules out density and pressure waves like sound or shock
Navier–Stokes_equations
Mathematics concept
sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface. Sometimes one considers only orientable Haken manifolds, in which
Haken_manifold
Topics referred to by the same term
see Incompressible surface a proof method in mathematics, see Incompressibility method a property of strings in computer science, see Incompressible string
Incompressibility
Fluid flow in which density remains constant
mechanics, incompressible flow is a flow in which the material density does not vary over time. Equivalently, the divergence of an incompressible flow velocity
Incompressible_flow
Mathematical space
special surfaces embedded in it. One can choose the surface to be nicely placed in the 3-manifold, which leads to the idea of an incompressible surface and
3-manifold
Set of quasilinear hyperbolic equations governing adiabatic and inviscid flow
conductivity. The Euler equations can be applied to incompressible and compressible flows. The incompressible Euler equations consist of Cauchy equations for
Euler equations (fluid dynamics)
Euler_equations_(fluid_dynamics)
Principle relating to fluid dynamics
Therefore, the fluid can be considered to be incompressible, and these flows are called incompressible flows. Bernoulli performed his experiments on
Bernoulli's_principle
Simplest non-trivial closed knot with three crossings
complement is also a Seifert fibred with boundary, it has a horizontal incompressible surface—this is also the fiber of the Milnor map. (This assumes the knot
Trefoil_knot
American mathematician
and Peter Scott, he proved foundational results on least-area incompressible surfaces in 3-manifolds. Hass also constructed examples of bounded 3-manifolds
Joel_Hass
American mathematician
a closed orientable surface, Topology 19 (1980), no. 3, 221–237. Allen Hatcher, On the boundary curves of incompressible surfaces, Pacific Journal of
Allen_Hatcher
Type of mathematical knot
mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement. Every knot is either
Satellite_knot
Function for incompressible divergence-free flows in two dimensions
incompressible (divergence-free), two-dimensional flows. The Stokes stream function, named after George Gabriel Stokes, is defined for incompressible
Stream_function
Vector field with zero divergence
solenoidal vector field, while in hydrodynamics it is commonly called an incompressible vector field, and in mathematical physics it is sometimes called an
Solenoidal_vector_field
Wave where gravity is the main restoring force
{\displaystyle (u'(x,z,t),w'(x,z,t)).} Because the fluid is assumed incompressible, this velocity field has the streamfunction representation u ′ = ( u
Gravity_wave
called sufficiently large if it contains a properly embedded 2-sided incompressible surface. This property is the main requirement for a 3-manifold to be called
Eventually_(mathematics)
Handlebody Incompressible surface Dehn's lemma Loop theorem (aka the Disk theorem) Sphere theorem Haken manifold JSJ decomposition Branched surface Lamination
List of geometric topology topics
List_of_geometric_topology_topics
Subfield of mathematical topology
3-manifold contains no incompressible surface has been algorithmically implemented by Burton, Rubinstein and Tillmann and based on normal surface theory. The Manning
Computational_topology
Polish American mathematician
Fields Mathematics Workplaces George Washington University Thesis Incompressible Surfaces in 3-Manifolds (1981) Doctoral advisor Joan Birman Website blogs
Józef_Przytycki
Type of fluid
does not depend on the velocity or stress state of the fluid. For an incompressible and isotropic Newtonian fluid in laminar flow only in the direction
Newtonian_fluid
Branch of fluid mechanics that studies fluids at rest
more often it includes both gases and liquids, whether compressible or incompressible. It encompasses the study of the conditions under which fluids are at
Hydrostatics
Elliptic partial differential equation
suggest implementing this technique with an adaptive octree. For the incompressible Navier–Stokes equations, given by ∂ v ∂ t + ( v ⋅ ∇ ) v = − 1 ρ ∇ p
Poisson's_equation
Type of mathematical link
Mathematical Society, ISBN 0-8050-7380-9. William Menasco (1984) "Closed incompressible surfaces in alternating knot and link complements", Topology 23(1):37–44
Hyperbolic_link
American mathematician (1946–2012)
knot complement. By utilizing Haken's normal surface techniques, he classified the incompressible surfaces in the knot complement. Together with his analysis
William_Thurston
Force distributed over an area
the energy per unit volume in an ideal, incompressible liquid is constant throughout its vessel. At the surface of a stationary liquid in a vessel gravitational
Pressure
Theorem in geometry
circle. The idea of the proof is to cut a Haken manifold M along an incompressible surface, to obtain a new manifold N. By induction one assumes that the interior
Hyperbolization_theorem
Dimensionless number describing relative pressures in a fluid flow field
or boat. The pressure coefficient is a parameter for studying both incompressible/compressible fluids such as water and air. The relationship between
Pressure_coefficient
Branch of dynamics concerned with studying the motion of air
fundamental relationship between pressure, density, and flow velocity for incompressible flow known today as Bernoulli's principle, which provides one method
Aerodynamics
Branch of physics
describe the force balance at a given point within a fluid. For an incompressible fluid with vector velocity field u {\displaystyle \mathbf {u} } , the
Fluid_mechanics
Velocity field as the gradient of a scalar function
gradient of a scalar always being equal to zero. In the case of an incompressible flow the velocity potential satisfies Laplace's equation, and potential
Potential_flow
manifold. Surgery theory Juhász, András (12 March 2008). "Floer homology and surface decompositions". msp.org. Mathematical Sciences Publishers, Geometry &
Manifold_decomposition
Free-diving self-propelled deep-sea submersible
available, buoyant in water, and, like any liquid, practically incompressible. The incompressibility of the gasoline means the tanks can be very lightly constructed
Bathyscaphe
Surface waves generated by wind on open water
two-dimensional parallel shear flow incompressible, inviscid water and wind irrotational water slope of the displacement of the water surface is small Generally, these
Wind_wave
Law describing the pressure drop in an incompressible and Newtonian fluid
Poiseuille equation, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical
Hagen–Poiseuille_equation
American mathematician
geometric group theory. Floyd and Allen Hatcher classified all the incompressible surfaces in punctured-torus bundles over the circle. In a 1980 paper Floyd
William_Floyd_(mathematician)
Type of fluid flow
\mathbf {u} } the fluid velocity. To obtain the equations of motion for incompressible flow, it is assumed that the density, ρ {\displaystyle \rho } , is a
Stokes_flow
Type of liquid flow within a conduit
equations, it is acceptable to make several assumptions: The flow is incompressible (this is not a good assumption for rapidly-varied flow) The Reynolds
Open-channel_flow
Hyperelastic material model
known neo-Hookean model. The strain energy density function for an incompressible Mooney–Rivlin material is W = C 10 ( I 1 − 3 ) + C 01 ( I 2 − 3 ) ;
Neo-Hookean_solid
Australian mathematician (born 1945)
755–833. Freedman, Michael; Hass, Joel; Scott, Peter. Least area incompressible surfaces in 3-manifolds. Invent. Math. 71 (1983), no. 3, 609–642. AMS (February
Leon_Simon
British-American mathematician (born 1979)
then his PhD at Stanford University (2007) for the dissertation "Incompressible Surfaces in Hyperbolic Punctured Torus Bundles are Strongly Detected" under
Henry_Segerman
In high-speed flight, the assumptions of incompressibility of the air used in low-speed aerodynamics no longer apply. In subsonic aerodynamics, the theory
High-speed_flight
Equations of fluid dynamics
generally accompanies the Navier–Stokes equation. In the case of an incompressible fluid, Dρ/Dt = 0 (the density following the path of a fluid element
Derivation of the Navier–Stokes equations
Derivation_of_the_Navier–Stokes_equations
Classical solution for inviscid, incompressible flow around a cylinder
vector normal to the cylinder surface. The upstream flow is uniform and has no vorticity. The flow is inviscid, incompressible and has constant mass density
Potential flow around a circular cylinder
Potential_flow_around_a_circular_cylinder
Principle in fluid mechanics
mechanics that states that a pressure change at any point in a confined incompressible fluid is transmitted throughout the fluid such that the same change
Pascal's_law
published it in 1849) states that the steady irrotational motion of an incompressible fluid occupying a simply connected region has less kinetic energy than
Kelvin's minimum energy theorem
Kelvin's_minimum_energy_theorem
Physical quantity
the following two assumptions. Since many liquids can be considered incompressible, a reasonable good estimation can be made from assuming a constant density
Hydrostatic_pressure
Standard surface gravity
planet grows in size, as they are not incompressible bodies. That is why the experimental relationship between surface gravity and mass does not grow as 1/3
Surface_gravity
Liquid, gas, or other continuously deforming and flowing material
pressure is applied to the fluid or when the fluid becomes supersonic. Incompressible fluid: A fluid that does not vary in volume with changes in pressure
Fluid
Layer of fluid in the immediate vicinity of a bounding surface
continuity and Navier–Stokes equations for a two-dimensional steady incompressible flow in Cartesian coordinates are given by ∂ u ∂ x + ∂ υ ∂ y = 0 {\displaystyle
Boundary_layer
Mathematical model of the time dependence of a point in space
[citation needed]. Another intuition comes from permutations which are "incompressible". One can interpret the set of all possible permutations of a set as
Dynamical_system
Millennium Prize Problem
proposed by the Clay Mathematics Institute is in three dimensions, for an incompressible and homogeneous fluid, only that case is considered below. Let v ( x
Navier–Stokes existence and smoothness
Navier–Stokes_existence_and_smoothness
Mathematical concept applicable to physics
the quantity in a control volume around a given point in space. For incompressible flow, the divergence of the volume flux is zero. As mentioned above
Flux
Mathematical technique in aerodynamics
certain compressible flow problems by incompressible-flow calculation methods. It also allows applying incompressible-flow data to compressible-flow cases
Prandtl–Glauert transformation
Prandtl–Glauert_transformation
Characterizes homeomorphisms of a compact orientable surface
periodic, then Mg has an H2 × R structure; If g is reducible, then Mg has incompressible tori, and should be cut along these tori to yield pieces that each have
Nielsen–Thurston classification
Nielsen–Thurston_classification
Geometric representation of material yield
A yield surface is a five-dimensional surface in the six-dimensional space of stresses. The yield surface is usually convex and the state of stress of
Yield_surface
Dimensionless number; ratio of a fluid's flow inertia to the external field
field) is thus notable and can be studied with perturbation theory. Incompressible Navier–Stokes momentum equation is a Cauchy momentum equation with the
Froude_number
Energy transfer between scales of motion
22–24. Kolmogorov, A. N. (1941). "Local structure of turbulence in an incompressible fluid at very high Reynolds numbers". Dokl. Akad. Nauk SSSR. 31: 99–101
Energy_cascade
space is not a Haken manifold, that is, it does not contain any incompressible surfaces; Burton, Rubinstein & Tillmann (2012) proved the conjecture with
Seifert–Weber_space
Aspects of fluid mechanics involving fluid flow
modelled as an incompressible flow. Otherwise the more general compressible flow equations must be used. Mathematically, incompressibility is expressed
Fluid_dynamics
Term used in fluid dynamics
Bernoulli's Equation. For incompressible liquids such as water, Static lift + Pressure head together equal the difference in fluid surface elevation between the
Total_dynamic_head
Wave on the surface of a fluid, dominated by surface tension
constant (i.e., incompressibility), and likewise g {\displaystyle g} (waves are not high enough for gravitation to change appreciably). For surface tension,
Capillary_wave
Society. ISBN 978-0-8218-3678-1. Menasco, William (1984). "Closed incompressible surfaces in alternating knot and link complements" (PDF). Topology. 23 (1):
Alternating_knot
Analysis and solving of problems that involve fluid flows
1965). "Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface". The Physics of Fluids. 8 (12): 2182–2189. Bibcode:1965PhFl
Computational_fluid_dynamics
Formula to approximate nuclear mass based on nucleon counts
Archibald Wheeler and Lise Meitner. It treats the nucleus as a drop of incompressible fluid of very high density, held together by the nuclear force (a residual
Semi-empirical_mass_formula
Fluid dynamics theory on gravity waves
the mean surface elevation. The impermeable bed underneath the fluid layer is at z = −h. Further, the flow is assumed to be incompressible and irrotational
Airy_wave_theory
Steam whose temperature can be decreased without immediately condensing
that water vapor containing entrained liquid droplets is generally incompressible at those pressures. In a reciprocating engine or turbine, if steam doing
Superheated_steam
Tunisian mathematician
hydrodynamic limit value of the Boltzmann equation, limit behavior to incompressibility, chemotaxis (Keller-Segel equations), the Ginsburg-Landau equation
Nader_Masmoudi
Generalized manifold
be small if it is closed, irreducible and does not contain any incompressible surfaces. Orbifold Theorem. Let M be a small 3-manifold. Let φ be a non-trivial
Orbifold
method Finite elements method Finite difference method We begin with the incompressible form of the momentum equation. The equation has been divided through
Discretization of Navier–Stokes equations
Discretization_of_Navier–Stokes_equations
Arrival of a spacecraft on the Moon's surface
exterior blanket of crushable balsa wood and an interior filled with incompressible liquid freon. A 42-kilogram (93 lb) 30-centimetre-diameter (0.98 ft)
Moon_landing
Laplacian vector field is a vector field which is both irrotational and incompressible. If the field is denoted as v, then it is described by the following
Laplacian_vector_field
Static exact solution in general relativity
which consists of an incompressible fluid (implying that density is constant throughout the body) and has zero pressure at the surface. This is a static
Interior_Schwarzschild_metric
Ratio of inertial to viscous forces acting on a liquid
uL\mu ^{-1}} , the Reynolds number. Alternatively, we can take the incompressible Navier–Stokes equations (convective form): ∂ u ∂ t + ( u ⋅ ∇ ) u − ν
Reynolds_number
Function in fluid dynamics
streamlines and flow velocity in a three-dimensional incompressible flow with axisymmetry. A surface with a constant value of the Stokes stream function
Stokes_stream_function
Field lines in a fluid flow
stream surface. In the case of a closed curve in a steady flow, fluid that is inside a stream surface must remain forever within that same stream surface, because
Streamlines, streaklines, and pathlines
Streamlines,_streaklines,_and_pathlines
Limit of sequence of smooth functions
S.O.; Tryggvason, G. (1992), "A front-tracking method for viscous, incompressible, multi-fluid flows" (PDF), Journal of Computational Physics, 100 (1):
Laplacian_of_the_indicator
Mathematics textbook
torus. However, for more complicated manifolds, cutting along incompressible surfaces can be used to construct the JSJ decomposition of a manifold. This
Introduction_to_3-Manifolds
Physical law relating heat loss to temperature difference
C = d U / d T {\displaystyle C=dU/dT} (in J/K), for the case of an incompressible material. The internal energy may be written in terms of the temperature
Newton's_law_of_cooling
Constitutive model for ideally elastic material
relationship can be defined as non-linearly elastic, isotropic and incompressible. Hyperelasticity provides a means of modeling the stress–strain behavior
Hyperelastic_material
Method for solving problems in continuum mechanics
was first used in Monaghan (1992) to simulate incompressible free surface flows by SPH. The incompressible limit is obtained by choosing a very large speed
Finite_pointset_method
British mathematician
Freedman, Michael; Hass, Joel; Scott, Peter (1983). "Least area incompressible surfaces in 3-manifolds". Inventiones Mathematicae. 71 (3): 609–642. Bibcode:1983InMat
G._Peter_Scott
Device for measuring or restricting fluid flow
obstruction and abnormalities in the flow. By assuming steady-state, incompressible (constant fluid density), inviscid, laminar flow in a horizontal pipe
Orifice_plate
State of matter
of the container in the direction of the force. Liquids are nearly incompressible, maintaining their volume even under pressure. The density of a liquid
Liquid
Dimensionless number in fluid mechanics
l^{2}\sigma } . The Weber number appears in the incompressible Navier-Stokes equations through a free surface boundary condition. For a fluid of constant
Weber_number
Paradox by d'Alembert on fluid dynamics
mathematician Jean le Rond d'Alembert. D'Alembert proved that – for incompressible and inviscid potential flow – the drag force is zero on a body moving
D'Alembert's_paradox
Equation describing the evolution of the vorticity of a fluid particle as it flows
constant density fluid (including incompressible fluid) where ∇ρ = 0. Note that this is not the same as an incompressible flow, for which the barotropic
Vorticity_equation
(MPS) method is a computational method for the simulation of incompressible free surface flows. It is a macroscopic, deterministic particle method (Lagrangian
Moving particle semi-implicit method
Moving_particle_semi-implicit_method
Body part type that consists mainly of muscles with no skeletal support
hydrostatic skeleton, relies on the fact that water is effectively incompressible at physiological pressures. In contrast to a hydrostatic skeleton, where
Muscular_hydrostat
Physical and chemical properties of pure water
and of water in particular, leads to their often being assumed as incompressible. The low compressibility of water means that even in the deep oceans
Properties_of_water
Tendency of a fluid jet to stay attached to a surface of any form
; Newmann, B. G. (August 1960). "Reattachment of a two-dimensional, incompressible jet to an adjacent flat Plate". The Aeronautical Quarterly. 11 (3):
Coandă_effect
Solution of Euler equations
equations for periodic surface gravity waves. It describes a progressive wave of permanent form on the surface of an incompressible fluid of infinite depth
Trochoidal_wave
Free-surface modelling technique
(1965-12-01). "Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface". The Physics of Fluids. 8 (12): 2182–2189. Bibcode:1965PhFl
Volume_of_fluid_method
Solid object used in fluid mechanics
the simplified Navier–Stokes equations, applicable when the fluid is incompressible. And since the effects of the compressibility of air at low speeds is
Foil_(fluid_mechanics)
Term in fluid dynamics
applications include rocket stage separation and aircraft store separation. In incompressible fluids (liquids) such as water, a bow wake is created when a watercraft
Wake_(physics)
Italian mathematician
aspects of the theory of hyperbolic systems of conservation laws and of incompressible fluid dynamics. In particular, together with László Székelyhidi Jr.
Camillo_De_Lellis
Theorem in calculus
fluid crossing this surface, i.e., the surface integral of the velocity over the surface. Under the assumption of an incompressible fluid, the amount of
Divergence_theorem
Infinitesimal volume of fluid within a fluid flow
distortion by the flow. In an incompressible flow, the volume of the fluid parcel is also a constant (isochoric flow). Material surfaces and material lines are
Fluid_parcel
Series of waves generated by distant weather systems
{\displaystyle Ua(y)} . Incompressible, inviscid water/wind. Irrotational water. Small slope of the displacement of the surface. Generally, these wave
Swell_(wave)
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