In fluid mechanics or more generally continuum mechanics, incompressible flow refers to a flow in which the material density is constant within a fluid parcel ¿ an infinitesimal volume that moves with the velocity of the fluid. An equivalent statement implying incompressibility is, that the divergence of the fluid velocity is zero (see the derivation below, which illustrates why these conditions are equivalent). Incompressible flow does not imply that the fluid itself is incompressible.
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Smoothed-particle hydrodynamics
Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating fluid flows. It was developed by Gingold and Monaghan (1977) and Lucy (1977) initially for astrophysical problems. It has been used in many fields of research, including astrophysics, ballistics, volcanology, and oceanography. It is a mesh-free Lagrangian method (where the coordinates move with the fluid), and the resolution of the method can easily be adjusted with respect to variables such as the density.
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Compressibility
In thermodynamics and fluid mechanics, compressibility is a measure of the relative volume change of a fluid or solid as a response to a pressure change. where V is volume and p is pressure Note: most textbooks use the notation for this quantity
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Poisson's equation
In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in electrostatics, mechanical engineering and theoretical physics. It is named after the French mathematician, geometer and physicist Siméon-Denis Poisson.
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Fluid
In physics, a fluid is a substance that continually deforms (flows) under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids. In common usage, "fluid" is often used as a synonym for "liquid", with no implication that gas could also be present. For example, "brake fluid" is hydraulic oil and will not perform its required function if there is gas in it.
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Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure (also spelled gage pressure) is the pressure relative to the local atmospheric or ambient pressure.
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Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ¿ (the lower case Greek letter rho). In some cases (for instance, in the United States oil and gas industry), density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight. Different materials usually have different densities, so density is an important concept regarding buoyancy, purity and packaging.
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Monte Carlo method
Monte Carlo methods (or Monte Carlo experiments) are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems. These methods are most suited to calculation by a computer and tend to be used when it is infeasible to compute an exact result with a deterministic algorithm. This method is also used to complement theoretical derivations.
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