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Incompressible
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An equivalent statement that implies incompressibility is that the divergence of the flow 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. It is shown in the derivation below that (under the right conditions) even compressible fluids can – to a good approximation – be modelled as an incompressible flow. Incompressible flow implies that the density remains constant within a parcel of fluid that moves with the flow velocity. Derivation The fundamental requirement for incompressible flow is that the density, \rho , is constant within a small element volume, ''dV'', which moves at the flow velocity u. Mathematica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Equations (fluid Dynamics)
In fluid dynamics, the Euler equations are a set of quasilinear partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular, they correspond to the Navier–Stokes equations with zero viscosity and zero thermal conductivity. The Euler equations can be applied to incompressible or compressible flow. The incompressible Euler equations consist of Cauchy equations for conservation of mass and balance of momentum, together with the incompressibility condition that the flow velocity is a solenoidal field. The compressible Euler equations consist of equations for conservation of mass, balance of momentum, and balance of energy, together with a suitable constitutive equation for the specific energy density of the fluid. Historically, only the equations of conservation of mass and balance of momentum were derived by Euler. However, fluid dynamics literature often refers to the full set of the compressible Euler equat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Navier–Stokes Equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes). The Navier–Stokes equations mathematically express conservation of momentum and conservation of mass for Newtonian fluids. They are sometimes accompanied by an equation of state relating pressure, temperature and density. They arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing ''viscous flow''. The difference between them and the closely related Euler equations is that Navier–Stokes equations take ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli's Principle
In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. The principle is named after the Swiss mathematician and physicist Daniel Bernoulli, who published it in his book ''Hydrodynamica'' in 1738. Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard Euler in 1752 who derived Bernoulli's equation in its usual form. The principle is only applicable for isentropic flows: when the effects of irreversible processes (like turbulence) and non-adiabatic processes (e.g. thermal radiation) are small and can be neglected. Bernoulli's principle can be applied to various types of fluid flow, resulting in various forms of Bernoulli's equation. The simple form of Bernoulli's equation is valid for incompressible flows (e.g. most liquid flows and gases moving at low Mach number). More advanced forms may be applied ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projection Method (fluid Dynamics)
In fluid dynamics, The projection method is an effective means of numerically solving time-dependent incompressible fluid-flow problems. It was originally introduced by Alexandre Chorin in 1967 as an efficient means of solving the incompressible Navier-Stokes equations. The key advantage of the projection method is that the computations of the velocity and the pressure fields are decoupled. The algorithm The algorithm of the projection method is based on the Helmholtz decomposition (sometimes called Helmholtz-Hodge decomposition) of any vector field into a solenoidal part and an irrotational part. Typically, the algorithm consists of two stages. In the first stage, an intermediate velocity that does not satisfy the incompressibility constraint is computed at each time step. In the second, the pressure is used to project the intermediate velocity onto a space of divergence-free velocity field to get the next update of velocity and pressure. Helmholtz–Hodge decomposition Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fluid Mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and biomedical engineering, geophysics, oceanography, meteorology, astrophysics, and biology. It can be divided into fluid statics, the study of fluids at rest; and fluid dynamics, the study of the effect of forces on fluid motion. It is a branch of continuum mechanics, a subject which models matter without using the information that it is made out of atoms; that is, it models matter from a ''macroscopic'' viewpoint rather than from ''microscopic''. Fluid mechanics, especially fluid dynamics, is an active field of research, typically mathematically complex. Many problems are partly or wholly unsolved and are best addressed by numerical methods, typically using computers. A modern discipline, called computational fluid dynamics (CFD), is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fluid Flow
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation. Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, such as flow velocity, pressure, density, and temperature, as functions of space and time. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compressible Fluid
Compressible flow (or gas dynamics) is the branch of fluid mechanics that deals with flows having significant changes in fluid density. While all flows are compressible, flows are usually treated as being incompressible when the Mach number (the ratio of the speed of the flow to the speed of sound) is smaller than 0.3 (since the density change due to velocity is about 5% in that case).Anderson, J.D., ''Fundamentals of Aerodynamics'', 4th Ed., McGraw–Hill, 2007. The study of compressible flow is relevant to high-speed aircraft, jet engines, rocket motors, high-speed entry into a planetary atmosphere, gas pipelines, commercial applications such as abrasive blasting, and many other fields. History The study of gas dynamics is often associated with the flight of modern high-speed aircraft and atmospheric reentry of space-exploration vehicles; however, its origins lie with simpler machines. At the beginning of the 19th century, investigation into the behaviour of fired bullets led to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mach Number
Mach number (M or Ma) (; ) is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a boundary to the local speed of sound. It is named after the Moravian physicist and philosopher Ernst Mach. : \mathrm = \frac, where: : is the local Mach number, : is the local flow velocity with respect to the boundaries (either internal, such as an object immersed in the flow, or external, like a channel), and : is the speed of sound in the medium, which in air varies with the square root of the thermodynamic temperature. By definition, at Mach1, the local flow velocity is equal to the speed of sound. At Mach0.65, is 65% of the speed of sound (subsonic), and, at Mach1.35, is 35% faster than the speed of sound (supersonic). Pilots of high-altitude aerospace vehicles use flight Mach number to express a vehicle's true airspeed, but the flow field around a vehicle varies in three dimensions, with corresponding variations in local Mach number. The loc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compressibility
In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a fluid or solid as a response to a pressure (or mean stress) change. In its simple form, the compressibility \kappa (denoted in some fields) may be expressed as :\beta =-\frac\frac, where is volume and is pressure. The choice to define compressibility as the negative of the fraction makes compressibility positive in the (usual) case that an increase in pressure induces a reduction in volume. The reciprocal of compressibility at fixed temperature is called the isothermal bulk modulus. Definition The specification above is incomplete, because for any object or system the magnitude of the compressibility depends strongly on whether the process is isentropic or isothermal. Accordingly, isothermal compressibility is defined: :\beta_T=-\f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flow Velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the flow velocity vector is the flow speed and is a scalar. It is also called velocity field; when evaluated along a line, it is called a velocity profile (as in, e.g., law of the wall). Definition The flow velocity ''u'' of a fluid is a vector field : \mathbf=\mathbf(\mathbf,t), which gives the velocity of an '' element of fluid'' at a position \mathbf\, and time t.\, The flow speed ''q'' is the length of the flow velocity vector :q = \, \mathbf \, and is a scalar field. Uses The flow velocity of a fluid effectively describes everything about the motion of a fluid. Many physical properties of a fluid can be expressed mathematically in terms of the flow velocity. Some common examples follow: Steady flow The flow of a fluid is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplacian Vector Field
In vector calculus, a 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 differential equations: :\begin \nabla \times \mathbf &= \mathbf, \\ \nabla \cdot \mathbf &= 0. \end From the vector calculus identity \nabla^2 \mathbf \equiv \nabla (\nabla\cdot \mathbf) - \nabla\times (\nabla\times \mathbf) it follows that :\nabla^2 \mathbf = 0 that is, that the field v satisfies Laplace's equation. However, the converse is not true; not every vector field that satisfies Laplace's equation is a Laplacian vector field, which can be a point of confusion. For example, the vector field = (xy, yz, zx) satisfies Laplace's equation, but it has both nonzero divergence and nonzero curl and is not a Laplacian vector field. A Laplacian vector field in the plane satisfies the Cauchy–Riemann equations: it is holomorphic. Since the curl of v is zero, it follows that (when the domain of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solenoidal
In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field: \nabla \cdot \mathbf = 0. A common way of expressing this property is to say that the field has no sources or sinks.This statement does not mean that the field lines of a solenoidal field must be closed, neither that they cannot begin or end. For a detailed discussion of the subject, see J. Slepian: "Lines of Force in Electric and Magnetic Fields", American Journal of Physics, vol. 19, pp. 87-90, 1951, and L. Zilberti: "The Misconception of Closed Magnetic Flux Lines", IEEE Magnetics Letters, vol. 8, art. 1306005, 2017. Properties The divergence theorem gives an equivalent integral definition of a solenoidal field; namely that for any closed surface, the net total flux through the surface must be zero: where d\mathbf is the outward normal to each surface el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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