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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 scalar, the ''flow speed''. 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 sai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuum Mechanics
Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles. Continuum mechanics deals with ''deformable bodies'', as opposed to rigid bodies. A continuum model assumes that the substance of the object completely fills the space it occupies. While ignoring the fact that matter is made of atoms, this provides a sufficiently accurate description of matter on length scales much greater than that of inter-atomic distances. The concept of a continuous medium allows for intuitive analysis of bulk matter by using differential equations that describe the behavior of such matter according to physical laws, such as mass conservation, momentum conservation, and energy conservation. Information about the specific material is expressed in constitutive relationships. Continuum mechanics treats the physical properties of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the nabla operator), or \Delta. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian of a function at a point measures by how much the average value of over small spheres or balls centered at deviates from . The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stream Function
In fluid dynamics, two types of stream function (or streamfunction) are defined: * The two-dimensional (or Lagrange) stream function, introduced by Joseph Louis Lagrange in 1781, is defined for incompressible flow, incompressible (divergence-free), two-dimensional fluid flow, flows. * The Stokes stream function, named after George Gabriel Stokes, is defined for incompressible, three-dimensional flows with axisymmetry. The properties of stream functions make them useful for analyzing and graphically illustrating flows. The remainder of this article describes the two-dimensional stream function. Two-dimensional stream function Assumptions The two-dimensional stream function is based on the following assumptions: * The flow field can be described as two-dimensional plane flow, with velocity vector : \quad \mathbf = \begin u (x,y,t) \\ v (x,y,t) \\ 0 \end. * The velocity satisfies the continuity equation for incompressible flow: : \quad \nabla \cdot \mathbf = 0. * The domain h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strain-rate Tensor
In continuum mechanics, the strain-rate tensor or rate-of-strain tensor is a physical quantity that describes the rate of change of the strain (i.e., the relative deformation) of a material in the neighborhood of a certain point, at a certain moment of time. It can be defined as the derivative of the strain tensor with respect to time, or as the symmetric component of the Jacobian matrix (derivative with respect to position) of the flow velocity. In fluid mechanics it also can be described as the velocity gradient, a measure of how the velocity of a fluid changes between different points within the fluid. Though the term can refer to a velocity profile (variation in velocity across layers of flow in a pipe), it is often used to mean the gradient of a flow's velocity with respect to its coordinates. The concept has implications in a variety of areas of physics and engineering, including magnetohydrodynamics, mining and water treatment. The strain rate tensor is a purely kinema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strain Rate
In mechanics and materials science, strain rate is the time derivative of strain of a material. Strain rate has dimension of inverse time and SI units of inverse second, s−1 (or its multiples). The strain rate at some point within the material measures the rate at which the distances of adjacent parcels of the material change with time in the neighborhood of that point. It comprises both the rate at which the material is expanding or shrinking (expansion rate), and also the rate at which it is being deformed by progressive shearing without changing its volume ( shear rate). It is zero if these distances do not change, as happens when all particles in some region are moving with the same velocity (same speed and direction) and/or rotating with the same angular velocity, as if that part of the medium were a rigid body. The strain rate is a concept of materials science and continuum mechanics that plays an essential role in the physics of fluids and deformable solids. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pressure Gradient
In hydrodynamics and hydrostatics, the pressure gradient (typically of air but more generally of any fluid) is a physical quantity that describes in which direction and at what rate the pressure increases the most rapidly around a particular location. The pressure gradient is a dimensional quantity expressed in units of pascals per metre (Pa/m). Mathematically, it is the gradient of pressure as a function of position. The gradient of pressure in hydrostatics is equal to the ''body force density'' (generalised Stevin's Law). In petroleum geology and the petrochemical sciences pertaining to oil wells, and more specifically within hydrostatics, pressure gradients refer to the gradient of vertical pressure in a column of fluid within a wellbore and are generally expressed in pounds per square inch per foot (psi/ft). This column of fluid is subject to the compound pressure gradient of the overlying fluids. The path and geometry of the column is totally irrelevant; only the vertical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle Velocity
Particle velocity (denoted or ) is the velocity of a particle (real or imagined) in a medium as it transmits a wave. The SI unit of particle velocity is the metre per second (m/s). In many cases this is a longitudinal wave of pressure as with sound, but it can also be a transverse wave as with the vibration of a taut string. When applied to a sound wave through a medium of a fluid like air, particle velocity would be the physical speed of a parcel of fluid as it moves back and forth in the direction the sound wave is travelling as it passes. Particle velocity should not be confused with the speed of the wave as it passes through the medium, i.e. in the case of a sound wave, particle velocity is not the same as the speed of sound. The wave moves relatively fast, while the particles oscillate around their original position with a relatively small particle velocity. Particle velocity should also not be confused with the velocity of individual molecules, which depends mostly on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Velocity
The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope (waves), envelope'' of the wave—propagates through space. For example, if a stone is thrown into the middle of a very still pond, a circular pattern of waves with a quiescent center appears in the water, also known as a capillary wave. The expanding ring of waves is the wave group or wave packet, within which one can discern individual waves that travel faster than the group as a whole. The amplitudes of the individual waves grow as they emerge from the trailing edge of the group and diminish as they approach the leading edge of the group. History The idea of a group velocity distinct from a wave's phase velocity was first proposed by William Rowan Hamilton, W.R. Hamilton in 1839, and the first full treatment was by John Strutt, 3rd Baron Rayleigh, Rayleigh in his "Theory of Sound" in 1877. Definition and interpretation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enstrophy
In fluid dynamics, the enstrophy \mathcal can be interpreted as another type of potential density; or, more concretely, the quantity directly related to the kinetic energy in the flow model that corresponds to dissipation effects in the fluid. It is particularly useful in the study of turbulent flows, and is often identified in the study of thrusters as well as in combustion theory and meteorology. Given a domain \Omega \subseteq \R^n and a once-weakly differentiable vector field u \in H^1(\R^n)^n which represents a fluid flow, such as a solution to the Navier-Stokes equations, its enstrophy is given by:where , \nabla \mathbf, ^2 = \sum_^n \left, \partial_i u^j \^2 . This quantity is the same as the squared seminorm , \mathbf, _^2of the solution in the Sobolev space H^1(\Omega)^n. Incompressible flow In the case that the flow is incompressible, or equivalently that \nabla \cdot \mathbf = 0 , the enstrophy can be described as the integral of the square of the vorticity \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Drift Velocity
Drift or Drifts may refer to: Geography * Drift or ford (crossing) of a river * Drift (navigation), difference between heading and course of a vessel * Drift, Kentucky, unincorporated community in the United States * In Cornwall, England: ** Drift, Cornwall, village ** Drift Reservoir, associated with the village Science, technology, and physics * Directional Recoil Identification from Tracks, a dark-matter experiment * Drift (video gaming), a typical game-controller malfunction * Drift pin, metalworking tool for localizing hammer blows and for aligning holes * Drift (geology), deposited material of glacial origin * drift (in mining), a roughly horizontal passage; an adit * Drift, linear term of a stochastic process * Drift (motorsport), the controlled sliding of a vehicle through a sharp turn, either via over-steering with sudden sharp braking, or counter-steering with a sudden "clutch kick" acceleration * Incremental changes: ** Drift (linguistics), a type of language ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Displacement Field (mechanics)
In mechanics, a displacement field is the assignment of displacement vectors for all points in a region or body that are displaced from one state to another. A displacement vector specifies the position of a point or a particle in reference to an origin or to a previous position. For example, a displacement field may be used to describe the effects of deformation on a solid body. Formulation Before considering displacement, the state before deformation must be defined. It is a state in which the coordinates of all points are known and described by the function: \vec_0: \Omega \to P where *\vec_0 is a placement vector *\Omega are all the points of the body *P are all the points in the space in which the body is present Most often it is a state of the body in which no forces are applied. Then given any other state of this body in which coordinates of all its points are described as \vec_1 the displacement field is the difference between two body states: \vec = \vec_1 - \vec_0 w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volume Flow Rate
In physics and engineering, in particular fluid dynamics, the volumetric flow rate (also known as volume flow rate, or volume velocity) is the volume of fluid which passes per unit time; usually it is represented by the symbol (sometimes \dot V). Its SI unit is cubic metres per second (m3/s). It contrasts with '' mass flow rate'', which is the other main type of fluid flow rate. In most contexts a mention of "rate of fluid flow" is likely to refer to the volumetric rate. In hydrometry, the volumetric flow rate is known as '' discharge''. The volumetric flow rate across a unit area is called '' volumetric flux'', as defined by Darcy's law and represented by the symbol . Conversely, the integration of a volumetric flux over a given area gives the volumetric flow rate. Units The SI unit is cubic metres per second (m3/s). Another unit used is standard cubic centimetres per minute (SCCM). In US customary units and imperial units, volumetric flow rate is often expressed as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
