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Lamb Vector
In fluid dynamics, Lamb vector is the cross product of vorticity vector and velocity vector of the flow field, named after the physicist Horace Lamb.Truesdell, C. (1954). The kinematics of vorticity (Vol. 954). Bloomington: Indiana University Press. The Lamb vector is defined as :\mathbf l = \mathbf \times \boldsymbol where \mathbf is the velocity field and \boldsymbol=\nabla\times\mathbf is the vorticity field of the flow. It appears in the Navier–Stokes equations through the material derivative term, specifically via convective acceleration term, : \mathbf u\cdot \nabla\mathbf u = \frac\nabla\mathbf u^2 - \mathbf\times\boldsymbol = \frac\nabla\mathbf u^2 - \mathbf l In irrotational flows, the Lamb vector is zero, so does in Beltrami flows. The concept of Lamb vector is widely used in turbulent flows. The Lamb vector is analogous to electric field, when the Navier–Stokes equation is compared with Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fluid Dynamics
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 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross Product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is denoted by the symbol \times. Given two linearly independent vectors and , the cross product, (read "a cross b"), is a vector that is perpendicular to both and , and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product). If two vectors have the same direction or have the exact opposite direction from each other (that is, they are ''not'' linearly independent), or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two per ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vorticity
In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along with the flow. It is an important quantity in the dynamical theory of fluids and provides a convenient framework for understanding a variety of complex flow phenomena, such as the formation and motion of vortex rings. Mathematically, the vorticity \vec is the curl of the flow velocity \vec: :\vec \equiv \nabla \times \vec\,, where \nabla is the nabla operator. Conceptually, \vec could be determined by marking parts of a continuum in a small neighborhood of the point in question, and watching their ''relative'' displacements as they move along the flow. The vorticity \vec would be twice the mean angular velocity vector of those particles relative to their center of mass, oriented according to the right-hand rule. In a two-dimensiona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies. Velocity is a physical vector quantity; both magnitude and direction are needed to define it. The scalar absolute value ( magnitude) of velocity is called , being a coherent derived unit whose quantity is measured in the SI ( metric system) as metres per second (m/s or m⋅s−1). For example, "5 metres per second" is a scalar, whereas "5 metres per second east" is a vector. If there is a change in speed, direction or both, then the object is said to be undergoing an ''acceleration''. Constant velocity vs acceleration To have a ''constant velocity'', an object must have a constant speed in a constant direction. Constant dire ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horace Lamb
Sir Horace Lamb (27 November 1849 – 4 December 1934)R. B. Potts,, '' Australian Dictionary of Biography'', Volume 5, MUP, 1974, pp 54–55. Retrieved 5 Sep 2009 was a British applied mathematician and author of several influential texts on classical physics, among them ''Hydrodynamics'' (1895) and ''Dynamical Theory of Sound'' (1910). Both of these books remain in print. The word vorticity was coined by Lamb in 1916. Biography Early life and education Lamb was born in Stockport, Cheshire, the son of John Lamb and his wife Elizabeth, ''née'' Rangeley. John Lamb was a foreman in a cotton mill who had gained some distinction by the invention of an improvement to spinning machines, he died when his son was a child. Lamb's mother married again, and shortly afterwards Horace went to live with his strict but maternal aunt, Mrs. Holland. He studied at Stockport Grammar School, where he made the acquaintance of a wise and kindly headmaster in the Rev. Charles Hamilton, and a ... [...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 equat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Material Derivative
In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material derivative can serve as a link between Eulerian and Lagrangian descriptions of continuum deformation. For example, in fluid dynamics, the velocity field is the flow velocity, and the quantity of interest might be the temperature of the fluid. In which case, the material derivative then describes the temperature change of a certain fluid parcel with time, as it flows along its pathline (trajectory). Other names There are many other names for the material derivative, including: *advective derivative *convective derivative *derivative following the motion *hydrodynamic derivative *Lagrangian derivative *particle derivative *substantial derivative *substantive derivative *Stokes derivative *total derivative, although the material derivat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beltrami Flow
In fluid dynamics, Beltrami flows are flows in which the vorticity vector \mathbf and the velocity vector \mathbf are parallel to each other. In other words, Beltrami flow is a flow where Lamb vector is zero. It is named after the Italian mathematician Eugenio Beltrami due to his derivation of the Beltrami vector field, while initial developments in fluid dynamics were done by the Russian scientist Ippolit S. Gromeka in 1881. Description Since the vorticity vector \boldsymbol and the velocity vector \mathbf are parallel to each other, we can write :\boldsymbol\times\mathbf=0, \quad \boldsymbol = \alpha(\mathbf,t) \mathbf, where \alpha(\mathbf,t) is some scalar function. One immediate consequence of Beltrami flow is that it can never be a planar or axisymmetric flow because in those flows, vorticity is always perpendicular to the velocity field. The other important consequence will be realized by looking at the incompressible vorticity equation :\frac + (\mathbf\cdot\nabla)\b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electric Field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field for a system of charged particles. Electric fields originate from electric charges and time-varying electric currents. Electric fields and magnetic fields are both manifestations of the electromagnetic field, one of the four fundamental interactions (also called forces) of nature. Electric fields are important in many areas of physics, and are exploited in electrical technology. In atomic physics and chemistry, for instance, the electric field is the attractive force holding the atomic nucleus and electrons together in atoms. It is also the force responsible for chemical bonding between atoms that result in molecules. The electric field is defined as a vector field that associates to each point in space the electrostatic (Coulomb) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maxwell's Equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. They describe how electric field, electric and magnetic fields are generated by electric charge, charges, electric current, currents, and changes of the fields.''Electric'' and ''magnetic'' fields, according to the theory of relativity, are the components of a single electromagnetic field. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon. The modern f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fluid Dynamics
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 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
