|
Kelvin's Circulation Theorem
In fluid mechanics, Kelvin's circulation theorem (named after William Thomson, 1st Baron Kelvin who published it in 1869) states:In a barotropic, ideal fluid with conservative body forces, the circulation around a closed curve (which encloses the same fluid elements) moving with the fluid remains constant with time. Stated mathematically: :\frac = 0 where \Gamma is the circulation around a material contour C(t). Stated more simply, this theorem says that if one observes a closed contour at one instant, and follows the contour over time (by following the motion of all of its fluid elements), the circulation over the two locations of this contour are equal. This theorem does not hold in cases with viscous stresses, nonconservative body forces (for example the Coriolis force) or non-barotropic pressure-density relations. Mathematical proof The circulation \Gamma around a closed material contour C(t) is defined by: :\Gamma(t) = \oint_C \boldsymbol \cdot \mathrm\boldsymbol where ... [...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 dev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The Last Universalist", since he excelled in all fields of the discipline as it existed during his lifetime. As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is also considered to be one of the founders of the field of topology. Poincaré made clear the importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the Lorentz transformations in their modern symmetrical form. Poincaré discove ... [...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 and time. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equations Of 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 and time. Bef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thermomagnetic Convection
Ferrofluids can be used to transfer heat, since heat and mass transport in such magnetic fluids can be controlled using an external magnetic field. B. A. Finlayson first explained in 1970 (in his paper "Convective instability of ferromagnetic fluids", ''Journal of Fluid Mechanics'', 40:753-767) how an external magnetic field imposed on a ferrofluid with varying magnetic susceptibility, e.g., due to a temperature gradient, results in a nonuniform magnetic body force, which leads to thermomagnetic convection. This form of heat transfer can be useful for cases where conventional convection fails to provide adequate heat transfer, e.g., in miniature microscale devices or under reduced gravity conditions. Ozoe group has studied thermomagnetic convection both experimentally and numerically. They showed how to enhance, suppress and invert the convection modes. They have also carried out scaling analysis for paramagnetic fluids in microgravity conditions. A comprehensive review of therm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmholtz's Theorems
In fluid mechanics, Helmholtz's theorems, named after Hermann von Helmholtz, describe the three-dimensional motion of fluid in the vicinity of vortex lines. These theorems apply to inviscid flows and flows where the influence of viscous forces are small and can be ignored. Helmholtz's three theorems are as follows: ;Helmholtz's first theorem: :The strength of a vortex line is constant along its length. ;Helmholtz's second theorem: :A vortex line cannot end in a fluid; it must extend to the boundaries of the fluid or form a closed path. ;Helmholtz's third theorem: :A fluid element that is initially irrotational remains irrotational. Helmholtz's theorems apply to inviscid flows. In observations of vortices in real fluids the strength of the vortices always decays gradually due to the dissipative effect of viscous forces. Alternative expressions of the three theorems are as follows: # The strength of a vortex tube does not vary with time. # Fluid elements lying on a vortex line at so ... [...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 equations – ... [...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] |
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-dimensional fl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vilhelm Bjerknes
Vilhelm Friman Koren Bjerknes ( , ; 14 March 1862 – 9 April 1951) was a Norwegian physicist and meteorologist who did much to found the modern practice of weather forecasting. He formulated the primitive equations that are still in use in numerical weather prediction and climate modeling, and he developed the so-called Bergen School of Meteorology, which was successful in advancing weather prediction and meteorology in the early 20th century. Life and career Born in Christiania (later renamed Oslo), Bjerknes enjoyed an early exposure to fluid dynamics, as assistant to his father, Carl Anton Bjerknes, who had discovered by mathematical analysis the apparent actions at a distance between pulsating and oscillating bodies in a fluid, and their analogy with the electric and magnetic actions at a distance. Apparently no attempt had been made to demonstrate experimentally the theories arrived at by the older professor until Vilhelm Bjerknes, then about 17 or 18 years of age, tur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient Theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space (generally ''n''-dimensional) rather than just the real line. For as a differentiable function and as any continuous curve in which starts at a point and ends at a point , then \int_ \nabla\varphi(\mathbf)\cdot \mathrm\mathbf = \varphi\left(\mathbf\right) - \varphi\left(\mathbf\right) where denotes the gradient vector field of . The gradient theorem implies that line integrals through gradient fields are path-independent. In physics this theorem is one of the ways of defining a ''conservative'' force. By placing as potential, is a conservative field. Work done by conservative forces does not depend on the path followed by the obje ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Thomson, 1st Baron Kelvin
William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy at the University of Glasgow for 53 years, he did important work in the mathematical analysis of electricity and formulation of the first and second laws of thermodynamics, and did much to unify the emerging discipline of physics in its contemporary form. He received the Royal Society's Copley Medal in 1883, was its president 1890–1895, and in 1892 was the first British scientist to be elevated to the House of Lords. Absolute temperatures are stated in units of kelvin in his honour. While the existence of a coldest possible temperature ( absolute zero) was known prior to his work, Kelvin is known for determining its correct value as approximately −273.15 degrees Celsius or −459.67 degrees Fahrenheit. The Joule–Thomson effect is also named in his honour. He worked closely with mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
