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Governing Equations
The governing equations of a mathematical model describe how the values of the unknown variables (i.e. the dependent variables) change when one or more of the known (i.e. independent) variables change. Mass balance A mass balance, also called a material balance, is an application of conservation of mass to the analysis of physical systems. It is the simplest governing equation, and it is simply a budget (balance calculation) over the quantity in question: \text + \text = \text + \text \ + \text Differential equation Physics The governing equations in classical physics that are lectured at universities are listed below. * balance of mass * balance of (linear) momentum * balance of angular momentum * balance of energy * balance of entropy * Maxwell-Faraday equation for induced electric field * Ampére-Maxwell equation for induced magnetic field * Gauss equation for electric flux * Gauss equation for magnetic flux Classical continuum mechanics The basic equations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, biology, earth science, chemistry) and engineering disciplines (such as computer science, electrical engineering), as well as in non-physical systems such as the social sciences (such as economics, psychology, sociology, political science). The use of mathematical models to solve problems in business or military operations is a large part of the field of operations research. Mathematical models are also used in music, linguistics, and philosophy (for example, intensively in analytic philosophy). A model may help to explain a system and to study the effects of different components, and to make predictions about behavior. Elements of a mathematical model Mathematical models can take many forms, including dynamical systems, statisti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hele-Shaw Flow
Hele-Shaw flow is defined as Stokes flow between two parallel flat plates separated by an infinitesimally small gap, named after Henry Selby Hele-Shaw, who studied the problem in 1898. Various problems in fluid mechanics can be approximated to Hele-Shaw flows and thus the research of these flows is of importance. Approximation to Hele-Shaw flow is specifically important to micro-flows. This is due to manufacturing techniques, which creates shallow planar configurations, and the typically low Reynolds numbers of micro-flows. The governing equation of Hele-Shaw flows is identical to that of the inviscid potential flow and to the flow of fluid through a porous medium (Darcy's law). It thus permits visualization of this kind of flow in two dimensions. Mathematical formulation of Hele-Shaw flows Let x, y be the directions parallel to the flat plates, and z the perpendicular direction, with H being the gap between the plates (at z=0, H). When the gap between plates is asymptotically sm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Large Eddy Simulation
Large eddy simulation (LES) is a mathematical model for turbulence used in computational fluid dynamics. It was initially proposed in 1963 by Joseph Smagorinsky to simulate atmospheric air currents, and first explored by Deardorff (1970). LES is currently applied in a wide variety of engineering applications, including combustion, acoustics, and simulations of the atmospheric boundary layer. The simulation of turbulent flows by numerically solving the Navier–Stokes equations requires resolving a very wide range of time and length scales, all of which affect the flow field. Such a resolution can be achieved with direct numerical simulation (DNS), but DNS is computationally expensive, and its cost prohibits simulation of practical engineering systems with complex geometry or flow configurations, such as turbulent jets, pumps, vehicles, and landing gear. The principal idea behind LES is to reduce the computational cost by ignoring the smallest length scales, which are the most co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Acoustics
Nonlinear acoustics (NLA) is a branch of physics and acoustics dealing with sound waves of sufficiently large amplitudes. Large amplitudes require using full systems of governing equations of fluid dynamics (for sound waves in liquids and gases) and elasticity (for sound waves in solids). These equations are generally nonlinear, and their traditional linearization is no longer possible. The solutions of these equations show that, due to the effects of nonlinearity, sound waves are being distorted as they travel. Introduction A sound wave propagates through a material as a localized pressure change. Increasing the pressure of a gas or fluid increases its local temperature. The local speed of sound in a compressible material increases with temperature; as a result, the wave travels faster during the high pressure phase of the oscillation than during the lower pressure phase. This affects the wave's frequency structure; for example, in an initially plain sinusoidal wave of a singl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel Function For Solving Integral Equation Of Surface Radiation Exchanges
In physics and engineering, the radiative heat transfer from one surface to another is the equal to the difference of incoming and outgoing radiation from the first surface. In general, the heat transfer between surfaces is governed by temperature, surface emissivity properties and the geometry of the surfaces. The relation for heat transfer can be written as an integral equation with boundary conditions based upon surface conditions. Kernel functions can be useful in approximating and solving this integral equation. Governing equation The radiative heat exchange depends on the local surface temperature of the enclosure and the properties of the surfaces, but does not depend upon the media. Because media neither absorb, emit, nor scatter radiation. Governing equation of heat transfer between two surface ''A''''i'' and ''A''''j'' \begin q(r_i) =& \int_^\infty \int_^ \int_^\frac \varepsilon_ (\lambda,\psi_i,\theta_i,r_i) I_(\cos\theta_i\sin\theta_i)\,d\theta_i\,d\psi_i\,d\l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kelvin's Circulation Theorem
In fluid mechanics, Kelvin's circulation theorem (named after William Thomson, 1st Baron Kelvin William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, Mathematical physics, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy (Glasgow), Professor of Natural Philoso ... who published it in 1869) states:In a barotropic, ideal fluid with Conservative force, conservative body forces, the Circulation (fluid dynamics), 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 (fluid dynamics), 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 equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Precipitation Hardening
Precipitation hardening, also called age hardening or particle hardening, is a heat treatment technique used to increase the yield strength of malleable materials, including most structural alloys of aluminium, magnesium, nickel, titanium, and some steels and stainless steels. In superalloys, it is known to cause yield strength anomaly providing excellent high-temperature strength. Precipitation hardening relies on changes in solid solubility with temperature to produce fine particles of an impurity phase, which impede the movement of dislocations, or defects in a crystal's lattice. Since dislocations are often the dominant carriers of plasticity, this serves to harden the material. The impurities play the same role as the particle substances in particle-reinforced composite materials. Just as the formation of ice in air can produce clouds, snow, or hail, depending upon the thermal history of a given portion of the atmosphere, precipitation in solids can produce many diff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acoustic Theory
Acoustic theory is a scientific field that relates to the description of sound waves. It derives from fluid dynamics. See acoustics for the engineering approach. For sound waves of any magnitude of a disturbance in velocity, pressure, and density we have : \begin \frac +\rho_0\nabla\cdot \mathbf + \nabla\cdot(\rho'\mathbf) & = 0 \qquad \text \\ (\rho_0+\rho')\frac + (\rho_0+\rho')(\mathbf\cdot\nabla)\mathbf + \nabla p' & = 0 \qquad \text \end In the case that the fluctuations in velocity, density, and pressure are small, we can approximate these as : \begin \frac +\rho_0\nabla\cdot \mathbf & = 0 \\ \frac + \frac\nabla p'& = 0 \end Where \mathbf(\mathbf,t) is the perturbed velocity of the fluid, p_0 is the pressure of the fluid at rest, p'(\mathbf,t) is the perturbed pressure of the system as a function of space and time, \rho_0 is the density of the fluid at rest, and \rho'(\mathbf, t) is the variance in the density of the fluid over space and tim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Volume Method For Unsteady Flow
Unsteady flows are characterized as flows in which the properties of the fluid are time dependent. It gets reflected in the governing equations as the time derivative of the properties are absent. For Studying Finite-volume method for unsteady flow there is some governing equations > Governing Equation The conservation equation for the transport of a scalar in unsteady flow has the general form as \frac + \operatorname\left(\rho \phi \upsilon\right) = \operatorname\left(\Gamma \operatorname \phi\right) + S_\phi \rho is density and \phi is conservative form of all fluid flow, \Gamma is the Diffusion coefficient and S is the Source term. \operatorname\left(\rho \phi \upsilon\right) is Net rate of flow of \phi out of fluid element(convection), \operatorname\left(\Gamma \operatorname \phi\right) is Rate of increase of \phi due to diffusion, S_\phi is Rate of increase of \phi due to sources. \frac is Rate of increase of \phi of fluid element(transient), The first t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Astronautics
Astronautics (or cosmonautics) is the theory and practice of travel beyond Earth's atmosphere into outer space. Spaceflight is one of its main applications and space science its overarching field. The term ''astronautics'' (originally ''astronautique'' in French) was coined in the 1920s by J.-H. Rosny, president of the Goncourt academy, in analogy with aeronautics. Because there is a degree of technical overlap between the two fields, the term aerospace is often used to describe both at once. In 1930, Robert Esnault-Pelterie published the first book on the new research field. The term ''cosmonautics'' (originally ''cosmonautique'' in French) was introduced in 1930s by Ary Sternfeld with his book ''Initiation à la Cosmonautique'' (Introduction to cosmonautics) (the book brought him the Prix REP-Hirsch, later known as the Prix d'Astronautique, of the French Astronomical Society in 1934.) As with aeronautics, the restrictions of mass, temperatures, and external forces require ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annular Fin
In thermal engineering, an annular fin is a specific type of fin used in heat transfer that varies, radially, in cross-sectional area. Adding an annular fin to an object increases the amount of surface area in contact with the surrounding fluid, which increases the convective heat transfer between the object and surrounding fluid. Because surface area increases as length from the object increases, an annular fin transfers more heat than a similar pin fin at any given length. Annular fins are often used to increase the heat exchange in liquid–gas heat exchanger systems. Governing Equation To derive the governing equation of an annular fin, certain assumptions must be made. The fin must have constant thermal conductivity and other material properties, there must be no internal heat generation, there must be only one-dimensional conduction, and the fin must be at steady state. Applying the energy conservation principle to a differential element between radii ''r'' and ''r''&nbs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vortex Shedding
In fluid dynamics, vortex shedding is an oscillating flow that takes place when a fluid such as air or water flows past a bluff (as opposed to streamlined) body at certain velocities, depending on the size and shape of the body. In this flow, vortices are created at the back of the body and detach periodically from either side of the body forming a Kármán vortex street. The fluid flow past the object creates alternating low-pressure vortices on the downstream side of the object. The object will tend to move toward the low-pressure zone. If the bluff structure is not mounted rigidly and the frequency of vortex shedding matches the resonance frequency of the structure, then the structure can begin to resonate, vibrating with harmonic oscillations driven by the energy of the flow. This vibration is the cause for overhead power line wires humming in the wind, and for the fluttering of automobile whip radio antennas at some speeds. Tall chimneys constructed of thin-walled st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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