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Topic: Euler equations

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Relativistic Euler equations

Fluid Dynamics

Incompressible flow

Van der Waals equation

Fluid Mechanics

Fluid power

Compressible flow

Reynolds decomposition

Computational Fluid Dynamics

Potential flow

Leonhard Euler

Hydrostatics

State space (controls)

Inviscid

Newtonian fluid

	Leonhard Euler - Wikipedia, the free encyclopedia
		Leonhard Euler was born in Basel, Switzerland, the son of Paul Euler, a Lutheran minister.
		In 1727 Euler was called to St. Petersburg by Catherine I of Russia and became professor of physics in 1730, with an additional mathematics appointment in 1733.
		Euler wrote Tentamen novae theoriae musicae in 1739 which was an attempt to combine mathematics and music; a biography comments that the work was "for musicians too advanced in its mathematics and for mathematicians too musical".
	en.wikipedia.org /wiki/Leonhard_Euler (1459 words)

	Euler equations - Wikipedia, the free encyclopedia
		In fluid dynamics, the Euler equations govern the motion of a compressible, inviscid fluid.
		Although the Euler equations formally reduce to potential flow in the limit of vanishing Mach number, this is not helpful in practice, essentially because the approximation of incompressibility is almost invariably very close.
		The better known Bernoulli's equation can be derived by integrating Euler's equation along a streamline under the assumption of constant density and a sufficiently stiff equation of state.
	en.wikipedia.org /wiki/Euler_equations (380 words)

	Euler Equations
		The equations are named in honor of Leonard Euler, who was a student with Daniel Bernoulli, and studied various fluid dynamics problems in the mid-1700's.
		The Euler equations neglect the effects of the viscosity of the fluid which are included in the Navier-Stokes equations.
		An equation of state relates the pressure and the density of a gas.
	www.lerc.nasa.gov /WWW/K-12/airplane/eulereqs.html (755 words)

	Euler
		Euler wrote an article on acoustics, which went on to become a classic, in his bid for selection to the post but he was not chosen to go forward to the stage where lots were drawn to make the final decision on who would fill the chair.
		Euler was also helped by two other members of the Academy, W L Krafft and A J Lexell, and the young mathematician N Fuss who was invited to the Academy from Switzerland in 1772.
		Euler was the first to appreciate the importance of introducing uniform analytic methods into mechanics, thus enabling its problems to be solved in a clear and direct way.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Euler.html (4046 words)

	Relativistic Euler equations - Wikipedia, the free encyclopedia
		In fluid mechanics and astrophysics, the relativistic Euler equations are a generalization of the Euler equations that account for the effects of special relativity.
		The equations of motion are contained in the continuity equation of the stress-energy tensor T
		The relativistic Euler equations may be applied to calculate the speed of sound in a fluid with a relativistic equation of state (that is, one in which the pressure is comparable with the internal energy density e, including the rest energy; e = ρc
	en.wikipedia.org /wiki/Relativistic_Euler_equations (209 words)

	Learn more about Fluid mechanics in the online encyclopedia. (Site not responding. Last check: 2007-10-07)
		The central equations for fluid mechanics are the Navier-Stokes equations, which are non-linear differential equations that describe fluid flow.
		The standard equations of inviscid flow are the Euler equations.
		Another often used model, especially in computational fluid mechanics, is to use the Euler equations far from the body and the boundary layer equations close to the body.
	www.onlineencyclopedia.org /f/fl/fluid_mechanics_1.html (961 words)

	Encyclopedia: Euler equations
		Athough the Euler equations formally reduce to potential flow in the limit of vanishing Mach number, this is not helpful in practice, essentially because the approximation of incompressibility is almost invariably very close.
		The second equation includes the divergence of a dyadic tensor, and may be clearer in subscript notation: In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point.
		In physics and thermodynamics, an equation of state is a constitutive equation describing the state of matter under a given set of physical conditions.
	www.nationmaster.com /encyclopedia/Euler-equations (735 words)

	LEONHARD EULER (Site not responding. Last check: 2007-10-07)
		Leonhard Euler (1707-1783) was arguably the greatest mathematician of the eighteenth century (His closest competitor for that title is Lagrange) and one of the most prolific of all time; his publication list of 886 papers and books may be exceeded only by Paul Erdös.
		But there is another "Euler's formula" that (to use the modern terminology adopted long after Euler's death) gives the values of the Riemann zeta function at positive even integers in terms of Bernoulli numbers.
		Euler's study of the bridges of Königsberg can be seen as the beginning of combinatorial topology (which is why the Euler characteristic bears his name).
	www.usna.edu /Users/math/meh/euler.html (402 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		In fluid mechanics, the Euler equations govern the motion of a compressible, inviscid fluid.
		Closing the system requires an equation of state; the most commonly used one would be the ideal gas law (ie).
		Indeed, it is not clear to this author that a shock is a solution: the Euler equations as stated above do not admit discontinuities; resolving the shock's structure requires the bulk viscosity, but Knudsen number effects clearly dominate at those lengthscales.
	www.informationgenius.com /encyclopedia/e/eu/euler_equations.html (381 words)

	Equations of motion for a rigid body
		Euler’s laws: The laws of motion for a rigid body are known as Euler’s laws.
		The second of Euler’s two laws describes how the change of angular momentum of the rigid body is controlled by the moment of forces and couples applied on the body.
		The laws of Euler are written for a body of fixed matter (i.e., matter can not be added to the body, matter can not be subtracted from the body, nor can matter be replaced by other matter).
	em-ntserver.unl.edu /NEGAHBAN/EM373/note19/note19.htm (1071 words)

	Multigrid Solution of the Euler Equations with Local Preconditioning - Lynn (ResearchIndex) (Site not responding. Last check: 2007-10-07)
		Introduction Recent advances in multigrid relaxation for the Euler equations have come from the understanding that the acoustic equations embedded in the system have to be isolated from the advection equations and treated differently.
		Ta'asan [8] suggests using a set of "canonical" equations based on the steady form of the Euler equations; Muller [7] uses a...
		8 Preconditioning for the Navier-Stokes equations with finite-..
	citeseer.ist.psu.edu /351702.html (494 words)

	Mathematical Methods (10/24.539) Linear Differential Equations Homogeneous 2nd Order Systems with Constant Coefficients
		However, there is one special case of a variable coefficient system, called the Euler-Cauchy equation, that occurs quite frequently in applications and has a solution scheme very similar to that of constant coefficient systems.
		These three cases, for the Euler Cauchy equation, are identified in detail in the remainder of this subsection.
		This is also true for particular solutions when the forcing function for the Euler-Cauchy equation is of the same form as the homogeneous solution (to be discussed shortly).
	gershwin.ens.fr /vdaniel/Doc-Locale/Cours-Mirrored/Methodes-Maths/white/math/s2/s2secor/s2secor.html (676 words)

	Euler Equations (Site not responding. Last check: 2007-10-07)
		The momentum equation is sometimes called Euler's equation.
		The equations are often solved by finite differences whereby the values of each velocity component, the density, and the internal energy are computed at each point in the flow.
		Since Euler equations permit rotational flow and enthalpy losses (through shock waves), they are very useful in solving transonic flow problems, propeller or rotor aerodynamics, and flows with vortical structures in the field.
	adg.stanford.edu /aa208/modeling/euler.html (188 words)

	Euler Equations (Site not responding. Last check: 2007-10-07)
		The resulting equations are not particularly useful in many practical problems and consequently, they are often transformed into the control volume or Eulerian viewpoint.
		In this approach, a control volume is identified, which may be in motion or stationary, and the flow of the fluid through this control volume is mathematically represented.
		This formula and Reynolds' transport theorem are of paramount importance in the development of the governing equations and as a result, the reader is urged to study the material on these topics before proceeding to the actual development of the three conservation laws.
	clue.eng.iastate.edu /~hindman/classes/446/webcourse/index/equationsets/euler.htm (271 words)

	ipedia.com: Fluid dynamics Article (Site not responding. Last check: 2007-10-07)
		The central equations for fluid dynamics are the Navier-Stokes equations, which are non-linear differential equations that describe the flow of a fluid whose stress depends linearly on velocity and on pressure.
		The unsimplified equations do not have a general closed-form solution, so they are only of use in computational fluid dynamics.
		Another often used model, especially in computational fluid dynamics, is to use the Euler equations far from the body and the boundary layer equations close to the body.
	www.ipedia.com /fluid_dynamics.html (950 words)

	Math Seminars (Site not responding. Last check: 2007-10-07)
		The Euler equations describe the motion of an ideal incompressible fluid.
		The Euler equations may be recast as some integral identities, expressing the local mass and momentum balance.
		Every solution of the Euler equations satisfies these identities (which may be called the weak Euler equations); but there may a priori exist very nonregular (merely square integrable) velocity fields, which satisfy the weak Euler equations.
	www.math.psu.edu /dynsys/abstracts/shnirelman1.html (405 words)

	FUB-HEP/94-5 Euler Equations for Rigid-Body -- a Case for Autoparallel Trajectories in Spaces with Torsion
		In the literature on gravity with curvature and torsion [1], there is a widespread belief that in spaces with torsion (for the geometry of such spaces see [2]), spinless particles move on shortest paths [3].
		Thus it contributes the equation for the autoparallel (1).
		We shall apply the variational principle of Ref. [5] and derive, within the body-fixed reference system both the Euler equations for the angular momentum and the equations for the translational motion and show, that the rigid body moves along autoparallel trajectories in the body-fixed reference system.
	www.physik.fu-berlin.de /~kleinert/kleiner_re224/euler.html (2135 words)

	Euler Equations, Navier Stokes Equations, and Singularities (Site not responding. Last check: 2007-10-07)
		We are organizing a conference at the Ohio State University with the title "The Euler Equations, Navier Stokes Equations, and Singularities" for the dates April 18-20, 1997.
		In this situation, an important regularizing effect such as, but not limited to, viscosity is usually omitted in the formulation of the problem in order to simplify the model.
		Therefore, a modeler is challenged to obtain the missing physical mechanism and to incorporate this missing mechanism into the system of equations which describe the problem.
	www.csc.fi /math_topics/Mail/NANET97-1/msg00236.html (285 words)

	New Factorizable Discretizations for the Euler Equations
		A multigrid method is defined as having textbook multigrid efficiency (TME) if solutions to the governing system of equations are attained in a computational work that is a small (less than 10) multiple of the operation count in one target-grid residual evaluation.
		A way to achieve TME for the Euler and Navier--Stokes equations is to apply the distributed relaxation method, thereby separating the elliptic and hyperbolic partitions of the equations.
		In particular, discrete schemes for the nonconservative Euler equations possessing properties (1) and (2) have been derived and analyzed.
	epubs.siam.org /sam-bin/dbq/article/40478 (282 words)

	En-Ez
		This is a linear equation, and although the addition of nonlinearities has brought model results and observations into closer concordance, it is thought that they are not essential for maintaining the undercurrent and serve only to modify the linear dynamics.
		The motion is unidirectional and parallel to the equator everywhere, and in each vertical plane parallel to the equator the motion is the same as for a nonrotating fluid.
		At the vernal equinox the Sun crosses the celestial equator from south to the north, and at the autumnal equinox from north to south, with the former being the zero point in celestial coordinate systems.
	www.ep.sci.hokudai.ac.jp /~minobe/baum_glosarry_ocean/node14.html (4633 words)

	Navier-Stokes equations - Science Articles :: Physics Post
		In fluid dynamics, the Navier-Stokes equations are a set of nonlinear partial differential equations that describe the flow of fluids such as liquids and gases.
		However, the Navier-Stokes equations are useful for a wide range of practical problems, providing their limitations are borne in mind.
		The Navier-Stokes equations with zero viscosity are known as the Euler equations; there, the emphasis is on compressible flow and shock waves.
	www.physicspost.com /science-article-195.html (785 words)

	The Maxwell-Vlasov Equations in Euler-Poincaré Form (Site not responding. Last check: 2007-10-07)
		Low's well known action principle for the Maxwell-Vlasov equations of ideal plasma dynamics was originally expressed in terms of a mixture of Eulerian and Lagrangian variables.
		By imposing suitable constraints on the variations and analyzing invariance properties of the Lagrangian, as one does for the Euler equations for the rigid body and ideal fluids, we first transform this action principle into purely Eulerian variables.
		Because of degeneracies in the Lagrangian, the Legendre transform is dealt with using the Dirac theory of constraints.
	www.cds.caltech.edu /~marsden/bib/1998/01-CeHoHoMa1998 (231 words)

	Properties of the Euler equations
		The Euler equations are a simpler version of the Navier-Stokes equations.
		From a mathematical point of view the Euler equations are a set of non-linear, coupled, hyperbolic differential equations.
		For those who are curious, some more background to both the physical and mathematical concepts behind the Euler equations are be found in the lecture notes compiled by Vincent Icke (ask me for copies).
	www.astro.su.se /~garrelt/education/num_exercise/node3.html (313 words)

	Re: Dimensionless formulation of Euler equations
		Keep in mind the dimensionless form of the Euler equations used in the code.
		One usually chooses reference quantities so that the freestream Mach number does not explicitly enter the dimensionless form of the Euler equations in the code.
		Re: Dimensionless formulation of Euler equations - jvn, Thu, 8 Jul 2004, 11:29 a.m.
	www.cfd-online.com /Forum/main_archive.cgi?read=32198 (216 words)

	Hydraulics Collection (Site not responding. Last check: 2007-10-07)
		For example, in 1822 Louis Marie Henri Navier (1785-1836), a bridge engineer, was the first to attempt the extension of the Euler equations of acceleration to include the flow of a viscous fluid.
		The same equations were developed with groater comprehension somewhat later by the mathematician Baron Augustin Louis de Cauchy (1789-1857) [30], next by the mechanician Simeon Denis Poisson (1781-1840) [31], and finally in 1845 by the Cambridge professor George Gabriel Stokes (1819 1903) [32], the latter eventually applying the equations to the resistance of small spheres.
		Ably begun by Euler and d'Alembert, the practice was continued by such equally famous men as Lagrange (1736-1813) [44], Laplace (1749-1827) [45], Helmholtz (1821-94) [46], Kelvin (1824-1907) [47], and Rayleigh (1842-1919) [48], as recorded in the many editions of the treatise Hydrodymimics [49] by the Manchester professor Horace Lamb (1849-1934).
	www.lib.uiowa.edu /spec-coll/bai/hydraul.htm (4472 words)

	Geometric Analysis of the Averaged Euler Equations (ResearchIndex) (Site not responding. Last check: 2007-10-07)
		Abstract: This report concerns the geometric analysis of the averaged Euler equations of ideal incompressible hydrodynamics.
		The equations are a system of conservative PDEs that model the motion of uids at length scales greater than some given scale in the problem (for instance, the smallest scale possible within the numerical discretization).
		2 The geometry and analysis of the averaged Euler equations an..
	citeseer.csail.mit.edu /135672.html (392 words)

	On the Two-Dimensional Gas Expansion for Compressible Euler Equations
		The flow is quasi-stationary, and using hodograph transform, we describe it by a partial differential equation of second order in the state space if it is irrotational initially.
		Furthermore, this equation is reduced to a linearly degenerate system of three partial differential equations with inhomogeneous source terms.
		These properties are used to prove that the flow is globally smooth when a wedge of gas expands into a vacuum, and to analyze that shocks may appear in the interaction of four planar rarefaction waves.
	epubs.siam.org /sam-bin/dbq/article/36134 (205 words)

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