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Fluid dynamics

In physics, physical chemistry 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.

Before the twentieth century, hydrodynamics was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability, both of which can also be applied to gases.

Equations
The foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of mass, conservation of linear momentum, and conservation of energy (also known as the First Law of Thermodynamics). These are based on classical mechanics and are modified in quantum mechanics and general relativity. They are expressed using the Reynolds transport theorem.

In addition to the above, fluids are assumed to obey the continuum assumption. At small scale, all fluids are composed of molecules that collide with one another and solid objects. However, the continuum assumption assumes that fluids are continuous, rather than discrete. Consequently, it is assumed that properties such as density, pressure, temperature, and flow velocity are well-defined at infinitesimally small points in space and vary continuously from one point to another. The fact that the fluid is made up of discrete molecules is ignored.

For fluids that are sufficiently dense to be a continuum, do not contain ionized species, and have flow velocities that are small in relation to the speed of light, the momentum equations for Newtonian fluids are the Navier–Stokes equations—which is a non-linear set of differential equations that describes the flow of a fluid whose stress depends linearly on flow velocity gradients and pressure. The unsimplified equations do not have a general closed-form solution, so they are primarily of use in computational fluid dynamics. The equations can be simplified in several ways, all of which make them easier to solve. Some of the simplifications allow some simple fluid dynamics problems to be solved in closed form.

In addition to the mass, momentum, and energy conservation equations, a thermodynamic equation of state that gives the pressure as a function of other thermodynamic variables is required to completely describe the problem. An example of this would be the perfect gas equation of state:

 * $$p= \frac{\rho R_u T}{M}$$

where $p$ is pressure, $ρ$ is density, and $T$ is the absolute temperature, while $R_{u}$ is the gas constant and $M$ is molar mass for a particular gas. A constitutive relation may also be useful.

Conservation laws
Three conservation laws are used to solve fluid dynamics problems, and may be written in integral or differential form. The conservation laws may be applied to a region of the flow called a control volume. A control volume is a discrete volume in space through which fluid is assumed to flow. The integral formulations of the conservation laws are used to describe the change of mass, momentum, or energy within the control volume. Differential formulations of the conservation laws apply Stokes' theorem to yield an expression that may be interpreted as the integral form of the law applied to an infinitesimally small volume (at a point) within the flow.

Mass continuity (conservation of mass):The rate of change of fluid mass inside a control volume must be equal to the net rate of fluid flow into the volume. Physically, this statement requires that mass is neither created nor destroyed in the control volume, and can be translated into the integral form of the continuity equation:
 * $\frac{\partial}{\partial t} \iiint_V \rho \, dV = - \, {} $ $$

Above, $ρ$ is the fluid density, $u$ is the flow velocity vector, and $t$ is time. The left-hand side of the above expression is the rate of increase of mass within the volume and contains a triple integral over the control volume, whereas the right-hand side contains an integration over the surface of the control volume of mass convected into the system. Mass flow into the system is accounted as positive, and since the normal vector to the surface is opposite to the sense of flow into the system the term is negated. The differential form of the continuity equation is, by the divergence theorem:
 * $\ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 $

Conservation of momentum: Newton's second law of motion applied to a control volume, is a statement that any change in momentum of the fluid within that control volume will be due to the net flow of momentum into the volume and the action of external forces acting on the fluid within the volume.
 * $ \frac{\partial}{\partial t} \iiint_{\scriptstyle V} \rho\mathbf{u} \, dV = -\, {} $ $$ $ (\rho\mathbf{u}\cdot d\mathbf{S}) \mathbf{u} -{}$ $$ $\displaystyle{}+ \iiint_{\scriptstyle V} \rho \mathbf{f}_\text{body} \, dV + \mathbf{F}_\text{surf}$

In the above integral formulation of this equation, the term on the left is the net change of momentum within the volume. The first term on the right is the net rate at which momentum is convected into the volume. The second term on the right is the force due to pressure on the volume's surfaces. The first two terms on the right are negated since momentum entering the system is accounted as positive, and the normal is opposite the direction of the velocity $u$ and pressure forces. The third term on the right is the net acceleration of the mass within the volume due to any body forces (here represented by $f_{body}$). Surface forces, such as viscous forces, are represented by $F_{surf}$, the net force due to shear forces acting on the volume surface. The momentum balance can also be written for a moving control volume.

The following is the differential form of the momentum conservation equation. Here, the volume is reduced to an infinitesimally small point, and both surface and body forces are accounted for in one total force, $F$. For example, $F$ may be expanded into an expression for the frictional and gravitational forces acting at a point in a flow.
 * $\ \frac{D \mathbf{u}}{D t} = \mathbf{F} - \frac{\nabla p}{\rho} $

In aerodynamics, air is assumed to be a Newtonian fluid, which posits a linear relationship between the shear stress (due to internal friction forces) and the rate of strain of the fluid. The equation above is a vector equation in a three-dimensional flow, but it can be expressed as three scalar equations in three coordinate directions. The conservation of momentum equations for the compressible, viscous flow case is called the Navier–Stokes equations. |undefined

Conservation of energy: Although energy can be converted from one form to another, the total energy in a closed system remains constant.
 * $\ \rho \frac{Dh}{Dt} = \frac{D p}{D t} + \nabla \cdot \left( k \nabla T\right) + \Phi $

Above, $h$ is the specific enthalpy, $k$ is the thermal conductivity of the fluid, $T$ is temperature, and $Φ$ is the viscous dissipation function. The viscous dissipation function governs the rate at which the mechanical energy of the flow is converted to heat. The second law of thermodynamics requires that the dissipation term is always positive: viscosity cannot create energy within the control volume. The expression on the left side is a material derivative.

Compressible versus incompressible flow
All fluids are compressible to an extent; that is, changes in pressure or temperature cause changes in density. However, in many situations the changes in pressure and temperature are sufficiently small that the changes in density are negligible. In this case the flow can be modelled as an incompressible flow. Otherwise the more general compressible flow equations must be used.

Mathematically, incompressibility is expressed by saying that the density $ρ$ of a fluid parcel does not change as it moves in the flow field, that is,
 * $$\frac{\mathrm{D} \rho}{\mathrm{D}t} = 0 \, ,$$

where $D⁄Dt$ is the material derivative, which is the sum of local and convective derivatives. This additional constraint simplifies the governing equations, especially in the case when the fluid has a uniform density.

For flow of gases, to determine whether to use compressible or incompressible fluid dynamics, the Mach number of the flow is evaluated. As a rough guide, compressible effects can be ignored at Mach numbers below approximately 0.3. For liquids, whether the incompressible assumption is valid depends on the fluid properties (specifically the critical pressure and temperature of the fluid) and the flow conditions (how close to the critical pressure the actual flow pressure becomes). Acoustic problems always require allowing compressibility, since sound waves are compression waves involving changes in pressure and density of the medium through which they propagate.

Newtonian versus non-Newtonian fluids

All fluids, except superfluids, are viscous, meaning that they exert some resistance to deformation: neighbouring parcels of fluid moving at different velocities exert viscous forces on each other. The velocity gradient is referred to as a strain rate; it has dimensions $T$. Isaac Newton showed that for many familiar fluids such as water and air, the stress due to these viscous forces is linearly related to the strain rate. Such fluids are called Newtonian fluids. The coefficient of proportionality is called the fluid's viscosity; for Newtonian fluids, it is a fluid property that is independent of the strain rate.

Non-Newtonian fluids have a more complicated, non-linear stress-strain behaviour. The sub-discipline of rheology describes the stress-strain behaviours of such fluids, which include emulsions and slurries, some viscoelastic materials such as blood and some polymers, and sticky liquids such as latex, honey and lubricants.

Inviscid versus viscous versus Stokes flow
The dynamic of fluid parcels is described with the help of Newton's second law. An accelerating parcel of fluid is subject to inertial effects.

The Reynolds number is a dimensionless quantity which characterises the magnitude of inertial effects compared to the magnitude of viscous effects. A low Reynolds number ($Re ≪ 1$) indicates that viscous forces are very strong compared to inertial forces. In such cases, inertial forces are sometimes neglected; this flow regime is called Stokes or creeping flow.

In contrast, high Reynolds numbers ($Re ≫ 1$) indicate that the inertial effects have more effect on the velocity field than the viscous (friction) effects. In high Reynolds number flows, the flow is often modeled as an inviscid flow, an approximation in which viscosity is completely neglected. Eliminating viscosity allows the Navier–Stokes equations to be simplified into the Euler equations. The integration of the Euler equations along a streamline in an inviscid flow yields Bernoulli's equation. When, in addition to being inviscid, the flow is irrotational everywhere, Bernoulli's equation can completely describe the flow everywhere. Such flows are called potential flows, because the velocity field may be expressed as the gradient of a potential energy expression.

This idea can work fairly well when the Reynolds number is high. However, problems such as those involving solid boundaries may require that the viscosity be included. Viscosity cannot be neglected near solid boundaries because the no-slip condition generates a thin region of large strain rate, the boundary layer, in which viscosity effects dominate and which thus generates vorticity. Therefore, to calculate net forces on bodies (such as wings), viscous flow equations must be used: inviscid flow theory fails to predict drag forces, a limitation known as the d'Alembert's paradox.

A commonly used model, especially in computational fluid dynamics, is to use two flow models: the Euler equations away from the body, and boundary layer equations in a region close to the body. The two solutions can then be matched with each other, using the method of matched asymptotic expansions.

Steady versus unsteady flow
A flow that is not a function of time is called steady flow. Steady-state flow refers to the condition where the fluid properties at a point in the system do not change over time. Time dependent flow is known as unsteady (also called transient ). Whether a particular flow is steady or unsteady, can depend on the chosen frame of reference. For instance, laminar flow over a sphere is steady in the frame of reference that is stationary with respect to the sphere. In a frame of reference that is stationary with respect to a background flow, the flow is unsteady.

Turbulent flows are unsteady by definition. A turbulent flow can, however, be statistically stationary. The random velocity field $U(x, t)$ is statistically stationary if all statistics are invariant under a shift in time. This roughly means that all statistical properties are constant in time. Often, the mean field is the object of interest, and this is constant too in a statistically stationary flow.

Steady flows are often more tractable than otherwise similar unsteady flows. The governing equations of a steady problem have one dimension fewer (time) than the governing equations of the same problem without taking advantage of the steadiness of the flow field.

Laminar versus turbulent flow
Turbulence is flow characterized by recirculation, eddies, and apparent randomness. Flow in which turbulence is not exhibited is called laminar. The presence of eddies or recirculation alone does not necessarily indicate turbulent flow—these phenomena may be present in laminar flow as well. Mathematically, turbulent flow is often represented via a Reynolds decomposition, in which the flow is broken down into the sum of an average component and a perturbation component.

It is believed that turbulent flows can be described well through the use of the Navier–Stokes equations. Direct numerical simulation (DNS), based on the Navier–Stokes equations, makes it possible to simulate turbulent flows at moderate Reynolds numbers. Restrictions depend on the power of the computer used and the efficiency of the solution algorithm. The results of DNS have been found to agree well with experimental data for some flows.

Most flows of interest have Reynolds numbers much too high for DNS to be a viable option, given the state of computational power for the next few decades. Any flight vehicle large enough to carry a human ($L$ > 3 m), moving faster than 20 m/s is well beyond the limit of DNS simulation ($Re$ = 4 million). Transport aircraft wings (such as on an Airbus A300 or Boeing 747) have Reynolds numbers of 40 million (based on the wing chord dimension). Solving these real-life flow problems requires turbulence models for the foreseeable future. Reynolds-averaged Navier–Stokes equations (RANS) combined with turbulence modelling provides a model of the effects of the turbulent flow. Such a modelling mainly provides the additional momentum transfer by the Reynolds stresses, although the turbulence also enhances the heat and mass transfer. Another promising methodology is large eddy simulation (LES), especially in the form of detached eddy simulation (DES) — a combination of LES and RANS turbulence modelling.

Other approximations
There are a large number of other possible approximations to fluid dynamic problems. Some of the more commonly used are listed below.
 * The Boussinesq approximation neglects variations in density except to calculate buoyancy forces. It is often used in free convection problems where density changes are small.
 * Lubrication theory and Hele–Shaw flow exploits the large aspect ratio of the domain to show that certain terms in the equations are small and so can be neglected.
 * Slender-body theory is a methodology used in Stokes flow problems to estimate the force on, or flow field around, a long slender object in a viscous fluid.
 * The shallow-water equations can be used to describe a layer of relatively inviscid fluid with a free surface, in which surface gradients are small.
 * Darcy's law is used for flow in porous media, and works with variables averaged over several pore-widths.
 * In rotating systems, the quasi-geostrophic equations assume an almost perfect balance between pressure gradients and the Coriolis force. It is useful in the study of atmospheric dynamics.

Flows according to Mach regimes
While many flows (such as flow of water through a pipe) occur at low Mach numbers (subsonic flows), many flows of practical interest in aerodynamics or in turbomachines occur at high fractions of $M = 1$ (transonic flows) or in excess of it (supersonic or even hypersonic flows). New phenomena occur at these regimes such as instabilities in transonic flow, shock waves for supersonic flow, or non-equilibrium chemical behaviour due to ionization in hypersonic flows. In practice, each of those flow regimes is treated separately.

Reactive versus non-reactive flows
Reactive flows are flows that are chemically reactive, which finds its applications in many areas, including combustion (IC engine), propulsion devices (rockets, jet engines, and so on), detonations, fire and safety hazards, and astrophysics. In addition to conservation of mass, momentum and energy, conservation of individual species (for example, mass fraction of methane in methane combustion) need to be derived, where the production/depletion rate of any species are obtained by simultaneously solving the equations of chemical kinetics.

Magnetohydrodynamics
Magnetohydrodynamics is the multidisciplinary study of the flow of electrically conducting fluids in electromagnetic fields. Examples of such fluids include plasmas, liquid metals, and salt water. The fluid flow equations are solved simultaneously with Maxwell's equations of electromagnetism.

Relativistic fluid dynamics
Relativistic fluid dynamics studies the macroscopic and microscopic fluid motion at large velocities comparable to the velocity of light. This branch of fluid dynamics accounts for the relativistic effects both from the special theory of relativity and the general theory of relativity. The governing equations are derived in Riemannian geometry for Minkowski spacetime.

Fluctuating hydrodynamics
This branch of fluid dynamics augments the standard hydrodynamic equations with stochastic fluxes that model thermal fluctuations. As formulated by Landau and Lifshitz, a white noise contribution obtained from the fluctuation-dissipation theorem of statistical mechanics is added to the viscous stress tensor and heat flux.

Terminology
The concept of pressure is central to the study of both fluid statics and fluid dynamics. A pressure can be identified for every point in a body of fluid, regardless of whether the fluid is in motion or not. Pressure can be measured using an aneroid, Bourdon tube, mercury column, or various other methods.

Some of the terminology that is necessary in the study of fluid dynamics is not found in other similar areas of study. In particular, some of the terminology used in fluid dynamics is not used in fluid statics.

Terminology in incompressible fluid dynamics
The concepts of total pressure and dynamic pressure arise from Bernoulli's equation and are significant in the study of all fluid flows. (These two pressures are not pressures in the usual sense—they cannot be measured using an aneroid, Bourdon tube or mercury column.) To avoid potential ambiguity when referring to pressure in fluid dynamics, many authors use the term static pressure to distinguish it from total pressure and dynamic pressure. Static pressure is identical to pressure and can be identified for every point in a fluid flow field.

A point in a fluid flow where the flow has come to rest (that is to say, speed is equal to zero adjacent to some solid body immersed in the fluid flow) is of special significance. It is of such importance that it is given a special name—a stagnation point. The static pressure at the stagnation point is of special significance and is given its own name—stagnation pressure. In incompressible flows, the stagnation pressure at a stagnation point is equal to the total pressure throughout the flow field.

Terminology in compressible fluid dynamics
In a compressible fluid, it is convenient to define the total conditions (also called stagnation conditions) for all thermodynamic state properties (such as total temperature, total enthalpy, total speed of sound). These total flow conditions are a function of the fluid velocity and have different values in frames of reference with different motion.

To avoid potential ambiguity when referring to the properties of the fluid associated with the state of the fluid rather than its motion, the prefix "static" is commonly used (such as static temperature and static enthalpy). Where there is no prefix, the fluid property is the static condition (so "density" and "static density" mean the same thing). The static conditions are independent of the frame of reference.

Because the total flow conditions are defined by isentropically bringing the fluid to rest, there is no need to distinguish between total entropy and static entropy as they are always equal by definition. As such, entropy is most commonly referred to as simply "entropy".

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Fluid dynamics


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Talk:Functional group

Please try to understand the questions of lower grade students
We want Wiki-the encyclo. to solve the basic problems and try to understand the questions not only in words of chemistry but in the combination of other languages too. for instance english. And the difficulties of students of lower grade should also be solved. That would be really nice of this site. Lots of thanks —Preceding unsigned comment added by 59.95.197.203 (talk) 19:13, 8 December 2007 (UTC) please give me the answer in simple words — Preceding unsigned comment added by 182.188.178.254 (talk) 07:20, 24 February 2014 (UTC)

Is moiety a synonym for functional group?
I would rather say moiety is what is marked as R in all formulas. See IUPAC definition at http://goldbook.iupac.org/M03968.html English is not my native tongue and at the moment I am struggling with the problem of how to name R groups, so I am not a definitive source ;) —Preceding unsigned comment added by Borek (talk • contribs) 10:28, 26 March 2008 (UTC)
 * 'R' comes from "radical", although bound R's in organic molecules are not radicals in the modern sense. The word "moiety" as itself means "a half", "a section", so it appears that IUPAC thinks that small parts should not be "moieties" since they're not halves or significant sections of the molecule. In IUPAC's example, notice that the "alcohol moiety" is not R, but RO. IUPAC's example is rather unusual since an ester does not have an alcohol functional group; it does have a "formal alkoxide" (and acyl) if you want to stretch the definition. Nevertheless, an alcohol functional group should be "-OH". IUPAC's definition appears to express that although "RO-" is not an alcohol, if we synthesized an ester from an alcohol and an acyl anion equivalent, then the "RO-" half or moiety would come from the alcohol. As such, the term "moiety" is not synonymous with "functional group". --Vuo (talk) 22:24, 28 March 2008 (UTC)

 * They may not be synonymous, but the explanation in the article leaves something to be desired.
 * "The word moiety (pronounced /ˈmɔɪəti/) is often used synonymously to "functional group," but, according to the IUPAC definition,[3] a moiety is a part of a molecule that may include functional groups as substructures. For example, an ester is a functional group composed of an alcohol moiety and an acyl moiety."
 * This suggests the difference is that a moiety may include functional groups as substructures... but the example is of a functional group which includes moieties as substructures. There isn't a proper distinction here, and the matter is confused further by the fact that Moiety (chemistry) redirects here.  I'm not competent enough to want to edit anything, here, but I think someone should.
 * -- Student Anselmus (talk) 01:21, 13 August 2011 (UTC)

 * From "Oxford English Dictionary" re "moiety":

 * c. Chem. and Biochem. A group of atoms forming a distinct part of a large molecule.

 * 1935  W. A. N. Dorland & E. C. L. Miller Med. Dict. (ed. 17) 842/1   Carbohydrate moiety, the non-nitrogenous residue of the amino acids resulting from deamination.
 * 1945  Jrnl. Biol. Chem. 159 311   The lactone moiety [of pantothenic acid] can replace pantothenic acid for growth of the above organisms.
 * 1954  A. White et al. Princ. Biochem. xii. 265   The other penicillins have the same type of structure but have different side chains replacing the benzyl (C6H5CH2—) moiety.
 * 1974  Nature 13 Dec. 586/2   Its molecular structure (containing both an indole and a phenylethylamine moiety) suggests the possibility of an interaction with brain monoamines.
 * 1993  Dyes & Pigments 21 16   The electron absorption spectra presented are due to the cationic moieties.

 * O.E.D. re "functional group":

 * functional group  n. Organic Chem. an atom or group of atoms which has a characteristic effect on the physical or chemical properties of the molecule to which it belongs.

 * 1906  Chem. News 28 Sept. 159/2   The authors find that the relations between functional groups in the same molecule are not a periodic function of the position of these functional groups.
 * 1933  Jrnl. Amer. Chem. Soc. 55 3913   When there is only one functional group, the fundamental chain will be selected so as to contain this group.
 * 1957  G. I. Brown Introd. Org. Chem. i. 17   In a compound containing two or more functional groups, e.g. glycine.., which contains an –NH2 and a –COOH group, one of the groups may affect the properties of the other.
 * 1983  R. O. C. Norman & D. J. Waddington Mod. Org. Chem. (ed. 4) xv. 235   Functional group isomerism, exhibited by isomers which have the same molecular formula but contain different functional groups.

 * -lifeform (talk) 21:40, 29 May 2014 (UTC)

Oximes and Hydrazones
Shouldn't we include oximes and hydrazones next to imines? --Kupirijo (talk) 23:14, 17 April 2008 (UTC)

I agree, those are missing. 83.13.239.255 (talk) 16:27, 25 January 2016 (UTC)

Acyl Chloride?
Is it just me or is the Acyl Chloride (R-COCl) group missing? Shouldn't it be in the "groups containing oxygen" category?

Megapanphilos (talk) 18:50, 17 May 2008 (UTC)

Good article!
I read it and thoroughly enjoyed it (how often can you say that about entries on chemistry?), and would submit that this article is of higher quality than just "start". —Preceding unsigned comment added by 213.243.137.56 (talk) 20:19, 16 April 2009 (UTC)
 * Upgraded to C-class! -- Beland (talk) 21:12, 31 August 2019 (UTC)

Has sulfonic ester reached the notability threshold to appear on this page?
Sulfonic ester: Sulfonate

Notability: Notability

The reason I make this request arises when I read the articles on Chondroitin sulfate and Heparin, and I try to find the names of the functional groups. For that matter, I would also appreciate if anyone could give the name of NH-SO3 functional group.

On a different note, it seems that there is a need for ranking and filtering by the "notability" of functional groups, so that grade school students can see an abridged list, while other mature readers can see a much longer list, and with more details.

75.142.97.184 (talk) 05:15, 24 November 2013 (UTC)
 * Well there is no standalone article yet, and only a short paragraph. We have fairly poor coverage of groups of chemicals or functional groups. but I don't see why you cannot have it. Earlier today I was looking for info on the more obscure -N=S=O group which might be called "iminooxosulphurane", but I am not too sure about that. We don't have any yet for silicon, arsenic or selenium.  Currently the order is reasonable going from well known to obscure. Graeme Bartlett (talk) 06:29, 24 November 2013 (UTC)
 * Some Si groups are covered in organosilicon. Off the top of my head I suppose the arsenic ones would be things like arsonic acids with As(V), since the As(III) compounds tend to be substituted things like R3As (I guess also the arsoles and arsabenzenes). I'd also add tellurium to the list. Double sharp (talk) 03:23, 8 December 2016 (UTC)

Wrong illus.
The illustration for Methyl ethyl ketone peroxide appears to be wrong. 206.130.136.162 (talk) 05:10, 15 December 2013 (UTC)
 * I replaced it with tert-Butyl hydroperoxide, as Methyl ethyl ketone peroxide forms a dimer - which would be an example of the next entry. Graeme Bartlett (talk) 00:40, 31 May 2014 (UTC)

Carbonyl is sp2, hydroxyl is sp3?
Under "Table of common functional groups", at the top of "Groups containing oxygen", should sp/sp2 be sp2/sp3? --lifeform (talk) 07:00, 31 May 2014 (UTC)

Prime notation and diagrams
Some of the formulas have primes and double primes on Rs that are not reflected in the visual diagrams. Some of the visual diagrams (and matching formulas) use subscripts to indicate the identity of Rs, but subscripts are usually used to indicate quantity (which they are in this article for atoms and confusingly, for some Rs). For clarity, it would be good to convert everything to use primes or superscript numbers instead of subscripts to indicate identity. (Hmm, and other articles might have the same problem.) -- Beland (talk) 21:26, 31 August 2019 (UTC)
 * Ah, Manual_of_Style/Chemistry specifies superscripts on Rs. -- Beland (talk) 21:30, 31 August 2019 (UTC)

Non-functional groups
Although not normally considered functional it would be good to include a list of common alkyl and aryl groups with names and abbreviations. By common I mean those that are common enough to have an abbreviation that may call for interpretation. In my case it is Bn- (benzyl?) that I am not familiar with. If they shouldn't be included here because they are not classified as functional (in certain respects they will have functions) please include a link to some other article.150.227.15.253 (talk) 09:16, 5 September 2019 (UTC)

A Commons file used on this page or its Wikidata item has been nominated for deletion
The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion: Participate in the deletion discussion at the. —Community Tech bot (talk) 12:06, 19 June 2022 (UTC)
 * Sulfonic-acid.svg

title: Talk:Functional group


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Fin

A fin is a thin component or appendage attached to a larger body or structure. Fins typically function as foils that produce lift or thrust, or provide the ability to steer or stabilize motion while traveling in water, air, or other fluids. Fins are also used to increase surface areas for heat transfer purposes, or simply as ornamentation.

Fins first evolved on fish as a means of locomotion. Fish fins are used to generate thrust and control the subsequent motion. Fish and other aquatic animals, such as cetaceans, actively propel and steer themselves with pectoral and tail fins. As they swim, they use other fins, such as dorsal and anal fins, to achieve stability and refine their maneuvering.

The fins on the tails of cetaceans, ichthyosaurs, metriorhynchids, mosasaurs and plesiosaurs are called flukes.

Thrust generation
Foil shaped fins generate thrust when moved, the lift of the fin sets water or air in motion and pushes the fin in the opposite direction. Aquatic animals get significant thrust by moving fins back and forth in water. Often the tail fin is used, but some aquatic animals generate thrust from pectoral fins. Fins can also generate thrust if they are rotated in air or water. Turbines and propellers (and sometimes fans and pumps) use a number of rotating fins, also called foils, wings, arms or blades. Propellers use the fins to translate torquing force to lateral thrust, thus propelling an aircraft or ship. Turbines work in reverse, using the lift of the blades to generate torque and power from moving gases or water.

Cavitation can be a problem with high power applications, resulting in damage to propellers or turbines, as well as noise and loss of power. Cavitation occurs when negative pressure causes bubbles (cavities) to form in a liquid, which then promptly and violently collapse. It can cause significant damage and wear. Cavitation damage can also occur to the tail fins of powerful swimming marine animals, such as dolphins and tuna. Cavitation is more likely to occur near the surface of the ocean, where the ambient water pressure is relatively low. Even if they have the power to swim faster, dolphins may have to restrict their speed because collapsing cavitation bubbles on their tail are too painful. Cavitation also slows tuna, but for a different reason. Unlike dolphins, these fish do not feel the bubbles, because they have bony fins without nerve endings. Nevertheless, they cannot swim faster because the cavitation bubbles create a vapor film around their fins that limits their speed. Lesions have been found on tuna that are consistent with cavitation damage.

Scombrid fishes (tuna, mackerel and bonito) are particularly high-performance swimmers. Along the margin at the rear of their bodies is a line of small rayless, non-retractable fins, known as finlets. There has been much speculation about the function of these finlets. Research done in 2000 and 2001 by Nauen and Lauder indicated that "the finlets have a hydrodynamic effect on local flow during steady swimming" and that "the most posterior finlet is oriented to redirect flow into the developing tail vortex, which may increase thrust produced by the tail of swimming mackerel".

Fish use multiple fins, so it is possible that a given fin can have a hydrodynamic interaction with another fin. In particular, the fins immediately upstream of the caudal (tail) fin may be proximate fins that can directly affect the flow dynamics at the caudal fin. In 2011, researchers using volumetric imaging techniques were able to generate "the first instantaneous three-dimensional views of wake structures as they are produced by freely swimming fishes". They found that "continuous tail beats resulted in the formation of a linked chain of vortex rings" and that "the dorsal and anal fin wakes are rapidly entrained by the caudal fin wake, approximately within the timeframe of a subsequent tail beat".

Motion control

Once motion has been established, the motion itself can be controlled with the use of other fins. Boats control direction (yaw) with fin-like rudders, and roll with stabilizer and keel fins. Airplanes achieve similar results with small specialised fins that change the shape of their wings and tail fins.

Stabilising fins are used as fletching on arrows and some darts, and at the rear of some bombs, missiles, rockets and self-propelled torpedoes. These are typically planar and shaped like small wings, although grid fins are sometimes used. Static fins have also been used for one satellite, GOCE.

Temperature regulation
Engineering fins are also used as heat transfer fins to regulate temperature in heat sinks or fin radiators.

Ornamentation and other uses
In biology, fins can have an adaptive significance as sexual ornaments. During courtship, the female cichlid, Pelvicachromis taeniatus, displays a large and visually arresting purple pelvic fin. "The researchers found that males clearly preferred females with a larger pelvic fin and that pelvic fins grew in a more disproportionate way than other fins on female fish."

Reshaping human feet with swim fins, rather like the tail fin of a fish, add thrust and efficiency to the kicks of a swimmer or underwater diver Surfboard fins provide surfers with means to maneuver and control their boards. Contemporary surfboards often have a centre fin and two cambered side fins.

The bodies of reef fishes are often shaped differently from open water fishes. Open water fishes are usually built for speed, streamlined like torpedoes to minimise friction as they move through the water. Reef fish operate in the relatively confined spaces and complex underwater landscapes of coral reefs. For this manoeuvrability is more important than straight line speed, so coral reef fish have developed bodies which optimize their ability to dart and change direction. They outwit predators by dodging into fissures in the reef or playing hide and seek around coral heads.

The pectoral and pelvic fins of many reef fish, such as butterflyfish, damselfish and angelfish, have evolved so they can act as brakes and allow complex maneuvers. Many reef fish, such as butterflyfish, damselfish and angelfish, have evolved bodies which are deep and laterally compressed like a pancake, and will fit into fissures in rocks. Their pelvic and pectoral fins are designed differently, so they act together with the flattened body to optimise maneuverability. Some fishes, such as puffer fish, filefish and trunkfish, rely on pectoral fins for swimming and hardly use tail fins at all.

Evolution

There is an old theory, proposed by anatomist Carl Gegenbaur, which has been often disregarded in science textbooks, "that fins and (later) limbs evolved from the gills of an extinct vertebrate". Gaps in the fossil record had not allowed a definitive conclusion. In 2009, researchers from the University of Chicago found evidence that the "genetic architecture of gills, fins and limbs is the same", and that "the skeleton of any appendage off the body of an animal is probably patterned by the developmental genetic program that we have traced back to formation of gills in sharks". Recent studies support the idea that gill arches and paired fins are serially homologous and thus that fins may have evolved from gill tissues.

Fish are the ancestors of all mammals, reptiles, birds and amphibians. In particular, terrestrial tetrapods (four-legged animals) evolved from fish and made their first forays onto land 400 million years ago. They used paired pectoral and pelvic fins for locomotion. The pectoral fins developed into forelegs (arms in the case of humans) and the pelvic fins developed into hind legs. Much of the genetic machinery that builds a walking limb in a tetrapod is already present in the swimming fin of a fish.

In 2011, researchers at Monash University in Australia used primitive but still living lungfish "to trace the evolution of pelvic fin muscles to find out how the load-bearing hind limbs of the tetrapods evolved." Further research at the University of Chicago found bottom-walking lungfishes had already evolved characteristics of the walking gaits of terrestrial tetrapods.

In a classic example of convergent evolution, the pectoral limbs of pterosaurs, birds and bats further evolved along independent paths into flying wings. Even with flying wings there are many similarities with walking legs, and core aspects of the genetic blueprint of the pectoral fin have been retained.

About 200 million years ago the first mammals appeared. A group of these mammals started returning to the sea about 52 million years ago, thus completing a circle. These are the cetaceans (whales, dolphins and porpoises). Recent DNA analysis suggests that cetaceans evolved from within the even-toed ungulates, and that they share a common ancestor with the hippopotamus. About 23 million years ago another group of bearlike land mammals started returning to the sea. These were the seals. What had become walking limbs in cetaceans and seals evolved further, independently in a reverse form of convergent evolution, back to new forms of swimming fins. The forelimbs became flippers and the hind limbs became a tail terminating in two fins, called a fluke in the case of cetaceans. Fish tails are usually vertical and move from side to side. Cetacean flukes are horizontal and move up and down, because cetacean spines bend the same way as in other mammals.

Ichthyosaurs are ancient reptiles that resembled dolphins. They first appeared about 245 million years ago and disappeared about 90 million years ago. "This sea-going reptile with terrestrial ancestors converged so strongly on fishes that it actually evolved a dorsal fin and tail in just the right place and with just the right hydrological design. These structures are all the more remarkable because they evolved from nothing — the ancestral terrestrial reptile had no hump on its back or blade on its tail to serve as a precursor." The biologist Stephen Jay Gould said the ichthyosaur was his favorite example of convergent evolution.

Robotics

The use of fins for the propulsion of aquatic animals can be remarkably effective. It has been calculated that some fish can achieve a propulsive efficiency greater than 90%. Fish can accelerate and maneuver much more effectively than boats or submarine, and produce less water disturbance and noise. This has led to biomimetic studies of underwater robots which attempt to emulate the locomotion of aquatic animals. An example is the Robot Tuna built by the Institute of Field Robotics, to analyze and mathematically model thunniform motion. In 2005, the Sea Life London Aquarium displayed three robotic fish created by the computer science department at the University of Essex. The fish were designed to be autonomous, swimming around and avoiding obstacles like real fish. Their creator claimed that he was trying to combine "the speed of tuna, acceleration of a pike, and the navigating skills of an eel".

The AquaPenguin, developed by Festo of Germany, copies the streamlined shape and propulsion by front flippers of penguins. Festo also developed AquaRay, AquaJelly and AiraCuda, respectively emulating the locomotion of manta rays, jellyfish and barracuda.

In 2004, Hugh Herr at MIT prototyped a biomechatronic robotic fish with a living actuator by surgically transplanting muscles from frog legs to the robot and then making the robot swim by pulsing the muscle fibers with electricity.

Robotic fish offer some research advantages, such as the ability to examine part of a fish design in isolation from the rest, and variance of a single parameter, such as flexibility or direction. Researchers can directly measure forces more easily than in live fish. "Robotic devices also facilitate three-dimensional kinematic studies and correlated hydrodynamic analyses, as the location of the locomotor surface can be known accurately. And, individual components of a natural motion (such as outstroke vs. instroke of a flapping appendage) can be programmed separately, which is certainly difficult to achieve when working with a live animal."

title: Fin


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Freyr

Freyr (Old Norse: 'Lord'), sometimes anglicized as Frey, is a widely attested god in Norse mythology, associated with kingship, fertility, peace, prosperity, fair weather, and good harvest. Freyr, sometimes referred to as Yngvi-Freyr, was especially associated with Sweden and seen as an ancestor of the Swedish royal house. According to Adam of Bremen, Freyr was associated with peace and pleasure, and was represented with a phallic statue in the Temple at Uppsala. According to Snorri Sturluson, Freyr was "the most renowned of the æsir", and was venerated for good harvest and peace.

In the mythological stories in the Icelandic books the Poetic Edda and the Prose Edda, Freyr is presented as one of the Vanir, the son of the god Njörðr and his sister-wife, as well as the twin brother of the goddess Freyja. The gods gave him Álfheimr, the realm of the Elves, as a teething present. He rides the shining dwarf-made boar Gullinbursti, and possesses the ship Skíðblaðnir, which always has a favorable breeze and can be folded together and carried in a pouch when it is not being used. Freyr is also known to have been associated with the horse cult. He also kept sacred horses in his sanctuary at Trondheim in Norway. He has the servants Skírnir, Byggvir and Beyla.

The most extensive surviving Freyr myth relates Freyr's falling in love with the female jötunn Gerðr. Eventually, she becomes his wife but first Freyr has to give away his sword, which fights on its own "if wise be he who wields it." Although deprived of this weapon, Freyr defeats the jötunn Beli with an antler. However, lacking his sword, Freyr will be killed by the fire jötunn Surtr during the events of Ragnarök.

Like other Germanic deities, veneration of Freyr was revived during the modern period through the Heathenry movement.

Name
The Old Norse name Freyr ('lord') is generally thought to descend from a Proto-Norse form reconstructed as *frawjaʀ, stemming from the Proto-Germanic noun *frawjaz ~ *fraw(j)ōn ('lord'), and cognate with Gothic frauja, Old English frēa, or Old High German frō, all meaning 'lord, master'. The runic form frohila, derived from an earlier *frōjila, may also be related. Recently, however, an etymology deriving the name of the god from a nominalized form of the Proto-Scandinavian adjective *fraiw(i)a- ('fruitful, generative') has also been proposed. According to linguist Guus Kroonen, "within Germanic, the attestation of ON frjar, frjór, frær, Icel. frjór adj. 'fertile; prolific' < *fraiwa- clearly seems to point to a stem *frai(w)- meaning 'fecund'. Both in form and meaning, fraiwa- ('seed') is reminiscent of Freyr 'fertility deity' < *frauja-. The possibility must be considered, therefore, that *fraiwa- was metathesized from *frawja-, a collective of some kind." Freyr is also known by a series of other names which describe his attributes and role in religious practice and associated mythology.

Adam of Bremen
Written c. 1080, one of the oldest written sources on pre-Christian Scandinavian religious practices is Adam of Bremen's Gesta Hammaburgensis ecclesiae pontificum. Adam claimed to have access to first-hand accounts on pagan practices in Sweden. He refers to Freyr with the Latinized name Fricco and mentions that an image of him at Skara was destroyed by the Christian missionary Bishop Egino. Adam's description of the Temple at Uppsala gives some details on the god.

Later in the account Adam states that when a marriage is performed a libation is made to the image of Fricco.

Historians are divided on the reliability of Adam's account.

Prose Edda
When Snorri Sturluson was writing in 13th century Iceland, the indigenous Germanic gods were still remembered although they had not been openly worshiped for more than two centuries.

Gylfaginning
In the Gylfaginning section of his Prose Edda, Snorri introduces Freyr as one of the major gods.

This description has similarities to the older account by Adam of Bremen but the differences are interesting. Adam assigns control of the weather and produce of the fields to Thor but Snorri says that Freyr rules over those areas. Snorri also omits any explicitly sexual references in Freyr's description. Those discrepancies can be explained in several ways. Adam and Snorri were writing with different goals in mind. It is possible that the Norse gods did not have exactly the same roles in Icelandic and Swedish paganism. Either Snorri or Adam may also have had distorted information.

The only extended myth related to Freyr in the Prose Edda is the story of his marriage.

The woman is Gerðr, a beautiful giantess. Freyr immediately falls in love with her and becomes depressed and taciturn. After a period of brooding, he consents to talk to Skírnir, his foot-page. He tells Skírnir that he has fallen in love with a beautiful woman and thinks he will die if he cannot have her. He asks Skírnir to go and woo her for him.

The loss of Freyr's sword has consequences. According to the Prose Edda, Freyr had to fight Beli without his sword, and slew him with an antler. But the result at Ragnarök, the end of the world, will be much more serious. Freyr is fated to fight the fire-giant Surtr, and since he does not have his sword he will be defeated. Even after the loss of his weapon Freyr still has two magical artifacts, both dwarf-made. One is the ship Skíðblaðnir, which will have favoring breeze wherever its owner wants to go and can also be folded together like a napkin and carried in a pouch. The other is the boar Gullinbursti whose mane glows to illuminate the way for his owner. No myths involving Skíðblaðnir have come down to us but Snorri relates that Freyr rode to Baldr's funeral in a wagon pulled by Gullinbursti.

Skaldic poetry
Freyr is referred to several times in skaldic poetry. In Húsdrápa, partially preserved in the Prose Edda, he is said to ride a boar to Baldr's funeral.

In a poem by Egill Skalla-Grímsson, Freyr is called upon along with Njörðr to drive Eric Bloodaxe from Norway. The same skald mentions in Arinbjarnarkviða that his friend has been blessed by the two gods.

Nafnaþulur
In Nafnaþulur Freyr is said to ride the horse Blóðughófi (Bloody Hoof).

Poetic Edda
Freyr is mentioned in several of the poems in the Poetic Edda. The information there is largely consistent with that of the Prose Edda while each collection has some details not found in the other.

Völuspá
Völuspá, the best known of the Eddic poems, describes the final confrontation between Freyr and Surtr during Ragnarök.

Some scholars have preferred a slightly different translation, in which the sun shines "from the sword of the gods". The idea is that the sword which Surtr slays Freyr with is the "sword of the gods" which Freyr had earlier bargained away for Gerðr. This would add a further layer of tragedy to the myth. Sigurður Nordal argued for this view but the possibility represented by Ursula Dronke's translation above is equally possible.

Grímnismál
Grímnismál, a poem which largely consists of miscellaneous information about the gods, mentions Freyr's abode.

A tooth-gift was a gift given to an infant on the cutting of the first tooth. Since Alfheimr or Álfheimr means "World of Álfar (Elves)" the fact that Freyr should own it is one of the indications of a connection between the Vanir and the obscure Álfar. Grímnismál also mentions that the sons of Ívaldi made Skíðblaðnir for Freyr and that it is the best of ships.

Lokasenna
In the poem Lokasenna, Loki accuses the gods of various misdeeds. He criticizes the Vanir for incest, saying that Njörðr had Freyr with his sister. He also states that the gods discovered Freyr and Freyja having sex together. The god Týr speaks up in Freyr's defense.

Lokasenna also mentions that Freyr has servants called Byggvir and Beyla. They seem to have been associated with the making of bread.

Skírnismál
The courtship of Freyr and Gerðr is dealt with extensively in the poem Skírnismál. Freyr is depressed after seeing Gerðr. Njörðr and Skaði ask Skírnir to go and talk with him. Freyr reveals the cause of his grief and asks Skírnir to go to Jötunheimr to woo Gerðr for him. Freyr gives Skírnir a steed and his magical sword for the journey.

When Skírnir finds Gerðr he starts by offering her treasures if she will marry Freyr. When she declines he forces her to accept by threatening her with destructive magic.

Ynglinga saga
Snorri Sturluson starts his epic history of the kings of Norway with Ynglinga saga, a euhemerized account of the Norse gods. Here Odin and the Æsir are men from Asia who gain power through their prowess in war and Odin's skills. But when Odin attacks the Vanir he bites off more than he can chew and peace is negotiated after the destructive and indecisive Æsir-Vanir War. Hostages are exchanged to seal the peace deal and the Vanir send Freyr and Njörðr to live with the Æsir. At this point the saga, like Lokasenna, mentions that incest was practised among the Vanir.

Odin makes Njörðr and Freyr priests of sacrifices and they become influential leaders. Odin goes on to conquer the North and settles in Sweden where he rules as king, collects taxes and maintains sacrifices. After Odin's death, Njörðr takes the throne. During his rule there is peace and good harvest and the Swedes come to believe that Njörðr controls these things. Eventually Njörðr falls ill and dies.

Freyr had a son named Fjölnir, who succeeds him as king and rules during the continuing period of peace and good seasons. Fjölnir's descendants are enumerated in Ynglingatal which describes the mythological kings of Sweden.

Ögmundar þáttr dytts
The 14th century Icelandic Ögmundar þáttr dytts contains a tradition of how Freyr was transported in a wagon and administered by a priestess, in Sweden. Freyr's role as a fertility god needed a female counterpart in a divine couple (McKinnell's translation 1987 ):

"Great heathen sacrifices were held there at that time, and for a long while Frey had been the god who was worshipped most there – and so much power had been gained by Frey's statue that the devil used to speak to people out of the mouth of the idol, and a young and beautiful woman had been obtained to serve Frey. It was the faith of the local people that Frey was alive, as seemed to some extent to be the case, and they thought he would need to have a sexual relationship with his wife; along with Frey she was to have complete control over the temple settlement and all that belonged to it."

In this short story, a man named Gunnar was suspected of manslaughter and escaped to Sweden, where Gunnar became acquainted with this young priestess. He helped her drive Freyr's wagon with the god effigy in it, but the god did not appreciate Gunnar and so attacked him and would have killed Gunnar if he had not promised himself to return to the Christian faith if he would make it back to Norway. When Gunnar had promised this, a demon jumped out of the god effigy and so Freyr was nothing but a piece of wood. Gunnar destroyed the wooden idol and dressed himself as Freyr, then Gunnar and the priestess travelled across Sweden where people were happy to see the god visiting them. After a while he made the priestess pregnant, but this was seen by the Swedes as confirmation that Freyr was truly a fertility god and not a scam. Finally, Gunnar had to flee back to Norway with his young bride and had her baptized at the court of Olaf Tryggvason.

Other Icelandic sources
Worship of Freyr is alluded to in several Icelanders' sagas.

The protagonist of Hrafnkels saga is a priest of Freyr. He dedicates a horse to the god and kills a man for riding it, setting in motion a chain of fateful events.

In Gísla saga a chieftain named Þorgrímr Freysgoði is an ardent worshipper of Freyr. When he dies he is buried in a howe.

Hallfreðar saga, Víga-Glúms saga and Vatnsdœla saga also mention Freyr.

Other Icelandic sources referring to Freyr include Íslendingabók, Landnámabók, and Hervarar saga.

Íslendingabók, written c. 1125, is the oldest Icelandic source that mentions Freyr, including him in a genealogy of Swedish kings. Landnámabók includes a heathen oath to be sworn at an assembly where Freyr, Njörðr, and "the almighty áss" are invoked. Hervarar saga mentions a Yuletide sacrifice of a boar to Freyr.

Gesta Danorum
The 12th Century Danish Gesta Danorum describes Freyr, under the name Frø, as the "viceroy of the gods".

That Freyr had a cult at Uppsala is well confirmed from other sources. The reference to the change in sacrificial ritual may also reflect some historical memory. There is archaeological evidence for an increase in human sacrifices in the late Viking Age though among the Norse gods human sacrifice is most often linked to Odin. Another reference to Frø and sacrifices is found earlier in the work, where the beginning of an annual blót to him is related. King Hadingus is cursed after killing a divine being and atones for his crime with a sacrifice.

The sacrifice of dark-coloured victims to Freyr has a parallel in Ancient Greek religion where the chthonic fertility deities preferred dark-coloured victims to lighter ones.

In book 9, Saxo identifies Frø as the "king of Sweden" (rex Suetiae):

The reference to public prostitution may be a memory of fertility cult practices. Such a memory may also be the source of a description in book 6 of the stay of Starcatherus, a follower of Odin, in Sweden.

Yngvi
A strophe of the Anglo-Saxon rune poem (c. 1100) records that:

 * Ing was first among the East Danes seen by men

This may refer to the origins of the worship of Ingui in the tribal areas that Tacitus mentions in his Germania as being populated by the Inguieonnic tribes. A later Danish chronicler lists Ingui was one of three brothers that the Danish tribes descended from. The strophe also states that "then he (Ingui) went back over the waves, his wagon behind him" which could connect Ingui to earlier conceptions of the wagon processions of Nerthus and the later Scandinavian conceptions of Freyr's wagon journeys.

Ingui is mentioned also in some later Anglo-Saxon literature under varying forms of his name, such as "For what doth Ingeld have to do with Christ" and the variants used in Beowulf to designate the kings as 'leader of the friends of Ing'. The compound Ingui-Frea (OE) and Yngvi-Freyr (ON) likely refer to the connection between the god and the Germanic kings' role as priests during the sacrifices in the pagan period, as Frea and Freyr are titles meaning 'Lord'.

The Swedish royal dynasty was known as the Ynglings from their descent from Yngvi-Freyr. This is supported by Tacitus, who wrote about the Germans: "In their ancient songs, their only way of remembering or recording the past they celebrate an earth-born god Tuisco, and his son Mannus, as the origin of their race, as their founders. To Mannus they assign three sons, from whose names, they say, the coast tribes are called Ingaevones; those of the interior, Herminones; all the rest, Istaevones".

Rällinge statuette
In 1904, a Viking Age statuette identified as a depiction of Freyr was discovered on the farm Rällinge in Lunda, Södermanland parish in the province of Södermanland, Sweden. The depiction features a cross-legged seated, bearded male with an erect penis. He is wearing a pointed cap or helmet and stroking his triangular beard. The seven-centimeter-tall statue is displayed at the Swedish Museum of National Antiquities.

Skog tapestry
A part of the Swedish Skog tapestry depicts three figures that have been interpreted as allusions to Odin, Thor, and Freyr, but also as the three Scandinavian holy kings Canute, Eric and Olaf. The figures coincide with 11th century descriptions of statue arrangements recorded by Adam of Bremen at the Temple at Uppsala and written accounts of the gods during the late Viking Age. The tapestry is originally from Hälsingland, Sweden but is now housed at the Swedish Museum of National Antiquities.

Gullgubber
Small pieces of gold foil featuring engravings dating from the Migration Period into the early Viking Age (known as gullgubber) have been discovered in various locations in Scandinavia, at one site almost 2,500. The foil pieces have been found largely on the sites of buildings, only rarely in graves. The figures are sometimes single, occasionally an animal, sometimes a man and a woman with a leafy bough between them, facing or embracing one another. The human figures are almost always clothed and are sometimes depicted with their knees bent. Scholar Hilda Ellis Davidson says that it has been suggested that the figures are taking part in a dance, and that they may have been connected with weddings, as well as linked to the Vanir group of gods, representing the notion of a divine marriage, such as in the Poetic Edda poem Skírnismál; the coming together of Gerðr and Freyr.

Toponyms
Norway Sweden Denmark
 * Freysakr ("Freyr's field") - name of two old farms in Gol and Torpa.
 * Freyshof ("Freyr's temple") - name of two old farms in Hole and Trøgstad.
 * Freysland ("Freyr's land/field") - name of six old farms in Feda, Halse, Førde, Sogndal, Søgne and Torpa.
 * Freyslíð ("Freyr's hill") - name of two old farms in Lunner and Torpa.
 * Freysnes ("Freyr's headland") - name of an old farm in Sandnes.
 * Freyssetr ("Freyr's farm") - name of two old farms in Masfjorden and Soknedal.
 * Freyssteinn ("Freyr's stone") - name of an old farm in Lista.
 * Freysteigr ("Freyr's field") - name of an old farm in Ramnes.
 * Freysvík ("Freyr's inlet/bay") - name of two old farms in Fresvik and Ullensvang.
 * Freysvin ("Freyr's meadow") - name of four old farms in Hole, Lom, Sunnylven and Østre Gausdal.
 * Freysvǫllr ("Freyr's field") - name of an old farm in Sør-Odal.
 * Freysþveit ("Freyr's thwaite") - name of an old farm in Hedrum.
 * Fröslunda ("Freyr's grove") - Uppland
 * Frösåker ("Freyr's field") - Uppland
 * Frösön ("Freyr's island") - Jämtland
 * Fröseke ("Freyr's oak forest") - Småland
 * Frösve ("Freyr's sanctuary") - Västergötland
 * Frösakull ("Freyr's hill") – Halland
 * Frøs Herred ("Freyr's Shire") - Southern Jutland

Modern influence
Freyr appears in numerous works of modern art and literature. He appears, for example, alongside numerous othe figures from Norse mythology in the Danish poet Adam Gottlob Oehlenschläger's Nordens Guder (1819). He also appears in Icelandic poet Gerður Kristný's Blóðhófnir (2010), a feminist retelling of the Eddic poem Skírnismál that won the 2010 Icelandic Literature Award.

title: Freyr


text:
Freyja

In Norse mythology, Freyja (Old Norse "(the) Lady") is a goddess associated with love, beauty, fertility, sex, war, gold, and seiðr (magic for seeing and influencing the future). Freyja is the owner of the necklace Brísingamen, rides a chariot pulled by two cats, is accompanied by the boar Hildisvíni, and possesses a cloak of falcon feathers. By her husband Óðr, she is the mother of two daughters, Hnoss and Gersemi. Along with her twin brother Freyr, her father Njörðr, and her mother (Njörðr's sister, unnamed in sources), she is a member of the Vanir. Stemming from Old Norse Freyja, modern forms of the name include Freya, Freyia, and Freja.

Freyja rules over her heavenly field, Fólkvangr, where she receives half of those who die in battle. The other half go to the god Odin's hall, Valhalla. Within Fólkvangr lies her hall, Sessrúmnir. Freyja assists other deities by allowing them to use her feathered cloak, is invoked in matters of fertility and love, and is frequently sought after by powerful jötnar who wish to make her their wife. Freyja's husband, the god Óðr, is frequently absent. She cries tears of red gold for him, and searches for him under assumed names. Freyja has numerous names, including Gefn, Hörn, Mardöll, Sýr, Vanadís, and Valfreyja.

Freyja is attested in the Poetic Edda, compiled in the 13th century from earlier traditional sources; in the Prose Edda and Heimskringla, composed by Snorri Sturluson in the 13th century; in several Sagas of Icelanders; in the short story "Sörla þáttr; in the poetry of skalds; and into the modern age in Scandinavian folklore.

Scholars have debated whether Freyja and the goddess Frigg ultimately stem from a single goddess common among the Germanic peoples. They have connected her to the valkyries, female battlefield choosers of the slain, and analyzed her relation to other goddesses and figures in Germanic mythology, including the thrice-burnt and thrice-reborn Gullveig/Heiðr, the goddesses Gefjon, Skaði, Þorgerðr Hölgabrúðr and Irpa, Menglöð, and the 1st century CE "Isis" of the Suebi. In Scandinavia, Freyja's name frequently appears in the names of plants, especially in southern Sweden. Various plants in Scandinavia once bore her name, but it was replaced with the name of the Virgin Mary during the process of Christianization. Rural Scandinavians continued to acknowledge Freyja as a supernatural figure into the 19th century, and Freyja has inspired various works of art.

Etymology
The name Freyja transparently means 'lady, mistress' in Old Norse. Stemming from the Proto-Germanic feminine noun *frawjōn ('lady, mistress'), it is cognate with Old Saxon frūa ('lady, mistress') or Old High German frouwa ('lady'; cf. modern German Frau). Freyja is also etymologically close to the name of the god Freyr, meaning 'lord' in Old Norse. The theonym Freyja is thus considered to have been an epithet in origin, replacing a personal name that is now unattested.

Alternative names
In addition to Freyja, Old Norse sources refer to the goddess by the following names:

Poetic Edda
In the Poetic Edda, Freyja is mentioned or appears in the poems Völuspá, Grímnismál, Lokasenna, Þrymskviða, Oddrúnargrátr, and Hyndluljóð.

Völuspá contains a stanza that mentions Freyja, referring to her as "Óð's girl"; Freyja being the wife of her husband, Óðr. The stanza recounts that Freyja was once promised to an unnamed builder, later revealed to be a jötunn and subsequently killed by Thor (recounted in detail in Gylfaginning chapter 42; see Prose Edda section below). In the poem Grímnismál, Odin (disguised as Grímnir) tells the young Agnar that every day Freyja allots seats to half of those that are slain in her hall Fólkvangr, while Odin owns the other half.

In the poem Lokasenna, where Loki accuses nearly every female in attendance of promiscuity or unfaithfulness, an aggressive exchange occurs between Loki and Freyja. The introduction to the poem notes that among other gods and goddesses, Freyja attends a celebration held by Ægir. In verse, after Loki has flyted with the goddess Frigg, Freyja interjects, telling Loki that he is insane for dredging up his terrible deeds, and that Frigg knows the fate of everyone, though she does not tell it. Loki tells her to be silent, and says that he knows all about her—that Freyja is not lacking in blame, for each of the gods and elves in the hall have been her lover. Freyja objects. She says that Loki is lying, that he is just looking to blather about misdeeds, and since the gods and goddesses are furious at him, he can expect to go home defeated. Loki tells Freyja to be silent, calls her a malicious witch, and conjures a scenario where Freyja was once astride her brother when all of the gods, laughing, surprised the two. Njörðr interjects—he says that a woman having a lover other than her husband is harmless, and he points out that Loki has borne children, and calls Loki a pervert. The poem continues in turn.

The poem Þrymskviða features Loki borrowing Freyja's cloak of feathers and Thor dressing up as Freyja to fool the lusty jötunn Þrymr. In the poem, Thor wakes up to find that his powerful hammer, Mjöllnir, is missing. Thor tells Loki of his missing hammer, and the two go to the beautiful court of Freyja. Thor asks Freyja if she will lend him her cloak of feathers, so that he may try to find his hammer. Freyja agrees: Loki flies away in the whirring feather cloak, arriving in the land of Jötunheimr. He spies Þrymr sitting on top of a mound. Þrymr reveals that he has hidden Thor's hammer deep within the earth and that no one will ever know where the hammer is unless Freyja is brought to him as his wife. Loki flies back, the cloak whistling, and returns to the courts of the gods. Loki tells Thor of Þrymr's conditions.

The two go to see the beautiful Freyja. The first thing that Thor says to Freyja is that she should dress herself and put on a bride's head-dress, for they shall drive to Jötunheimr. At that, Freyja is furious—the halls of the gods shake, she snorts in anger, and from the goddess the necklace Brísingamen falls. Indignant, Freyja responds: The gods and goddesses assemble at a thing and debate how to solve the problem. The god Heimdallr proposes to dress Thor up as a bride, complete with bridal dress, head-dress, jingling keys, jewelry, and the famous Brísingamen. Thor objects but is hushed by Loki, reminding him that the new owners of the hammer will soon be settling in the land of the gods if the hammer is not returned. Thor is dressed as planned and Loki is dressed as his maid. Thor and Loki go to Jötunheimr.

In the meantime, Thrym tells his servants to prepare for the arrival of the daughter of Njörðr. When "Freyja" arrives in the morning, Thrym is taken aback by her behavior; her immense appetite for food and mead is far more than what he expected, and when Thrym goes in for a kiss beneath "Freyja's" veil, he finds "her" eyes to be terrifying, and he jumps down the hall. The disguised Loki makes excuses for the bride's odd behavior, claiming that she simply has not eaten or slept for eight days. In the end, the disguises successfully fool the jötnar and, upon sight of it, Thor regains his hammer by force.

In the poem Oddrúnargrátr, Oddrún helps Borgny give birth to twins. In thanks, Borgny invokes vættir, Frigg, Freyja, and other unspecified deities.

Freyja is a main character in the poem Hyndluljóð, where she assists her faithful servant Óttar in finding information about his ancestry so that he may claim his inheritance. In doing so, Freyja turns Óttar into her boar, Hildisvíni, and, by means of flattery and threats of death by fire, Freyja successfully pries the information that Óttar needs from the jötunn Hyndla. Freyja speaks throughout the poem, and at one point praises Óttar for constructing a hörgr (an altar of stones) and frequently making blót (sacrifices) to her:

Prose Edda
Freyja appears in the Prose Edda books Gylfaginning and Skáldskaparmál. In chapter 24 of Gylfaginning, the enthroned figure of High says that after the god Njörðr split with the goddess Skaði, he had two beautiful and mighty children (no partner is mentioned); a son, Freyr, and a daughter, Freyja. Freyr is "the most glorious" of the gods, and Freyja "the most glorious" of the goddesses. Freyja has a dwelling in the heavens, Fólkvangr, and that whenever Freyja "rides into battle she gets half the slain, and the other half to Odin [...]". In support, High quotes the Grímnismál stanza mentioned in the Poetic Edda section above.

High adds that Freyja has a large, beautiful hall called Sessrúmnir, and that when Freyja travels she sits in a chariot and drives two cats, and that Freyja is "the most approachable one for people to pray to, and from her name is derived the honorific title whereby noble ladies are called fruvor [noble ladies]". High adds that Freyja has a particular fondness for love songs, and that "it is good to pray to her concerning love affairs".

In chapter 29, High recounts the names and features of various goddesses, including Freyja. Regarding Freyja, High says that, next to Frigg, Freyja is highest in rank among them and that she owns the necklace Brísingamen. Freyja is married to Óðr, who goes on long travels, and the two have a very fair daughter by the name of Hnoss. While Óðr is absent, Freyja stays behind and in her sorrow she weeps tears of red gold. High notes that Freyja has many names, and explains that this is because Freyja adopted them when looking for Óðr and traveling "among strange peoples". These names include Gefn, Hörn, Mardöll, Sýr, and Vanadís.

Freyja plays a part in the events leading to the birth of Sleipnir, the eight-legged horse. In chapter 42, High recounts that, soon after the gods built the hall Valhalla, a builder (unnamed) came to them and offered to build for them in three seasons a fortification so solid that no jötunn would be able to come in over from Midgard. In exchange, the builder wants Freyja for his bride, and the sun and the moon. After some debate the gods agree, but with added conditions. In time, just as he is about to complete his work, it is revealed that the builder is, in fact, himself a jötunn, and he is killed by Thor. In the meantime, Loki, in the form of a mare, has been impregnated by the jötunn's horse, Svaðilfari, and so gives birth to Sleipnir. In support, High quotes the Völuspá stanza that mentions Freyja. In chapter 49, High recalls the funeral of Baldr and says that Freyja attended the funeral and there drove her cat-chariot, the final reference to the goddess in Gylfaginning.

At the beginning of the book Skáldskaparmál, Freyja is mentioned among eight goddesses attending a banquet held for Ægir. Chapter 56 details the abduction of the goddess Iðunn by the jötunn Þjazi in the form of an eagle. Terrified at the prospect of death and torture due to his involvement in the abduction of Iðunn, Loki asks if he may use Freyja's "falcon shape" to fly north to Jötunheimr and retrieve the missing goddess. Freyja allows it, and using her "falcon shape" and a furious chase by eagle-Þjazi, Loki successfully returns her.

In chapter 6, a means of referring to Njörðr is provided that refers to Frejya ("father of Freyr and Freyja"). In chapter 7, a means of referring to Freyr is provided that refers to the goddess ("brother of Freyja"). In chapter 8, ways of referring to the god Heimdallr are provided, including "Loki's enemy, recoverer of Freyja's necklace", inferring a myth involving Heimdallr recovering Freyja's necklace from Loki.

In chapter 17, the jötunn Hrungnir finds himself in Asgard, the realm of the gods, and becomes very drunk. Hrungnir boasts that he will move Valhalla to Jötunheimr, bury Asgard, and kill all of the gods—with the exception of the goddesses Freyja and Sif, who he says he will take home with him. Freyja is the only one of them that dares to bring him more to drink. Hrungnir says that he will drink all of their ale. After a while, the gods grow bored of Hrungnir's antics and invoke the name of Thor. Thor immediately enters the hall, hammer raised. Thor is furious and demands to know who is responsible for letting a jötunn in to Asgard, who guaranteed Hrungnir safety, and why Freyja "should be serving him drink as if at the Æsir's banquet".

In chapter 18, verses from the 10th century skald's composition Þórsdrápa are quoted. A kenning used in the poem refers to Freyja. In chapter 20, poetic ways to refer to Freyja are provided; "daughter of Njörðr", "sister of Freyr", "wife of Óðr", "mother of Hnoss", "possessor of the fallen slain and of Sessrumnir and tom-cats", possessor of Brísingamen, "Van-deity", Vanadís, and "fair-tear deity". In chapter 32, poetic ways to refer to gold are provided, including "Freyja's weeping" and "rain or shower [...] from Freyja's eyes".

Chapter 33 tells that once the gods journeyed to visit Ægir, one of whom was Freyja. In chapter 49, a quote from a work by the skald Einarr Skúlason employs the kenning "Óðr's bedfellow's eye-rain", which refers to Freyja and means "gold".

Chapter 36 explains again that gold can be referring to as Freyja's weeping due to her red gold tears. In support, works by the skalds Skúli Þórsteinsson and Einarr Skúlason are cited that use "Freyja's tears" or "Freyja's weepings" to represent "gold". The chapter features additional quotes from poetry by Einarr Skúlason that references the goddess and her child Hnoss. Freyja receives a final mention in the Prose Edda in chapter 75, where a list of goddesses is provided that includes Freyja.

Heimskringla
The Heimskringla book Ynglinga saga provides a euhemerized account of the origin of the gods, including Freyja. In chapter 4, Freyja is introduced as a member of the Vanir, the sister of Freyr, and the daughter of Njörðr and his sister (whose name is not provided). After the Æsir–Vanir War ends in a stalemate, Odin appoints Freyr and Njörðr as priests over sacrifices. Freyja becomes the priestess of sacrificial offerings and it was she who introduced the practice of seiðr to the Æsir, previously only practiced by the Vanir.

In chapter 10, Freyja's brother Freyr dies, and Freyja is the last survivor among the Æsir and Vanir. Freyja keeps up the sacrifices and becomes famous. The saga explains that, due to Freyja's fame, all women of rank become known by her name—frúvor ("ladies"), a woman who is the mistress of her property is referred to as freyja, and húsfreyja ("lady of the house") for a woman who owns an estate.

The chapter adds that not only was Freyja very clever, but that she and her husband Óðr had two immensely beautiful daughters, Gersemi and Hnoss, "who gave their names to our most precious possessions".

Other
Freyja is mentioned in the sagas Egils saga, Njáls saga, Hálfs saga ok Hálfsrekka, and in Sörla þáttr.

In Egils saga, when Egill Skallagrímsson refuses to eat, his daughter Þorgerðr (here anglicized as "Thorgerd") says she will go without food and thus starve to death, and in doing so will meet the goddess Freyja: Thorgerd replied in a loud voice, "I have had no evening meal, nor will I do so until I join Freyja. I know no better course of action than my father's. I do not want to live after my father and brother are dead."
 * Egils saga

In the first chapter of the 14th century legendary saga Hálfs saga ok Hálfsrekka, King Alrek has two wives, Geirhild and Signy, and cannot keep them both. He tells the two women that he would keep whichever of them that brews the better ale for him by the time he has returned home in the summer. The two compete and during the brewing process Signy prays to Freyja and Geirhild to Hött ("hood"), a man she had met earlier (earlier in the saga revealed to be Odin in disguise). Hött answers her prayer and spits on her yeast. Signy's brew wins the contest.
 * Hálfs saga ok Hálfsrekka

In Sörla þáttr, a short, late 14th century narrative from a later and extended version of the Óláfs saga Tryggvasonar found in the Flateyjarbók manuscript, a euhemerized account of the gods is provided. In the account, Freyja is described as having been a concubine of Odin, who bartered sex to four dwarfs for a golden necklace. In the work, the Æsir once lived in a city called Asgard, located in a region called "Asialand or Asiahome". Odin was the king of the realm, and made Njörðr and Freyr temple priests. Freyja was the daughter of Njörðr, and was Odin's concubine. Odin deeply loved Freyja, and she was "the fairest of woman of that day". Freyja had a beautiful bower, and when the door was shut no one could enter without Freyja's permission.
 * Sörla þáttr

Chapter 1 records that one day Freyja passed by an open stone where dwarfs lived. Four dwarfs were smithying a golden necklace, and it was nearly done. Looking at the necklace, the dwarfs thought Freyja to be most fair, and she the necklace. Freyja offered to buy the collar from them with silver and gold and other items of value. The dwarfs said that they had no lack of money, and that for the necklace the only thing she could offer them would be a night with each of them. "Whether she liked it better or worse", Freyja agreed to the conditions, and so spent a night with each of the four dwarfs. The conditions were fulfilled and the necklace was hers. Freyja went home to her bower as if nothing happened.

As related in chapter 2, Loki, under the service of Odin, found out about Freyja's actions and told Odin. Odin told Loki to get the necklace and bring it to him. Loki said that since no one could enter Freyja's bower against her will, this would not be an easy task, yet Odin told him not to come back until he had found a way to get the necklace. Howling, Loki turned away and went to Freyja's bower but found it locked, and that he could not enter. So Loki transformed himself into a fly, and after having trouble finding even the tiniest of entrances, he managed to find a tiny hole at the gable-top, yet even here he had to squeeze through to enter.

Having made his way into Freyja's chambers, Loki looked around to be sure that no one was awake, and found that Freyja was asleep. He landed on her bed and noticed that she was wearing the necklace, the clasp turned downward. Loki turned into a flea and jumped onto Freyja's cheek and there bit her. Freyja stirred, turning about, and then fell asleep again. Loki removed his flea's shape and undid her collar, opened the bower, and returned to Odin.

The next morning Freyja woke and saw that the doors to her bower were open, yet unbroken, and that her precious necklace was gone. Freyja had an idea of who was responsible. She got dressed and went to Odin. She told Odin of the malice he had allowed against her and of the theft of her necklace, and that he should give her back her jewelry.

Odin said that, given how she obtained it, she would never get it back. That is, with one exception: she could have it back if she could make two kings, themselves ruling twenty kings each, battle one another, and cast a spell so that each time one of their numbers falls in battle, they will again spring up and fight again. And that this must go on eternally, unless a Christian man of a particular stature goes into the battle and smites them, only then will they stay dead. Freyja agreed.

Later Scandinavian folklore

Although the Christianization of Scandinavia sought to demonize the native gods, belief and reverence in the gods, including Freyja, persisted throughout the modern period and melded into Scandinavian folklore. comments that Freyja became a particular target under Christianization: Freyja's erotic qualities became an easy target for the new religion, in which an asexual virgin was the ideal woman [...] Freyja is called "a whore" and "a harlot" by the holy men and missionaries, whereas many of her functions in the everyday lives of men and women, such as protecting the vegetation and supplying assistance in childbirth were transferred to the Virgin Mary.

However, Freyja did not disappear. In Iceland, Freyja was called upon for assistance by way of Icelandic magical staves as late as the 18th century; and as late as the 19th century, Freyja is recorded as retaining elements of her role as a fertility goddess among rural Swedes.

The Old Norse poem Þrymskviða (or its source) continued into Scandinavian folk song tradition, where it was euhemerized and otherwise transformed over time. In Iceland, the poem became known as Þrylur, whereas in Denmark the poem became Thor af Havsgaard and in Sweden it became Torvisan or Hammarhämtningen. A section of the Swedish Torvisan, in which Freyja has been transformed into "the fair" (den väna) Frojenborg, reads as follows:

In the province of Småland, Sweden, an account is recorded connecting Freyja with sheet lightning in this respect. Writer Johan Alfred Göth recalled a Sunday in 1880 where men were walking in fields and looking at nearly ripened rye, where Måns in Karryd said: "Now Freyja is out watching if the rye is ripe". Along with this, Göth recalls another mention of Freyja in the countryside: When as a boy I was visiting the old Proud-Katrina, I was afraid of lightning like all boys in those days. When the sheet lightning flared at the night, Katrina said: "Don't be afraid little child, it is only Freyja who is out making fire with steel and flintstone to see if the rye is ripe. She is kind to people and she is only doing it to be of service, she is not like Thor, he slays both people and livestock, when he is in the mood" [...] I later heard several old folks talk of the same thing in the same way.

In Värend, Sweden, Freyja could also arrive at Christmas night and she used to shake the apple trees for the sake of a good harvest and consequently people left some apples in the trees for her sake. However, it was dangerous to leave the plough outdoors, because if Freyja sat on it, it would no longer be of any use.

Many Asatru practitioners today mostly honor Freyja as a goddess of fertility, abundance and beauty. A common rite for modern Freya worshippers is to bake foods that have some connection to love in one way or another, such as chocolate. Freyja is also called upon for protection, usually when it comes to a domestic violence situation.

Eponyms

Several plants were named after Freyja, such as Freyja's tears and Freyja's hair (Polygala vulgaris), but during the process of Christianization, the name of the goddess was replaced with that of the Virgin Mary. In the pre-Christian period, the Orion constellation was called either Frigg's distaff or Freyja's distaff (Swedish Frejerock).

Place names in Norway and Sweden reflect devotion to the goddess, including the Norwegian place name Frøihov (originally *Freyjuhof, literally "Freyja's hof") and Swedish place names such as Frövi (from *Freyjuvé, literally "Freyja's vé"). In a survey of toponyms in Norway, M. Olsen tallies at least 20 to 30 location names compounded with Freyja. Three of these place names appear to derive from *Freyjuhof ('Freyja's hof'), whereas the goddess's name is frequently otherwise compounded with words for 'meadow' (such as -þveit, -land) and similar land formations. These toponyms are attested most commonly on the west coast though a high frequency is found in the southeast.

Place names containing Freyja are yet more numerous and varied in Sweden, where they are widely distributed. A particular concentration is recorded in Uppland, among which a number derive from the above-mentioned *Freyjuvé and also *Freyjulundr ('Freyja's sacred grove'), place names that indicate public worship of Freyja. A variety of place names (such as Frøal and Fröale) have been seen as containing an element cognate to Gothic alhs and Old English ealh ("temple"), although these place names may be otherwise interpreted. In addition, Frejya appears as a compound element with a variety of words for geographic features such as fields, meadows, lakes and natural objects such as rocks.

The Freyja name Hörn appears in the Swedish place names Härnevi and Järnevi, stemming from the reconstructed Old Norse place name *Hörnar-vé (meaning "Hörn's vé").

Archaeological record and historic depictions

A priestess was buried c. 1000 with considerable splendour in Hagebyhöga in Östergötland. In addition to being buried with her wand, she had received great riches which included horses, a wagon and an Arabian bronze pitcher. There was also a silver pendant, which represents a woman with a broad necklace around her neck. This kind of necklace was only worn by the most prominent women during the Iron Age and some have interpreted it as Freyja's necklace Brísingamen. The pendant may represent Freyja herself.

A 7th-century phalara found in a "warrior grave" in what is now Eschwege in northwestern Germany features a female figure with two large braids flanked by two "cat-like" beings and holding a staff-like object. This figure has been interpreted as Freyja. This image may be connected to various B-type bracteates, referred to as the Fürstenberg-type, that may also depict the goddess; they "show a female figure, in a short skirt and double-looped hair, holding a stave or sceptre in her right hand and a double-cross feature in the left".

Upon its discovery, the 10th century Oseberg ship burial was found to contain a ceremonial wagon. One side of the ornate wagon features a depiction of nine cats. Scholars have linked this depiction to Freyja's cat-led chariot and a broader associations between the Vanir and wagons.

A 12th century depiction of a cloaked but otherwise nude woman riding a large cat appears on a wall in the Schleswig Cathedral in Schleswig-Holstein, Northern Germany. Beside her is similarly a cloaked yet otherwise nude woman riding a distaff. Due to iconographic similarities to the literary record, these figures have been theorized as depictions of Freyja and Frigg respectively.

Relation to Frigg and other goddesses and figures
Due to numerous similarities, scholars have frequently connected Freyja with the goddess Frigg. The connection with Frigg and question of possible earlier identification of Freyja with Frigg in the Proto-Germanic period (Frigg and Freyja common origin hypothesis) remains a matter of scholarly discourse. Regarding a Freyja-Frigg common origin hypothesis, scholar Stephan Grundy comments, "the problem of whether Frigg or Freyja may have been a single goddess originally is a difficult one, made more so by the scantiness of pre-Viking Age references to Germanic goddesses, and the diverse quality of the sources. The best that can be done is to survey the arguments for and against their identity, and to see how well each can be supported."

Like the name of the group of gods to which Freyja belongs, the Vanir, the name Freyja is not attested outside of Scandinavia, as opposed to the name of the goddess Frigg, who is attested as a goddess common among the Germanic peoples, and whose name is reconstructed as Proto-Germanic *Frijjō. Similar proof for the existence of a common Germanic goddess from which Freyja descends does not exist, but scholars have commented that this may simply be due to lack of evidence.

In the Poetic Edda poem Völuspá, a figure by the name of Gullveig is burnt three times yet is three times reborn. After her third rebirth, she is known as Heiðr. This event is generally accepted as precipitating the Æsir–Vanir War. Starting with scholar Gabriel Turville-Petre, scholars such as Rudolf Simek, Andy Orchard, and John Lindow have theorized that Gullveig/Heiðr is the same figure as Freyja, and that her involvement with the Æsir somehow led to the events of the Æsir–Vanir War.

Outside of theories connecting Freyja with the goddess Frigg, some scholars, such as Hilda Ellis Davidson and, have theorized that other goddesses in Norse mythology, such as Gefjon, Gerðr, and Skaði, may be forms of Freyja in different roles or ages.

Receiver of the slain
Freyja and her afterlife field Fólkvangr, where she receives half of the slain, have been theorized as connected to the valkyries. Scholar Britt-Mari Näsström points out the description in Gylfaginning where it is said of Freyja that "whenever she rides into battle she takes half of the slain", and interprets Fólkvangr as "the field of the Warriors". Näsström notes that, just like Odin, Freyja receives slain heroes who have died on the battlefield, and that her house is Sessrumnir (which she translates as "filled with many seats"), a dwelling that Näsström posits likely fills the same function as Valhalla. Näsström comments that "still, we must ask why there are two heroic paradises in the Old Norse view of afterlife. It might possibly be a consequence of different forms of initiation of warriors, where one part seemed to have belonged to Óðinn and the other to Freyja. These examples indicate that Freyja was a war-goddess, and she even appears as a valkyrie, literally 'the one who chooses the slain'."

Siegfried Andres Dobat comments that "in her mythological role as the chooser of half the fallen warriors for her death realm Fólkvangr, the goddess Freyja, however, emerges as the mythological role model for the Valkyrjar [sic] and the dísir."

The Oriental hypothesis
Gustav Neckel, writing in 1920, connects Freyja to the Phrygian goddess Cybele. According to Neckel, both goddesses can be interpreted as "fertility goddesses" and other potential resemblances have been noted. Some scholars have suggested that the image of Cybele subsequently influenced the iconography of Freyja, the lions drawing the former's chariot becoming large cats. These observations became an extremely common observation in works regarding Old Norse religion until at least the early 1990s. In her book-length study of scholarship on the topic of Freyja, Britt-Mari Näsström (1995) is highly critical of this deduction; Näsström says that "these 'parallels' are due to sheer ignorance about the characteristics of Cybele; scholars have not troubled to look into the resemblances and differences between the two goddesses, if any, in support for their arguments for a common origin."

In art and literature

Into the modern period, Freyja was treated as a Scandinavian counterpart to the Roman Venus in, for example, Swedish literature, where the goddess may be associated with romantic love or, conversely, simply as a synonym for "lust and potency". In the 18th century, Swedish poet Carl Michael Bellman referred to Stockholm prostitutes in his Fredman's Epistles as "the children of Fröja". In the 19th century, Britt-Mari Näsström observes, Swedish Romanticism focused less on Freyja's erotic qualities and more on the image of "the pining goddess, weeping for her husband".

Freyja is mentioned in the first stanza ("it is called old Denmark and it is Freja's hall") of the civil national anthem of Denmark, Der er et yndigt land, written by 19th century Danish poet Adam Gottlob Oehlenschläger in 1819. In addition, Oehlenschläger wrote a comedy entitled Freyjas alter (1818) and a poem Freais sal featuring the goddess.

The 19th century German composer Richard Wagner's Der Ring des Nibelungen opera cycle features Freia, the goddess Freyja combined with the apple-bearing goddess Iðunn.

In late 19th century and early 20th century Northern Europe, Freyja was the subject of numerous works of art, including Freyja by H. E. Freund (statue, 1821–1822), Freja sökande sin make (painting, 1852) by Nils Blommér, Freyjas Aufnahme uner den Göttern (charcoal drawing, 1881), and Frigg; Freyja (drawing, 1883) by, Freyja (1901) by Carl Emil Doepler d. J., and Freyja and the Brisingamen by J. Doyle Penrose (painting, 1862–1932). Like other Norse goddesses, her name was applied widely in Scandinavia to, for example, "sweetmeats or to stout carthorses".

Vanadís, one of Freyja's names, is the source of the name of the chemical element vanadium, so named because of its many colored compounds.

In popular culture
Frigga (sometimes called Freyja) is a fictional character appearing in Marvel Comics starting in 1963. The character in particular is based on goddess Freyja of the Norse mythology.

Starting in the early 1990s, derivatives of Freyja began to appear as a given name for girls. According to the Norwegian name database from the Central Statistics Bureau, around 500 women are listed with the first name Frøya (the modern Norwegian spelling of the goddess's name) in the country. There are also several similar names, such as the first element of the dithematic personal name Frøydis.

Freyja is featured in several video games including the 2002 Ensemble Studios game Age of Mythology, the 2014 third-person multiplayer online battle arena game Smite, the 2018 Santa Monica Studio game God of War, and in its 2022 sequel God of War Ragnarök.

"Freya" is a song by American heavy metal band The Sword from their 2006 debut album Age of Winters. A playable cover version was featured in Guitar Hero II, released the same year.

title: Freyja