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In [[fluid mechanics]], '''added mass''' or '''virtual mass''' is the [[inertia]] added to a system because an accelerating or decelerating body must move (or deflect) some [[volume]] of surrounding [[fluid]] as it moves through it. Added mass is a common issue because the object and surrounding fluid cannot occupy the same physical space simultaneously. For simplicity this can be modeled as some volume of fluid moving with the object, though in reality "all" the fluid will be accelerated, to various degrees. | |||
The [[dimensionless]] '''added mass coefficient''' is the added mass divided by the displaced fluid mass – i.e. divided by the fluid [[density]] times the volume of the body. In general, the added mass is a second-order [[tensor]], relating the fluid acceleration [[Euclidean vector|vector]] to the resulting [[force]] vector on the body.<ref>{{Cite book | last=Newman | first=John Nicholas | authorlink=John Nicholas Newman | title=Marine hydrodynamics | year=1977 | publisher=[[MIT Press]] | location=Cambridge, Massachusetts | isbn=0-262-14026-8 | at=§4.13, p. 139 }}</ref> | |||
==Background== | |||
[[Friedrich Bessel]] proposed the concept of added mass in 1828 to describe the motion of a [[pendulum]] in a fluid. The period of such a pendulum increased relative to its period in a vacuum (even after accounting for [[buoyancy]] effects), indicating that the surrounding fluid increased the effective mass of the system.<ref>{{cite journal | authorlink=George Gabriel Stokes | first=G. G. | last=Stokes | journal=Transactions of the Cambridge Philosophical Society | title=On the effect of the internal friction of fluids on the motion of pendulums | volume=9 | pages=8–106 | year=1851|bibcode = 1851TCaPS...9....8S }}</ref> | |||
The concept of added mass is arguably the first example of renormalization in physics.<ref>{{Cite book | publisher = Springer | isbn = 978-3-540-60503-4 | last1 = González | first1 = José | last2 = Martín-Delgado| first2=Miguel A. | last3 = Sierra | first3 = Germán | last4 = Vozmediano | first4 = Angeles H. | title = Quantum electron liquids and high-T<sub>c</sub> superconductivity | year = 1995 | page =32 }}</ref><ref name=Falkovich>{{Cite book | last=Falkovich | first=Gregory | year=2011 | title=Fluid Mechanics, a short course for physicists |at=Section 1.3|url=http://www.cambridge.org/gb/knowledge/isbn/item6173728/?site_locale=en_GB | publisher=Cambridge University Press | isbn=978-1-107-00575-4 }}</ref><ref>{{cite journal |title=The added mass coefficient of a dispersion of spherical gas bubbles in liquid |journal=International Journal of Multiphase Flow | year=1989 | volume=15 |issue=6 |pages=911–924 |first1=A. |last1=Biesheuvel |first2=S. |last2=Spoelstra | doi=10.1016/0301-9322(89)90020-7 }}</ref> | |||
The concept can also be thought of as a classical physics analogue of the quantum mechanical concept of [[quasiparticle]]s. It is, however, not to be confused with [[relativistic mass]] increase. | |||
It is often erroneously stated that the added mass is determined by the momentum of the fluid. That it is not so is clear from considering the case of the fluid in a large box where the fluid momentum is exactly zero at every moment of time. The added mass is actually determined by the quasi-momentum: the added mass times the body acceleration is equal to the time derivative of the fluid quasi-momentum.<ref name=Falkovich/> | |||
==Virtual mass force== | |||
Unsteady forces due to a change of the relative velocity of a body submerged in a fluid can be divided into two parts: the virtual mass effect and the [[Basset force]]. | |||
The origin of the force is that the fluid will gain kinetic energy at the expense of the work done by an accelerating submerged body. | |||
It can be shown that the virtual mass force, for a spherical particle submerged in an inviscid, incompressible fluid is<ref>{{cite book |first1=Clayton T. |last1=Crowe |first2=Martin |last2=Sommerfeld |first3=Yutaka |last3=Tsuji |title=Multiphase flows with droplets and particles |publisher=CRC Press |year=1998 |isbn=0-8493-9469-4 |page=81}}</ref> | |||
::<math>\mathbf{F}=\frac{\rho_\mathrm{c}V_\mathrm{p}}{2}\left(\frac{\mathrm{D}\mathbf{u}}{\mathrm{D}t}-\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}\right),</math> | |||
where bold symbols denote vectors, <math>\mathbf{u}</math> is the fluid [[flow velocity]], <math>\mathbf{v}</math> is the spherical particle velocity, <math>\rho_\mathrm{c}</math> is the [[mass density]] of the [[fluid]] (continuous phase), <math>V_\mathrm{p}</math> is the volume of the particle, and D/D''t'' denotes the [[material derivative]]. | |||
The origin of the notion "virtual mass" becomes evident when we take a look at the momentum equation for the particle. | |||
::<math>m_\mathrm{p}\frac{\mathrm{d}\mathbf v_\mathrm{p}}{\mathrm{d}t}=\sum\mathbf F + \frac{\rho_\mathrm{c}V_\mathrm{p}}{2}\left(\frac{\mathrm{D}\mathbf u}{\mathrm{D}t}-\frac{\mathrm{d}\mathbf v}{\mathrm{d}t}\right),</math> | |||
where <math>\sum\mathbf F</math> is the sum of all other force terms on the particle, such as [[gravity]], [[pressure gradient]], [[Drag (physics)|drag]], [[Lift (force)|lift]], [[Basset force]], etc. | |||
Moving the derivative of the particle velocity from the right hand side of the equation to the left we get | |||
::<math>\left(m_\mathrm{p}+\frac{\rho_\mathrm{c}V_\mathrm{p}}{2}\right)\frac{\mathrm{d}\mathbf v_\mathrm{p}}{\mathrm{d}t}=\sum\mathbf F + \frac{\rho_\mathrm{c}V_\mathrm{p}}{2}\frac{\mathrm{D}\mathbf u}{\mathrm{D}t},</math> | |||
so the particle is accelerated as if it had an added mass of half the fluid it displaces, and there is also an additional force contribution on the right hand side due to acceleration of the fluid. | |||
==Applications== | |||
The added mass can be incorporated into most physics equations by considering an effective mass as the sum of the mass and added mass. This sum is commonly known as the "virtual mass". | |||
A simple formulation of the added mass for a spherical body permits Newton's classical second law to be written in the form | |||
:<math>F = m\,a</math> {{pad|2em}} becomes {{pad|2em}} <math>F = (m + m_\text{added})\,a.</math> | |||
One can show that the added mass for a sphere (of radius <math>r</math>) is <math>\tfrac{2}{3} \pi r^3 \rho_\text{fluid}.</math> For a general body, the added mass becomes a [[tensor]] (referred to as the induced mass tensor), with components depending on the direction of motion of the body. Not all elements in the added mass tensor will have dimension mass, some will be mass × length and some will be mass × length<sup>2</sup>. | |||
All bodies accelerating in a fluid will be affected by added mass, but since the added mass is dependent on the density of the fluid, the effect is often neglected for dense bodies falling in much less dense fluids. For situations where the density of the fluid is comparable to or greater than the density of the body, the added mass can often be greater than the mass of the body and neglecting it can introduce significant errors into a calculation. | |||
For example, a spherical air bubble rising in water has a mass of <math>\tfrac{4}{3} \pi r^3 \rho_\text{air}</math> but an added mass of <math>\tfrac{2}{3} \pi r^3 \rho_\text{water}.</math> Since water is approximately 800 times denser than air (at [[Standard conditions for temperature and pressure|RTP]]), the added mass in this case is approximately 400 times the mass of the bubble. | |||
===Naval architecture=== | |||
These principles also apply to ships, submarines, and offshore platforms. In ship design, the energy required to accelerate the added mass must be taken into account when performing a sea keeping analysis. For ships, the added mass can easily reach ¼ or ⅓ of the mass of the ship and therefore represents a significant [[inertia]], in addition to frictional and wavemaking [[drag force]]s. | |||
In aircraft (other than lighter-than-air balloons and blimps), the added mass is not usually taken into account because the density of the air is so small. | |||
== See also == | |||
* [[Basset force]] for describing the effect of the body's relative motion history on the [[viscosity|viscous]] forces in a [[Stokes flow]] | |||
* [[Basset–Boussinesq–Oseen equation]] for the description of the motion of – and forces on – a particle moving in an [[unsteady flow]] at low Reynolds numbers | |||
* [[Darwin drift]] for the relation between added mass and the Darwin drift volume | |||
* [[Response amplitude operator|Response Amplitude Operator]] for the use of added mass in ship design | |||
* [[Keulegan–Carpenter number]] for a dimensionless parameter giving the relative importance of the [[drag (physics)|drag]] force to inertia in [[wave loading]] | |||
* [[Morison equation]] for an empirical force model in wave loading, involving added mass and drag | |||
== References == | |||
{{reflist}} | |||
== External links == | |||
*[http://web.mit.edu/2.016/www/labs/L01_Added_Mass_050915.pdf MIT OpenCourse Ware] | |||
*[http://authors.library.caltech.edu/233/01/BRE052.pdf Naval Civil Engineering Laboratory] | |||
{{DEFAULTSORT:Added mass}} | |||
[[Category:Fluid dynamics]] | |||
Latest revision as of 20:10, 13 November 2013
In fluid mechanics, added mass or virtual mass is the inertia added to a system because an accelerating or decelerating body must move (or deflect) some volume of surrounding fluid as it moves through it. Added mass is a common issue because the object and surrounding fluid cannot occupy the same physical space simultaneously. For simplicity this can be modeled as some volume of fluid moving with the object, though in reality "all" the fluid will be accelerated, to various degrees.
The dimensionless added mass coefficient is the added mass divided by the displaced fluid mass – i.e. divided by the fluid density times the volume of the body. In general, the added mass is a second-order tensor, relating the fluid acceleration vector to the resulting force vector on the body.[1]
Background
Friedrich Bessel proposed the concept of added mass in 1828 to describe the motion of a pendulum in a fluid. The period of such a pendulum increased relative to its period in a vacuum (even after accounting for buoyancy effects), indicating that the surrounding fluid increased the effective mass of the system.[2]
The concept of added mass is arguably the first example of renormalization in physics.[3][4][5] The concept can also be thought of as a classical physics analogue of the quantum mechanical concept of quasiparticles. It is, however, not to be confused with relativistic mass increase.
It is often erroneously stated that the added mass is determined by the momentum of the fluid. That it is not so is clear from considering the case of the fluid in a large box where the fluid momentum is exactly zero at every moment of time. The added mass is actually determined by the quasi-momentum: the added mass times the body acceleration is equal to the time derivative of the fluid quasi-momentum.[4]
Virtual mass force
Unsteady forces due to a change of the relative velocity of a body submerged in a fluid can be divided into two parts: the virtual mass effect and the Basset force.
The origin of the force is that the fluid will gain kinetic energy at the expense of the work done by an accelerating submerged body.
It can be shown that the virtual mass force, for a spherical particle submerged in an inviscid, incompressible fluid is[6]
where bold symbols denote vectors, is the fluid flow velocity, is the spherical particle velocity, is the mass density of the fluid (continuous phase), is the volume of the particle, and D/Dt denotes the material derivative.
The origin of the notion "virtual mass" becomes evident when we take a look at the momentum equation for the particle.
where is the sum of all other force terms on the particle, such as gravity, pressure gradient, drag, lift, Basset force, etc.
Moving the derivative of the particle velocity from the right hand side of the equation to the left we get
so the particle is accelerated as if it had an added mass of half the fluid it displaces, and there is also an additional force contribution on the right hand side due to acceleration of the fluid.
Applications
The added mass can be incorporated into most physics equations by considering an effective mass as the sum of the mass and added mass. This sum is commonly known as the "virtual mass".
A simple formulation of the added mass for a spherical body permits Newton's classical second law to be written in the form
- Template:Pad becomes Template:Pad
One can show that the added mass for a sphere (of radius ) is For a general body, the added mass becomes a tensor (referred to as the induced mass tensor), with components depending on the direction of motion of the body. Not all elements in the added mass tensor will have dimension mass, some will be mass × length and some will be mass × length2.
All bodies accelerating in a fluid will be affected by added mass, but since the added mass is dependent on the density of the fluid, the effect is often neglected for dense bodies falling in much less dense fluids. For situations where the density of the fluid is comparable to or greater than the density of the body, the added mass can often be greater than the mass of the body and neglecting it can introduce significant errors into a calculation.
For example, a spherical air bubble rising in water has a mass of but an added mass of Since water is approximately 800 times denser than air (at RTP), the added mass in this case is approximately 400 times the mass of the bubble.
These principles also apply to ships, submarines, and offshore platforms. In ship design, the energy required to accelerate the added mass must be taken into account when performing a sea keeping analysis. For ships, the added mass can easily reach ¼ or ⅓ of the mass of the ship and therefore represents a significant inertia, in addition to frictional and wavemaking drag forces.
In aircraft (other than lighter-than-air balloons and blimps), the added mass is not usually taken into account because the density of the air is so small.
See also
- Basset force for describing the effect of the body's relative motion history on the viscous forces in a Stokes flow
- Basset–Boussinesq–Oseen equation for the description of the motion of – and forces on – a particle moving in an unsteady flow at low Reynolds numbers
- Darwin drift for the relation between added mass and the Darwin drift volume
- Response Amplitude Operator for the use of added mass in ship design
- Keulegan–Carpenter number for a dimensionless parameter giving the relative importance of the drag force to inertia in wave loading
- Morison equation for an empirical force model in wave loading, involving added mass and drag
References
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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534