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In a body submerged in a [[fluid]], unsteady forces due to [[acceleration]] of that body with respect to the fluid, can be divided into two parts: the [[virtual mass effect]] and the '''Basset force'''. | |||
The Basset force term describes the force due to the lagging [[boundary layer]] development with changing relative velocity (acceleration) of bodies moving through a fluid.<ref>C. Crowe et al., Multiphase flows with droplets and particles, CRC Press, 1998, ISBN 0-8493-9469-4, p. 81</ref> | |||
The Basset term accounts for [[viscosity|viscous]] effects and addresses the temporal delay in boundary layer development as the relative velocity changes with time. It is also known as the "history" term. | |||
The Basset force is difficult to implement and is commonly neglected for practical reasons; however, it can be substantially large when the body is accelerated at a high rate.<ref>R.W. Johnson, The handbook of fluid dynamics, CRC Press, 1998, ISBN 0-8493-2509-9, pp. 18–3</ref> | |||
== | This force in an accelerating [[Stokes flow]] has been proposed by [[Joseph Valentin Boussinesq]] in 1885 and [[Alfred Barnard Basset]] in 1888. Consequently, it is also referred to as the '''Boussinesq–Basset force'''.<ref name=CAS2004>{{citation | author1 = F. Candelier | author2 = J. R. Angilella | author3 = M. Souhar | title = On the effect of the Boussinesq–Basset force on the radial migration of a Stokes particle in a vortex | year = 2004 | journal = Physics of Fluids | volume = 16 | issue = 5 | pages = 1765–1776 | doi = 10.1063/1.1689970 |bibcode = 2004PhFl...16.1765C }}</ref><ref name=Michaelides2003>{{ citation | author = E. E. Michaelides | title = Hydrodynamic force and heat/mass transfer from particles, bubbles, and drops—The Freeman Scholar Lecture | year = 2003 | journal = Journal of Fluids Engineering | volume = 125 | issue = 2 | pages = 209–238 | doi = 10.1115/1.1537258 }}</ref> | ||
==Acceleration of a flat plate== | |||
Consider an infinitely large plate started impulsively with a step change in velocity—from 0 to ''u<sub>0</sub>''—in the direction of the plate–fluid interface plane. | |||
The equation of motion for the fluid—[[Stokes flow]] at low [[Reynolds number]]—is | |||
= | :<math>\frac{\partial u}{\partial t}=\nu_c\,\frac{\partial^2u}{\partial y^2},</math> | ||
where ''u''(''y'',''t'') is the velocity of the fluid, at some time ''t'', parallel to the plate, at a distance ''y'' from the plate, and ''ν<sub>c</sub>'' is the [[kinematic viscosity]] of the fluid (c~continuous phase). | |||
The solution to this equation is,<ref>{{citation | author=F. M. White | title=Viscous fluid flow | at=Chapter 3 | origyear=2006 | publisher=McGraw Hill | publication-place=New York | year=2006 }}</ref> | |||
== | :<math>u=u_0 - u_0\, \operatorname{erf}\left(\frac{y}{2\sqrt{\nu_ct}}\right) = u_0 \operatorname{erfc}\left(\frac{y}{2\sqrt{\nu_ct}}\right),</math> | ||
where erf and erfc denote the [[error function]] and the [[complementary error function]], respectively. | |||
Assuming that an acceleration of the plate can be broken up into a series of such step changes in the velocity, it can be shown that the cumulative effect on the shear stress on the plate is | |||
:<math>\tau=\sqrt{\frac{\rho_c\mu_c}{\pi}}\int\limits_0^t\frac{\frac{\partial u_p}{\partial t'}}{\sqrt{t-t'}} \, dt', </math> | |||
where ''u<sub>p</sub>(t)'' is the velocity of the plate, ''ρ<sub>c</sub>'' is the [[mass density]] of the fluid, and ''μ<sub>c</sub>'' is the [[viscosity]] of the fluid. | |||
==Acceleration of a spherical particle== | |||
[[Joseph Valentin Boussinesq|Boussinesq]] (1885) and Basset (1888) found that the force '''''F''''' on an accelerating spherical particle in a viscous fluid is<ref name=CAS2004/><ref name=Michaelides2003/><ref>{{citation | author=J. V. Boussinesq |title=Sur la résistance qu’oppose un fluide indéfini au repos, sans pesanteur, au mouvement varié d’une sphère solide qu’il mouille sur toute sa surface, quand les vitesses restent bien continues et assez faibles pour que leurs carrés et produits soient négligeables | journal=Comptes Rendu de l'Academie des Sciences | volume=100 | pages=935–937 | year=1885 }}</ref><ref>{{citation | author=A. B. Basset | title=Treatise on hydrodynamics | at=Chapter 22 | volume=2 | origyear=1888 | publisher=Deighton, Bell and Co. | publication-place=Cambridge | year=1961 }}</ref> | |||
:<math>\mathbf{F}=\frac{3}{2}D^2\sqrt{\pi\rho_c\mu_c}\int\limits_0^t\frac{\frac{D\mathbf{u}}{Dt'}-\frac{\partial\mathbf{v}}{\partial t'}}{\sqrt{t-t'}}dt',</math> | |||
where ''D'' is the particle diameter, {{frac|D|Dt'}} is the [[material derivative]], and '''''u''''' and '''''v''''' are the fluid and particle velocity vectors, respectively. | |||
==See also== | |||
* [[Basset–Boussinesq–Oseen equation]] | |||
* [[Stokes boundary layer]] | |||
==References== | |||
<references/> | |||
{{DEFAULTSORT:Basset Force}} | |||
[[Category:Fluid dynamics]] | |||
Revision as of 14:44, 31 January 2014
In a body submerged in a fluid, unsteady forces due to acceleration of that body with respect to the fluid, can be divided into two parts: the virtual mass effect and the Basset force.
The Basset force term describes the force due to the lagging boundary layer development with changing relative velocity (acceleration) of bodies moving through a fluid.[1] The Basset term accounts for viscous effects and addresses the temporal delay in boundary layer development as the relative velocity changes with time. It is also known as the "history" term. The Basset force is difficult to implement and is commonly neglected for practical reasons; however, it can be substantially large when the body is accelerated at a high rate.[2]
This force in an accelerating Stokes flow has been proposed by Joseph Valentin Boussinesq in 1885 and Alfred Barnard Basset in 1888. Consequently, it is also referred to as the Boussinesq–Basset force.[3][4]
Acceleration of a flat plate
Consider an infinitely large plate started impulsively with a step change in velocity—from 0 to u0—in the direction of the plate–fluid interface plane.
The equation of motion for the fluid—Stokes flow at low Reynolds number—is
where u(y,t) is the velocity of the fluid, at some time t, parallel to the plate, at a distance y from the plate, and νc is the kinematic viscosity of the fluid (c~continuous phase). The solution to this equation is,[5]
where erf and erfc denote the error function and the complementary error function, respectively.
Assuming that an acceleration of the plate can be broken up into a series of such step changes in the velocity, it can be shown that the cumulative effect on the shear stress on the plate is
where up(t) is the velocity of the plate, ρc is the mass density of the fluid, and μc is the viscosity of the fluid.
Acceleration of a spherical particle
Boussinesq (1885) and Basset (1888) found that the force F on an accelerating spherical particle in a viscous fluid is[3][4][6][7]
where D is the particle diameter, Template:Frac is the material derivative, and u and v are the fluid and particle velocity vectors, respectively.
See also
References
- ↑ C. Crowe et al., Multiphase flows with droplets and particles, CRC Press, 1998, ISBN 0-8493-9469-4, p. 81
- ↑ R.W. Johnson, The handbook of fluid dynamics, CRC Press, 1998, ISBN 0-8493-2509-9, pp. 18–3
- ↑ 3.0 3.1 Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - ↑ 4.0 4.1 Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010