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Irrotational vector fields

The vector field a is irrotational if its curl vanishes everywhere. By Stokes' theorem the circulation around any closed curve also vanishes. Also, an irrotational vector field can be expressed as the gradient of a scalar.

× a = 0 C a t d s = 0 a = ϕ a an irrotational field

The velocity field of motions where the viscous effects are insignificant compared to inertial effects and the flow is initially irrotational can be approximated as an irrotational velocity field.

Solenoidal vector fields

A solenoidal vector field is defined as one in which the divergence vanishes. This implies that the flux across a closed surface must also vanish. A vector identity states that the divergence of the curl of a vector is zero. Thus a continuously differentiable solenoidal vector field has the following three equivalent characteristics.

a = 0 S a n d S = 0 a = × A a a solenoidal field

The velocity field of motions where the effects of compressibility are insignificant can be approximated as a solenoidal vector field. The surface integral of velocity vanishing over any closed surface means that the net volumetric flow across closed surfaces is zero. Incompressible flow fields can be expressed as the curl of a vector potential. Two-dimensional, incompressible flows have only one nonzero component of the vector potential and this is identified as the stream function.

Helmholtz' representation

We found that an irrotational vector is the gradient of a scalar potential and a solenoidal vector is the curl of a vector potential. Here we show that any vector field with sufficient continuity is divisible into irrotational and solenoidal parts, and so is expressible in terms of a scalar and a vector potential. The fundamental problem in the analysis of a vector field is the determination of these potentials and their expression in terms of the essential characteristics of the vector, namely divergence, curl, discontinuities, and boundary values. For when the potentials are known the vector itself can be determined by differentiation. The following analysis is taken from H. B. Phillips, Vector Analysis , John Wiley&Sons, 1933. The following nomenclature will differ somewhat in that the vector is expressed as the negative of the gradient of a scalar. Also, the vector field of interest will be denoted as m a t h b f F . Bold face capital letters will also be used for other vector quantities. Also the equations have the 4 π factor of electromagnetism in mks units rather than the factors ε o and ε o c 2 of the SI units.

Let V be a region of space where the vector field F has piecewise continuous second derivatives, S 1 be surfaces of discontinuity of F , and S be the bounding surface of V . The Helmholtz's theorem states that F can be expressed in terms of the potentials.

F = - ϕ + × A where ϕ ( x P ) = V ρ d V r + S 1 σ d S r - 1 4 π S n F d S r A ( x P ) = V I d V r + S 1 J d S r - 1 4 π S n × F d S r F = - 2 ϕ = 4 π ρ × F = × × A = 4 π I n Δ F = 4 π σ n × Δ F = 4 π J r = x P - x Q where x Q is coordinate of integrand

The vectors I and J are not arbitrary. They are subject to the equation of continuity A = 0 . The effect of this condition is to make I and J behave like space and surface currents of something which is nowhere created or destroyed. In the electromagnetic field they usually represent currents of electricity. In hydrodynamic fields they represent vorticity. If A is everywhere solenoidal, the following three equations must then be everywhere satisfied.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
?
Kyle
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Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
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Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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Source:  OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
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