Maxwell Stress Tensor In Hydrodynamics
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1) A stress tensor is defined for vorticity fields in hydrodynamics, analogous to Maxwell stress tensor in electromagnetism. This vorticity stress tensor represents forces on surfaces in a vorticity field. 2) The vorticity stress tensor can be decomposed into terms representing tension along vortex lines and pressure normal to vortex lines. 3) This vorticity stress tensor establishes a theoretical framework for studying forces between vortex lines and deformation of fluid volumes due to vorticity.
![IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. V (Jan - Feb. 2015), PP 58-60
www.iosrjournals.org
DOI: 10.9790/5728-11155860 www.iosrjournals.org 58 | Page
Maxwell Stress Tensor In Hydrodynamics
M. J. Vedan, M. B. Rajeswari Devi, Susan Mathew Panakkal
1
Department of Computer Applications, Cochin University of Science and Technology, Cochin, India.
2
Department of Mathematics, Cochin University of Science and Technology, Cochin, India.
3
Department of Mathematics, St.Teresa’s College, Cochin, India.
Abstract: Analogous to Maxwell stress tensor in electric and magnetic fields, a stress tensor is defined in a
vorticity field. Thus by treating vortices as physical structures, it is possible to study the forces on a surface
element in it. Based on this the force between vortex lines, the pressure and the shearing stress that deform the
volume element can also be defined.
Keywords: Enstrophy, Maxwell stress tensor, static pressure, vorticity equation, vorticity stress tensor.
I. Introduction
Though the electromagnetic theory was developed more than a century back, the practice of calculating
forces on magnetic media remained ambiguous (Casperson, 2002)[1]. In particular, the application of Maxwell
stress tensor gave way to other methods based on energy variation principle (Shwarz, 1963) [2](Sher, 1968) [3]
(Pohl and Crane, 1972)[4] and effective dipole or multipole method (Wang, 1996)[5]. The method was revived
by Washizu and Jones (1996) [6], Sauer and Schlogl (1985) [7] etc.
Wang, Wang and Gascoyne (1997)[8] explored the application of the Maxwell Stress Tensor (MST)
method to dielectrophoresis (DEP) and electrorotation (ROT) studies. By integrating Maxwell stress tensor over
the surface of a dielectric particle, they derived the general expressions for DEP and ROT generated by electric
fields of arbitrary configurations. As they claim, this was the first time that, such complete expressions were
derived from the first principles using MST formalism.
The analogy between the electromagnetic and fluid dynamic equations was first noted by Maxwell
(1861) [9]. He suggested that the vector potential A of the magnetic induction B represents some kind of a
fluid velocity field. This was interpreted in the light of Fizeau’s experiment by Cook, Fearn and Milonni
(1995)[10]. The analogy between Navier Stoke’s equation and Maxwell’s equations was used by Marmanis
(1997-1998) [11] in the development of his metafluid dynamics for the study of turbulence. This marked the
beginning of introducing new flow parameters (Lamb vectors). To complete the analogy between
electromagnetic and fluid dynamic equation, Scofield and Huq (2008, 2009, 2010)[12] [13] [14] introduced the
concept of vortex field and developed a uniform theory for electro dynamic, fluid dynamic and gravitational
fields. A main source of interest in the study of the analogy was the theoretical developments in ferrofluids
following its synthesis in 1960’s.” The Maxwell Stress Tensor and the forces in magnetic liquids” by Klaus
Stierstadt and Mario Liu (2014)[15] can be considered as the present state of the theory. In this paper we explore
yet another analogy between Maxwell stress tensor in electromagnetic continuum and a stress in incompressible
inviscid fluids whose origin is vorticity.
II. Vorticity Stress Tensor
In the case of incompressible flow of an inviscid fluid, the vorticity field is frozen-in and satisfies the equation:
∂𝛚
∂t
= ∇ × (𝛝 × 𝛚) (1)
Here ϑ is the velocity and 𝛚 = ∇ × 𝛝 is the vorticity. This equation admits the solution given by:
ωi X, t = ωj(x, 0)
∂Xi
∂xj
(2)
This is the well-known Cauchy’s equation, which relates the current vorticity (at Xi) to the initial
velocity (at xi), and thus it establishes a topological equivalence between them. Corresponding to the magnetic
energy, the energy associated with the vorticity field is defined by
1
2
ω2
. Thus the rate of change of energy is:
∂
∂t
ω2
2
= 𝛚 ∙
∂𝛚
∂t
= 𝛚 ∙ ∇ × (𝛝 × 𝛚) [from (1)]
= −𝛝 ∙ ∇ × 𝛚 × 𝛚 + ∇ ∙ 𝛝 × 𝛚 × 𝛚
= −𝛝 ∙ 𝐟 × 𝛚 + ∇ ∙ 𝛝 × 𝛚 × 𝛚 (3)](https://image.slidesharecdn.com/i011155860-151126040518-lva1-app6892/85/Maxwell-Stress-Tensor-In-Hydrodynamics-1-320.jpg)
![Maxwell Stress Tensor in Hydrodynamics
DOI: 10.9790/5728-11155860 www.iosrjournals.org 59 | Page
where 𝐟 = ∇ × 𝛚 is the flexion field.
If the vorticity is confined to a sub-domain of the fluid, the divergence term vanishes on integrating
over the entire volume.
Thus we get:
dM
dt
= − 𝛝 ∙ 𝐅 dVV
(4)
Here M =
ω2
2
dVV
is the total enstrophy and 𝐅 = 𝐟 × 𝛚 is a force analogous to Lorentz force.
The ith
component of this force 𝐅 is:
Fi =
∂
∂xj
ij , where ij = ωiωj −
ω2
2
δij (5)
and δij is the kronecker delta.
Thus analogous to the Maxwell Stress tensor associated to Lorentz force (Ferraro and Plumpton 1966)
[16] we get ij as the vorticity stress tensor. This tensor is related to the enstrophy in the same way, as
magnetic stress tensor is associated to magnetic energy. If we consider a vortex, the normal component of this
stress represents the tension in the line vortices and the terms ωiωj are the shearing forces between adjacent
vortex lines of the filament whose limiting case is the line vortex.
ij being a symmetric matrix can be diagonalized . Choosing the principal axis OX1 in the
direction of vorticity we get:
=
ω2
2
0 0
0
−ω2
2
0
0 0
−ω2
2
ij (6)
Thus the principal stress tensor constitutes a tension
1
2
ω2
along the line vortex and an equal pressure
normal to it.
We can rewrite this tensor as:
ij
ω2
0 0
0 0 0
0 0 0
+
−ω2
2
0 0
0
−ω2
2
0
0 0
−ω2
2
(7)
where the first matrix gives the effect of the force 𝐅 and the second matrix represents the static pressure.
The above expressions for the vorticity stress tensor can be compared to the magnetic stress tensor as
discussed by Steirstadt and Liu (2014)[17]. So the figures given by them apply to vortex stress tensor also. They
consider electromagnetic stress tensor (EMST) and discuss magnetic stress tensor. What we find here is that a
similar stress tensor exists in the case of vortex fields.
In the context of magneto hydrodynamics the analogy between the induction equation:
∂𝐁
∂t
= ∇ × 𝛝 × 𝐁 + λ∇2
𝐁 ∇ ∙ 𝐁 = 0 (8)
and the vorticity equation:
∂𝛚
∂t
= ∇ × 𝛝 × 𝛚 + ν∇2
𝛚 ∇ ∙ 𝛚 = 0 (9)
for barotropic flow of a fluid was first pointed out by Elsasser (1946)[18]. In this equation λ is the
magnetic diffusivity of the fluid and ν is the kinematic viscosity. As pointed out by Moffat[19] the analogy has
limitations as 𝛝 and 𝐁 are not related. This analogy has been the basis of many studies on vorticity especially
vortex knots.
III. Conclusion
While computing the force on a surface, the surface over which the stress tensor is integrated need not
correspond to a physical surface. This leads to the question of how the electromagnetic force is transmitted to
the physical matter inside the surface. The answer given is, via the electromagnetic field that enter into the stress
tensor. In the days of Maxwell, more explanations that are physical were considered necessary which led to the](https://image.slidesharecdn.com/i011155860-151126040518-lva1-app6892/85/Maxwell-Stress-Tensor-In-Hydrodynamics-2-320.jpg)
![Maxwell Stress Tensor in Hydrodynamics
DOI: 10.9790/5728-11155860 www.iosrjournals.org 60 | Page
concept of ether, whose velocity acts as the vector potential for magnetic induction. This explanation perhaps
applies more to the vorticity stress tensor, since the origin of vorticity is velocity.
It is difficult to take into account the stretching of vortex lines in three dimensions. Most of these
studies make use of local induction approximation (LIA) or perturbation methods. But the stress tensor
associated with vorticity can be made use of in such studies.
References
[1]. Casperson, L. W. Forces on permeable conductors in magnetic fields. Am. J. Phys, 70(2), 2002,163-168.
[2]. Shwarz G., General equation for the mean electrical energyof a dielectric body in an alternating electrical field ,J.Chem.Phys 39,
1963, 2387-2388.
[3]. Sher L.D, Dielectrophoresis in lossy dielectric media, Nature 220, 1968, 695-696 .
[4]. Pohl H.A., Crane J.S., Dielectrophoretic force, J.Theor. Bio 37, 1979,69
[5]. Wang,H.Y.,LeSar, An efficient fast multipole algorithm based on an expression in the solid harmonics J.Chem.Phys 104,
1996,4173-4179.
[6]. Jones T.B ,Washizu M., Multipolar dielectrophoretic and electrorotation theory, J.Electrostatics 37, 1996,121-134.
[7]. Sauer F.A., Schlogl R.W, Chiabrera. A, .Nicolini C., Schwan H.P , (eds), Interaction between electromagnetic fields and cells,
Plenum Publ. Co, Newyork, 1985,203-252.
[8]. Wang X., Wang X-B., Gascoyne P. R.C., General expressions for dielectricphoretic force and electrorotational torque derived using
the Maxwell Stress Tensor method,J.Electrostatics 39, 1997, 277-295.
[9]. Maxwell, J.C., “On Physical lines of force, Part I: The theory of molecular vortices applied to magnetic phenomena”. Philosophical
Magazine and Journal of Science, 21 No 139 , 1861,161-175.
[10]. Cook K.J, Fearn H. and Milonni P.,Fizeau’s experiment and the Aharonov- Bohm effect. Am. J. Phys 63(8), 1995,705-710.
[11]. Marmanis H., Analogy between Navier- Stokes equations and Maxwell’s equations: Applications to turbulence, Phy. Fluids 10(6),
1998,1428-1437.
[12]. Scofield D.F, Huq. P, Vortex Fields and the Lamb Stokes dissipation relation of Fluid dynamics. Phys. Lett A 372, 2008,4474-
4477.
[13]. Scofield D.F, Huq.,P. Transverse waves and vortex fields in non-relativistic fluid flows. Phys. Lett. A 373, 2009,1155-1158.
[14]. Scofield D.F, Huq.P, Concordances among electromagnetic, fluid dynamical and gravitational field theories. Phys. Lett A 374,
2010, 3476-3482.
[15]. Scofield D.F, Huq.P, Evolution of helicity in fluid flows, J. Math.. Phys 51, 2010, 033520,.
[16]. Ferraro, V. C. A. ,Plumpton, C. An Introduction to Magneto Fluid Mechanics. Clarandon Press, Oxford.,1966.
[17]. Klaus Stierstadt and Mario Liu, Maxwell's stress tensor and the forces in magnetic liquids , ZAMM - Journal of Applied
Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik. Volume 95, Issue 1, January 2015 4–37.
[18]. Elsasser, W.M., Induction effects in terrestrial magnetism, Part I: Theory. Phys. Rev. 69, 1946,106-16.
[19]. Moffat, H.K., Magnetic Field Generation in Electrically Conducting Fluids, Cambridge University Press, 1978.](https://image.slidesharecdn.com/i011155860-151126040518-lva1-app6892/85/Maxwell-Stress-Tensor-In-Hydrodynamics-3-320.jpg)
Maxwell Stress Tensor In Hydrodynamics
- 1. IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. V (Jan - Feb. 2015), PP 58-60 www.iosrjournals.org DOI: 10.9790/5728-11155860 www.iosrjournals.org 58 | Page Maxwell Stress Tensor In Hydrodynamics M. J. Vedan, M. B. Rajeswari Devi, Susan Mathew Panakkal 1 Department of Computer Applications, Cochin University of Science and Technology, Cochin, India. 2 Department of Mathematics, Cochin University of Science and Technology, Cochin, India. 3 Department of Mathematics, St.Teresa’s College, Cochin, India. Abstract: Analogous to Maxwell stress tensor in electric and magnetic fields, a stress tensor is defined in a vorticity field. Thus by treating vortices as physical structures, it is possible to study the forces on a surface element in it. Based on this the force between vortex lines, the pressure and the shearing stress that deform the volume element can also be defined. Keywords: Enstrophy, Maxwell stress tensor, static pressure, vorticity equation, vorticity stress tensor. I. Introduction Though the electromagnetic theory was developed more than a century back, the practice of calculating forces on magnetic media remained ambiguous (Casperson, 2002)[1]. In particular, the application of Maxwell stress tensor gave way to other methods based on energy variation principle (Shwarz, 1963) [2](Sher, 1968) [3] (Pohl and Crane, 1972)[4] and effective dipole or multipole method (Wang, 1996)[5]. The method was revived by Washizu and Jones (1996) [6], Sauer and Schlogl (1985) [7] etc. Wang, Wang and Gascoyne (1997)[8] explored the application of the Maxwell Stress Tensor (MST) method to dielectrophoresis (DEP) and electrorotation (ROT) studies. By integrating Maxwell stress tensor over the surface of a dielectric particle, they derived the general expressions for DEP and ROT generated by electric fields of arbitrary configurations. As they claim, this was the first time that, such complete expressions were derived from the first principles using MST formalism. The analogy between the electromagnetic and fluid dynamic equations was first noted by Maxwell (1861) [9]. He suggested that the vector potential A of the magnetic induction B represents some kind of a fluid velocity field. This was interpreted in the light of Fizeau’s experiment by Cook, Fearn and Milonni (1995)[10]. The analogy between Navier Stoke’s equation and Maxwell’s equations was used by Marmanis (1997-1998) [11] in the development of his metafluid dynamics for the study of turbulence. This marked the beginning of introducing new flow parameters (Lamb vectors). To complete the analogy between electromagnetic and fluid dynamic equation, Scofield and Huq (2008, 2009, 2010)[12] [13] [14] introduced the concept of vortex field and developed a uniform theory for electro dynamic, fluid dynamic and gravitational fields. A main source of interest in the study of the analogy was the theoretical developments in ferrofluids following its synthesis in 1960’s.” The Maxwell Stress Tensor and the forces in magnetic liquids” by Klaus Stierstadt and Mario Liu (2014)[15] can be considered as the present state of the theory. In this paper we explore yet another analogy between Maxwell stress tensor in electromagnetic continuum and a stress in incompressible inviscid fluids whose origin is vorticity. II. Vorticity Stress Tensor In the case of incompressible flow of an inviscid fluid, the vorticity field is frozen-in and satisfies the equation: ∂𝛚 ∂t = ∇ × (𝛝 × 𝛚) (1) Here ϑ is the velocity and 𝛚 = ∇ × 𝛝 is the vorticity. This equation admits the solution given by: ωi X, t = ωj(x, 0) ∂Xi ∂xj (2) This is the well-known Cauchy’s equation, which relates the current vorticity (at Xi) to the initial velocity (at xi), and thus it establishes a topological equivalence between them. Corresponding to the magnetic energy, the energy associated with the vorticity field is defined by 1 2 ω2 . Thus the rate of change of energy is: ∂ ∂t ω2 2 = 𝛚 ∙ ∂𝛚 ∂t = 𝛚 ∙ ∇ × (𝛝 × 𝛚) [from (1)] = −𝛝 ∙ ∇ × 𝛚 × 𝛚 + ∇ ∙ 𝛝 × 𝛚 × 𝛚 = −𝛝 ∙ 𝐟 × 𝛚 + ∇ ∙ 𝛝 × 𝛚 × 𝛚 (3)
- 2. Maxwell Stress Tensor in Hydrodynamics DOI: 10.9790/5728-11155860 www.iosrjournals.org 59 | Page where 𝐟 = ∇ × 𝛚 is the flexion field. If the vorticity is confined to a sub-domain of the fluid, the divergence term vanishes on integrating over the entire volume. Thus we get: dM dt = − 𝛝 ∙ 𝐅 dVV (4) Here M = ω2 2 dVV is the total enstrophy and 𝐅 = 𝐟 × 𝛚 is a force analogous to Lorentz force. The ith component of this force 𝐅 is: Fi = ∂ ∂xj ij , where ij = ωiωj − ω2 2 δij (5) and δij is the kronecker delta. Thus analogous to the Maxwell Stress tensor associated to Lorentz force (Ferraro and Plumpton 1966) [16] we get ij as the vorticity stress tensor. This tensor is related to the enstrophy in the same way, as magnetic stress tensor is associated to magnetic energy. If we consider a vortex, the normal component of this stress represents the tension in the line vortices and the terms ωiωj are the shearing forces between adjacent vortex lines of the filament whose limiting case is the line vortex. ij being a symmetric matrix can be diagonalized . Choosing the principal axis OX1 in the direction of vorticity we get: = ω2 2 0 0 0 −ω2 2 0 0 0 −ω2 2 ij (6) Thus the principal stress tensor constitutes a tension 1 2 ω2 along the line vortex and an equal pressure normal to it. We can rewrite this tensor as: ij ω2 0 0 0 0 0 0 0 0 + −ω2 2 0 0 0 −ω2 2 0 0 0 −ω2 2 (7) where the first matrix gives the effect of the force 𝐅 and the second matrix represents the static pressure. The above expressions for the vorticity stress tensor can be compared to the magnetic stress tensor as discussed by Steirstadt and Liu (2014)[17]. So the figures given by them apply to vortex stress tensor also. They consider electromagnetic stress tensor (EMST) and discuss magnetic stress tensor. What we find here is that a similar stress tensor exists in the case of vortex fields. In the context of magneto hydrodynamics the analogy between the induction equation: ∂𝐁 ∂t = ∇ × 𝛝 × 𝐁 + λ∇2 𝐁 ∇ ∙ 𝐁 = 0 (8) and the vorticity equation: ∂𝛚 ∂t = ∇ × 𝛝 × 𝛚 + ν∇2 𝛚 ∇ ∙ 𝛚 = 0 (9) for barotropic flow of a fluid was first pointed out by Elsasser (1946)[18]. In this equation λ is the magnetic diffusivity of the fluid and ν is the kinematic viscosity. As pointed out by Moffat[19] the analogy has limitations as 𝛝 and 𝐁 are not related. This analogy has been the basis of many studies on vorticity especially vortex knots. III. Conclusion While computing the force on a surface, the surface over which the stress tensor is integrated need not correspond to a physical surface. This leads to the question of how the electromagnetic force is transmitted to the physical matter inside the surface. The answer given is, via the electromagnetic field that enter into the stress tensor. In the days of Maxwell, more explanations that are physical were considered necessary which led to the
- 3. Maxwell Stress Tensor in Hydrodynamics DOI: 10.9790/5728-11155860 www.iosrjournals.org 60 | Page concept of ether, whose velocity acts as the vector potential for magnetic induction. This explanation perhaps applies more to the vorticity stress tensor, since the origin of vorticity is velocity. It is difficult to take into account the stretching of vortex lines in three dimensions. Most of these studies make use of local induction approximation (LIA) or perturbation methods. But the stress tensor associated with vorticity can be made use of in such studies. References [1]. Casperson, L. W. Forces on permeable conductors in magnetic fields. Am. J. Phys, 70(2), 2002,163-168. [2]. Shwarz G., General equation for the mean electrical energyof a dielectric body in an alternating electrical field ,J.Chem.Phys 39, 1963, 2387-2388. [3]. Sher L.D, Dielectrophoresis in lossy dielectric media, Nature 220, 1968, 695-696 . [4]. Pohl H.A., Crane J.S., Dielectrophoretic force, J.Theor. Bio 37, 1979,69 [5]. Wang,H.Y.,LeSar, An efficient fast multipole algorithm based on an expression in the solid harmonics J.Chem.Phys 104, 1996,4173-4179. [6]. Jones T.B ,Washizu M., Multipolar dielectrophoretic and electrorotation theory, J.Electrostatics 37, 1996,121-134. [7]. Sauer F.A., Schlogl R.W, Chiabrera. A, .Nicolini C., Schwan H.P , (eds), Interaction between electromagnetic fields and cells, Plenum Publ. Co, Newyork, 1985,203-252. [8]. Wang X., Wang X-B., Gascoyne P. R.C., General expressions for dielectricphoretic force and electrorotational torque derived using the Maxwell Stress Tensor method,J.Electrostatics 39, 1997, 277-295. [9]. Maxwell, J.C., “On Physical lines of force, Part I: The theory of molecular vortices applied to magnetic phenomena”. Philosophical Magazine and Journal of Science, 21 No 139 , 1861,161-175. [10]. Cook K.J, Fearn H. and Milonni P.,Fizeau’s experiment and the Aharonov- Bohm effect. Am. J. Phys 63(8), 1995,705-710. [11]. Marmanis H., Analogy between Navier- Stokes equations and Maxwell’s equations: Applications to turbulence, Phy. Fluids 10(6), 1998,1428-1437. [12]. Scofield D.F, Huq. P, Vortex Fields and the Lamb Stokes dissipation relation of Fluid dynamics. Phys. Lett A 372, 2008,4474- 4477. [13]. Scofield D.F, Huq.,P. Transverse waves and vortex fields in non-relativistic fluid flows. Phys. Lett. A 373, 2009,1155-1158. [14]. Scofield D.F, Huq.P, Concordances among electromagnetic, fluid dynamical and gravitational field theories. Phys. Lett A 374, 2010, 3476-3482. [15]. Scofield D.F, Huq.P, Evolution of helicity in fluid flows, J. Math.. Phys 51, 2010, 033520,. [16]. Ferraro, V. C. A. ,Plumpton, C. An Introduction to Magneto Fluid Mechanics. Clarandon Press, Oxford.,1966. [17]. Klaus Stierstadt and Mario Liu, Maxwell's stress tensor and the forces in magnetic liquids , ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik. Volume 95, Issue 1, January 2015 4–37. [18]. Elsasser, W.M., Induction effects in terrestrial magnetism, Part I: Theory. Phys. Rev. 69, 1946,106-16. [19]. Moffat, H.K., Magnetic Field Generation in Electrically Conducting Fluids, Cambridge University Press, 1978.