CC BY
37
Bulletin of PFUR Series Mathematics. Information Sciences. Physics. No 2 (2). 2010. Pp. 157-160
UDC 537.84
Hamiltonian Formulation of Ideal Ferrohydrodynamics
P. A. Eminov, V. V. Sokolov
Department of Mathematics Moscow State University of Instrumentation and Information Science Stromynka 20,107996 Moscow, Russia
The Hamiltonian description of ferrohydrodynamics equations for a ideal nonconducting compressible magnetic fluid with frozen magnetization is presented.
Key words and phrases: Hamiltonian formalism, ferrohydrodynamics, magnetic fluid.
1. Introduction
An interesting class of intelligent materials are magnetic fluids, or ferrofluids, which without magnetic field are homogeneous colloidal suspensions of ferromagnetic nanoparticles coated with surface-active dispersive medium (typical diameters of particles range from 5 to 10 nm) in a carrier liquids [1]. Ferrohydrodynamics describes evolution of a magnetic fluid, carrying a magnetic field. The continuous models of ferrohydrodynamics have been studied in recent years from different points of view. Dynamic processes in a magnetic fluid can be described in continual approximation for two limiting cases. One of them corresponds to the equilibrium magnetization of the magnetic fluid [2], i.e., the case where the relaxation time characterizing the magnetization relaxation to the equilibrium value is infinitely small. The other limiting case corresponds to the situation where the magnetic fluid possesses a frozen magnetization [3], i.e., the case where the relaxation time is infinitely large. A complete system of equations describing an ideal non-conducting magnetic fluid with the frozen magnetization was obtained in [3]. As shown in [4,5], the linear approximation of ferro-hydrodynamics equations for the fluid with frozen magnetization describes adequately experimental data for the anisotropy of ultrasonic velocity in magnetized magnetic fluids based on the various liquids. The complete system of equations describing an ideal nonconducting magnetic fluid of density p with the frozen-in magnetization M = prn has the form [3]:
9p + d (pvj) = 0 dt dxi '
pdv1 = _dP +_Hi) d(№) +Mk9HT ,
dt dxi 1 1 dxj dxk '
dmi dvi ds ds
—rr = m^—- , — +Vi— = 0 ,
d d xj d d xi
2deA TT — v72lT,_,„d (Pmj )
(1)
i \dmJp,s , P V dp)s,rn , i dxi , 4
d xj
The system of equations is closed by setting a specific form of the internal energy density per unit mass e = e(p,s,mi) which depends on fluid density p = p (x, t), the specific entropy s = s (x, t), and the components of the magnetization per unit mass mi = mi (x, t). The specific feature of system (1) is the equation for the magnetization that express the condition of the magnetization is frozen in a magnetic fluid. The latter
Received 28th November, 2009.
This work was the part of a study supported by the Ministry of Education and Science RF (project No. 1.2.06).
two equations of system (1) are the Maxwell magnetostatic equations, where ^ is a scalar potential of the magnetic field.
The purpose of this work is to obtain the Hamiltonian equations of ferrohydrody-namics with frozen magnetization.
2. Hamiltonian Description of the Ideal Nonconducting Magnetic Fluid with Frozen
Magnetization
The functional of the total energy of the magnetized nonconducting fluid is represented in the form
^ (p,s,m,v) = j àx
.,2
V2 r^ tt, H
PY + pi (p, s, rn) - p(m, H) - —
(2)
Then, the action functional with the Lagrangian with constraints is determined by the formula
S = Ldt =
/( V]2 / -A H''
dtdr < p—— pe (p, s, m} — p ymHj + + a
+ P
dp d . . m + dx;(pVfc )
+ K
2
8n dmn
d d dt Vk dxk
+
d
+ vk-
dmn dxk
mk
d vn dxk
where the functions a ,<p and Xn, (n = 1,2,3) are the Lagrangian multipliers, and by the twice repeating index, summation from 1 to 3 is performed.
The extended Lagrangian set of equations [6,7] follow upon setting the functional derivative of the action functional to zero:
f = 0,
OCM,
(3)
where ai = {^i, Vi,Pi}, Pi = (p,s,mn), Pi = (p,a,\n).
Clebsch representation for the hydrodynamic momentum density is determined by the formula
ÔS dip ds dmn
t— = 0 nk = pvk = p---a---\n——
dvk dxk dxk dxk
d (\nmn) dxk '
(4)
If we then introduce a generalized momentum conjugate to generalized coordinates pi = (p, s, mn) and construct the extended Hamiltonian of the system as the Legendre transformation, then we obtain that (i) the Lagrange multipliers in the extended Hamiltonian formalism play the role of generalized momentum pi and (ii) the extended Hamiltonian does not contain nonphysical variables, or Lagrangian multipliers, and coincides with the functional of the total energy (2). As a result, taking into account (4), the system (3) is equivalent to the extended Hamiltonian system
dpi = ¿h dEi dt ôpi' dt
SH
Spi'
(5)
3. Poisson Brackets Method in Ferrohydrodynamics
Let us show the way in which the Hamiltonian equations for physical variables is obtained in the context of the method of Poisson brackets. The results presented below are obtained using formula (2) for the Hamiltonian of ferrohydrodynamics, formula (4)
Hamiltonian Formulation of Ideal Ferrohydrodynamics
159
for the hydrodynamic momentum, and a known property of the Poisson bracket (see, e.g., [6,8]):
h
{H (F1F2,...,Fn),Fk] = dV^^y {F(xJ ),Fk ]. (6)
Calculating the reciprocal Poisson brackets for physical fields = (p, s, mn) taking into account (4) and requiring the resultant density of hydrodynamic forces to be independent of velocity, we obtain (see also [6,8,9]):
3
{P,p] = 0 , {p,s'] = 0 , {mn,mk] = 0 , {p,^k] = p (x') 5(x' -x) , 3 s(x')
{S^'k ] =--5(x' -x) , {7Ti,^'k] = 3'i (n'k &) -3k s),
- mn -
{mn, kk] = S (x' - x) —x~ - ^nk-x7" [ma (x') 5 (x' - x)] .
Therefore, the Hamiltonian equations of motion are formulated taking into account (2) and the set of commutation relations presented above. They have the following final form:
I = {H'P] = --x (P' k) •
- mn - mn - -
— = {H,mn) = V— , - = {H, s] = -vk3xx~k ,
(7)
3= {h,k] =- ikif1)-3k (niVkH
3 / 3e \ / de \ dpmj pma dxa\dmi) \dmi J
, + -Hi)
It is easy to notice that these Hamiltonian equations coincide with the equations of system (1).
4. Conclusion
Therefore, in this work, the Hamiltonian set of equations of ferrohydrodynamics (5) is for the first time constructed with the use of Hamiltonian (2) of the system and of the method of Poisson brackets.
We can apparently affirm that the experimental verification of the suggested Hamil-tonian theory will allow one to create the complete theory of ferrohydrodynamics.
References
1. Rosensweig R. E. Ferrohydrodynamics. — M.: Mir, 1989.
2. Sokolov V. V., Tolmachev V. V. On Employment of the Generalized Virtual Work Principle in Ferrohydrodynamics. 1. Magnetic Fluid with Free Magnetization // Magnetohydrodynamics. — 1996. — Vol. 32, No 3. — Pp. 286-290.
3. Sokolov V. V., Tolmachev V. V. On Employment of the Generalized Virtual Work Principle in Ferrohydrodynamics. 1. Magnetic Fluid With Frozen Magnetization // Magnetohydrodynamics. — 1996. — Vol. 32, No 3. — Pp. 291-294.
4. Sokolov V. V., Tolmachev V. V. Anisotropy of Sound Propagation Velocity in Magnetic Fluid // Acoust. Phys. — 1997. — Vol. 43, No 1. — Pp. 92-97.
5. Ovchinnikov I. E., Sokolov V. V. Effect of an External Magnetic Field on the Ropagation Velocities of Magnetoacoustic Waves in a Magnetic Fluid // Acoust. Phys. — 2009. — Vol. 55, No 3. — Pp. 359-364.
6. Zakharov V. E., Kuznetsov E. A. Hamiltonian Formalism for Nonlinear Waves // Usp. Fiz. Nauk. — 1997. — Vol. 167, No 11. — Pp. 1137-1167.
7. Gitman D. M., Tyutin I. V. Classical Quantization of Fields with Bonds. — M.:Nauka, 1986.
8. Goncharov V. P., Pavlov V. I. Hamiltonian Vortex and Wave Dynamics. — M.: Geos, 2008.
9. Dzyaloshinskii I. E., Volovic G. E. Poisson Brackets in Condensed Matter Physics // Ann. Phys. — 1979. — Vol. 125. — Pp. 67-97.
УДК 537.84
Гамильтонова формулировка идеальной феррогидродинамики
П. А. Эминов, В. В. Соколов
Кафедра высшей математики Московский государственный университет приборостроения и информатики ул. Стромынка, д.20, Москва, 107999, Россия
Дано гамильтоново описание уравнений феррогидродинамики для идеальной непроводящей сжимаемой магнитной жидкости с вмороженной намагниченностью.
Ключевые слова: формализм Гамильтона, феррогидродинамика, магнитная жидкость.
