The document investigates heat dissipation in a fuel drop containing magnetic nanoparticles when subjected to a circularly polarized magnetic field and fluid vorticity. It presents a theoretical model describing the magnetization of ferrofluids under these conditions. The model accounts for Brownian and Néel relaxation mechanisms and viscous and magnetic torques. It determines an expression for the heating rate of the drop as a function of field frequency, fluid vorticity, and model parameters. Simulations show the heating rate increases with greater difference between field and vorticity frequencies, reaching a constant saturation value determined by low vorticities.

1. Heat dissipation in a fuel drop with magnetic nanoparticles in the presence of a
circular polarized eld and vorticity
A. C. Seridonio and F. F. Fachini
INPE-Instituto Nacional de Pesquisas Espaciais,
Rodovia Presidente Dutra Km 40, CEP 12630-000, Cachoeira Paulista, SP, Brazil
This work investigates the thermal response of an external magnetic eld on magnetic nanoparti-
cles embedded in a fuel drop. We adopt as theoretical approach the well-established magnetization
equation of ferrohydrodynamics proposed by M. I. Shiliomis et al, which describes the motion of free
dipoles in a viscous uid. Taking into account vorticity eects, we show that new features arise when
a circular polarized eld is considered. The interplay between the Brown and Néel mechanisms of
relaxation times also presents an important role in the heating process of the drop. Thus the friction
of the particle rotation, due to magnetic torques, with the viscous medium of the drop, constitutes
an extra source of heat. Our treatment allows the determination of the drop temperature prole
as a function of the model parameters, which is necessary to characterize completely its thermal
properties. Such prole is of fundamental interest in this emerging eld of research of fuel drops
with magnetic nanoparticles.
I. INTRODUCTION
The use of alternating magnetic elds on uids with magnetic nanoparticles (MNPs) allows a production of heat
in the host system [1]. Such power loss is due to a friction eect mediated by the viscous and magnetic torques on
the nanoparticles, being the former opposite to the latter. We propose that this mechanism of heating in ferrouids
can be also used to heat fuel drops with MNPs. To that end, a theoretical study must be done to establish the
basis of future applications. As the ferrouid community has restricted the determination of the system heating
rate to quiescent uids with linearly polarized magnetic elds [13], we investigate this quantity for rotating elds
in uids with nite ow vorticities.
Taking into account these eects in our calculations of the heating rate function, we show that the heating
process is favored and becomes more ecient when performed with circularly polarized magnetic elds. This paper
is organized as follows. In Sec. II, the phenomenological model of M. I. Shiliomis et al [46] for ferrouids is
discussed and particular attention is given to the relaxation mechanisms of Brown and Néel [7]. In Sec. III, using
the formalism of the complex frequency dependent susceptibility [1, 8, 9], we determine the net magnetization in
the system due to a circular counterclockwise polarization. The heating rate function for an adiabatic process is
derived in Sec. IV, and in Sec. V, numerical results for the frequency dependence of this function are presented.
We summarize our ndings in Sec. VI.
II. THEORETICAL MODEL FOR FERROFLUIDS
The phenomenological model of M. I. Shliomis et al [46] describes the net magnetization M of a viscous uid
with rotating magnetic MNPs characterized by a macroscopic angular velocity Ωp due to an applied magnetic eld
H.
Magnetic and viscous torques on these particles embedded in a uid with vorticity ow Ω are considered in this
model. The external eld interacts with the MNP dipoles m forcing an alignement with the eld direction. As the
particles are in contact with a thermal bath of nite temperature T, the equilibrium magnetization
ϕ ξ
M0 = |m| L ξ (1)
V ξ
is reached. The constant ϕ is the volume fraction of the MNPs, V is the MNP single volume and L ξ is the
Langevin function
L (ξ) = coth ξ − ξ −1 (2)
that depends on the magnitude of

2. 2
|m| H
ξ= . (3)
kB T
This ratio measures the competition between the magnetic energy |m| H of the oriented MNPs and the thermal
energy kB T of the ferrouid. Due to a mechanism of relaxation for the dipoles, a magnetization rate of change
appears until the equilibrium magnetization is established. In the rotating frame of the MNPs, this relaxation
process is assumed to be governed by the equation [7]
dM 1
=− M − M0 , (4)
dt τ
whose solution is
M = [1 − exp (−t/τ )] M0 (5)
to ensure the expected value M = M0 for t τ. In the laboratory frame we have to add to the right side of the
Eq. (4) the contribution ΩP × M due to the magnetic torque
τmag = M × H (6)
to derive
dM 1
= ΩP × M − M − M0 (7)
dt τ
as the equation of motion for the out-of-equilibrium magnetization. The relaxation parameter τ consists of two
processes in parallel [1, 7], which is expressed as
1 1 1
= + , (8)
τ τB τN
with
3ηV
τB = (9)
kB T
as the Brownian time of rotational diusion and
exp (KVM /kB T )
τN = τ0 (10)
KVM /kB T
the Néel relaxation. The parameters are the uid viscosity η, the hydrodynamic volume V of a single MNP, the
constant of proportionality τ0 , the anisotropy constant K and the magnetic volume VM of a MNP. In the Brownian
relaxation, the magnetic dipoles rotate together with the particles. In the Néel case, the MNPs are frozen and the
magnetic dipoles rotate exclusively in respect to the crystal axes of the particles. Thus considering the viscosity η
of the uid, the viscous torque
τvisc = Γ ΩP − Ω (11)
on the rotating MNPs is opposite to the magnetic torque of Eq. (6) and is determined by the moment of inertia per
unit time Γ = 6ϕη, the macroscopic angular velocity ΩP and the vorticity Ω. As a result, the macroscopic angular
acceleration for the MNPs obeys the following rotational equation
dΩp
I = τmag − τvisc , (12)
dt

3. 3
where I is the total rotation moment of inertia. The model considered here treats the steady state of Eq. (12),
which gives
1
ΩP = Ω + M ×H (13)
6ϕη
as the macroscopic angular velocity for the MNPs. Thus the magnetization rate of change in the laboratory frame
is given by
dM 1 1
=− M − M0 + Ω × M − M × M ×H . (14)
dt τ 6ϕη
According to Ref. [6], the dynamics of Eq. (14) agrees very well with numerical results extracted from a Focker-
Planck analysis for the limit Ωτ 1. To apply this model, this paper investigates the heating process in such
range.
III. THE NET MAGNETIZATION AND THE CIRCULAR POLARIZATION
The heating rate of the system is determined by the magnetization of the ferrouid due to an alternate magnetic
eld. In particular, we choose a circular counterclockwise polarization
H = H0 cos ωtˆ + H0 sin ωty,
x ˆ (15)
where H0 and ω are the amplitude and the angular frequency of the eld, respectively. The solution of Eq. (14)
for the magnetization can be obtained numerically, but a guessed solution based on the expected behavior of the
magnetization can be used instead . To that end, we remind the role of the relaxation process in the system. In
the absence of a relaxation mechanism, the magnetization is in-phase with the alternating magnetic eld, which is
described by the instantaneous version of Eq. (1). Thus, a nite relaxation time introduces a phase shift in the
magnetization, i.e.,
M = M cos (ωt − δ) x + M sin (ωt − δ) y .
ˆ ˆ (16)
Making the denitions
χ´ = M cos δ, (17)
¨
χ = M sin δ, (18)
the magnetization can be presented in terms of the components of the frequency dependent complex susceptibility
χ (ω) = χ´ − iχ¨ [1, 8, 9],
M = χ´ cos ωt + χ¨ sin ωt x + χ´ sin ωt − χ¨ cos ωt y.
ˆ ˆ (19)
To determine the unknown variables of Eqs. (17) and (18) as functions of the model parameters, we consider the
low-amplitude approximation for the magnetic eld |m| H /kB T 1, which leads to the instantaneous equilibrium
magnetization
M0 (t) = χ0 H (t) , (20)
ϕ 2
with χ0 = V |m| /3kB T as the equilibrium susceptibility. Performing the substitutions of Eqs. (15) and (16) in
Eqs. (7) and (13), we show that
M = χ0 H0 cos δ (21)

4. 4
and
tan δ = τ (ω − Ω) . (22)
Combining these results, the complex susceptibility for the system becomes
χ0
χ (ω) = = χ´ − iχ¨ , (23)
1 + iτ (ω − Ω)
with the magnetization
χ0 H0
M= [cos (ωt − δ) x + sin (ωt − δ) y ]
ˆ ˆ (24)
2
1 + τ 2 (ω − Ω)
phase shifted to the external magnetic eld by
{χ (ω)}
δ = arctan − . (25)
{χ (ω)}
IV. THE HEATING RATE
According to the rst law of thermodynamics dE = δQ − δW per unit volume, the dissipation energy density
E in a cycle T = 2π/ω for an adiabatic process δQ = 0 is the work density δW = −H.dB done by the magnetic
eld on the ferrouid. Using the constitutive equation B = µ0 H + M for the magnetic induction and the cycle
¸
condition H.dH = 0, we derive for the limit Ωτ 1, the following dissipation energy density
ˆ T
dM 2
E = µ0 dtH. = −2πµ0 H0 {χ (ω)} , (26)
0 dt
where µ0 is the magnetic permeability of the free space. Taking into account Eq. (23) in Eq. (26), this energy
density becomes
2 τ (ω − Ω)
E = 2πµ0 χ0 H0 2. (27)
1 + τ 2 (ω − Ω)
For a ow vorticity Ω = 0, our expression for the density has the same functional dependence as the result derived
by [1] for a magnetic eld with a linear polarization, except for a factor of two. This factor arises from the degrees
of freedom of the alternating magnetic eld, while the circular polarization has two components in the xOy plane,
the linear case is xed to only one direction. However, both cases introduce a shift of Ω in the frequency dependence
of the energy density for Ω = 0. Thus, our expression allows an enhanced heating process for external elds that
satisfy the condition ω Ω.
The characterization of the system temperature prole for a constant relaxation can be performed dening the
heating rate function [1, 2]
∆T ωE
= , (28)
∆t 2πρc
where ∆T and ∆t are the temperature rise and the duration of the heating, respectively. For the ferrouid, ρ is the
density and c the specic heat. According to Eq. (26), the heating rate is governed by the behavior of the minus
part of the complex frequency dependent susceptibility given by Eq. (23). So this quantity becomes important to
investigate the features of the heating process as a function of the model parameters. To explicit such dependencies,
we prefer to dene the dimensionless variables ∆ = τ (ω − Ω) and ε = Ωτ to derive

5. 5
Figure 1: Heating rate HC due to a circular polarization as a function of the dimensionless relative frequency ∆ = τ (ω − Ω)
with dierent dimensionless vorticities ε = Ωτ. For external frequencies near the ow vorticity, the heating rate displays a
linear behavior. A crossover region occurs until a saturation value is reached for large relative frequencies. Constant heating
rates dependent on the uid vorticity appear. A comparison between the linear and the circular heating for a quiescent uid
(Ω = 0) is presented in the inset. As it can be observed, the circular case is more ecient.
τ ρc ∆T ∆2 ∆
HC = 2 = +ε (29)
µ 0 χ 0 H0 ∆t 1 + ∆2 1 + ∆2
as the renormalized heating rate due to the circular polarization. Keeping ε 1 to ensure the validity of the model,
Eq. (29) allows the knowledge of the dependence of the ferrouid heating rate in terms of the dimensionless relative
eld frequency and the ow vorticity given by and ε, respectively. According to Ref. [1], the heating rate due to
a linear polarization is the half of a circular,
1
HL = HC . (30)
2
V. RESULTS
In Fig. (1) we present simulations for the heating rate function. They show a linear behavior for small deviations
of the relative frequency , which is ow vorticity dependent. For intermediate deviations, this behavior disappears
and the slopes of the curves start to decrease. Such decrease stabilizes and constant heating rates occurs for large
deviations, whose intensities are determined by low vorticities. Ecient heating rates for circular polarization are
produced, see the comparison between the circular with linear cases in the inset. These behaviors are consequence
of the Eqs. (29) and (30).
VI. CONCLUSIONS
Based on a phenomenological model for ferrouids, we derived the heating rate function due to a magnetic eld
with a circular counterclockwise polarization. For a xed relaxation parameter, we demonstrated that the heating

6. 6
process is more ecient for large dierences between the external and vorticity frequencies with a saturation value
determined by low vorticities. This mechanism can be applied to heat fuel drops with MNPs.
Acknowledgments
This work was supported by the brazilian agency CAPES.
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