Numerical Modelling of Magnetic Shielding by a Cylindrical Ferroﬂuid Layer

Home , Magnetostatics

Mathematical Modelling and Analysis http://mma.vgtu.lt Volume 24, Issue 2, 155–170, 2019 ISSN: 1392-6292 https://doi.org/10.3846/mma.2019.011 eISSN: 1648-3510

Olga Lavrovaa, Viktor Polevikovb and Sergei Polevikovc aFaculty of Mechanics and Mathematics, Belarusian State University Independence Ave. 4, 220030 Minsk, Belarus bFaculty of Applied Mathematics and Informatics, Belarusian State University Independence Ave. 4, 220030 Minsk, Belarus cOppenheimerFunds, Inc 225 Liberty Street, 2WFC, 16 Floor, New York, NY 10281, USA E-mail(corresp.): [email protected] E-mail: [email protected] E-mail: [email protected]

Received May 7, 2018; revised December 15, 2018; accepted December 27, 2018

Abstract. A coupled method of finite differences and boundary elements is applied to solve a nonlinear transmission problem of magnetostatics. The problem describes an interaction of a uniform magnetic field with a cylindrical ferrofluid layer. Fer- rofluid magnetisations, based on expansions over the Langevin law, are considered to model ferrofluids with a different concentration of ferroparticles. The shielding effectiveness factor of the cylindrical thick-walled ferrofluid layer is calculated depending on intensities of the uniform magnetic field and on thickness of the ferrofluid layer. Keywords: transmission magnetostatics problem, finite difference method, boundary element method, magnetic fluid, shielding. AMS Subject Classification: 35Q60; 65N06; 65N38.

1 Introduction

Currently, one of the most important fields of scientific research is a study of electromagnetic properties of composite materials used in radio and electronic engineering. It requires development of modelling methods and computational techniques for processes of interaction between electromagnetic fields and ob- jects of different geometries and material structures which are important for

Copyright c 2019 The Author(s). Published by VGTU Press This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. 156 O. Lavrova, V. Polevikov and S. Polevikov further engineering of technical devices. A special challenge is studying materials with nonlinear physical properties [8,11], in particular soft magnetic materials [28]. Computational approaches of handling equilibrium states of ferrofluid systems with different geometries and free surfaces – drop, layer, capillary – have been developed in [16, 17, 18, 19, 23] for uniform applied magnetic fields. Recent results related to shells and membrane theories of composite materials in mechanics and biology are presented in [1]. In particular shells are applied for suppression of vibrations by means of magnetic fields, see e.g. [21, 22] for thin sandwich-like shells containing magnetorheological layers. One of considered research directions is a study of transmission processes for electromagnetic fields through shields and shells, see e.g. [7]. Active research is performed on shielding problems of electromagnetic fields by solid magnetized thin-walled layers and films, see e.g. [2,10,12,13,30,31,32,33,34]. Current paper is devoted to studying shielding properties of cylindrical thick-walled ferrofluid layers pro- tecting against uniform magnetic fields, where ferrofluids of various concentration of ferroparticles are considered. Numerical solutions of the problem have been obtained for a simpler mathematical model in [24], neglecting the effect of magnetodipole interparticle interactions in ferrofluids.

2 Mathematical model for the magnetostatics problem

Magnetic fluids (ferrofluids) are stable colloidal suspensions of ferromagnetic nanoparticles in a nonmagnetic carrier-liquid (such as water, kerosene, trans- former oil, organic compounds and others). The ferroparticles of size 10 nm are in Brownian motion inside the carried-liquid. The fluid composition results in magnetic properties of magnetic fluids affected by magnetic fields [3,4,6,28]. The magnetic permeability of ferrofluids µf is defined by the magnetisation law M(H)

µf = µ0µ, µ = µ(H) = 1 + M(H)/H,

−7 where µ0 = 4π · 10 H/m denotes the vacuum permeability. H = |H| is the intensity of the magnetic field. Different models of the ferrofluid magnetisation should be used to model fluids with different concentration of ferroparticles [14]. Namely, the following relations are suggested in [14] based on expansions over the Langevin law ML(H) = MsL(H/H∗) within the framework of the modified mean-field theory,

M(1)(H) = ML(H), 1 M (H) = M H + M (H) , (2) L 3 L 1 1 dM (H) M (H) = M H + M (H) + M (H) L . (3) L 3 L 144 L dH

Here Ms is saturation magnetisation of the ﬂuid, L(t) = coth (t) − 1/t is the −23 Langevin function, H∗ = kT/(µ0m), k = 1.38 · 10 J/K is the Boltzmann constant, T is the absolute temperature of the ﬂuid, m is magnetic moment of a ferropaticle. Numerical Modelling of Magnetic Shielding 157

In contrast to M(1)(H), the magnetisation laws M(2)(H) and M(3)(H) take into account the magnetodipole interparticle interactions [14]. The magnetisation M(1)(H) describes diluted or weakly-concentrated ferrofluids (the magnetic phase concentration φm < 10 %), M(2)(H) describes moderately- concentrated ferrofluids (φm ∼ 10–12 %), and M(3)(H) describes dense or highly-concentrated ferrofluids (φm ∼ 12–18 %) [14]. It is assumed that the ferroparticles are uniformly distributed inside the ferrofluid. Every magnetisation law is characterized by initial magnetic susceptibility χ(i) = limH→0 (dM(i)/dH), i = 1, 3:

Ms 1 2 1 2 1 3 χL = , χ(1) = χL, χ(2) = χL + χL, χ(3) = χL + χL + χL. 3H∗ 3 3 144

χ(i) is a dimensionless parameter of the ferroﬂuid. We formulate expressions for the relative magnetic permeability in terms of the dimensionless ﬁeld intensity h = H/H∗:

L(h) µ (H) =µ ¯ (h) = 1 + 3χ , (2.1) (1) (1) L h L (h + χ L(h)) µ (H) =µ ¯ (h) = 1 + 3χ L , (2.2) (2) (2) L h 1 2 dL(h) L h + χLL(h) + 16 χLL(h) dh µ (H) =µ ¯ (h) = 1 + 3χ . (2.3) (3) (3) L h

2.1 Diﬀerential statement of the problem Let us introduce a Cartesian coordinate system OXYZ and the corresponding cylindrical one OρϕZ in R3. We consider a cylindrical layer, inﬁnite in Z direction and described by its cross-section

D2 = {(ρ, ϕ): R1 < ρ < R2, 0 ≤ ϕ < 2π}, with layer boundaries Γ1 and Γ2 such as

Γ1 = {(ρ, ϕ): ρ = R1, 0 ≤ ϕ < 2π},Γ2 = {(ρ, ϕ): ρ = R2, 0 ≤ ϕ < 2π}, see Figure1. The interfaces Γ1 and Γ2 are assumed to be absolutely permeable for magnetic fields. The inner domain D1 = {(ρ, ϕ) : 0 ≤ ρ < R1, 0 ≤ ϕ < 2π} and the outer domain D3 = {(ρ, ϕ): R2 < ρ < ∞, 0 ≤ ϕ < 2π} are filled with vacuum of permeability µ0, whereas layer D2 is filled with magnetic fluid of permeability µf . The layer D2 is considered under the effect of a uniform magnetic field H0 = H0eY with constant intensity H0, where eY = (0, 1, 0), see Figure1. Let us denote Hi the magnetic field inside the domain Di and represent magnetic field of the external domain as H3 = H0 + H˜ 3. The applied field H0 does not depend on Z coordinate. That is why the problem can be treated as two-dimensional in the polar coordinate system Oρϕ. Let us introduce

Math. Model. Anal., 24(2):155–170, 2019. 158 O. Lavrova, V. Polevikov and S. Polevikov

R R

2 1

Figure 1. The problem setting: D1 corresponds to the internal nonmagnetic medium, D2 corresponds to the ferroﬂuid, and D3 corresponds to the external nonmagnetic medium.

magnetostatic potential functions Ui(ρ, ϕ) such that ∂U 1 ∂U H = ∇U = i , i , i = 0, 3. i i ∂ρ ρ ∂ϕ

Now let us formulate the boundary value problem for unknown potentials U1,U2 and U˜3 = U3 − U0, see e.g. [24], with the Laplace equations in the nonmagnatic domains D1 and D3: 1 ∂ ∂U 1 ∂2U ∇2U = ρ 1 + 1 = 0 in D , (2.4) 1 ρ ∂ρ ∂ρ ρ2 ∂ϕ2 1 ! 1 ∂ ∂U˜ 1 ∂2U˜ ∇2U˜ = ρ 3 + 3 = 0 in D , (2.5) 3 ρ ∂ρ ∂ρ ρ2 ∂ϕ2 3 and the nonlinear Laplace-type equation in the ferroﬂuid domain D2: 1 ∂ ∂U 1 ∂ ∂U ∇ · (µ∇U ) = µρ 2 + µ 2 = 0 in D , (2.6) 2 ρ ∂ρ ∂ρ ρ2 ∂ϕ ∂ϕ 2 with transmission conditions on interfaces Γ1 and Γ2: ∂U ∂U U = U , 1 = µ 2 on Γ , (2.7) 1 2 ∂ρ ∂ρ 1 ∂U ∂U ∂U˜ U = U , µ 2 = 0 + 3 on Γ , (2.8) 2 3 ∂ρ ∂ρ ∂ρ 2 and a radiation condition lim U˜3 = 0. (2.9) ρ→∞

Here the potential of the external applied field is given as U0 = H0ρ sin ϕ. The permeability of the ferrofluid µ = µ(|∇U2|) is defined by one of the relations (2.1)–(2.3). The Laplace equation (2.5) for the potential U˜3 = U3 − U0 follows 2 from the Laplace equation for U3 and the fact that ∇ U0 = 0. The mathematical model is described by the Laplace equations (2.4) and (2.5), the nonlinear Laplace-type equation (2.6), transmission conditions (2.7), (2.8) and radiation condition (2.9). Numerical Modelling of Magnetic Shielding 159

2.2 Integral reformulation Let us now introduce dimensionless variables and parameters, where space variables are dimensionless over R1, and the magnetic ﬁeld is dimensionless over H∗:

ρ X Y Ui(R1r, ϕ) r = , x = , y = , ui(r, ϕ) = , R1 R1 R1 R1H∗ s 2 2 Hi ∂ui 1 ∂ui H0 R2 hi = = + , u0 = h0r sin ϕ, h0 = , δ = . H∗ ∂r r ∂ϕ H∗ R1

The corresponding dimensionless geometry is

Ω1 = {(r, ϕ) : 0 < r < 1, 0 ≤ ϕ < 2π},Ω2={(r, ϕ) : 1 < r < δ, 0 ≤ ϕ < 2π},

Ω3 = {(r, ϕ): δ < r < ∞, 0 ≤ ϕ < 2π},

γ1 = {(r, ϕ): r = 1, 0 ≤ ϕ < 2π}, γ2 = {(r, ϕ): r = δ, 0 ≤ ϕ < 2π}.

The dimensionless analog of the Laplace equation (2.4) in the bounded domain Ω1 can be reformulated as a boundary integral equation on the interface γ1, based on Green’s representation formula, see e.g. [5]: Z ∗ ∗ πu1(ξ0) + [u1(ξ)q (ξ0, ξ) − q1(ξ)u (ξ0, ξ)] dγ1(ξ) = 0, ∀ξ0 ∈ γ1. (2.10) γ1

Here ξ0 = (r0, ϕ0) is a ﬁxed source point, ξ = (r, ϕ) is integration variable and q1(ξ) = ∂u1/∂r is normal derivative of the potential function on γ1. The ∗ fundamental solution u (ξ0, ξ) for the plane Laplace equation is of the following form, see e.g. [29]:

∂u∗(ξ , ξ) u∗(ξ , ξ) = − ln |ξ − ξ |, q∗(ξ , ξ) = 0 , (2.11) 0 0 0 ∂r

1/2 2 2 where |ξ − ξ0| = (r cos ϕ − r0 cos ϕ0) + (r sin ϕ − r0 sin ϕ0) is the dis- tance between points ξ and ξ0. Similarly, the dimensionless form of the Laplace equation (2.5) in the unbounded domain Ω3 with the radiation condition (2.9) can be reformulated as a boundary integral equation on the interface γ2: Z ∗ ∗ πu˜3(ξ0) − [˜u3(ξ)q (ξ0, ξ) − q˜3(ξ)u (ξ0, ξ)] dγ2(ξ) = 0, ∀ξ0 ∈ γ2, (2.12) γ2 whereu ˜3 = u3 − u0, andq ˜3(ξ) = ∂u˜3/∂r is normal derivative of the potential functionu ˜3 on γ2. For the problem under study,

∗ ϕ − ϕ0 ∗ 1 u (ξ0, ξ) = − ln 2 sin , q (ξ0, ξ) = − , ∀ξ0, ξ ∈ γ1, (2.13) 2 2

∗ ϕ − ϕ0 ∗ 1 u (ξ0, ξ) = − ln 2δ sin , q (ξ0, ξ) = − , ∀ξ0, ξ ∈ γ2. (2.14) 2 2δ

Math. Model. Anal., 24(2):155–170, 2019. 160 O. Lavrova, V. Polevikov and S. Polevikov

∗ As a result, an integration of the term u1(ξ)q (ξ0, ξ) in the boundary integral ∗ equation (2.10) and the termu ˜3(ξ)q (ξ0, ξ) in (2.12) turns out to be zero due ∗ to the constant value of q (ξ0, ξ), see (2.13), (2.14), and the symmetry of the potential functions Z Z ∗ 1 u1(ξ)q (ξ0, ξ)dγ1(ξ) = − u1(ξ)dγ1(ξ) = 0, ∀ξ0 ∈ γ1, γ1 2 γ1 Z Z ∗ 1 u˜3(ξ)q (ξ0, ξ)dγ2(ξ) = − u˜3(ξ)dγ2(ξ) = 0, ∀ξ0 ∈ γ2. (2.15) γ2 2δ γ2

The mathematical model (2.4)–(2.9) will be redeﬁned by means of boundary integral equations (2.10) and (2.12), whereas the radiation condition (2.9) is exactly fulﬁlled due to integral formulation (2.12). A symmetry of the problem setting with respect to the coordinate axes (see Figure1) allows us to restrict the dimensionless computational domain to the positive quadrant, see Figure2. Hereafter,

Ω1={(r, ϕ) : 0 < r < 1, 0 < ϕ < π/2},Ω2={(r, ϕ) : 1 < r < δ, 0 < ϕ < π/2},

Ω3={(r, ϕ): δ < r < ∞, 0 < ϕ < π/2}, and the layer boundaries are

γ1 = {(r, ϕ): r = 1, 0 < ϕ < π/2}, γ2 = {(r, ϕ): r = δ, 0 < ϕ < π/2}.

We introduce additional notation for space variables on interfaces γ1 and γ2 to handle the symmetry in the integral equations:

[σ] [σ] [σ] [σ] ξ1 = (σ, ϕ), ξ2 = (σ, π−ϕ), ξ3 = (σ, π+ϕ), ξ4 = (σ, 2π−ϕ) for σ ∈ {1, δ}.

Due to symmetry of the potential functions and relations (2.15), we reformulate the boundary integral equations (2.10), (2.12) as

Z π/2 [1] h ∗ [1] ∗ [1] ∗ [1] ∗ [1]i q1 ξ1 u ξ0, ξ1 + u ξ0, ξ2 − u ξ0, ξ3 − u ξ0, ξ4 dϕ 0 = πu1(ξ0), ∀ξ0 ∈ γ1, (2.16) π/2 Z [δ] h ∗ [δ] ∗ [δ] ∗ [δ] ∗ [δ]i q˜3 ξ1 u ξ0, ξ1 + u ξ0, ξ2 − u ξ0, ξ3 − u ξ0, ξ4 dϕ 0 π = − u˜ (ξ ), ∀ξ ∈ γ . (2.17) δ 3 0 0 2

The mathematical model in dimensionless variables (see Figure2) consists of boundary integral equations (2.16) and (2.17), together with the nonlinear Laplace-type equation

1 ∂ ∂u 1 ∂ ∂u µ¯(h )r 2 + µ¯(h ) 2 = 0 in Ω , (2.18) r ∂r 2 ∂r r2 ∂ϕ 2 ∂ϕ 2 Numerical Modelling of Magnetic Shielding 161

(0,h )

3 /2

2 =

1 1

x = 0

0 1

Figure 2. Computational domain for the problem under study.

as well as transmission conditions on interfaces γ1 and γ2:

u1 = u2, q1 =µ ¯(h2)q2 on γ1, (2.19)

u2 =u ˜3 + h0δ sin ϕ, µ¯(h2)q2 =q ˜3 + h0 sin ϕ on γ2, (2.20) and the symmetry conditions

u1|ϕ=0 = u2|ϕ=0 =u ˜3|ϕ=0 = 0, (2.21)

∂u1 ∂u2 ∂u˜3 = = = 0. (2.22) ∂ϕ π ∂ϕ π ∂ϕ π ϕ= 2 ϕ= 2 ϕ= 2

Here the normal derivatives of potential functions on γ1 and γ2 are denoted as q1 = ∂u1/∂r, q2 = ∂u2/∂r andq ˜3 = ∂u˜3/∂r. The unknown quantities of the mathematical model (2.16)–(2.22) are the volume potential u2(r, ϕ) and the boundary fluxes q1(1, ϕ) andq ˜3(δ, ϕ). The mathematical model has three dimensionless parameters: applied field intensity h0, layer thickness δ and magnetic susceptibility χ. Relative magnetic permeability of the ferrofluidµ ¯ could take different forms such asµ ¯(i), i = 1, 3, see (2.1)–(2.3). The corresponding model formulations are denoted as models i, whereas the model 1 describes weakly-concentrated ferrofluids, the model 2 describes moderately-concentrated ones, and the model 3 corresponds to dense ferrofluids.

3 Computational procedure

The problem under study is not only of practical interest. It is also unique from the point of view of its numerical modelling. The computational algorithm for the integro-differential model (2.16)–(2.22) is based on the combination of the boundary element method in the nonmagnetic domains Ω1 and Ω3 and the finite-difference method in the ferrofluid domain Ω2. Due to the boundary element approach, a solution of the plane Laplace equation with known fundamental solution (see e.g. [29]) is replaced in the nonmagnetic domains with an

Math. Model. Anal., 24(2):155–170, 2019. 162 O. Lavrova, V. Polevikov and S. Polevikov equivalent problem of solving boundary integral equations (2.16) and (2.17). These equations are considered the Fredholm integral equations of the ﬁrst kind with respect to the normal derivative q1 on the interface γ1 and to the normal derivativeq ˜3 on the interface γ2, respectively. An advantage of the boundary element method is that no meshes are required in the domains Ω1 and Ω3, and the radiation condition (2.9) is exactly fulﬁlled by the integral reformulation. We apply a collocation approach with constant elements to approximate boundary integral equations (2.16) and (2.17) on uniform interface meshes {(1, ϕj)} and {(δ, ϕj)} with respect to the integration variable ϕ, where ϕj = j∆ϕ, j = 0,Nϕ, and the mesh step ∆ϕ = π/ (2Nϕ). For the given approxi- (1) (3) mate values of the interface potentials uj ≈ u1(1, ϕj) andu ˜j ≈ u˜3(δ, ϕj), j = 1,Nϕ, we get two independent systems of linear algebraic equations rela- (1) (3) tive to the derivatives qj ≈ q1(1, ϕj) = ∂u1(1, ϕj)/∂r andq ˜j ≈ q˜3(δ, ϕj) = ∂u˜3(δ, ϕj)/∂r:

Nϕ X [1] (1) (1) (1) (1) aij qj = πui , i = 1,Nϕ; u0 = q0 = 0, (3.1) j=1

Nϕ X [δ] (3) π (3) (3) (3) a q˜ = − u˜ , i = 1,N ;u ˜ =q ˜ = 0, (3.2) ij j δ i ϕ 0 0 j=1 where

Z ϕj+1/2 [σ] h ∗ [σ] ∗ [σ] ∗ [σ] ∗ [σ]i aij = u ξ0, ξ1 +u ξ0, ξ2 − u ξ0, ξ3 −u ξ0, ξ4 dϕ, ϕj−1/2 for σ ∈ {1, δ} and ξ0 = (σ, ϕi). Due to a special form of the fundamental [1] [δ] solution on γ1 and γ2, see (2.16), (2.17), we get aij = aij . Hereafter we omit the superscript for aij:

Z ϕj+1/2 tan ((ϕ + ϕ )/2) tan ((ϕ + ϕ )/2) a = ln i dϕ ≈ ∆ϕ ln j i ij tan (|ϕ − ϕ |/2) tan (|ϕ − ϕ |/2) ϕj−1/2 i j i for i = 1,Nϕ, j = 1,Nϕ − 1, i 6= j,

Z ϕi+1/2 tan ((ϕ + ϕ )/2) 4 tan ϕ a = ln i dϕ ≈ ∆ϕ 1+ ln i , i = 1,N − 1, ii tan (|ϕ − ϕ |/2) ∆ϕ ϕ ϕi−1/2 i ϕ Z Nϕ+1/2 ϕ − ϕ π/2 − ϕ a = − ln tan i dϕ ≈ −∆ϕ ln tan i , i = 1,N − 1, i,Nϕ 2 2 ϕ ϕNϕ−1/2 ϕ Z Nϕ+1/2 |π/2 − ϕ| ∆ϕ a = − ln tan dϕ ≈ ∆ϕ 1 − ln . (3.3) Nϕ,Nϕ 2 4 ϕNϕ−1/2

Note that singular integrals for the coefficients aii are computed similarly to the approach in [5]. The nonlinear problem (2.18), (2.21),(2.22) is approximated by the finite- differences of the second order on a uniform mesh along r and ϕ. The con- structed finite-difference scheme is presented by a system of nonlinear algebraic Numerical Modelling of Magnetic Shielding 163 equations. It satisfies the maximum principle in linear approximation at any mesh step, i.e. the finite-difference scheme is monotonic. Computational process has been organized in the form of an iterative algorithm, where three mesh problems for the potential are solved independently of each other at each iteration.

1. At ﬁrst, two linear BEM systems (3.1), (3.3) and (3.2), (3.3) are solved by the Gauss elimination method with respect to mesh values of q(1) andq ˜(3), respectively, for the given mesh values of the potential u(1) and u˜(3) from the previous iteration. In spite of the fact that the problem is ill-posed in the theory of integral equations, we obtain only correct numerical solutions at diﬀerent meshes.

(2) 2. Then, the interface mesh-values of q ≈ q2 on γ1 at r = 1 and on γ2 at r = δ are computed from the transmission conditions (2.19) and (2.20) for the new mesh ﬂuxes q(1) andq ˜(3).

3. The nonlinear finite-difference problem with respect to the potential u2 in the ferrofluid domain Ω2 is solved by the iterative Seidel-type method. (2) As a result, new mesh values of the potential u ≈ u2 are determined not only at the internal nodes of the domain Ω2 but also at the interface nodes. New mesh values of the magnetic permeabilityµ ¯(h2) are recomputed using these interface nodes.

4. At last, the mesh functions u(1) andu ˜(3) are recomputed at the interface nodes, using transmission conditions (2.19) and (2.20). It completes the iteration of the computational process.

When the iterative process is ﬁnished, the potential u1(ξ) at any internal point ξ ∈ Ω1 and the potential u3(ξ) at any internal point ξ ∈ Ω3 are calculated (1) (3) by explicit formulas, based on the interface values of the potentials uj , uj (1) (3) and the normal derivatives qj ,q ˜j at the interface nodes {(1, ϕj)}, {(δ, ϕj)} for j = 1,Nϕ:

Nϕ 1 X [1] (1) [1] (1) u (ξ ) = a q − b u , ∀ξ ∈ Ω , 1 0 2π j j j j 0 1 j=1

Nϕ δ X [δ] (δ) [δ] (3) u (ξ )= − a q˜ −b u −h δ sin ϕ +h r sin ϕ , ∀ξ ∈ Ω , 3 0 2π j j j j 0 j 0 0 0 0 3 j=1 where for σ ∈ {1, δ},

[σ] h ∗ [σ] ∗ [σ] ∗ [σ] ∗ [σ]i aj = ∆ϕ u ξ0, ξ1,j + u ξ0, ξ2,j − u ξ0, ξ3,j − u ξ0, ξ4,j ,

[σ] h ∗ [σ] ∗ [σ] ∗ [σ] ∗ [σ]i bj = ∆ϕ q ξ0, ξ1,j + q ξ0, ξ2,j − q ξ0, ξ3,j − q ξ0, ξ4,j , [σ] [σ] [σ] [σ] ξ1,j = (σ, ϕj), ξ2,j = (σ, π − ϕj), ξ3,j = (σ, π + ϕj), ξ4,j = (σ, 2π − ϕj).

Math. Model. Anal., 24(2):155–170, 2019. 164 O. Lavrova, V. Polevikov and S. Polevikov

The computations were carried out on a uniform mesh with the number of partitions Nr = 10 for δ ≤ 1.1, and Nr = 100 for δ > 1.1 for the variable r ∈ [1, δ], and Nϕ = 100 for the variable ϕ ∈ [0, π/2]. The test calculations on a finer mesh with twice the number of partitions for both variables did not reveal any significant changes in the solution. An undisturbed magnetic potential u2 = u0 was chosen as starting configuration for the iterative process.

4 Results of computations

The initial magnetic susceptibility χ lies in the range 0 < χ ≤ 1 for the weakly- concentrated ferrofluids, and in the range 1 < χ ≤ 10 for the moderately- concentrated ones. It reaches values χ ∼ 50–80 for the dense ferrofluids under the room-temperature conditions [20, 26, 27]. For instance, the dense decane- based ferrofluid with magnetic phase φm = 22.6 % has initial susceptibility χ = 51 under the room temperature, see e.g. [9]. Under low temperatures (T ∼ 230–240 K), the value of the initial magnetic susceptibility of dense laboratory synthesized ferrofluids increases up to χ ∼ 100–120 for the preserved fluidity, see e.g. [20, 26, 27]. We take the initial magnetic susceptibility from the range 10 ≤ χ ≤ 50 for the test computations, namely χ = {10, 30, 50}. The value χ = 10 corresponds to moderately-concentrated ferrofluids, whereas χ = {30, 50} describes dense ferrofluids. Shielding effectiveness factor of the uniform magnetic field H0 by the cylindrical thick-walled ferrofluid layer D2 is defined by the formula s 2 2 |H0(0, ϕ)| h0 ∂u1 1 ∂u1 Kef = = =h0/ + . (4.1) |H1(0, ϕ)| |∇u1(0, ϕ)| ∂r r ∂ϕ r=0

Shielding eﬀectiveness factor Kef estimates the decay factor of the externally applied magnetic ﬁeld intensity H0 transmitted into the internal cylinder D1. We compute (4.1) at a point where ϕ = π/2, lying on the axis Oy, with symmetry condition (2.22), hence

∂u1(r, π/2) Kef = h0/ lim . r→0 ∂r

Figure3 demonstrates the shielding effectiveness factor Kef dependent on the applied field for models 1–3. The geometric configuration of the problem is specified by the layer thickness δ = 1.1. The ferrofluid properties are defined two ways: by the initial magnetic susceptibility χ = {10, 30, 50} with χ(i) = χ, see Figure3a, and by the Langevin susceptibility χL = {4.06, 7.61, 9.97} with χ(i) = χ(i)(χL), see Figure3b. The values of the susceptibilities are taken in such a way that χ(3)(χL) = χ, it leads to the identical numerical results for model 3 (solid lines) in Figure3a and Figure3b. Figure3a illustrates the comparison of models 1–3, aligned for weak magnetic fields by the same values of χ, whereas Figure3b shows the comparison of models 1–3 with analogous physical ferrofluid properties, fixed by the same values of χL. Numerical results 2 in Figure3 suggest that in weak fields ( h0 < 0.1,H0 < 10 A/m) the shielding factor preserves a constant maximum value, and the shielding effect is higher Numerical Modelling of Magnetic Shielding 165 for more concentrated ferrofluids. Namely, Kef ≈ 1.4 for the moderately- concentrated ferrofluid with χ = 10 and Kef ≈ 3.1 for the dense ferrofluid with χ = 50, see Figure3a. We note that the numerical results of model 1–3 for the same χ, shown in Figure3a, coincide in weak magnetic fields due to the identical linear behavior of magnetisation laws M(i) = χ(i)H → χH in the limit case H → 0. As magnetic fields increase, the shielding effectiveness factor Kef monotonically decreases to 1 and the shielding effect is almost absent for 2 6 h0 > 10 (H0 > 10 A/m). We conclude that the ferrofluid shield is effective in weak magnetic fields, whereas the concentration of magnetic particles could be considered as a way to control the intensity of the externally applied magnetic field, penetrating to the inner domain of the shield. For instance, geomagnetic storms on the Earth surface have intensity H0 < 200 A/m and could be effec- tively shielded by ferrofluid layers. We note that numerical results in Figure3a for the model 1 have been originally published in [24].

K K

ef ef

= 50 3 3

= 9.97

= 7.61

= 30

2 2

= 4.06

= 10

1 1

0.01 0.1 1 10 h 0.01 0.1 1 10 h

0 0

a) b)

Figure 3. Dependence of the shielding effectiveness factor Kef on the dimensionless applied magnetic-field intensity h0 for the dimensionless layer thickness δ = 1.1. The ferrofluid properties are defined two ways: by the initial magnetic susceptibility χ = {10, 30, 50} (a) and by the Langevin permeability χL = {4.06, 7.61, 9.97} (b). Dotted lines correspond to model 1, dashed lines to model 2, solid lines to model 3. On the scale of the graph, the results for model 2 are indistinguishable from those of model 3 in (a). Lines for model 1 in (b) correspond to increasing values of χL bottom-up.

Figure3b allows to compare numerical results with respect to models 1–3, where the ferrofluid properties are specified by the model-independent parameter χL, in contrast to the parameter χ in Figure3a. Figure3b shows that different models produce quantitatively different results for the same ferrofluids. Namely, model 1 does not suit for describing moderately- and strongly- concentrated ferrofluids (χL > 1) for the problem under study. The shielding effect is underestimated for χL > 1 in the frame of model 1. The results of model 2 and model 3 are quantitatively comparable for moderately- concentrated ferrofluids (χL = 4.06), but the shielding effect is underestimated for strongly-concentrated ferrofluids (χL = 7.61 and χL = 9.97) in the frame of model 2. We note that the numerical results of model 1–3 for the same χL, shown in Figure3b, coincide for magnetic fields h0 > 10 due to the same behavior of magnetisation laws in saturation M(i) = χ(i)H → Ms in the limit

Math. Model. Anal., 24(2):155–170, 2019. 166 O. Lavrova, V. Polevikov and S. Polevikov case H → ∞. The shielding effectiveness factor has an explicit analytical representation in a weak field limit, see e.g. [15, equation (72)] or [7, equation (B.34)] 1 2 1 2 Kef = µ(0) + 1 − 2 µ(0) − 1 . 4µ(0) δ This formula corresponds to a linear magnetisation law of the ferrofluid M(0)(H) = χLH with the initial susceptibility χ(0) = χL and the relative magnetic permeability µ(0) = 1 + χL. To verify the computations, we compare the numerical results from Figure3a for h0 = 0.01 and χ = {10, 30, 50} with the analytical ones for χ = χL, δ = 1.1 and observe the agreement between them up to the fourth significant digit. Figure4 shows the shielding effectiveness factor Kef dependent on the layer thickness at the applied field h0 = 1 for different values of the Langevin susceptibility χL = {4.06, 7.61, 9.97} and for models 2–3. The layer thickness is presented here by the difference between the outer and the inner layer radii as δ − 1 = (R2 − R1)/R1 and specified by the values δ − 1 ∈ (0.01, 10).

= 9.97

L 13

h = 1

= 7.61

= 4.06

0.01 0.1 1 10

-1

Figure 4. Dependence of the shielding eﬀectiveness factor Kef on the dimensionless layer thickness δ − 1 = (R2 − R1)/R1 for the dimensionless applied magnetic-ﬁeld intensity h0 = 1 and the Langevin susceptibility χL = {4.06, 7.61, 9.97}. Dashed lines correspond to model 2. Solid lines correspond to model 3.

Numerical results demonstrate the monotone increase of the shielding effect as the layer thickness increases. The strongest growth of shielding effectiveness factor is shown for δ − 1 ∈ (0.01, 3) with tendency to reach a constant value of Kef for thick layers with δ − 1 > 5. This qualitative behavior is present for ferrofluids of different concentrations, namely, for the moderately-concentrated ferrofluid with χL = 4.06 and for the highly-concentrated ferrofluids with χL = 7.61 and χL = 9.97. A stronger shielding effect is achieved for ferrofluids with a larger value of χL or a higher ferroparticle concentration, similarly to the observation in Figure3. The shielding effectiveness factor is higher for model 3 than for model 2 at different layer thicknesses, whereas the difference between numerical results of models 2 and 3 increases with increasing the ferroparticle concentration. Results in Figure4 do support the statement in [14] Numerical Modelling of Magnetic Shielding 167 that model 2 should be applied for the modelling of moderately-concentrated ferrofluids, whereas model 3 should be taken for the modelling of strongly- concentrated ferrofluids.

y y

u/h = 1.2

1.0

0.8

1 1

0.8

0.6

0.4

0.2

x x 0 1 0 1

a) b)

Figure 5. Isolines of the dimensionless magnetostatic potential u/h0 for the dimensionless layer thickness δ = 1.1, the Langevin susceptibility χL = 7.61, the dimensionless applied magnetic-ﬁeld intensity h0 = 1 (a) and h0 = 3 (b). Dashed isolines correspond to model 2. Solid isolines correspond to model 3.

Figure5 illustrates magnetic field structure in the computational domain by means of isolines of the magnetostatic potential. The isolines for model 2 and model 3 are close to each other for χL = 7.61. For any value of χL and h0 ≤ 1, the magnetic field in the inner domain Ω1 is nearly uniform and vertically directed, as shown in Figure5a. As expected, the inner magnetic field has lower intensity than the externally applied field (h1 < h0). The computations, discussed in the paper, are made under assumption of a uniform ferroparticle distribution inside the ferrofluid. We have made test computations for the magnetostatics problem (model 1) coupled with the diffusion problem for ferromagnetic particles inside the ferrofluid in the case of a linear dependence of the magnetisation law on the concentration, when the diffusion problem is analytically solvable, see [25]. The numerical results for the coupled magnetostatics and diffusion problems, compared with the results of only model 1 for χL ≤ 1, are very close to each other.

5 Conclusions

Shielding properties of the cylindrical thick-walled ferrofluid layer, protect- ing against uniform magnetic fields, have been numerically investigated for the weakly-, moderately- and highly-concentrated ferrofluids. Computational procedure for the magnetic field, penetrating through the ferrofluid layer to the inner nonmagnetic domain, is developed. The computational algorithm is based on a coupled method of finite differences and boundary elements. The shielding effectiveness factor, estimating the decay factor of the externally applied magnetic field into the inner domain, is calculated for different ferrofluid properties, different layer thicknesses and different intensities of the applied

Math. Model. Anal., 24(2):155–170, 2019. 168 O. Lavrova, V. Polevikov and S. Polevikov

2 field. Numerical results demonstrate that in weak fields (H0 < 10 A/m) the shielding effectiveness factor practically preserves a constant value, which de- pends on the initial susceptibility χ of the ferrofluid. Namely, Kef ≈ 1.4 for the moderately-concentrated ferrofluid with χ = 10 and Kef ≈ 3.1 for the dense ferrofluid with χ = 50. The shielding effect is almost absent in strong fields 6 (H0 > 10 A/m). With respect to models 1, 2 and 3, one can conclude that the magnetodipole interparticle interactions in ferrofluids have significant influence to the shielding effectiveness factor and the mathematical formulation should depend on the ferrofluid under study. Namely, model 1 could be applied for the mathematical modelling of only weakly-concentrated ferrofluids, model 2 suits for moderately-concentrated ferrofluids, whereas model 3 should be used for the problem under study with application of strongly-concentrated ferrofluids.

Acknowledgements

The authors would like to thank Prof. G. Vainikko for useful comments con- cerning numerical solution of the system of integral Fredholm first-kind equations in the BEM algorithm. The authors are grateful to the reviewers for the valuable comments and helpful suggestions. In particular, we thank for the references to the publications, where analytical solutions for the shielding effectiveness factor in weak magnetic fields are discussed. We appreciate the suggestion of the reviewer to compare models 1–3 for the same values of the Langevin susceptibility. The authors are grateful to the Belarusian State Re- search Program “Convergence-2020” for the financial support of the project 1.5.01.3.

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