Multiscale modeling of magnetorheological suspensions
MMultiscale modeling of magnetorheological suspensions
Grigor Nika ∗ and Bogdan Vernescu Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39,10117 Berlin, Germany Department of Mathematical Sciences, Worcester Polytechnic Institute, 100Institute Rd., Worcester, MA 01601June 18, 2019
Abstract
We develop a multiscale approach to describe the behavior of a suspension of solidmagnetizable particles in a viscous non-conducting fluid in the presence of an exter-nally applied magnetic field. By upscaling the quasi-static Maxwell equations coupledwith the Stokes’ equations we are able to capture the magnetorheological effect. Themodel we obtain generalizes the one introduced by Neuringer & Rosensweig [24], [27]for quasistatic phenomena. We derive the macroscopic constitutive properties ex-plicitly in terms of the solutions of local problems. The effective coefficients have anonlinear dependence on the volume fraction when chain structures are present. Thevelocity profiles computed for some simple flows, exhibit an apparent yield stress andthe flow profile resembles a
Bingham fluid flow.
Keywords:
Magnetorheological fluids, homogenization, chain structures, Poiseuille, Cou-ette
MSC:
Primary 35M10, 35M12, 35M30; Secondary 76D07, 76T20
Magnetorhelogical fluids are a suspension of non–colloidal, ferromagnetic particles in anon–magnetizable carrier fluid. The particles are often of micron size ranging anywherefrom 0 . − µm with particle volume fraction from 10 −
40 %. They were discovered by J.Rabinow in 1948 [26]. Around the same time W. Winslow [34] discovered electrorheologicalfluids, a closely related counterpart [3], [12], [25], [33].Magnetorheological fluids respond to an external magnetic field by a rapid, reversiblechange in their properties. They can transform from a liquid to a semi solid state in amatter of milliseconds. Upon the application of a magnetic field, the dipole interaction of ∗ Corresponding author: [email protected] a r X i v : . [ m a t h . A P ] J un djacent particles aligns the particles in the direction of the magnetic field lines. Namelyparticles attract one another along the magnetic field lines and repel one another in thedirection perpendicular to them. This leads to the formation of aggregate structures or chain structures . Once these chain structures are formed, the magnetorheological fluidexhibits a higher viscosity and yield stress that can now be triggered and dynamicallycontrolled by an applied external magnetic field [3], [6], [9], [19].The presence of these chain structures leads to a non-Newtonian behavior of the fluid.In many works, the Bingham constitutive law is used as an approximation to modelthe response of the magnetorheological and electrorheological fluids, particularly in shearexperiments [7], [12], [25], [8]. Although the Bingham model has proven itself useful incharacterizing the behavior of magnetorheological fluids, it is not always sufficient. Recentexperimental data show that true magnetorheological fluids exhibit departures from theBingham model [12], [35], [36].Another member of the magnetic suspensions family are ferrofluids. Ferrofluids are stablecolloidal suspensions of nanoparticles in a non-magnetizable carrier fluid. The initiationinto the hydrodynamics of ferrofluids began with Neuringer and Rosensweig in 1964 [24]and by a series of works by Rosensweig and co-workers summarized in [27]. The modelintroduced in [24] assumes that the magnetization is collinear with the magnetic field andhas been very useful in describing quasi-stationary phenomena. This work was extendedby Shliomis [31] by avoiding the collinearity assumption of the magnetization and themagnetic field and by considering the rotation of the nanoparticles with respect to thefluid they are suspended in.Most of the models characterizing magnetorheological suspensions are derived phenomeno-logically. The first attempt to use homogenization mechanics to describe the behavior ofmagnetorheological / electrorheological fluids was carried out in [14], [15] and [25]. Inthe works [14], [15] the influence of the external magnetic field is introduced as a volumicdensity force acting on each particle and as a surface density force acting on the boundaryof each particle. The authors in [25] extend the work in [15], for electrorheological fluids,by presenting a more complete model that one way couples the conservation of mass andmomentum equations with Maxwell’s equations through the Maxwell stress tensor . Asan application they consider a uniform shearing of the electrorheological fluid submittedto a uniform electric field boundary conditions in a two dimensional slab and they re-cover a stress tensor, at the macroscopic scale, that has exactly the form of the Binghamconstitutive equation.The authors in [25], [27], [28], [33], [10] use models that decouple the conservation of massand momentum equations from the Maxwell equations. Thus, in principle, one can solvethe Maxwell equations and use the resulting magnetic or electric field as a force in theconservation of mass and momentum equations.The present work focuses on a suspension of rigid magnetizable particles in a Newtonianviscous fluid with an applied external magnetic field. We assume the fluid to be electricallynon-conducting. We use the homogenization method to upscale the quasi-static Maxwellequations coupled with the Stokes equations through Ohm’s law to capture the magne-torheological effect. In doing so we extend the model of [25], [33]. Thus, the Maxwellequations, and the balance of mass and momentum equations must be simultaneously2olved. Additionally, the model is able to capture the added effect particle chain struc-tures have on the effective coefficients. We demonstrate this added effect by carrying outexplicit computations of the effective coefficients using the finite element method.The paper is organized in the following way. In Section 2. we introduce the problem inthe periodic homogenization framework. The particles are periodically distributed andthe size of the period is of the same order as the characteristic length of the particles. Weassume the fluid velocity is continuous across the particle interface and that the particlesare in equilibrium in the presence of the magnetic field.In Section 3. we use two-scale expansions to obtain a one way coupled set of local problemsat order O ( (cid:15) − ). One problem characterizes the effective viscosity of the magnetorheolog-ical fluid while the other local problem represents the magnetic field contribution.Section 4. and Section 5. are devoted to the study of the local problems that arise fromthe contribution of the bulk magnetic field as well as the bulk velocity and we provide newconstitutive laws for Maxwell’s equations.In Section 6. we provide the governing effective equations of the magnetorheological fluidwhich include, in addition to the viscous stresses, a “ Maxwell type ” stress of second order inthe magnetic field. Furthermore, we provide formulas for the effective viscosity and threedifferent effective magnetic permeabilities for the “
Maxwell type ” stress that generalizethose in [15].Section 7. is devoted to numerical results for a suspension of circular iron particles ofdifferent volume fractions using the finite element method. Moreover, we explore theeffect that chain structures have in the effective coefficients and compare the results withthe absence of chain structures . Moreover, we compute the velocity profiles for Poiseuilleand Couette flows of the magnetorheological fluid and we plot the shear stress versus theshear rate curve for different values of the applied magnetic field to obtain a yield stresscomparable to the one observed experimentally (e.g. [35]). Finally Section 8. containsconclusions and perspectives on the study.
Notation
Throughout the paper we are going to be using the following notation: I indicates the n × n identity matrix, bold symbols indicate vectors in two or three dimensions, regular symbolsindicate tensors, e ( uuu ) indicates the strain rate tensor defined by e ( uuu ) = 12 (cid:16) ∇ uuu + ∇ uuu (cid:62) (cid:17) ,where often times we will use subscript to indicate the variable of differentiation. The innerproduct between matrices is denoted by A : B = tr ( A (cid:62) B ) = (cid:80) ij A ij B ji and throughoutthe paper we employ the Einstein summation notation for repeated indices. For the homogenization setting of the suspension problem we define Ω ⊂ R n , n = 2 ,
3, tobe a bounded open set with sufficiently smooth boundary ∂ Ω, Y = (cid:18) − , (cid:19) n is the unit3ube in R n , and Z n is the set of all n –dimensional vectors with integer components. Forevery positive (cid:15) , let N (cid:15) be the set of all points (cid:96) ∈ Z n such that (cid:15) ( (cid:96) + Y ) is strictly includedin Ω and denote by | N (cid:15) | their total number. Let T be the closure of an open connected setwith sufficiently smooth boundary, compactly included in Y . For every (cid:15) > (cid:96) ∈ N (cid:15) we consider the set T (cid:15)(cid:96) ⊂⊂ (cid:15) ( (cid:96) + Y ), where T (cid:15)(cid:96) = (cid:15) ( (cid:96) + T ). The set T (cid:15)(cid:96) represents one ofthe rigid particles suspended in the fluid, and S (cid:15)(cid:96) = ∂T (cid:15)(cid:96) denotes its surface (see Figure 1).Based on the above setting we define the following subsets of Ω:Ω (cid:15) = (cid:91) (cid:96) ∈ N (cid:15) T (cid:15)(cid:96) , Ω (cid:15) = Ω \ Ω (cid:15) . In what follows T (cid:15)(cid:96) will represent the magnetizable rigid particles, Ω (cid:15) is the domainoccupied by the rigid particles and Ω (cid:15) the domain occupied by the surrounding fluid. By nnn we indicate the unit normal on the particle surface pointing outwards and by (cid:74) · (cid:75) weindicate the jump discontinuity between the fluid and the rigid part.ΩΩ (cid:15) Ω (cid:15) x (cid:96)c T (cid:15)(cid:96) Y (cid:15)(cid:96) (cid:15)(cid:15) Figure 1: Schematic of the periodic suspension of rigid magnetizable particles in a non-magnetizable fluid. Ω (cid:15) represents the domain occupied by the rigid particles and Ω (cid:15) represents the domain occupied by the surrounding fluid. xxx (cid:96)c represents the center of massof the particle inside the cell of size (cid:15) , Y (cid:15)(cid:96) .We consider the Navier-Stokes equations coupled with the quasistastic Maxwell equations, ρ ∂ vvv (cid:15) ∂ t + ρ ( vvv (cid:15) · ∇ ) vvv (cid:15) − div σ (cid:15) = ρ fff , where σ (cid:15) = 2 ν e ( vvv (cid:15) ) − p (cid:15) I in Ω (cid:15) , (1a)div vvv (cid:15) = 0 , div BBB (cid:15) = 0 , curl HHH (cid:15) = 000 in Ω (cid:15) , (1b) e ( vvv (cid:15) ) = 0 , div BBB (cid:15) = 0 , curl HHH (cid:15) = η vvv (cid:15) × BBB (cid:15) in Ω (cid:15) , (1c)where BBB (cid:15) = µ (cid:15) HHH (cid:15) , with boundary conditions on the surface of each particle T (cid:15)(cid:96) , (cid:74) vvv (cid:15) (cid:75) = 000 , (cid:74) BBB (cid:15) · nnn (cid:75) = 0 , (cid:74) nnn × HHH (cid:15) (cid:75) = 000 on S (cid:15)(cid:96) , (2)and outer boundary conditions vvv (cid:15) = 000 , HHH (cid:15) = bbb on ∂ Ω , (3)4here ρ is the density of the fluid, ν is the viscosity, vvv (cid:15) represents the velocity field, p (cid:15) thepressure, e ( vvv (cid:15) ) the strain rate, fff the body forces, nnn the exterior normal to the particles, HHH (cid:15) the magnetic field, µ (cid:15) is the magnetic permeability of the material, µ (cid:15) ( xxx ) = µ if xxx ∈ Ω (cid:15) and µ (cid:15) ( xxx ) = µ if xxx ∈ Ω (cid:15) , η the electric conductivity of the rigid particles, and bbb is anapplied constant magnetic field on the exterior boundary of the domain Ω.In order to obtain the balance of forces and torques for the particles, let us observe when themagnetorheological fluid is submitted to a magnetic field, the rigid particles are subjectedto a force that makes them behave like a dipole aligned in the direction of the magneticfield. This force can be written in the form, FFF (cid:15) = − | HHH (cid:15) | ∇ µ (cid:15) , where | · | represents the standard Euclidean norm. The force can be written in terms ofthe Maxwell stress τ (cid:15)ij = µ (cid:15) H (cid:15)i H (cid:15)j − µ (cid:15) H (cid:15)k H (cid:15)k δ ij as FFF (cid:15) = div τ (cid:15) + BBB (cid:15) × curl HHH (cid:15) . Sincethe magnetic permeability is considered constant in each phase, it follows that the forceis zero in each phase. Therefore, we deduce thatdiv τ (cid:15) = (cid:40) xxx ∈ Ω (cid:15) − BBB (cid:15) × curl HHH (cid:15) if xxx ∈ Ω (cid:15) . (4)Lastly, we remark that unlike the viscous stress σ (cid:15) , the Maxwell stress is present in theentire domain Ω. Hence, we can write the balance of forces and torques in each particleas, (cid:90) T (cid:15)(cid:96) ρ duuu (cid:15) dt dxxx = (cid:90) S (cid:15)(cid:96) ( σ (cid:15) nnn + (cid:74) τ (cid:15) nnn (cid:75) ) ds + (cid:90) T (cid:15)(cid:96) BBB (cid:15) × curl HHH (cid:15) dxxx + (cid:90) T (cid:15)(cid:96) ρ fff dxxx, (cid:90) T (cid:15)(cid:96) ρ ( xxx − xxx (cid:96)c ) × duuu (cid:15) dt dxxx = (cid:90) S (cid:15)(cid:96) ( σ (cid:15) nnn + (cid:74) τ (cid:15) nnn (cid:75) ) × ( xxx − xxx (cid:96)c ) ds + (cid:90) T (cid:15)(cid:96) ( BBB (cid:15) × curl HHH (cid:15) ) × ( xxx − xxx (cid:96)c ) dxxx + (cid:90) T (cid:15)(cid:96) ρ fff × ( xxx − xxx (cid:96)c ) dxxx, (5)where xxx (cid:96)c is the center of mass of the rigid particle T (cid:15)(cid:96) . Equations (1), (5) together with boundary conditions (2), (3) describe the behavior of themagnetorheological suspension.
Before we proceed further we non-dimensionalize the problem. Denote by t ∗ = t/ LV , x ∗ = x/L , vvv ∗ = vvv/V , p ∗ = p/ν VL , HHH ∗ = HHH/H , fff ∗ = fff / V L , and µ (cid:15) ∗ = µ (cid:15) /µ . Here L isa characteristic length, V is a characteristic velocity, p is a characteristic pressure, fff is acharacteristic force and H is a characteristic unit of the magnetic field. Substituting theabove expressions into (1) as well as in the balance of forces and torques, and using the5act that the flow is assumed to be at low Reynolds numbers, we obtain Re (cid:18) ∂vvv (cid:15) ∗ ∂t + ( vvv (cid:15) ∗ · ∇ ) vvv (cid:15) ∗ (cid:19) − div ∗ σ (cid:15) ∗ = Refff ∗ , where σ (cid:15) ∗ = 2 e ( vvv (cid:15) ∗ ) − p (cid:15) ∗ I in Ω (cid:15) , div ∗ vvv (cid:15) ∗ = 0 , div ∗ BBB (cid:15) ∗ = 0 , curl ∗ HHH (cid:15) ∗ = 000 in Ω (cid:15) ,e ∗ ( vvv (cid:15) ∗ ) = 0 , div ∗ BBB (cid:15) ∗ = 0 , curl HHH (cid:15) ∗ = R m vvv (cid:15) ∗ × BBB (cid:15) ∗ in Ω (cid:15) , where BBB (cid:15) ∗ = µ (cid:15) ∗ HHH (cid:15) ∗ and with boundary conditions on the surface of each particle T (cid:15)(cid:96) , (cid:74) vvv (cid:15) ∗ (cid:75) = 000 , (cid:74) BBB (cid:15) ∗ · nnn (cid:75) = 0 , (cid:74) nnn × HHH (cid:15) ∗ (cid:75) = 000 on S (cid:15)(cid:96) ,vvv (cid:15) ∗ = 000 , HHH (cid:15) ∗ = bbb ∗ on ∂ Ω . together with the balance of forces and torques,Re (cid:90) T (cid:15)(cid:96) duuu (cid:15) ∗ dt ∗ dxxx ∗ = (cid:90) S (cid:15)(cid:96) σ (cid:15) ∗ nnnds ∗ + α (cid:90) S (cid:15)(cid:96) (cid:74) τ (cid:15) ∗ nnn (cid:75) ds ∗ + α (cid:90) T (cid:15)(cid:96) BBB (cid:15) ∗ × curl HHH (cid:15) ∗ dxxx ∗ + Re (cid:90) T (cid:15)(cid:96) fff ∗ dxxx ∗ , Re (cid:90) T (cid:15)(cid:96) ( xxx ∗ − xxx (cid:96) ∗ c ) × duuu (cid:15) ∗ dt ∗ dxxx ∗ = (cid:90) S (cid:15)(cid:96) σ (cid:15) ∗ nnn × ( xxx ∗ − xxx (cid:96)c ∗ ) ds ∗ + α (cid:90) S (cid:15)(cid:96) (cid:74) τ (cid:15) ∗ nnn (cid:75) × ( xxx ∗ − xxx (cid:96)c ∗ ) ds ∗ + α (cid:90) T (cid:15)(cid:96) ( BBB (cid:15) ∗ × curl HHH (cid:15) ∗ ) × ( xxx ∗ − xxx (cid:96)c ∗ ) dxxx ∗ + Re (cid:90) T (cid:15)(cid:96) fff ∗ × ( xxx ∗ − xxx (cid:96)c ∗ ) dxxx ∗ , where Re = ρ V Lν is the Reynolds number, α = µ H Lν V is the Alfven number, and R m = η µ L V is the magnetic Reynolds number.In what follows we drop the star for simplicity. Moreover, for low Reynolds numbers thepreceding equations become, − div σ (cid:15) = 000 , where σ (cid:15) = 2 e ( vvv (cid:15) ) − p (cid:15) I in Ω (cid:15) , (6a)div vvv (cid:15) = 0 , div HHH (cid:15) = 0 , curl HHH (cid:15) = 000 in Ω (cid:15) , (6b) e ( vvv (cid:15) ) = 0 , div HHH (cid:15) = 0 , curl HHH (cid:15) = R m vvv (cid:15) × BBB (cid:15) in Ω (cid:15) , (6c)with boundary conditions (cid:74) vvv (cid:15) (cid:75) = 000 , (cid:74) BBB (cid:15) · nnn (cid:75) = 0 , (cid:74) nnn × HHH (cid:15) (cid:75) = 000 on S (cid:15)(cid:96) ,vvv (cid:15) = 000 , HHH (cid:15) = bbb on ∂ Ω , (7)together with the balance of forces and torques,0 = (cid:90) S (cid:15)(cid:96) σ (cid:15) nnn ds + α (cid:90) S (cid:15)(cid:96) (cid:74) τ (cid:15) nnn (cid:75) ds + α (cid:90) T (cid:15)(cid:96) BBB (cid:15) × curl HHH (cid:15) dxxx, (cid:90) S (cid:15)(cid:96) σ (cid:15) nnn × ( xxx − xxx (cid:96)c ) ds + α (cid:90) S (cid:15)(cid:96) (cid:74) τ (cid:15) nnn (cid:75) × ( xxx − xxx (cid:96)c ) ds + α (cid:90) T (cid:15)(cid:96) ( BBB (cid:15) × curl HHH (cid:15) ) × ( xxx − xxx (cid:96)c ) dxxx. (8)6 Two scale expansions
We assume the particles are periodically distributed in Ω and thus consider the two scaleexpansion on vvv (cid:15) , HHH (cid:15) and p (cid:15) [1], [2], [4], [20], [29], [30], vvv (cid:15) ( xxx ) = + ∞ (cid:88) i =0 (cid:15) i vvv i ( xxx, yyy ) , HHH (cid:15) ( xxx ) = + ∞ (cid:88) i =0 (cid:15) i HHH i ( xxx, yyy ) , p (cid:15) ( xxx ) = + ∞ (cid:88) i =0 (cid:15) i p i ( xxx, yyy ) with yyy = xxx(cid:15) . where xxx ∈ Ω and yyy ∈ R n . One can show that vvv is independent of yyy and can thus obtainthe following problem at order (cid:15) − , − ∂σ ij ∂y j = 0 in Y f , (9a) σ ij = − p δ ij + 2 ν ( e ijx ( vvv ) + e ijy ( vvv )) (9b) ∂v j ∂x j + ∂v j ∂y j = 0 in Y f , (9c) e ijx ( vvv ) + e ijy ( vvv ) = 0 in T, (9d) ∂B j ∂y j = 0 , (cid:15) ijk ∂H k ∂y j = 0 where B i = µH i in Y, (9e)with boundary conditions (cid:113) vvv (cid:121) = 000 , (cid:113) BBB · nnn (cid:121) = 0 , (cid:113) nnn × HHH (cid:121) = 000 on S ,vvv , HHH are Y − periodic . (10)Here Y f and T denote the fluid, respectively the particle part of Y ; and S denotes thesurface of T . At order of (cid:15) and (cid:15) we obtain from (8) the balance of forces and torquesfor the particle T respectively,0 = (cid:90) S σ nnn ds + α (cid:90) S (cid:113) τ nnn (cid:121) ) ds − α (cid:90) T BBB × curl y HHH dyyy, (cid:90) S yyy × σ nnn ds + α (cid:90) S yyy × (cid:113) τ nnn (cid:121) ds − α (cid:90) T yyy × (cid:0) BBB × curl y ( HHH ) (cid:1) dyyy, (11)where τ ij is the Maxwell stress at order (cid:15) τ ij = µ H i H j − µ H k H k δ ij , (12)We remark that since from (9e) curl y ( HHH ) = 000 in Y , the balance of forces and torquessimplify to the following,0 = (cid:90) S σ nnn + α (cid:90) S (cid:113) τ nnn (cid:121) ds and 0 = (cid:90) S yyy × σ nnn ds + α (cid:90) S yyy × (cid:113) τ nnn (cid:121) ds. (13) Remark 1
At first order, the problem (9) - (13) becomes one way coupled, as one couldsolve the Maxwell equations (9e) independently. Once a solution is obtained, the Stokesproblem (9a) - (9c) , can be solved with a known magnetic force added to the balance of forcesand torques (13) . Constitutive relations for Maxwell’s equations
Using the results from the two scale expansions, (9e), we can see that curl y ( HHH ) = 000 in Y and thus there exists a function ψ = ψ ( xxx, yyy ) with average (cid:101) ψ = 0 such that H i = − ∂ψ ( xxx, yyy ) ∂y i + (cid:101) H i ( xxx ) , (14)where (cid:101) · = 1 | Y | (cid:90) Y · dyyy . Using the fact div y BBB = 0 in Y , B i = µ H i and the boundaryconditions (7) we have, − ∂∂y i (cid:18) µ (cid:18) − ∂ψ∂y i + (cid:101) H i (cid:19)(cid:19) = 0 in Y , (cid:115) µ (cid:18) − ∂ψ∂y i + (cid:101) H i (cid:19) n i (cid:123) = 0 on S ,ψ is Y − periodic , (cid:101) ψ = 0 . (15)Introducing the space of periodic functions, with zero average W = (cid:8) w ∈ H per ( Y ) | (cid:101) w = 0 (cid:9) , then the variational formulation of (15) isFind ψ ∈ W such that (cid:90) Y µ ∂ψ∂y i ∂v∂y i dyyy = (cid:101) H i ( xxx ) (cid:90) Y µ ∂v∂y i dyyy for any v ∈ W . (16)Since we have imposed that ψ has zero average over the unit cell Y , the solution to (16)can be determined uniquely by a simple application of the Lax-Milgram lemma.Let φ k be the unique solution ofFind φ k ∈ W such that (cid:90) Y µ ∂φ k ∂y i ∂v∂y i dyyy = (cid:90) Y µ ∂v∂y k dyyy for any v ∈ W . (17)By virtue of linearity of (16) we can write, ψ ( xxx, yyy ) = φ k ( yyy ) (cid:101) H k ( xxx ) + C ( xxx ) . In principle, once (cid:101) H k is known, we can determine ψ up to an additive function of xxx. Hence,combining (14) and the above relationship between ψ and φ k we obtain the followingconstitutive law between the magnetic induction and the magnetic field,8 B i = µ Hik (cid:101) H k , where µ Hik = (cid:90) Y µ (cid:18) − ∂φ k ∂y i + δ ik (cid:19) dyyy. (18)One can show (see [29]) that the homogenized magnetic permeability tensor is symmetric, µ Hik = µ Hki . Moreover, if we denote by A i(cid:96) ( yyy ) = (cid:16) − ∂φ (cid:96) ( yyy ) ∂y i + δ i(cid:96) (cid:17) one can see from (14) that H i = A i(cid:96) (cid:101) H (cid:96) and thus the Maxwell stress (12) takes the following form, τ ij = µ A i(cid:96) A jm (cid:101) H (cid:96) (cid:101) H m − µ A mk A (cid:96)k δ ij (cid:101) H m (cid:101) H (cid:96) = µ A m(cid:96)ij (cid:101) H m (cid:101) H (cid:96) . Here A m(cid:96)ij = ( A i(cid:96) A jm + A j(cid:96) A im − A mk A (cid:96)k δ ij ) and has the following symmetry, A m(cid:96)ij = A m(cid:96)ji = A (cid:96)mij . Recall that the div τ (cid:15) = 0 in Ω (cid:15) and div τ (cid:15) = − BBB (cid:15) × curl HHH (cid:15) in Ω (cid:15) . Fromthe two scale expansion, at order (cid:15) − from equation (4) we obtain,div y τ = 0 in Y. (19) Problem (9)-(10), (13) is an elliptic problem in the variable yyy ∈ Y with forcing terms vvv ( xxx )and (cid:101) HHH ( xxx ) at the macroscale. We can decouple the contributions of vvv ( xxx ) and (cid:101) H ( xxx ) andsplit vvv and p in two parts: a part that is driven by the bulk velocity, and a part thatcomes from the bulk magnetic field. v k ( xxx, yyy ) = χ m(cid:96)k ( yyy ) e m(cid:96) ( vvv ) + ξ m(cid:96)k ( yyy ) (cid:101) H m (cid:101) H (cid:96) + A k ( xxx ) , (20) p ( xxx, yyy ) = p m(cid:96) ( yyy ) e m(cid:96) ( vvv ) + π m(cid:96) ( yyy ) (cid:101) H m (cid:101) H (cid:96) + ¯ p ( xxx ) , (21)where (cid:90) Y f p m(cid:96) ( yyy ) dyyy = 0 and (cid:90) Y f π m(cid:96) ( yyy ) dyyy = 0 . Here, χχχ ml satisfies − ∂∂y j ε m(cid:96)ij = 0 in Y f ,ε m(cid:96)ij = − p m(cid:96) δ ij + 2 ( C ijm(cid:96) + e ijy ( χχχ m(cid:96) )) − ∂χ m(cid:96)i ∂y i = 0 in Y f , (cid:114) χχχ m(cid:96) (cid:122) = 0 on S ,C ijm(cid:96) + e ijy ( χχχ m(cid:96) ) = 0 in T ,χχχ m(cid:96) is Y − periodic , (cid:101) χχχ m(cid:96) = 000 in Y, (22)9ogether with the balance of forces and torques, (cid:90) S ε m(cid:96)ij n j ds = 0 and (cid:90) S (cid:15) ijk y j ε m(cid:96)kp n p ds = 0 , (23)where C ijm(cid:96) = 12 ( δ im δ j(cid:96) + δ i(cid:96) δ jm ) − n δ ij δ m(cid:96) . Equation (22) is well known having beenobtained by many authors [16, 17, 18] among others.The variational formulation of (22)-(23) is:Find χχχ m(cid:96) ∈ U such that (cid:90) Y f e ij ( χχχ m(cid:96) ) e ij ( φφφ − χχχ m(cid:96) ) dyyy = 0 , for all φφφ ∈ U , (24)where U is the closed, convex, non-empty subset of H per ( Y ) n defined by U = (cid:8) uuu ∈ H per ( Y ) n | div uuu = 0 in Y f , e ij ( uuu ) = − C ijm(cid:96) in T, (cid:74) uuu (cid:75) = 000 on S, (cid:101) uuu = 000 in Y } . Remark 2
We remark that if we define B ijk = ( y i δ jk + y j δ ik ) − n y k δ ij , then it imme-diately follows that e ij ( BBB m(cid:96) ) = C ijm(cid:96) . Existence and uniqueness of a solution follows from classical theory of variational inequal-ities [13]. In similar fashion we can derive the local problem for ξξξ ml , − ∂∂y j Σ m(cid:96)ij = 0 in Y f , Σ m(cid:96)ij = − π m(cid:96) δ ij + 2 e ijy ( ξξξ m(cid:96) ) − ∂ξ m(cid:96)i ∂y i = 0 in Y f , (cid:114) ξξξ m(cid:96) (cid:122) = 0 on S ,e ijy ( ξξξ m(cid:96) ) = 0 in T ,ξξξ m(cid:96) is Y − periodic , (cid:101) ξξξ m(cid:96) = 0 . (25)Using (19) the balance of forces reduces to, (cid:90) S Σ m(cid:96)ij n j ds = 0 , (26)together with the balance of torques 10 S (cid:15) ijk y j (cid:16) Σ m(cid:96)kp + α (cid:114) µA m(cid:96)kp (cid:122) (cid:17) n p ds = 0 . (27)We can formulate (25)–(27) variationally as,Find ξξξ m(cid:96) ∈ V such that (cid:90) Y f e ijy ( ξξξ m(cid:96) ) e ijy ( φφφ ) dyyy + (cid:90) Y A m(cid:96)ij e ijy ( φφφ ) dyyy = 0 , for all φφφ ∈ V , (28)where V = (cid:8) vvv ∈ H per ( Y ) n | div vvv = 0 in Y f , e y ( uuu ) = 0 in T, (cid:74) vvv (cid:75) = 000 on S, (cid:101) vvv = 000 in Y (cid:9) , is a closed subspace of H per ( Y ) n . Existence and uniqueness follows from an application ofthe Lax-Milgram lemma. These equations indicate the contribution of the magnetic fieldand the solution ξξξ m(cid:96) depends, through the balance of forces and torques on the solutionof the local problem (17) and the effective magnetic permeability of the composite. Remark 3
We remark that the only driving force that makes the solution ξξξ m(cid:96) non trivialin (28) is the rotation induced by the magnetic field through the fourth order tensor A m(cid:96)ij . The two-scale expansion at the (cid:15) order yields the following problems: − div x σ − div y σ = 000 in Y f , (29a)div x vvv + div y vvv = 0 in Y f , (29b)div x BBB + div y BBB = 0 in Y , (29c)curl x HHH + curl y HHH = 0 in Y f , (29d)curl x HHH + curl y HHH = R m vvv × BBB in T , (29e)with boundary conditions (cid:113) vvv (cid:121) = 000 , (cid:113) BBB · nnn (cid:121) = 0 (cid:113) nnn × HHH (cid:121) = 000 on S ,vvv , HHH are Y − periodic . (30)In each period, we consider a Taylor expansion, around the center of mass of the rigidparticle, both of the viscous stress and the Maxwell stress of the form (see [18]), σ (cid:15) ( xxx ) = σ ( xxx (cid:96)c , yyy ) + ∂σ ( xxx (cid:96)c , yyy ) ∂x α ( x α − x (cid:96)c,α ) + (cid:15) σ ( xxx (cid:96)c , yyy ) + (cid:15) ∂σ ( xxx (cid:96)c , yyy ) ∂x α ( x α − x (cid:96)c,α ) + · · · (cid:15) ( xxx ) = τ ( xxx (cid:96)c , yyy ) + ∂τ ( xxx (cid:96)c , yyy ) ∂x α ( x α − x (cid:96)c,α ) + (cid:15) τ ( xxx (cid:96)c , yyy ) + (cid:15) ∂τ ( xxx (cid:96)c , yyy ) ∂x α ( x α − x (cid:96)c,α ) + · · · where the expansion of the Maxwell stress occurs both inside the rigid particle and thefluid. Using this method we can expand the balance of forces, (8), and obtain at order (cid:15) ,0 = (cid:90) S (cid:32) ∂σ ij ∂x k y k + σ ij (cid:33) n j ds + α (cid:90) S (cid:116) (cid:32) ∂τ ij ∂x k y k + τ ij (cid:33) n j (cid:124) ds − α (cid:90) T ( BBB × (curl x HHH + curl y HHH )) i dyyy. (31)Integrate (29a) over Y f and add to (31) obtain the following,0 = (cid:90) Y f ∂σ ij ∂x j dyyy + (cid:90) S ∂σ ij ∂x k y k n j ds + α (cid:90) S (cid:74) ( ∂τ ij ∂x k y k + τ ij ) n j (cid:75) ds − α (cid:90) T ( BBB × (curl x HHH + curl y HHH )) i dyyy. (32)At order (cid:15) we obtain, div x τ + div y τ = 0 in Y f and div x τ + div y τ = − BBB × (curl x HHH + curl y HHH ) in T . Combining the aforementioned results and the divergencetheorem we can rewrite (32) the following way,0 = (cid:90) Y f ∂σ ij ∂x j dyyy + (cid:90) S ∂σ ik ∂x j y j n k ds + α (cid:90) S (cid:115) ∂τ ik ∂x j y j n k (cid:123) ds + α (cid:90) Y ∂τ ij ∂x j dyyy. (33)Using the decomposition of vvv and p in (20) and (21) we can re-write σ ij and τ ij , σ ij = − ¯ p δ ij + ε m(cid:96)ij e mlx ( vvv ) + Σ m(cid:96)ij (cid:101) H m (cid:101) H (cid:96) , τ ij = µ A m(cid:96)ij (cid:101) H m (cid:101) H (cid:96) . Moreover, equations (9b), (12), (22) and (25) allow us to retain the only symmetric partof (33). Hence the homogenized fluid equations (33) become,0 = ∂∂x j (cid:16) − ¯ p δ ij + (cid:110)(cid:90) Y f e ijy ( BBB m(cid:96) + χχχ m(cid:96) ) dyyy + (cid:90) S ε m(cid:96)pk B ijp n k ds (cid:111) e m(cid:96)x ( vvv )+ (cid:40)(cid:90) Y f e ijy ( ξξξ m(cid:96) ) dyyy + (cid:90) S Σ m(cid:96)pk B ijp n k ds + α (cid:90) Y µA m(cid:96)ij dyyy + α (cid:90) S (cid:114) µA m(cid:96)pk (cid:122) B ijp n k (cid:27) (cid:101) H m (cid:101) H (cid:96) (cid:17) . (34)Furthermore, using (9c)–(9d) and the divergence theorem we can obtain the incompress-ibility condition, div x vvv = 0.Denote by 12 Hijm(cid:96) = (cid:40)(cid:90) Y f e ijy ( BBB m(cid:96) + χχχ m(cid:96) ) dyyy + (cid:90) S ε m(cid:96)pk B ijp n k ds (cid:41) and β Hijm(cid:96) = (cid:40)(cid:90) Y f e ijy ( ξξξ m(cid:96) ) dyyy + (cid:90) S Σ m(cid:96)pk B ijp n k ds + α (cid:90) Y µA m(cid:96)ij dyyy + α (cid:90) S (cid:114) µA m(cid:96)pk (cid:122) B ijp n k (cid:41) then the homogenized equation (34) becomes0 = ∂∂x j (cid:16) − ¯ p δ ij + ν ijm(cid:96) e m(cid:96)x ( vvv ) + β ijm(cid:96) (cid:101) H m (cid:101) H (cid:96) (cid:17) . Using local problem (22) we can re-write the ν ijm(cid:96) the following way, ν Hijm(cid:96) = (cid:90) Y f e ( BBB ml + χχχ ml ) : e ( BBB ij + χχχ ij ) dyyy. (35)which is a well known formula derived in [16], [17], [29] as well as its generalizations derivedin [18], [23]. In a similar fashion, using local problem (25) and the kinematic condition in(22) we can re-write β Hijm(cid:96) as follows, β Hijm(cid:96) = (cid:90) Y f e ( ξξξ ml ) : e ( BBB ij + χχχ ij ) dyyy + α (cid:90) Y f µA m(cid:96) : e ( BBB ij + χχχ ij ) dyyy + α (cid:90) Y µA m(cid:96)ij dyyy. (36)It is now clear that ν Hijm(cid:96) possesses the following symmetry, ν Hijm(cid:96) = ν Hjim(cid:96) = ν Hm(cid:96)ij . Whilefor β Hijm(cid:96) , we have β Hijm(cid:96) = β Hjim(cid:96) = β Hij(cid:96)m . To obtain the homogenized Maxwell equations, average (29c), (29d), and (29e) over Y , Y f , and T respectively and use equation (18) to obtain, ∂ ( µ Hik (cid:101) H k ) ∂x j = 0 , (cid:15) ijk ∂ (cid:101) H k ∂x j = R m (cid:15) ijk v j µ HSkp (cid:101) H p in Ω , where µ HSik = (cid:90) T µ (cid:18) − ∂φ k ∂y i + δ ik (cid:19) dyyy (37)with boundary conditions, (cid:101) H i = b i , v i = 0 on ∂ Ω . ν Hijm(cid:96) and β Hijm(cid:96) . This is done by introducing the projection on hydrostatic fields, P b , and the pro-jection on shear fields P s (see [20], [21], [22]). The components of the projections in threedimensional space are given by:( P b ) ijk(cid:96) = 1 n δ ij δ k(cid:96) , ( P s ) ijk(cid:96) = 12 ( δ ik δ j(cid:96) + δ i(cid:96) δ jk ) − n δ ij δ k(cid:96) . Let us fix the following notation: ν b = tr ( P b ν H ) = 1 n ν Hppqq , ν s = tr ( P s ν H ) = (cid:18) ν Hpqpq − n ν Hppqq (cid:19) ,β Hb = tr ( P b β H ) = 1 n β Hppqq , β s = tr ( P s β H ) = (cid:18) β Hpqpq − n β Hppqq (cid:19) . we can re-write the homogenized coefficients ν Hijm(cid:96) and β Hijm(cid:96) as follows, ν Hijm(cid:96) = 1 n ( ν b − ν s ) δ ij δ m(cid:96) + 12 ν s ( δ ik δ j(cid:96) + δ i(cid:96) δ jk ) ,β Hijm(cid:96) = 1 n ( β b − ν s ) δ ij δ m(cid:96) + 12 β s ( δ ik δ j(cid:96) + δ i(cid:96) δ jk ) . Gathering all the equations we have that the homogenized equations governing the mag-netorheological fluid form the following coupled system between the Stokes equations andthe quasistatic Maxwell equations,div (cid:0) σ H + τ H (cid:1) = 000 in Ω ,σ H + τ H = − ¯ p I + ν s e ( vvv ) + 1 n ( β b − β s ) (cid:12)(cid:12)(cid:12) (cid:101) HHH (cid:12)(cid:12)(cid:12) I + β s (cid:101) HHH ⊗ (cid:101) HHH , div vvv = 0 in Ω , div (cid:16) µ H (cid:101) HHH (cid:17) = 0 in Ω , curl (cid:101) HHH = R m vvv × µ HS (cid:101) HHH in Ω ,vvv = 000 , on ∂ Ω , (cid:101) HHH = bbb on ∂ Ω . (38)Equation (38) generalizes the quasistatic set of equations introduced in [24], [27] in twoways: first by providing exact formulas for the effective coefficients which consist of thehomogenized viscosity, ν H , and three homogenized magnetic permeabilities, µ H , µ HS , and β H , which all depend on the geometry of the suspension, the volume fraction, the magneticpermeability µ , the Alfven number α , and the particles distribution. Second, by couplingthe fluid velocity field with the magnetic field through Ohm’s law.14 Numerical results for a suspension of iron particles in aviscous non-conducting fluid
The goal of this section is to carry out calculations, using the finite element method, of theeffective viscosity ν H and the effective magnetic coefficients β H , µ H , and µ HS that describethe behavior of a magnetorheological fluid in (38). To achieve this we need to computethe solutions φ k , χ m(cid:96) , and ξ m(cid:96) of the local problems (17), (22), and (25) respectively.Unlike regular suspensions for which the effective properties are dependent only on fluidviscosity, particle geometry, and volume fraction, for magnetorheological fluids of signifi-cance is also the particles’ distribution. The magnetic field polarizes the particles whichalign in the field direction to form chains and columns and that contributes significantlyto the increase of the yield stress [3], [32], [34].The choice of the periodic unit cell, as well as the geometry and distribution of particles,can lead to different chain structures and hence different effective properties. We areconsidering here a uniform distribution of particles as well as a chain distribution; toachieve this we change the aspect ratio of the unit cell from a 1 × × / × × P P × periodic unit cell, we used 100 × P P chain structures of ellipses angle orientation must be taken into account. Wedo not explore such cases in the current work and we consider only the case of circularparticles.We remark that in the two dimensional setting, the tensors entries C ijmm = 0 and BBB mm =000. As a consequence, of the linearity of the local problem (22), we have χχχ mm = 000. Hence, ν mmii = 0 which implies that ν b = 0. Using a similar argument, we can similarly showthat β mmii = 0 which implies that β b = 0.The relative magnetic permeability of the iron particle (99 .
95% pure) was fixed throughout to be 2 × while that of the fluid was set to 1 [5]. All the calculations were carriedout using the software FreeFem++ [11].We compute the solutions of the local problems both for a uniform distribution of particles,Figure 7, and for particles distributed in chains , Figure 7.
Here we are interested in exploring the influence chain structures have on the effectivecoefficients. In Figure 4 we plot the effective coefficients β s and ν s versus different volumefractions for the two types of particle distributions.15 χ χ ξ ξ ξ Figure 2: Streamlines of the solution of the local problems for circular iron particles of19% volume fraction and α = 1 for a uniform distribution of particles. The top rowshowcases the streamlines for the local solution χχχ m(cid:96) in equation (24) while the bottomrow showcases the streamlines for the local solution ξξξ m(cid:96) in equation (28). χ ξ χ ξ χ ξ Figure 3: Streamlines of the solution of the local problems for circular iron particles of19% volume fraction and α = 1 for particles distributed in chains . The left row showcasesthe streamlines for the local solution χχχ m(cid:96) in (24) while the row on the right showcases thestreamlines for the local solution ξξξ m(cid:96) in (28).16 a) (b) Figure 4: Effective magnetic coefficient β s and effective viscosity ν s plotted against volumefractions of 5%, 10%, 15%, and 19% for circular iron particles. On image (a), the red color curve showcases the increase of the effective magnetic coefficient β s under uniformparticle distribution while the blue color curve showcases the increase of the effectivemagnetic coefficient β s in the presence of chain structures . Similarly, on image (b) the red color curve showcases the increase of the effective viscosity ν s under uniform particledistribution while the blue color curve showcases the increase of the effective viscosity ν s in the presence of chain structures .In Figure 4 we see that the presence of chain structures rapidly increase the effectiveviscosity ν s and the effective magnetic coefficient β s as the volume fraction increases ina non-linear fashion. This is in stark contrast to the case where no particle chains arepresent where we notice a linear increase of the effective coefficients. For a particle of 19%volume fraction the presence of chain structures leads to roughly a 104% increase of theeffective viscosity ν s and a 246% increase of the magnetic coefficient β s . In this section we compute the cross sectional velocity profiles of Poiseuille and Couetteflow for circular suspensions of rigid particles. We denote by vvv = ( v , v ) the two dimen-sional velocity and by HHH = ( H , H ) the two dimensional magnetic field. Hence, the twodimensional stress of equation (38) reduces to, σ H + τ H = − ¯ p I + ν s e ( vvv ) − β s (cid:12)(cid:12)(cid:12) (cid:101) HHH (cid:12)(cid:12)(cid:12) I + β s (cid:101) HHH ⊗ (cid:101) HHH and the two dimensional magnetorheological equations in (38) reduce to the following:17 s (cid:18) ∂ v ∂x + ∂ v ∂x (cid:19) − ∂ π ∂x + β s ∂ ( H − H ) ∂x + β s ∂ ( H H ) ∂x = 0 , (39a) ν s (cid:18) ∂ v ∂x + ∂ v ∂x (cid:19) − ∂ π ∂x + β s ∂ ( H H ) ∂x + β s ∂ ( H − H ) ∂x = 0 , (39b) ∂∂x (cid:0) µ H H (cid:1) + ∂∂x (cid:0) µ H H (cid:1) = 0 , (39c) ∂H ∂x − ∂H ∂x = R m ( µ HS v H − µ HS v H ) , (39d) ∂v ∂x + ∂v ∂x = 0 . (39e) We consider the problem of a steady flow due to a pressure gradient between two infinite,parallel, stationary plates that are non-conducting and non-magnetizable with one platealigned along the x –axis while the second plate is of distance one unit apart. We applya stationary magnetic field HHH on the bottom plate. Since we are dealing with infiniteplates, the velocity vvv depends only on x . Using (39e) we immediately obtain that v isconstant and since the plates are stationary v = 0. Since the flow is unidirectional, weexpect that the the magnetic field will depend only on the height x . Hence, using (39c)we obtain H ( x ) = K , while the component parallel to the flow depends on the fluidvelocity. Therefore the equations in (39) reduce to the following, ν s ∂ v ∂x + β s K ∂H ∂x = ∂ π ∂x , (40a) − ∂ π ∂x − β s ∂H ∂x = 0 , (40b) − ∂H ∂x = R m µ HS K v . (40c)Making use of (40b) we obtain that π ( x , x ) + β s H ( x ) is a function of only x andtherefore by differentiating the expression with respect to x we get that ∂ π ∂x is a functiononly x . Therefore, on (40a) the left hand side is a function of x and the right hand sideis a function of x . Thus, they have to be constant. Substituting (40c) in (40a) we obtainthe following differential equations, d v d x − λ v = C p , ∂ π ∂x = C p with λ = (cid:115) R m µ HS β s ν s K (41)The general solution of (41) is, v ( x ) = c e λ x + c e − λ x + C p ν s λ . v (0) = v (1) = 0 we have, v ( x ) = C p ν s λ (cid:18) sinh( λ x ) − sinh( λ ( x − λ ) − (cid:19) . (42) Remark 4 As λ tends to zero we have lim λ → v ( x ) = C p ν s x ( x − , which is preciselythe profile of Poiseuille flow. Once the velocity v ( x ) is known, we can use (40c) to compute H ( x ) with boundarycondition H (0) = K and obtain, H ( x ) = R m µ HS K C p ν s λ sinh( λ ) ( − cosh( λ x ) + cosh( λ ( x − − cosh( λ ) + 1) + K . The setting and calculations for the unidirectional Couette flow are the same as Poiseuilleflow. In a similar way, we can carry out computations for the plane Couette flow. Forsimplicity we assumed the bottom plate is the x axis and the top plate is at x = 1 andthe pressure gradient is zero. A shear stress γ is applied to the top plate while the bottomplate remains fixed. Thus, we solve (41) with initial conditions v (0) = 0 and v (cid:48) (1) = γ and obtain v ( x ) = − ( C p e − λ − γν s ) e λx λ ( e λ + e − λ ) ν s − ( C p e λ + γν s ) e − λx λ ( e λ + e − λ ) ν s + C p ν s λ (43) Remark 5
We remark that for a zero pressure gradient the limit as λ approaches zerothe limit of v ( x ) = γ x , the profile of regular Couette flow. To compute H we use (40c) to obtain, H ( x )( R m µ HS K ) − = ( C p e − λ − γν s ) e λx λ ( e λ + e − λ ) ν s − ( C p e λ + γν s ) e − λx λ ( e λ + e − λ ) ν s − C p x ν s λ − ( C p e − λ − γν s ) λ ( e λ + e − λ ) ν s + ( C p e λ + γν s ) λ ( e λ + e − λ ) ν s + K ( R m µ HS K ) − (44)We plot the velocity profiles for Poiseuille and Couette flows, computed in the previousparagraph, for a magnetorheological suspension of iron particles for different magneticfield intensities. The volume fraction of the particles is set to 19%. Carrying out explicitcomputations of the effective coefficients in (35), (36) and (37) we obtain, ν s β s µ HS × / × chainstructures (red color) turns the magnetorheological fluid into a stiffer gel-like structure [3]at lower intensity magnetic fields. This phenomenon results in a Poiseuille flow that ismuch slower in the presence of chain structures . H = 50 H = 100 H = 200 Figure 5: Velocity profile of
Poiseuille flow for a magnetorheological suspension of circu-lar iron particles of 19% volume fraction in a viscous non-conducting fluid with α = 1 and R m = 10 − plotted against different values of the magnetic field. The blue color curverepresents the velocity profile for a uniform particle distribution while the red color curverepresents the velocity profile for a particle distribution in chains .Figure 6 showcases the velocity profile of magnetorheological Couette flow for zero pressuregradient and three different values of the magnetic field. We can observe that as themagnetic filed increases the magnetorheological fluid becomes harder to shear. Hence, an apparent yield stress is present. The apparent yield stress is larger in the presence of chainstructures (red color) than in the absence of chain structures (blue color). However, thevelocity profile is not linear like in the case of Bingham fluids.Figure. 7 showcases the shear stress τ vs shear rate γ relationship measured at x = 1.When K = 0 there is no yield stress present. However, for very small non-zero valuesof K we obtain results similar to [35] for the linear portion of the shear stress versusshear rate curve. Additionally, we can see that the presence of chain structures (red color)produce a higher yield stress as well as a steeper slope that gives higher shear stress valuesas the shear rate increases.For shear experiments, the response of magneto-rheological fluids is often modeled us-ing a Bingham constitutive law [3], [7], [9], [25]. Although the Bingham constitutive lawmeasures the response of the magnetorheological fluid quite reasonably, actual magne-torheological fluid behavior exhibits departures from the Bingham model [12], [35]. InFigure 5 and Figure 6 we see that for low intensity magnetic fields the Bingham constitu-tive law is not adequate, however, it appears that for higher intensity magnetic fields theflow gets closer to resembling a Bingham fluid behavior.20 = 50 H = 100 H = 200 Figure 6: Velocity profile of
Couette flow for a magnetorheological suspension of ironparticles of 19% volume fraction in a viscous non-conducting fluid with α = 1 and R m =10 − plotted against different values of the magnetic field. The blue color curve representsthe velocity profile for a uniform particle distribution while the red color curve representsthe velocity profile for a particle distribution of chains . We considered a suspension of magnetizable iron particles in a non-magnetizable, non-conducting aqueous viscous fluid. We obtained the effective equations governing the be-havior of the magnetorheological fluid presented in equation (38). The material parame-ters can be computed from the local problems (17), (24), (28), derived from the balanceof mass, momentum, and Maxwell equations.The proposed model generalizes the model put forth in [24], [27] in two ways: • First by providing exact formulas for the effective coefficients which consist of the ho-mogenized viscosity, ν H , and three homogenized magnetic permeabilities, µ H , µ HS ,and β H , which all depend on the geometry of the suspension, the volume fraction,the magnetic permeability µ , the Alfven number α , and the particles distribution. • Second, by coupling the fluid velocity field with the magnetic field through Ohm’slaw.Using the finite element method we carried out explicit computations of the effectivecoefficients for spherical iron particles of different volume fractions both under uniformparticle distribution and particle distribution in chains and showcased the nonlinear effectparticle chain structures have in the effective coefficients as the volume fraction increases.In the case of
Poiseuille flows we calculated the velocity profile explicitly; for small intensitymagnetic fields the velocity profile is close to parabolic while for large intensity magneticfields an apparent yield stress is present and the flow profile approaches a
Bingham flow profile. The magnetorheological effect is significantly higher when chain structures arepresent. A similar analysis has been done for the
Couette flows.21 = 50 H = 100 H = 200 H = 500 Figure 7:
Shear stress τ versus shear rate γ for a magnetorheological suspension ofcircular iron particles of 19% volume fraction in a viscous non-conducting fluid with α = 1, R m = 10 − , and K = 10 − . The red color curve represents particle distribution in chains while the blue color curve represents a uniform particle distribution. One can observethat the yield stress of the magnetorheological fluid is consistently higher in the presenceof chain structures (red color). Acknowledgments
G.N. was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Re-search Foundation) under Germany’s Excellence Strategy The Berlin Mathematics Re-search Center MATH+ (EXC-2046/1, project ID: 390685689) and would like to express hisgratitude to Konstantinos Danas and Andrei Constantinescu for their fruitful discussionsand suggestions. 22 eferences [1] G. Allaire,
Shape optimization by the homogenization methods , Springer-Verlag NewYork, 2002.[2] N. Bakhvalov and G. Panasenko,
Homogenisation: averaging processes in periodicmedia: mathematical problems in the mechanics of composite materials , Kluwer Aca-demic Publishers, 1989.[3] G. Bossis, S. Lacis, A. Meunier, and O. Volkova,
Magnetorheological fluids , J MagnetMagnetic Mat (2002), 224–228.[4] C. Cioranescu and P. Donato,
An introduction to homogenization , Oxford LectureSeries in Mathematics and Its Applications, Oxford University Press, 2000.[5] E. H. Condon and H. Odishaw,
Handbook phys , NY, McGraw-Hill, 1958.[6] K. Danas,
Effective response of classical, auxetic and chiral magnetoelastic materialsby use of a new variational principle. , J Mech Phys Solids (2017), 25–53.[7] J. Goldasz and B. Sapinski,
Insight into magnetorheological shock absorbers , Springer,2015.[8] C. Graczykowski and P. Pawlowski,
Exact physical model of magnetorheologicaldamper , Applied Mathematical Modelling (2017), 400–424.[9] T. C. Halsey, Electrorheological fluids , Science (1992), 761–766.[10] M.R. Hashemi, M.T. Manzari, and R. Fatehi,
A sph solver for simulating param-agnetic solid fluid interaction in the presence of an external magnetic field , AppliedMathematical Modelling (2016), no. 7, 4341–4369.[11] F. Hecht, New development in freefem++ . , J Numer Math (2012), 251–265.[12] R.W. Hoppe and W. G. Litvinov, Modeling, simulation and optimization of elec-trorheological fluids , In: Glowinski, R., Xu, J. (eds.) Num Meth Non-Newt Fluids.Hand Num Analysis (2011), 719–793.[13] D. Kinderlehrer and G. Stampacchia,
An introduction to variational inequalities andtheir applications , Classics in applied mathematics, SIAM, 2000.[14] T. Levy,
Suspension de particules solides soumises ´a des couples , J M´ech Th´eor App(1985), 53–71.[15] T. Levy and R. K. T. Hsieh,
Homogenization mechanics of a non-dilute suspensionof magnetic particles , Int J Engng Sci (1988), 1087–1097.[16] T. Levy and E. Sanchez-Palencia, Einstein-like approximation for homogenizationwith small concentration. i. elliptic problems. , Nonlinear Analysis (1985), 1243–1254. 2317] , Einstein-like approximation for homogenization with small concentration. ii.navier-stokes equations. , Nonlinear Analysis (1985), 1255–1268.[18] R. Lipton and B. Vernescu, Homogenization of two-phase emulsions , Proc Roy SocEdinburgh (1994), 1119–1134.[19] M. T. Lopez-Lopez, P. Kuzhir, S. Lacis, F. Gonzalez-Caballero, and J.D.G. Duran,
Magnetorheology for suspensions of solid particles dispersed in ferrofluids , J Phys:Condens Matter (2006), 2803–2813.[20] C. C. Mei and B. Vernescu, Homogenization methods for multiscale mechanics , WorldScientific, 2010.[21] G. Milton,
The theory of composites , Cambridge Monographs on Applied and Com-putational Mathematics, Cambridge University Press, 2002.[22] G. Milton and R.V. Kohn,
Variational bounds on the effective modulii of anisotropiccomposites. , J. Mech. Phys. Solids (2017), 597–629.[23] G. Nika and B. Vernescu, Asymptotics for dilute emulsions with surface tension , JElliptic Parabol Equ (2015), 215–230.[24] J. L. Nueringer and R. E. Rosensweig, Ferrohydrodynamics , Physics of Fluids (1964), 1927–1937.[25] J. Perlak and B. Vernescu, Constitutive equations for electrorheological fluids , RevRoumaine Math (2000).[26] J. Rabinow, The magnetic fluid clutch , AIEE Trans. (1948), no. 17-18, 1308.[27] R. E. Rosensweig, Ferrohydrodynamics , Dover Publications, 2014.[28] M. Ruzicka,
Electrorheological fluids: Modeling and mathematical theory , LectureNotes in Mathematics, Springer-Verlag Berlin Heidelberg, 2000.[29] E. Sanchez-Palencia,
Non-homogeneous media and vibration theory , Lecture Notes inPhysics, Springer-Verlag Berlin Heidelberg, 1980.[30] E. Sanchez-Palencia and A. Zaoui,
Homogenization techniques for composite media.lectures delivered at the cism international center for mechanical sciences, udine,italy, july 1-5, 1985 , Lecture Notes in Physics, Springer-Verlag Berlin Heidelberg,1987.[31] M. I. Shliomis,
Effective viscosity of magnetic suspensions , Sov J Exp Theor Physics (1972), 1291–1294.[32] R. Tao, Super-strong magnetorheological fluids , J Phys: Condens Matter (2001),979–999.[33] B. Vernescu, Multiscale analysis of electrorheological fluids , Inter J Modern PhysicsB (2002), 2643–2648. 2434] W.M. Winslow, Induced fibration of suspensions , J Appl Phys (1949), 1137–1140.[35] G. Chen Y. Yang, L. Lin and W. Li, Magnetorheological properties of aqueous fer-rofluids , Soc Rheol Japan (2005), 25–31.[36] J. Zapomel, P. Ferfecki, and P. Forte, A new mathematical model of a short mag-netorheological squeeze film damper for rotordynamic applications based on a bilinearoil representation derivation of the governing equations , Applied Mathematical Mod-elling52