Mathematical modeling of magnetorheological fluids - Springer Link

Continuum Mech. Thermodyn. (2005) 17: 29–42 Digital Object Identifier (DOI) 10.1007/s00161-004-0185-1

Original article

Mathematical modeling of magnetorheological fluids I.A. Brigadnov1 , A. Dorfmann2 1 2

Department of Computer Science, North-Western State Technical University, Millionnaya 5, St. Petersburg, 191186, Russia Institute of Structural Engineering, Peter Jordan Str. 82, 1190 Vienna, Austria

Received July 14, 2003 / Accepted May 18, 2004 Published online February 22, 2005 – © Springer-Verlag 2005 Communicated by K.R. Rajagopal

Abstract. Magnetorheological (MR) fluids are a class of smart materials whose rheological properties may be rapidly modified by the application of a magnetic field. These materials typically consist of micron-sized ferrous particles dispersed in a fluid. In the present paper, we consider the full system of equations as well as the Clausius–Duhem inequality for moving isotropic MR fluids in an electro-magnetic field. We present the material constitutive relations for a non-Newtonian incompressible MR fluid. To illustrate the validity of the constitutive relations, the flow of a MR fluid between two parallel fixed plates under the influence of a constant magnetic field perpendicular to the flow direction is considered. Key words: Maxwell field equations, laws of thermodynamic, constitutive equations, incompressible MR fluid Introduction Here we consider magnetorheological (MR) fluids that respond to an applied magnetic field with a rapid change in their rheological properties. For details see, for example, Berkovsky et al. [1], Berkovsky [2], Blums et al. [3] and Rosensweig [31]. Typically, the change in properties is manifested by the development of a yield stress that monotonically increases with the magnetic field. Interest in MR fluids derives from their ability to provide simple, quiet, rapid-response interfaces between electronic controls and mechanical systems. That MR fluids have the potential to radically change the way electromechanical devices are designed and operated has long been recognized. A wide range of potential applications is presumably the reason for the intense research in recent years (Carlson et al. [6], Felt et al. [12], Jolly et al. [15,17], Mohebi et al. [23], Lazareva et al. [20], Nakato and Yamamoto [24], Weiss et al. [37]). Magnetorheological fluids and their electrorheological (ER) fluid counterparts are both non-colloidal suspensions of polarizable particles with a size on the order of a few microns (typically from 10−7 to 10−5 m). The initial discovery and development of MR fluids and devices can be attributed to Jacob Rabinow at the US National Bureau of Standards in the late 1940s, see Rabinow [27]. Interestingly, this work was almost concurrent with Winslow’s ER fluid work (Winslow [39]). The late 1940s and early 1950s actually saw more patents and publications relating to MR than to ER fluids. Except for a flurry of interest after their initial discovery, there has been little new published information on MR fluids. Only recently has a resurgence in interest in MR fluids been seen (Carlson and Jolly [7] and Kordonsky [18]). While the commercial breakthrough of ER fluids has Correspondence to: A. Dorfmann ([email protected])

30

I.A. Brigadnov, A. Dorfmann

remained elusive, MR fluids have enjoyed recent successes. A number of MR fluids and various MR fluid-based systems have been commercialized including a MR fluid brake system for use in physical exercise equipment, a controllable MR fluid damper for use in truck seat suspensions (Carlson et al. [6]) and a MR fluid damper for seismic application (Dyke et al. [9]). The magnetorheological effect is optimized by choosing a particle material with a high magnetic saturation Js . Cobalt has the largest magnetic saturation of known elements with Js = 2.4 Tesla, however cobalt is not commonly used. In general an alloy of iron with Js = 2.1 Tesla is used in MR fluids. Typical particle volume fractions are between 0.1 and 0.5. Carrier fluids are chosen based upon their rheological and tribological properties and on their temperature stability. Examples are silicone, petroleum based oils, mineral oils, polyesters, polyethers, water and synthetic hydrocarbon oils. The magnetorheological response of MR fluids results from the polarization of the suspended particles by the application of an external magnetic field. The interaction between the resulting induced dipoles causes the particles to form columnar structures, parallel to the applied magnetic field. These chain-like structures hinder the motion of the fluid, thereby increasing the viscous characteristics of the suspension. The mechanical energy needed to yield these chain-like structures increases proportional to the applied magnetic field resulting in a field dependent yield stress. The field-dependent behaviour of MR fluids is often represented as a Bingham material having variable yield strength. While the Bingham visco-plastic model has proven useful in the design and characterization of MR fluid-based devices, true MR fluid behaviour exhibits some significant departures from this simple model. Perhaps the most significant of these departures involves the non-Newtonian behaviour of these fluids (Jolly et al. [16]). Our analysis here builds upon a series of recent publications on the mathematical modeling of ER fluids (Rajagopal et al. [28], Rajagopal and R˚ uˇziˇcka [29], Wineman and Rajagopal [38]). We consider the MR medium as a homogenized single non-polar isotropic continuum. Section 1 starts with Pao’s classical work [26] summarizing the full system of equations for a moving isotropic non-polar continuous medium in an electromagnetic field such as the Maxwell field equations, the mechanical and thermodynamical balance laws as well as the Clausius–Duhem inequality. In Sect. 2 we derive the basic system of constitutive equations for MR fluids using a phenomenological approach based on experimental data. The reduced system of constitutive equations is complemented by the system of initial and boundary conditions. In Sect. 3 we present the material constitutive relation for nonNewtonian incompressible MR fluids. To illustrate the usefulness of the derived mathematical model, in Sect. 4 we examine the basic operational system for controllable fluid devices. The final section is devoted to concluding remarks. 1 Physical laws for moving continuous media in an electromagnetic field Let a continuous media in the reference configuration occupy a domain Ω ⊂ R3 . In the deformed configuration each point X ∈ Ω moves into the position x = χ(t, X) ∈ R3 where χ is the mapping and t is the parameter describing the motion of the fluid (usually, this is the physical time). For fluids the Eulerian or spatial description is used. We consider a laminar flow, i.e., a one-to-one or locally invertible and orientation-preserving mapping in Ω for every t > 0. In this paper standard mathematical notations are used, see for example, Lurie [21]. Namely, for the scalar ϕ, vectors a, b and second-order tensors A, B the algebraic rules are: a·b = ai bi ,

|a|2 = a·a ,

a×b = {εijk ai bj }k=1,2,3 , A:B = Aij Bij ,

|A|2 = A:A ,

ab = {ai bj }i,j=1,2,3 , A·b = {Aij bj }i=1,2,3 ,

and the spatial differential operators of divergence, gradient and curl are: ∂ai ∂ϕ ∂Aij ∂ϕ div a = = , div A = , grad ϕ = , ∂xi ∂xj i=1,2,3 ∂x ∂xi i=1,2,3 ∂aj ∂a ∂ai , curl a = εijk , grad a = = ∂x ∂xj i,j=1,2,3 ∂xi k=1,2,3

Mathematical modeling of magnetorheological fluids

31

where over repeated indices the summation rule applies. For the basis vectors ei the permutation or the Levi– Civita symbol is εijk = (ei × ej ) · ek with εijk = 1 or −1 according to whether the indices are in a cyclic or an anticyclic order, respectively, and εijk = 0 otherwise. The velocity, its spatial gradient and its deformation tensor are defined, respectively, by 1 ∂χ , l = grad v , d= l + lT , ∂t 2 where the superscript T denotes the transpose of a tensor. For the total time derivative of a scalar-valued function ϕ = ϕ(t, x) and a vector-valued function a = a(t, x) the classic formulas of Euler are used, respectively, v=

ϕ˙ =

∂ϕ dϕ = + (grad ϕ)·v , dt ∂t

da ∂a = + (grad a)·v . dt ∂t The most general case, the moving of non-polar fluid in the spatial (current) configuration, is described by: a˙ =

– electromagnetic variables such as the electric field intensity E, the magnetic field intensity H, the electric induction or displacement D, the electric polarization density P (electric moment per unit volume), the magnetic flux density or magnetic induction B, the magnetic polarization (magnetization) density or intrinsic induction M (magnetic moment per unit volume), the free electric current density J and the free electric charge density; – mechanical variables such as the velocity v and its spatial gradient l, the density ρ, the stress tensor σ and the external mechanical body force density f per unit mass; – thermodynamic variables such as the absolute temperature θ and the Cauchy heat flux vector Q. For these variables the appropriate physical laws are well known. The Maxwell field equations for continuous media are presented here in the standard meter-kilogram-secondCoulomb units. For full details of the relevant theory summarized in this section, the reader is referred to, for example, Pao [26], the book by Brown [5] and Kovetz [19] and the monographs by Hutter [13] and Eringen and Maugin [11]. The electric field intensity E and the magnetic flux density B are regarded as the basic variables in a vacuum. For condensed media additional variables, such as the electric polarization density P and the magnetic polarization density M are introduced. These two variables are used in the following relations1 : D = ε0 E + P ,

B = µ0 H + M ,

(1.1)

where ε0 ≈ 8.85·10−12 and µ0 = 4π·10−7 ≈ 1.26·10−6 are two universal constants such that ε0 µ0 = c−2 . The Gauss, Faraday and Amper laws are given, respectively, by ε0 div (E) = q − div P , curl E + µ−1 0 curl B = ε0

∂B =0, ∂t

∂E ∂P + + µ−1 0 curl M + J , ∂t ∂t

(1.2) (1.3) (1.4)

where −div P is the polarization charge, ∂P/∂t is the polarization current, and µ−1 0 curl M is the magnetization current. The conservation of magnetic flux is given by div B = 0 .

(1.5)

Note, that the equation for the conservation of electric charge ∂q/∂t + div J = 0 is not independent because it follows from (1.2), (1.4) and the identity div curl (∗) ≡ 0. 1

Note, that the relation B = µ0 (H + M) has also been used (Pao [26]). Here we replace M by µ−1 0 M in all relations.

32


Mechanical balance laws for fluids are given in local form and are the conservation of mass and the balance of linear momentum, which are given respectively by ∂ρ + ρ div v = 0 , (1.6) ∂t ρv˙ = div σ + ρf + fe , (1.7) where fe is the electromagnetic force. From the Dipole–Current Circuit Model for moving continuous media this force is given by ∂ (P×B) + div [vP×B] . (1.8) fe = q E + J×B + µ−1 0 (grad B)·M + (grad E)·P + ∂t Details on electromagnetic forces in deformable continua are given by Pao [26]; for the mechanical equations we refer to, for example, Eringen [10] and Truesdell [35]. The balance of angular momentum takes the local form ε:σ + (µ−1 (1.9) 0 M + v×P)×B + P×E = 0 , where ε denotes the third-order permutation tensor with the Levi–Civita symbol as its the components. The laws of thermodynamic have the following form (see, for example, Truesdell [34,35], Truesdell and Noll [36], Pao [26]). The first lawis the balance of energy in the local form 1 2 d U + |v| + div Q = div (σ·v) + ρ f ·v + ρ R + we , (1.10) ρ 2 dt where U and R denote, respectively, the specific2 internal energy and the radiant heating. Here, we is the electromagnetic power defined as dB d P ·Ee , (1.11) we = fe ·v + Je ·Ee − Me · +ρ dt dt ρ where Je is the effective conduction current, Ee is the effective electric field intensity and Me is the effective magnetization in the rest frame, which according to Minkowski’s theory (Minkowski [22]) are related to the not rotating laboratory frame variables by Ee = E + v×B , Me = µ−1 (1.12) Je = J − qv , 0 M + v×P . We note that all relativistic effects as well as the effect of Earth’s spin are neglected, (details see in Rajagopal, Tao [30]). Using (1.6) and (1.7) we get the reduced form of (1.10) ˙ + Je ·Ee + P·E ˙ e + (P·Ee ) div v . (1.13) ρ U˙ + div Q = σ:l + ρ R − Me ·B The second law of thermodynamics in the local formulation is given by the Clausius–Duhem inequality R dS Q −ρ ≥0, ρ + div (1.14) dt θ θ where S is the specific entropy . The law expresses the fact that the internal entropy increases at least as fast as the sum of heat flux and radiant heating, each divided by the temperature θ. We follow the classical approach used in continuum thermodynamic of eliminating the radiation from the entropy inequality. We are aware of the controversy and associated limitations of applying this approach to continua receiving energy in the form of electromagnetic radiation (see, for example, Rajagopal and Tao [30]). The dissipation inequality so obtained places no restriction on the constitutive representation for the radiant heating or for the electromagnetic energy received or absorbed. However based on the first law of thermodynamic, the radiant heating can not be arbitrary. To overcome this problem, Rajagopal and Tao suggest to include the radiant heating as part of an internal source of entropy in order to place stringent restrictions on the form of heating. Introducing the specific Helmholtz free-energy Ψ through 1 Ψ := U − θ S − Ee ·P , (1.15) ρ and substituting (1.15) and (1.10) into (1.14) we obtain the dissipation inequality 1 ˙ + Je · Ee − P · E ˙e ≥0. −ρ Ψ˙ + θ˙ S + σ : l − Q · grad θ − Me · B (1.16) θ We shall identify the system of equations (1.2)–(1.7), (1.9), (1.13) and the inequality (1.16) as the basic system. 2

As usual, for fluids all specific values are defined per unit mass.


33

2 The basic system for MR fluids It can be verified that the basic system is indeterminant, i.e., there are more unknown variables than equations. The system is rendered determinate by providing a sufficient number of constitutive material equations. From a thermodynamic point of view, the variables (ρ, v, d, Ee , B, θ, grad θ) are independent quantities (Truesdell [35], Truesdell and Noll [36]) and we, therefore, need to define constitutive relations for (U, S, σ, P, Je , Me , Q). The formulation of constitutive theories requires enforcement of the material frame indifference as described in detail by, for example, Truesdell and Toupin [33]. On the other hand, the less restrictive principle of Galilean invariance and proper Galilean transformations are applied to electromagnetic quantities as discussed by, for example, Kovetz [19]. There is, however, ongoing controversy, addressed among others by Rajagopal and Tao [30], on the use of the requirement of material frame indifference in favor of the less restrictive Galilean invariance. Rajagopal and Tao provide a physical perspective and some justification on the adoption of Galilean invariance in the formulation of constitutive relations on electromagnetic and thermodynamic quantities. Therefore, for consistency, we adopt the principle of Galilean invariance in this paper. The heat flux vector Q is given by the Fourier law of heat conduction Q = −k grad θ ,

(2.1)

where k > 0 is the thermal conductivity, which can be assumed constant for most engineering materials. In the remaining constitutive relations we drop the dependence on the temperature gradient. From the principle of Galilean invariance it follows that all constitutive relations are independent on v. As a result, we have ˆ ρ, d, Ee , B) , Φ = Φ(θ, (2.2) where Φ = (U, S, σ, P, Je , Me ) is the generalized vector of unknown variables. Using (2.1), (2.2) and computing Ψ˙ in (1.16) we obtain the following form of the Clausius–Duhem inequality ∂Ψ ∂Ψ ˙ ∂Ψ ∂Ψ ˙ e− − S θ˙ − ρ ρ˙ − ρ :d− ρ −ρ −P ·E ∂θ ∂ρ ∂d ∂Ee 2 ∂Ψ ˙ + σ : l + k |grad θ| + Je · Ee ≥ 0 . − ρ (2.3) − Me · B ∂B θ This inequality is expected to hold for real materials at all times and at every fixed point in space for a certain class of admissible thermodynamic processes, i.e., processes compatible with the balance laws and the constitutive ˙ d, ˙ E ˙ e and B ˙ are independent (Truesdell and Noll [36]) we obtain response functions. Since the quantities θ, ∂Ψ ∂Ψ ∂Ψ ∂Ψ , Me = −ρ , =0, P = −ρ , ∂θ ∂d ∂Ee ∂B and also by using (1.6), the reduced dissipation inequality becomes |grad θ|2 2 ∂Ψ σ+ρ I :l+k + Je ·Ee ≥ 0 , ∂ρ θ S=−

(2.4)

(2.5)

where I is the second-order identity tensor and I : l = div v. Isotropic continuous media are described by the following well known experimental laws for electro-magnetic variables introduced in (1.1) (Jackson [14], Pao [26]) J = ηE,

D = εε0 E ,

B = µµ0 H ,

(2.6)

where the scalar functions η = η(x, |E|) ≥ 0, ε = ε(x, |E|) ≥ 1 and µ = µ(x, |H|) ≥ 1 are the electric conductivity, the relative dielectric permittivity and the relative magnetic permeability, respectively. In vacuum η ≡ 0, ε ≡ 1 and µ ≡ 1. From (1.1), (1.12) and (2.6) we obtain the constitutive equations for the effective conduction current Je and the effective magnetization Me as Je = η E − q v ,

Me = γ B + (ε − 1)ε0 v × E ,

(2.7)

where γ = (µ − 1)(µ0 µ)−1 ≥ 0 is the function of magnetic saturation which is defined from the standard experimental curve µ0 |H| → |B| (Jolly et al. [15]).

34


In this and the following sections some of our assumptions are based on experimental data (Felt et al. [12], Jolly et al. [15,17], Lazareva et al. [20], Mohebi et al. [23], Nakato and Yamamoto [24]). The main assumption is that for MR materials the electric polarization is negligible P=0,

(2.8)

i.e., ε ≡ 1 in all relationships. From (2.72 ) we have Me = µ−1 0 M = γB. As a result, from the law of the balance of angular momentum in the form (1.9) it follows that the stress tensor σ is symmetric, i.e., it is the Cauchy stress tensor and thus the equality σ:l = σ:d therefore holds. Defining the thermodynamical pressure as p := −ρ2

∂Ψ , ∂ρ

and considering the above assumptions, the basic system can now be re-written as q , div E = ε0 ∂B =0, ∂t 1 ∂E + µ0 η E , curl µ−1 B = 2 c dt curl E +

(2.9)

(2.10) (2.11) (2.12)

div B = 0 ,

(2.13)

ρ˙ + ρ div v = 0 ,

(2.14)

ρv˙ = div σ + ρ f + fe ,

(2.15)

˙ + ρ R + E·G , ρ U˙ − k∆θ = σ:d − γ B·B

(2.16)

(σ − p I):d + k

|grad θ|2 + E·G ≥ 0 , θ

(2.17)

where fe and G are given, respectively, by fe = q E + η E×B + γ (grad B)·B , G = η (E + v×B) − q v .

(2.18)

In the system of constitutive relations (2.10)–(2.17) some variables and expressions differ by several orders of magnitude in problems of practical interest. Of course, we could repeat the non-dimensional analysis of constitutive relations (Rajagopal and R˚ uˇziˇcka [29]), but we prefer evaluating experimental data of commercial MR fluids (Carlson et al. [6], Felt et al. [12], Jolly et al. [15,17], Lazareva et al. [20], Mohebi et al. [23], Nakato and Yamamoto [24], Weiss et al. [37]). We find that for commercially available MR fluids, the electric charge and the electric conductivity are very small, and, therefore, we shall assume that q ≡ 0 and η ≡ 0. As a result, from (2.71 ) and (2.18) it follows that Je = 0 and G = 0. Second, experimental data for MR fluids show that the influence of the electric field is non-essential, therefore we re-write (2.2) as ˆ ρ, d, B) , Φ = Φ(θ, (2.19) where the generalized vector of unknown values for MR fluids now becomes Φ = (U, p, σ). From (2.42 ), (2.43 ) and (2.8) it follows that the free-energy function Ψ is independent in d and Ee . Moreover, we assume that the Helmholtz free-energy Ψ = Ψ (T, ρ, B) is a smooth function. Then, using (1.15) with P = 0, (2.4), (2.7) with Me = γB, (1.6) and (2.9), we obtain the following relation for the rate of the internal energy: ∂Ψ ∂Ψ ∂Ψ ˙ ∂ dΨ d ˙ Ψ −θ =ρ ρ˙ + ρ · B − ρθ = ρU = ρ dt ∂θ ∂ρ ∂B ∂θ dt 2 2 ∂p ˙ + ρ −θ ∂ Ψ θ˙ − ρθ ∂ Ψ · B ˙ . = p−θ div v − γB·B (2.20) 2 ∂θ ∂θ ∂θ∂B


35

The specific heat capacity is defined as cv := −θ

∂2Ψ >0, ∂θ2

(2.21)

and using (2.20) we re-write (2.16) in the form

∂2Ψ ∂p ˙ ˙ . I : d + ρθ cv ρ θ − k∆θ − ρ R = σ − p − θ ·B ∂θ ∂θ∂B The density ρ and magnetic field intensity H do not depend on the temperature θ, therefore, from (2.4), (2.6) and (2.7) we have ∂ ∂(µ − 1) θ ∂µ ∂2Ψ = −θ (γB) = −θ H=− B. ρθ ∂θ∂B ∂θ ∂θ µ0 µ ∂θ As a result, the final reduction of the basic system now gives div E = 0 , curl E +

(2.22)

∂B =0, ∂t

div B = 0 ,

curl µ−1 B =

(2.23) (2.24)

1 ∂E , c2 dt

(2.25)

ρ˙ + ρ div v = 0 ,

(2.26)

ρ v˙ = div σ + ρ f + γ (grad B)·B ,

∂p ˙ ˙ , cv ρ θ − k∆θ − ρ R = σ − p − θ I : d − ϕ B·B ∂θ

(2.27)

(σ − p I):d + k where functions γ=

µ−1 , µ0 µ

(2.28)

|grad θ|2 ≥0, θ

(2.29)

θ ∂µ µ0 µ ∂θ

(2.30)

ϕ=

are defined from experimental constitutive curves µ0 |H| → |B| for different temperatures3 . Every set of fields (E, B, v, σ, ρ, T ) satisfying the system of constitutive equations (2.22)–(2.28) is named admissible. The set of true fields is picked from the admissible ones by the main thermodynamic inequality (2.29). Assuming that the inside and surface electric charge and the electric current are absent, the solution of the system (2.22)–(2.28) is the set of true fields satisfying the jump conditions (Pao [26]): (i) for electro-magnetic fields E and B n·[E] = 0 ,

n·[B] = 0 ,

n × [E + v × B] = 0 , n × µ−1 B − c−2 v × E = 0 ,

(2.31)

(ii) for the velocity v, the Cauchy stress tensor σ and heat flux Q = −k grad θ [v] = 0 , n·[σ + τ ] = 0 ,

1 2 n· (σ + γBB)·v + ε0 |E|2 + µ−1 v − ε |B| E × B + k grad θ =0, 0 0 2 where τ is the electro-dynamic stress tensor defined by 1 2 ε0 |E|2 + µ−1 I − ε0 EE + (µ0 µ)−1 BB . τ = 0 − 2γ |B| 2 3

Of course, if these experimental data really exist.

(2.32)

36


Here n denotes the unit normal vector to the material surface in the current configuration. The quantities enclosed by square brackets can in general be subjected to a discontinuity at the boundary surface of a moving material volume. The initial conditions are: E|t=0 = E0 , B|t=0 = B0 , v|t=0 = v0 , 0

ρ|t=0 = ρ ,

σ |t=0 = σ 0 ,

p|t=0 = p0 , 0

(2.33) 0

−k grad θ|t=0 = Q ,

θ|t=0 = θ ,

where the functions with superscript zero are known. It is easily seen that the system of constitutive equations (2.22)–(2.28) is fully interconnected with the complex system of boundary conditions (2.31) and (2.32).

3 The constitutive relation for isotropic non-Newtonian incompressible MR fluids An important simple fluid is the linear Newtonian viscous fluid, or Navier–Stokes fluid (Eringen [10], Truesdell [35]), whose Cauchy stress tensor takes the form σ = (−p + λ tr d)I + 2ν d ,

(3.1)

in which p, λ and ν are functions of ρ. In the case of an incompressible linear viscous fluid tr d = 0. We recall that the Cauchy stress tensor σ is an objective tensor quantity. In the preceding section we have shown that for an isotropic MR fluid the stress tensor is symmetric. Now, using the Rivlin–Ericksen representation theorem (Drozdov [8], Lurie [21]) and some general results of the invariant theory (Spencer [32]), from (2.19) we write the constitutive relation for isotropic MR fluids in the most general from σ = α1 I + α2 BB + α3 d + α4 (d·BB + BB·d) + α5 d2 + α6 (d2 ·BB + BB·d2 ) ,

(3.2)

where αi (i = 1, 6) are functions of the following invariants: θ, ρ, tr d, |d|, det d, |B|, d:BB, d2 :BB .

(3.3)

Other second-order symmetric objective pseudo-tensors based on combinations of tensor d and vector B do not exist. As it follows from experimental data, commercially available MR fluids are practically incompressible and show essentially nonlinear behaviour in their response. Therefore, we shall consider non-Newtonian incompressible MR fluids only. First, for an incompressible MR fluid only an isochoric motion is possible since div v = tr d = 0 .

(3.4)

Therefore, in the representation (3.2) for the stress tensor σ we have α1 = −p, where p is the indeterminate hydrostatic part of the stress given by the constraint (3.4). Secondly, by analogy with the Navier–Stokes fluid we shall assume that the Cauchy stress tensor of the MR fluid is linear in d and quadratic4 in B (Rajagopal et al. [28], Wineman and Rajagopal [38]). For the remaining material functions α2 , α3 and α4 , we assume the combination of a power-like (shear dependence) and Newtonian behaviour, i.e., q−1 α2 = α20 + α21 |d| , q−2

α3 = α30 + α31 |d|

2

2

q−2

+ α32 |B| + α33 |B| |d| q−2

α4 = α40 + α41 |d|

,

(3.5)

,

α5 = α6 = 0 , where αij are functions of θ only, and, in the general case, q = q(|B|2 ) > 1 (Rajagopal and R˚ uˇziˇcka [29]). This structure of material functions may be used to formulate a number of MR fluid models characterizing different 4

This assumption is based on the power-like behaviour of MR fluids with mechanical properties dependent on the magnetic field energy, which is proportional to |B|2 (Pao [26]).


37

Fig. 1. Unidirectional flow of incompressible MR fluid between two infinite parallel fixed plates subjected to a magnetic field perpendicular to flow direction

material properties. For example, setting q = 2 and α21 = 0 in (3.5), we obtain the model for an Newtonian incompressible MR fluid (Carlson and Jolly [7]). Now, holding the temperature fixed we obtain from (3.2), (3.4) and (2.29) the reduced form of the Clausius– Duhem inequality q−1 2 2 α20 + α21 |d| d:BB + α30 + α32 |B| |d| + 2 q q−2 2 |d·B| ≥ 0 . + α31 + α33 |B| |d| + 2 α40 + α41 |d| (3.6) This inequality is expected to hold for all admissible d and B. Using the standard methods of specifying and ˇ scaling some of the variables (Neˇcas and Silhav´ y [25], Rajagopal and R˚ uˇziˇcka [29]) we obtain restrictions on αij (details in the Appendix). As a result, for isotropic non-Newtonian incompressible MR fluids we have the constitutive relation σ = −p I + α21 |d|q−1 BB + α30 + α32 |B|2 d+ + α31 + α33 |B|2 |d|q−2 d + α40 + α41 |d|q−2 (d·BB + BB·d) , (3.7) where the coefficients αij depending on θ only, have to satisfy the following conditions α30 ≥ 0 ,

α31 ≥ 0 ,

α32 ≥ 0 ,

α33 ≥ 0 ,

4 4 α40 ≥ 0 , α33 + α41 ≥ 0 , 3 3

3 4 α32 + α33 + (α40 + α41 ) . |α21 | ≤ 2 3 α32 +

(3.8)

Note, that for the parameter q ∈ (1, 2) the equivalent viscosity terms, corresponding to zero velocity can be infinite, which may correspond to the yield behaviour of MR fluids. However, this condition does not accurately represent experimental data of commercially used MR fluids (Jolly et al. [16]). This discrepancy can be reconciled by using a regularization technique in the material constitutive relation. We propose the weak regularization replacing |d|q−2 by (ω + |d|)q−2 , where 0 < ω 1 is the regularization parameter. But in this case the appropriate BVP is ill-conditioned and needs special preconditioned numerical methods (Brigadnov [4]).

4 Example: Flow between parallel fixed plates Let us consider the problem of an unidirectional steady flow of a MR fluid between infinite parallel fixed plates in the xy-plane along the x-direction, see Fig. 1. Suppose that the velocity field associated with the fluid and the magnetic field intensity, which is perpendicular to the flow, have the forms v = u(z)i ,

H = Hk ,

z ∈ [0, h] ,

where i, j, k are the Cartesian basis vectors, and h is the distance between the two parallel plates. We shall assume that: 1) 2) 3) 4)

the flow is isothermal, i.e., θ = const , ∂/∂θ = 0 and R ≡ 0; the MR fluid is incompressible; for the unidirectional flow, the variables do not depend on the y-coordinate, i.e. ∂/∂y = 0; for the steady flow d/dt = 0;

(4.1)

38

I.A. Brigadnov, A. Dorfmann 1

0.9

Normalized distance between plates

0.8

0.7

0.6

4

3

2

1

0

0.5

0.4

0.3

0.2

Fig. 2. The flow of the Newtonian incompressible MR fluid.Velocity profiles 0, 1, 2, 3 and 4 correspond to the magnetic flux density B0 = 0.0, 0.5, 1.0, 1.5 and 2.0 Tesla, respectively

0.1

0

5) 6) 7) 8)

0

0.1

0.2

0.3

0.4 0.5 0.6 Normalized Velocity

0.7

0.8

0.9

1

the body force is absent, i.e., f ≡ 0; the flow is due to a constant pressure gradient ∆p = −∂p/∂x per unit length along the x-direction; the fluid adheres to the surfaces of plates; the magnetic field intensity is constant, i.e., H = H0 = const and B = B0 k with B0 = µµ0 H0 = const .

Note, that for the velocity (4.1), the condition of incompressibility is fulfilled automatically. Within the framework of our assumptions, the constitutive equations (2.22)–(2.26) are satisfied for E ≡ 0 and ρ = const . The power balance equation (2.28) is not applicable because from a physical point of view the considered application represents an open system, which needs an infinite external energy supply. We shall assume that an incompressible MR fluid with shear dependent viscosity is described by the powerlike constitutive relation (3.7) with the coefficients q = const , α21 = 0, α31 = 0 and α32 + α40 = 0. In this case the Cauchy stress tensor is written as σ = −p I + 2ν d , (4.2) where the velocity deformation tensor d = form: (i) for the parameter q ≥ 2

1 2

u (ik+ki). Here ν is the equivalent viscosity having the following q−2

ν = a0 + a1 B02 |u |

,

(4.3)

(ii) for the parameter q ∈ (1, 2) with the regularization parameter 0 < ω 1 q−2

ν = a0 + a1 B02 (ω + |u |)

,

(4.4)

where a0 = α30 /2 ≥ 0 and a1 = 2−q/2 (α33 + α41 ) ≥ 0. We shall assume that the viscosity corresponding to zero magnetic flux density a0 > 0 and the field dependence ratio a1 > 0. In all numerical experiments we used data from the commercial MR fluid MRF-132LD (Carlson et al. [7]). The carrier fluid is hydrocarbon with 32% (volume) iron particles (∼ 3 µm), the maximum magnetic induction is B0max ∼ 2 Tesla, the viscosity corresponding to zero magnetic flux density a0 ≈ 0.1 Pa·sec and the field dependence ratio a1 ≈ 0.1. The equilibrium equation (2.27) is transformed into the following second order non-linear differential equation: (ν u ) = −∆p , z ∈ (0, h) . (4.5)


39

1

0.9


0.8

0.7

0.6

4

3

2

1

0

0.5

0.4

0.3

0.2

Fig. 3. The flow of the non-Newtonian incompressible MR fluid with the parameter q = 2.5. Velocity profiles 0, 1, 2, 3 and 4 correspond to the magnetic flux density B0 = 0.0, 0.5, 1.0, 1.5 and 2.0 Tesla, respectively

0.1

0

0

0.1

0.2

0.3


0.7

0.8

0.9

1

For the Newtonian MR fluid with the parameter q = 2 in the relation for the equivalent viscosity (4.3) we have the linear BVP describing the classic Poiseille flow u = −∆p/ν , z ∈ (0, h) , (4.6) u(0) = 0 , u(h) = 0 . The solution of this problem has the simple parabolic form z2 h2 ∆p z − 2 . (4.7) u(z) = 2ν h h In Fig. 2 the Poiseille velocity profiles are presented. The curves 0, 1, 2, 3 and 4 correspond, respectively, to B0 = 0.0, 0.5, 1.0, 1.5 and 2.0 Tesla. For the non-Newtonian MR fluid the appropriate non-linear BVP was solved numerically. The velocity profile is symmetric with respect to the mid-plane because the flow between parallel fixed plates is steady and gravity effects are neglected. As a result, using the condition u (h/2) = 0 the BVP is transformed into the following Cauchy problem: z ∈ (0, h) , u (z) = − ν(|u∆p (z)|) (z − h/2) , (4.8) u(0) = 0 , which was solved numerically by the method of simple iterations with an explicit Euler scheme of 1000 steps in z for every iteration. For the model (4.3) this procedure had good convergence (maximum of 14 iterations) for a relative precision of 10−4 . In Fig. 3 the velocity profiles are presented for the parameter q = 2.5. The curves 0, 1, 2, 3 and 4 correspond, respectively, to B0 = 0.0, 0.5, 1.0, 1.5 and 2.0 Tesla. For the parameter q = 1.1 the problem (4.8) was solved for the weakly regularized model (4.4) with the parameter ω = 10−4 . In this case the Cauchy problem (4.8) is rigid, i.e., the appropriate BVP is ill-conditioned. Therefore, the iteration method had a satisfactory convergence (maximum of 84 iterations for B0 = 2.0 Tesla) with the relative precision of 10−4 only for 10000 steps in the Euler scheme. In Fig. 4 the appropriate velocity profiles are presented. The curves 0, 1, 2, 3 and 4 correspond, respectively, to B0 = 0.0, 0.5, 1.0, 1.5 and 2.0 Tesla. It is easily seen that the central part of the flow moves as a solid for the large magnetic flux density. Moreover, for B0 = 2.0 Tesla the flow is practically fixed. In Fig. 5 the velocity profiles are presented for B0 = 2.0 Tesla and different parameters q. The curves 1, 2, 3, 4 and 5 correspond, respectively, to q = 2.8, 2.4, 2.0, 1.6 and 1.2.

40

I.A. Brigadnov, A. Dorfmann 1

0.9


0.8

0.7

0.6

4

3

2

1

0

0.5

0.4

0.3

0.2

Fig. 4. The flow of the non-Newtonian incompressible MR fluid with the parameter q = 1.1. Velocity profiles 0, 1, 2, 3 and 4 correspond to the magnetic flux density B0 = 0.0, 0.5, 1.0, 1.5 and 2.0 Tesla, respectively

0.1

0

0

0.1

0.2

0.3


0.7

0.8

0.9

2

1

1

1

0.9


0.8

0.7

0.6

5

4

3

0.5

0.4

0.3

0.2

Fig. 5. The flow of the incompressible MR fluid for the magnetic flux density B0 = 2.0 Tesla. Velocity profiles 1, 2, 3, 4 and 5 correspond to the parameter q = 2.8, 2.4, 2.0, 1.6 and 1.2, respectively

0.1

0

0

0.05

0.1

0.15 0.2 Normalized Velocity

0.25

0.3

From the numerical experiments it follows that for every parameter q > 1, increasing the magnetic field B0 results in an increase of the viscosity ν and thus in a decrease of the velocity u. The maximum effect is reached for the parameter q ∈ (1, 2). The presented numerical results fully corresponds to experimental data for commercially used MR fluids.


41

5 Conclusions In this paper, we have summarized the complete system of constitutive equations for an isotropic magnetorheological fluid within the framework of the electro-dynamical and thermo-mechanical theories. For non-Newtonian incompressible MR fluids the constitutive relation for the Cauchy stress tensor has been considered. This model allows the fluid to have shear dependence. In the future we intend to study the presented model to address issues related to the mathematical correctness and numerical solution of simple and complex boundary value problems. From our point of view the presented constitutive relation can be effectively used for computer simulation of commercial MR fluids. But the choice of the most realistic model must be determined by experiments only.

Appendix Here the standard analysis of the reduced Clausius–Duhem inequality (3.6) for the parameter q > 1 is presented. Rescaling (3.6) through d → d, multiplying by −1 and letting → 0 we get α20 d:BB ≥ 0 , which gives by changing the sign of d α20 = 0 . Setting B = 0 in (3.6), we have the inequality 2

q

α30 |d| + α31 |d| ≥ 0 , which gives (on rescaling d → d, multiplying by −2 or −q , letting → 0 or → ∞) α30 ≥ 0 ,

α31 ≥ 0 .

Setting d·B = 0 in (3.6), we have the inequality (on rescaling B → ωB, multiplying by ω −2 , letting ω → ∞) 2

q

α32 |d| + α33 |d| ≥ 0 , which gives as in the preceding case α32 ≥ 0 ,

α33 ≥ 0 .

Setting d:B = 0 in (3.6), we have inequalities (on rescaling d → d, multiplying by −2 or −q , letting → 0 or → ∞) 2 2 2 α30 + α32 |B| |d| + 2 α40 |d·B| ≥ 0 ,

2

α31 + α33 |B|

2

2

|d| + 2 α41 |d·B| ≥ 0 ,

which give (on rescaling B → ωB, multiplying by ω −2 , letting ω → ∞) α32 +

4 α40 ≥ 0 , 3

α33 + 2

4 α41 ≥ 0 3 2

2

because for any d withtr d = 0  and any B the estimation |d·B| ≤ 23 |d| |B| is optimal. 1 0 0 Setting d = 6−1/2  0 −2 0 , B = (0, 1, 0)T in (3.6), we obtain (on rescaling B → ωB, multiplying by 0 0 1 ω −2 , letting ω → ∞, changing the sign of d)

3 4 α32 + α33 + (α40 + α41 ) . |α21 | ≤ 2 3 Acknowledgements. The research was partially supported by the Research Directorates General of the European Commission (through project GRD1-1999-11095). The authors gratefully acknowledge this support.

42


References 1. Berkovsky, B.M., Medvedev, V.F., Krakov, M.S.: Magnetic Fluids – Engineering Applications. Oxford: Oxford University Press 1993 2. Berkovsky, B.M.: Magnetic Fluids and Application Handbook. New York: Begell House 1996 3. Blums, E., Cebers, A., Maiorov. M.M.: Magnetic Fluids. Berlin: Walter de Gruyter 1997 4. Brigadnov, I.A.: Mathematical correctness and numerical methods for solution of the plasticity initial-boundary value problems. Mech. Solids 4, 62–74 (1996) 5. Brown, W.F.: Magnetoelastic Interactions. Springer, Berlin 1966 6. Carlson, J.D., Catanzarite, D.M., St. Clair, K.A.: Commercial magnetorheological fluid devices. Int. J. Mod. Phys. B 10, 2857–2865 (1996) 7. Carlson, J.D., Jolly, M.R.: MR fluid, foam and elastomer devices. Mechatronics 10, 555–569 (2000) 8. Drozdov, A.D.: Finite Elasticity and Viscoelasticity: A Course in the Nonlinear Mechanics of Solids. Singapure: World Scientific 1996 9. Dyke, S.J., Spencer, B.F. Jr., Sain, M.K., Carslon, J.D.: Modeling and Control of Magnetorheological Dampers for Seimic Response Reduction. Smart Mat. Struct. 5, 565–575 (1996) 10. Eringen, A.C.: Mechanics of Continua. New York: Wiley 1980 11. Eringen, A.C., Maugin, G.: Electrodynamics of Continua, Vol. I and II. New York: Springer 1989 12. Felt, D.W., Hagenbüchle, M., Liu, J.: Rheology of a magnetorheological fluid. J. Intelligent Mater. Syst. Struct. 7, 589–593 (1996) 13. Hutter, K., van de Ven, A.A.F.:Field Matter Interactions in Thermoelastic Solids. Lecture Notes in Physics, vol. 88, Springer, Berlin 1978 14. Jackson, J.D.: Klassische Elektrodynamik. Berlin: Walter de Gruyter (1983) 15. Jolly, M.R., Carlson, J.D., Muñoz, B.C.: A model of the behaviour of magnetorheological materials. Smart Mater. Struct. 5, 607–614 (1996) 16. Jolly, M.R., Bender, J.W., Carlson, J.D.: Properties and application of commercial magnetorheological fluids. SPIE 5th Annual Int. Symp. on Smart Structures and Materials, San Diego, CA 1998 17. Jolly, M.R., Bender, J.W., Mathers, R.T.: Indirect measurements of microstructure development in magnetorheological fluids. Int. J. Modern Phys. B 13, 2036–2043 (1999) 18. Kordonsky, W.: Magnetorheological effects as a base of new devices and technologies. J. Mag. Mag. Mater. 122, 395–398 (1993) 19. Kovetz, A.: Electromagnetic Theory. New York, Oxford: University Press 2000 20. Lazareva, T.G., Kabisheva, I.V., Prodan, S.A.: Homogeneous magnetorheological materials. J. Intelligent Mater. Syst. Struct. 9, 655–661 (1998) 21. Lurie, A.I.: Nonlinear Theory of Elasticity. Amsterdam: North-Holland 1990 22. Minkowski, H.: Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern. Göttinger Nach., 53–111 (1908) 23. Mohebi, M., Jamashi, N., Flores, G.A., Jing Liu: Numerical study of the role of magnetic field ramping rate on the structure formation in magnetorheological fluids, Int. J. Modern Phys. 13, 2060–2067 (1999) 24. Nakato, M., Yamamoto, H.: Dynamic viscoelasticity of a magnetorheological fluid in oscillatory slit flow. Int. J. Modern Phys. 13, 2068–2076 (1999) ˇ y, M.: Viscous Multipolar Fluids. Quart. Appl. Math. 49, 247–265 (1991) 25. Neˇcas, J., Silhav´ 26. Pao, Y.H.: Electromagnetic Forces in Deformable Continua. In: Nevat–Nasser, S. (ed.): Mechanics Today. Vol. 4. Pergamon Press, 1978, pp. 209–306 27. Rabinow, J.: The magnetic fluid clutch. AIEE Trans. 67, 1308–1315 (1948) 28. Rajagopal, K.R., Yalamanchili, R.C., Wineman, A.S.: Modeling electro-rheological materials through mixture theory. Int. J. Engng. Sci. 32(3), 481–500 (1994) uˇziˇcka, M.: Mathematical modeling of electrorheological materials. Continuum Mech. Thermodyn. 13, 29. Rajagopal, K.R., R˚ 59–78 (2001) 30. Rajagopal, K.R., Tao, L.: Modeling of the microwave drying process of aqueous dielectrics. Z. angev. Math. Phys. 53, 923–948 (2002) 31. Rosensweig, R.E.: Ferrohydrodynamics. Cambridge: Cambridge University Press 1985 32. Spencer, A.J.M.: Theory of Invariants. In: Eringen, A.c. (ed.): Continuum Physics Vol. 1. New York: Academic Press 1971 33. Truesdell, C., Toupin, R.: The Classical Field Theories. In: Flügge, W. (ed.): Handbuch der Physik, Vol. III/1. New York: Springer 1960 34. Truesdell, C.: Rational Thermodynamics. New York: Springer (1984) 35. Truesdell, C.: First Course in Rational Continuum Mechanics, Vol. I. New York: Academic Press 1991 36. Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics, 2nd edn. Berlin: Springer 1992 37. Weiss, K.D., Carlson, J.D., Nixon, D.A.: Viscoelastic properties of magneto- and electrorheological fluids. J. Intell. Mater. Syst. & Struct. 5, 772–775 (1994) 38. Wineman, A.S., Rajagopal, K.R.: On constitutive equations for electrorheological materials. Continuum Mech. Thermodyn. 7, 1–22 (1995) 39. Winslow, W.M.: Induced vibration of suspensions. J. Appl. Phys. 20, 1137–1140 (1949)