A magnetic-dipoles-based micro–macro constitutive model for MRFs subjected to shear deformation

Abstract

A micro–macro description for the constitutive behavior of magnetorheological fluids (MRFs) under shear deformation is formulated based on a more exact magnetic-dipolar model and a statistical approach. The conventional Bingham’s model of viscoplasticity and the dual-viscosity model for MRFs can be obtained from the proposed model as the special cases. This model can take into account the effect of each of the main influencing factors, such as the intensity of magnetic induction, the size, and the volume fraction of particles, shear strain and shear strain rate, saturated magnetization, on the yield shear stress of MRFs. The satisfactory agreement with the experimental results demonstrates the validity of the proposed model. The effect of light weight coating on the sedimentation velocity of the suspended particles is also investigated. This model can evaluate comprehensively the overall property of an MRF and the effects of different main influencing factors; therefore, it may also be of help for the initial design and optimization of high-performance MRFs.

Appendix

The intensity of magnetic field at point P (Fig. 1) caused by a magnetic pole Q _m can be expressed as

$$ \label{eq33} {\tilde {\bf H}}=\frac{1}{4\pi \mu _0 }\frac{Q_m }{r^2}{\hat {\bf {r}}},\quad \quad \quad \mbox{with}\quad {\hat {\bf {r}}}=\frac{{\rm {\bf r}}}{r}. $$

(33)

Thus, the contribution of a dipole with two magnetic poles $Q_m^+$ and $Q_m^-$ (Fig. 1) can be expressed as

$$ \label{eq34} {\tilde {\bf {H}}}=\frac{Q_m }{4\pi \mu _0 }\left( {\frac{{\hat {\bf {r}}}_1 }{r_1^2 }-\frac{{\hat {\bf {r}}}_2 }{r_2^2 }} \right), $$

(34)

$$ \begin{array}{lll} \label{eq35} \mbox{with}\;{\hat {\bf {r}}}_1 &=&\frac{{\rm {\bf r}}_1 }{r_1 }=\frac{1}{r_1 }\left( {{\rm {\bf r}}-\frac{1}{2}{\rm {\bf a}}} \right),\\ {\hat {\bf {r}}}_2 &=&\frac{{\rm {\bf r}}_2 }{r_2 }=\frac{1}{r_2 }\left( {{\rm {\bf r}}+\frac{1}{2}{\rm {\bf a}}} \right), \mbox{and}\;{\rm {\bf a}}=a{\hat {\bf {n}}}. \end{array} $$

(35)

Substituting Eq. 35 into 34 gives

$$ \label{eq36} {\tilde {\bf {H}}}=\frac{Q_m }{4\pi \mu _0 }\left( {\frac{r_2^3 -r_1^3 }{r_1^3 r_2^3 }{\rm {\bf r}}-\frac{r_1^3 +r_2^3 }{2r_1^3 r_2^3 }{\rm {\bf a}}} \right). $$

(36)

Making use of the following definitions

$$ \label{eq37} {\rm {\bf j}}=Q_m {\rm {\bf a}}\;\;\mbox{and}\;\;{\rm {\bf j}}=\mu _0 {\rm {\bf m}}, $$

(37)

where j is the magnetic dipolar moment of a magnetized dipole, Eq. 36 can be rewritten as

$$ \label{eq38} {\tilde {\bf {H}}}=\frac{m}{4\pi a}\left( {\frac{r_2^3 -r_1^3 }{r_1^3 r_2^3 }{\rm {\bf r}}-\frac{r_1^3 +r_2^3 }{2r_1^3 r_2^3 }{\rm {\bf a}}} \right) $$

(38)

In free space

$$ \label{eq39} {\tilde {\bf {B}}}=\mu _0 {\tilde {\bf {H}}}=\frac{\mu _0 m}{4\pi a}\left( {\frac{r_2^3 -r_1^3 }{r_1^3 r_2^3 }{\rm {\bf r}}-\frac{r_1^3 +r_2^3 }{2r_1^3 r_2^3 }{\rm {\bf a}}} \right). $$

(39)

It should be noted that, in the existing work, it is usually assumed that || r ₁ || ≫ a and || r ₂ || ≫ a. Making use the condition $\frac{a}{\| {{\rm {\bf r}}_1 } \|}\approx 0$ and $\frac{a}{\| {{\rm {\bf r}}_2} \|}\approx 0$, Eq. 39 are reduced to (Fang and Zhang 2001)

$$ \label{eq40} {\tilde {\bf {B}}}=\frac{\mu _0 }{4\pi r^3}\left[ {3({\rm {\bf m}}\cdot {\hat {\bf {r}}}){\hat {\bf {r}}}-{\rm {\bf m}}} \right]. $$

(40)

Ignoring the additional magnetization of a particle caused by the other magnetized particles, considering a constant applied magnetic field, and making use of the relationship (Fang and Zhang 2001)

$$ \begin{array}{lll} \label{eq41} {\rm {\bf F}}_i &=&-{\rm {\bf m}}\cdot \nabla \Big( {{\rm {\bf B}}_0 +\mathop {{\tilde {\bf {B}}}}\limits^i } \Big)=-{\rm {\bf m}}\cdot \nabla \bigg( {{\rm {\bf B}}_0 +\sum\limits_{j\ne i} {\mathop {{\tilde {\bf {B}}{\it_j}}}\limits^i } } \bigg)\\ &=&-{\rm {\bf m}}\cdot \nabla \sum\limits_{j\ne i} {\mathop {{\tilde {\bf {B}}{\it _j}}}\limits^i } , \end{array} $$

(41)

the magnetic force on the ith particle, F _i , can be derived as

$$ \begin{array}{lll} \label{eq42} \par {{\rm {\bf F}}_i } &=& \frac{\mu _0 }{4\pi }\sum\limits_{j\ne i} {\frac{m}{2R}{\rm {\bf m}}}\cdot {{\rm {\bf i}}_k \frac{\partial }{\partial x_k }\!\left\{\! {\frac{{\rm {\bf r}}_{ij} -R{\hat {\bf {n}}}}{\big[ {r_{ij}^2 \!+\!R^2\!-\!2Rr_{ij} \cos \theta }\! \big]^{\frac{3}{2}}}}\right.}\\ &&{\left.{-\frac{{\rm {\bf r}}_{ij} +R{\hat {\bf {n}}}}{\big[ {r_{ij}^2 \!+\!R^2\!+\!2Rr_{ij} \cos \theta } \big]^{\frac{3}{2}}}} \right\}} \\ &=& \frac{\mu _0 }{4\pi }\sum\limits_{j\ne i} {\frac{m}{2R}\left\{ {\frac{{\rm {\bf m}}}{r_1^3 }-\frac{m( {3r_{ij} \cos \theta -3R} )( {{\rm {\bf r}}_{ij} -R{\hat {\bf {n}}}} )}{r_1^5 }}\right.}\\&&{\left.{ -\frac{{\rm {\bf m}}}{r_2^3 }\!+\!\frac{m({3r_{ij} \cos \theta \!+\!3R})( {{\rm {\bf r}}_{ij} \!+\!R{\hat {\bf {n}}}} )}{r_2^5 }} \right\}} \\ &=& \frac{3\mu _0 }{8\pi }\!\sum\limits_{j\ne i}\! {\frac{m^2}{Rr_1^5 r_2^5 }\!\left\{ \vphantom{{\left.{\left[ {\frac{1}{3}\left( {r_1^2 r_2^5 -r_1^5 r_2^2 } \right)+Rr_{ij} \cos \theta \big(r_1^5 +r_2^5 \big)+R^2\big(r_1^5 -r_2^5 \big)} \right]{\rm {\bf \hat {n}}}} \right\}}} {\left[ {r_{ij} \cos \theta \left(r_1^5 -r_2^5 \right)+R\left(r_1^5 +r_2^5 \right)} \right]{\rm {\bf r}}_{ij}}\right.}\\ &&{\left.{ +\!\left[ {\frac{1}{3}\!\left( {r_1^2 r_2^5 \!-\!r_1^5 r_2^2 } \right)\!+\!Rr_{ij} \cos \theta \!\left(r_1^5 \!+\!r_2^5 \right)}\right.}\right.}\!\!\!\\&&{\left.{\left.{ +R^2\left(r_1^5 -r_2^5 \right)} \vphantom{{\left.{\left[ {\frac{1}{3}\left( {r_1^2 r_2^5 -r_1^5 r_2^2 } \right)+Rr_{ij} \cos \theta \left(r_1^5 +r_2^5 \right)}\right.}\right.}}\right]{\hat {\bf {n}}}} \right\}} \par \par \end{array} $$

(42)

where a _i  =  a _j  =  2R, ${\rm {\bf m}}_i\, \approx\, {\rm {\bf m}}_j \,\approx\, {\rm {\bf m}}\,=\,\frac{4}{3}\pi R^3\chi {\rm {\bf H}}\,\approx$ $\frac{4}{3}\pi R^3\chi {\rm {\bf H}}_0$ and m _i  ≈ m _j  ≈ m have been employed, ignoring the disturbance of magnetic dipoles ${\tilde {\bf {H}}}$ to the magnetic field since ${\tilde {\bf {H}}}$ ≪ H ₀ , and $r_1 =\sqrt {r_{ij}^2 + R^2 - 2Rr_{ij} \cos \theta}$ and $r_2 =\sqrt {r_{ij}^2 +R^2+2Rr_{ij} \cos \theta}$.