Maximising the electrorheological effect for bidisperse nanofluids from the electrostatic force between two particles

Abstract

A multipole re-expansion solution for two nonidentical dielectric spheres in a parallel electric field is used to determine the critical ratio of particle radii which leads to the strongest force of attraction between the spheres at various interstices and under varying dielectric properties. These critical ratios provide genuine optimal dimensions, in the sense that the force of attraction decreases for both increasing and decreasing ratios. Numerical results are compared with experimental results from the literature and discussed from the perspective of the impact on the design of electrorheological nanofluids.

Appendix

A derivation of the re-expansion formula

The re-expansion method was suggested by Washizu (1992) and was employed by Matsuyama et al. (1995) to derive an axisymmetric re-expansion at another point on the axis of symmetry. In this appendix, we give a clearer view of the derivation of the re-expansion formula (Eq. 4). The method of the derivation follows that employed by Matsuyama et al. (1995). The result of Eq. 3 can be quickly derived using the same argument as that below and has not been included in this appendix.

Consider a Cartesian coordinate system with a potential which is axisymmetric around the z-axis. Now take two distinct points on the z-axis and centre separate axisymmetric spherical coordinate systems (r, θ) and (R, Θ) around these two points, where the first coordinate system is to the left (negative z direction) of the second, and the angles are measured in an anti-clockwise direction from the positive z direction. Figure 6 shows a common point being referenced in either polar coordinate system.

Fig. 6

Geometry showing origins of two different axisymmetric spherical coordinate systems being referenced from an arbitrary point P

Let there be potentials centred on each of the polar origins, each with a singularity at r→∞ and R=0, respectively. These potentials can be expanded with a Legendre expansions as

$$\Phi = {\sum\limits_{n = 0}^\infty {a_{n} \phi _{{\text{n}}} } },\quad \Psi = {\sum\limits_{n = 0}^\infty {A_{n} \psi _{n} ,} }$$

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where $\phi _{n} = r^{n} {\bf P}_{n} {\left( {\cos \Theta } \right)},\psi _{n} = R^{{ - {\left( {n + 1} \right)}}} {\bf P}_{n} {\left( {\cos \Theta } \right)},$and a _n and A _n are the coefficients of the expansion.

Now, differentiating φ _n and ψ _n with respect to z and noting that $\partial \mathord{\left/ {\vphantom {\partial {\partial z}}} \right. \kern-\nulldelimiterspace} {\partial z} = \cos \theta \,\partial \mathord{\left/ {\vphantom {\partial {\partial r}}} \right. \kern-\nulldelimiterspace} {\partial r} - {\left( {\sin \theta \mathord{\left/ {\vphantom {\theta r}} \right. \kern-\nulldelimiterspace} r} \right)}\partial \mathord{\left/ {\vphantom {\partial {\partial \theta }}} \right. \kern-\nulldelimiterspace} {\partial \theta }{\kern 1pt}$we yield

$$ \frac{{\partial \phi _{n} }} {{\partial z}} = r^{{n - 1}} {\left[ {{\left( {1 - \cos ^{2} \theta } \right)}{\mathbf{P}}^{\prime }_{n} {\left( {\cos \theta } \right)} + n\,\,\cos \theta {\mathbf{P}}_{n} {\left( {\cos \theta } \right)}} \right]}, $$

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$$ \psi _{a} = {\sum\limits_{n = 0}^\infty {\,{\left[ {a_{n} {\left( {\frac{{r_{a} }} {r}} \right)}^{{n + 1}} + b_{n} {\left( {\frac{r} {{r_{a} }}} \right)}^{n} } \right]}\,\,{\mathbf{P}}_{n} {\left( {\cos \theta } \right)} - Er\,\cos \theta ,} } $$

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and using following identities of Legendre polynomials

$${\left( {1 - x^{2} } \right)}{\bf P}^{\prime }_{n} {\left( x \right)} + nx{\bf P}_{n} {\left( x \right)} = n{\bf P}_{{n - 1}} {\left( x \right)},$$

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$$ {\left( {1 - x^{2} } \right)}{\mathbf{P}}^{\prime }_{n} {\left( x \right)} - {\left( {n + 1} \right)}x{\mathbf{P}}_{n} {\left( x \right)} = - {\left( {n + 1} \right)}{\mathbf{P}}_{{n + 1}} {\left( x \right)}, $$

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we can state

$$ \frac{{\partial \phi _{n} }} {{\partial z}} = n\phi _{{n - 1}} ,\quad \frac{{\partial \psi _{n} }} {{\partial z}} = - {\left( {n + 1} \right)}\psi _{{n + 1}} . $$

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Now, considering the figure above and applying the cosine rule, we can immediately see that $R^{2} = d^{2} - 2rd\,\,\,\,\cos \theta + r^{2} ,$which allows us to quickly derive

$$\frac{d} {R} = \frac{1} {{{\sqrt {1 - 2{\left( {r \mathord{\left/ {\vphantom {r d}} \right. \kern-\nulldelimiterspace} d} \right)}\cos \theta + {\left( {r \mathord{\left/ {\vphantom {r d}} \right. \kern-\nulldelimiterspace} d} \right)}^{2} } }}}.$$

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Noting that ψ ₀=1/R and that a generating function for the Legendre polynomials is

$$\frac{1} {{{\sqrt {1 - 2ux + u^{2} } }}} = {\sum\limits_{m = 0}^\infty {u^{m} } }{\bf P}_{m} {\left( x \right)},$$

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we can therefore state

$$\psi _{0} = {\sum\limits_{m = 0}^\infty {d^{{ - {\left( {m + 1} \right)}}} } }\phi _{m} .$$

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Now, considering the recurrence relationships (Eq. 34), we can make the following rearrangements:

$$ \begin{array}{*{20}c} {\psi _{n} = \frac{{{\left( { - 1} \right)}^{n} }} {{n!}}\frac{{\partial ^{n} }} {{\partial z^{n} }}\psi _{0} ,} \\ { = \frac{{{\left( { - 1} \right)}^{n} }} {{n!}}{\sum\limits_{m = 0}^\infty {d^{{ - {\left( {m + 1} \right)}}} } }\frac{{\partial ^{n} }} {{\partial z^{n} }}\phi _{m} ,} \\ { = \frac{{{\left( { - 1} \right)}^{n} }} {{n!}}{\sum\limits_{m = 0}^\infty {d^{{ - {\left( {m + 1} \right)}}} m{\left( {m - 1} \right)}{\left( {m - 2} \right)} \ldots {\left( {m - n + 1} \right)}\phi _{{m - n}} ,} }} \\ { = \frac{{{\left( { - 1} \right)}^{n} }} {{n!}}{\sum\limits_{m = - n}^\infty {d^{{ - {\left( {m + n + 1} \right)}}} {\left( {m + n} \right)}{\left( {m + n - 1} \right)}{\left( {m + n - 2} \right)} \ldots {\left( {m + 1} \right)}\phi _{m} ,} }} \\ { = {\left( { - 1} \right)}^{n} {\sum\limits_{m = 0}^\infty {d^{{ - {\left( {m + n + 1} \right)}}} \frac{{{\left( {m + n} \right)}!}} {{m!n!}}\phi _{m} } }.} \\ \end{array} $$

Therefore, by substituting in for ϕ _m , ψ _n , multiplying both sides by $r^{{n + 1}}_{{\text{b}}}$ and rearranging, we yield the required re-expansion

$${\left( {\frac{{r_{{\text{b}}} }} {R}} \right)}^{{n + 1}} {\bf P}_{n} {\left( {\cos \Theta } \right)} = {\left( { - 1} \right)}^{n} {\left( {\frac{{r_{{\text{b}}} }} {d}} \right)}^{{n + 1}} {\sum\limits_{m = 0}^\infty {\frac{{{\left( {m + n} \right)}!}} {{m!n!}}{\left( {\frac{r} {d}} \right)}^{m} } }{\bf P}_{n} {\left( {\cos \theta } \right)}.$$

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