Analysis of the Viscosity of Dilute Suspensions Beyond Einstein’s Formula

Abstract

We provide a mathematical analysis of the effective viscosity of suspensions of spherical particles in a Stokes flow, at low solid volume fraction $\phi$. Our objective is to go beyond Einstein’s approximation $\mu _{eff} = (1+\frac{5}{2}\phi ) \mu$. Assuming a lower bound on the minimal distance between the N particles, we are able to identify the $O(\phi ^2)$ correction to the effective viscosity, which involves pairwise particle interactions. Applying the methodology developped over the last years on Coulomb gases, we are able to tackle the limit $N \rightarrow +\infty$ of the $O(\phi ^2)$-correction, and provide an explicit formula for this limit when the particles centers can be described by either periodic or stationary ergodic point processes.

A Proof of Lemma 2.4

For any open set U, we denote $\fint _U = \frac{1}{|U|} \int _U$. By (H2), we have

$$\begin{aligned} d := \frac{c}{4} N^{-1/3} \leqq \min _{i \ne j} \frac{|x_i - x_j|}{4}. \end{aligned}$$

We write

$$\begin{aligned} A'_i = A'_{i,1} + A'_{i,2} + A'_{i,3}, \end{aligned}$$

with

$$\begin{aligned} A'_{i,1}&= \sum _{j\ne i} \fint _{B(x_j,d)}\Big ( D(v[A_j])(x_i - x_j) - D(v[A_j])(x_i - x') \Bigr ) \mathrm{d}x', \\ A'_{i,2}&= \sum _{j\ne i} \fint _{B(x_j,d)} \Big ( D(v[A_j])(x_i - x') - \fint _{B_i} D(v[A_j])(x - x') \mathrm{d}x \Bigr ) \mathrm{d}x', \\ A'_{i,3}&= \sum _{j\ne i} \fint _{B(x_j,d)} \fint _{B_i} D(v[A_j])(x - x') \mathrm{d}x \mathrm{d}x'. \end{aligned}$$

Setting $y_i = N^{-1/3} x_i$, using that for $i \ne j$, $|y_i - y_j| \geqq \frac{1}{2} (c + |y_i - y_j| ) \geqq c$,

$$\begin{aligned} |A'_{i,1}| \leqq C a^3 \sum _{j\ne i} \frac{d}{|x_i - x_j|^4} |A_j| \leqq C' \phi \sum _{j} \frac{|A_j|}{(c + |y_i - y_j|)^4}. \end{aligned}$$

From the inequality (2.35), applied with $a_{ij} = \frac{1}{(c + |y_i - y_j|)^4}$ and $b_j = A_j$, we deduce

$$\begin{aligned} \sum _{i} |A'_{i,1}|^q \, \leqq \mathcal {C} \phi ^q \sum _{j} |A_j|^q. \end{aligned}$$

Similarly,

$$\begin{aligned} |A'_{i,2}| \leqq C a^3 \sum _{j\ne i} \frac{a}{|x_i - x_j|^4} |A_j| \leqq C' \phi ^{\frac{4}{3}} \sum _{j} \frac{|A_j|}{c + |y_i - y_j|^4}. \end{aligned}$$

This leads to

$$\begin{aligned} \sum _{i} |A'_{i,2}|^q \, \leqq \mathcal {C} \phi ^{\frac{4q}{3}} \sum _{j} |A_j|^q. \end{aligned}$$

The last term is the most difficult. We follow [21]. Let us remind ourselves that

$$\begin{aligned} v[A] =-\frac{5}{2} A : (x \otimes x) \frac{a^3 x}{|x|^5}. \end{aligned}$$

Let $\chi _d(x) = \chi (x/d)$ a smooth function that is 0 in B(0, d), 1 outside B(0, 2d). Introducing the function $F_A = \sum _{j} A_j 1_{B(x_j,d)}$, using that $d \leqq \min _{i \ne j} \frac{|x_i - x_j|}{4}$, we can write that

$$\begin{aligned} A'_{i,3} = \frac{1}{d^3} \int _{B_i} \int _{\mathbb {R}^3} \chi _d(x_i-x') \mathbf {K}(x-x')F_A(x') \mathrm{d}x' \mathrm{d}x, \end{aligned}$$

where $\mathbf {K}(x)$ is an endomorphism of the space of symmetric matrices, defined by

$$\begin{aligned} \mathbf {K}(x)A = -\frac{5}{2} \Big ( \frac{4\pi }{3}\Big )^{-2} D \Big ( A : (x \otimes x) \frac{x}{|x|^5} \Big ). \end{aligned}$$

We then split $A'_{i,3} = M_i + N_i$, with

$$\begin{aligned} M_i&= \frac{1}{d^3} \int _{B_i} \int _{\mathbb {R}^3} \chi _d(x-x') \mathbf {K}(x-x') F_A(x') \mathrm{d}x' \mathrm{d}x, \\ N_i&= \frac{1}{d^3} \int _{B_i} \int _{\mathbb {R}^3} (\chi _d(x_i-x') - \chi _d(x-x')) \mathbf {K}(x-x')F_A(x') \mathrm{d}x' \mathrm{d}x. \end{aligned}$$

By Hölder inequality,

$$\begin{aligned} |M_i|^q \leqq \frac{1}{d^{3q}}a^{\frac{3q}{p}} \Vert \big ( \chi _d \mathbf {K}\bigr ) \star F_A \Vert _{L^q(B_i)}^q, \end{aligned}$$

and so

$$\begin{aligned} \sum _i |M_i|^q \leqq \frac{1}{d^{3q}}a^{\frac{3q}{p}} \Vert \big ( \chi _\mathrm{d}\mathbf {K}\bigr ) \star F_A \Vert _{L^q(\mathbb {R}^3)}^q. \end{aligned}$$

The kernel $\chi _d \mathbf {K}$ enters the framework of the Calderón–Zygmund theorem, see for instance [31, Chapters 4 and 5]: for all $1< q <+\infty$, the operator $\big ( \chi _\mathrm{d}\mathbf {K}\bigr ) \, \star$ is continuous from $L^q(\mathbb {R}^3)$ to $L^q(\mathbb {R}^3)$, with

$$\begin{aligned} \Vert \big ( \chi _\mathrm{d}\mathbf {K}\bigr ) \star \Vert _{\mathcal {L}(L^q, L^q)} \leqq C_q. \end{aligned}$$

We stress that the constant $C_q$ depends only on q, and not on d, as can be seen from the rescaling $x' := x'/d$. It follows that

$$\begin{aligned} \sum _i |M_i|^q \leqq \frac{C}{d^{3q}}a^{\frac{3q}{p}} \Vert F_A\Vert _{L^q(\mathbb {R}^3)}^q. \end{aligned}$$

As the balls $B(x_j,d)$ are disjoint, $|\sum A_j 1_{B(x_j,d)}|^q = \sum |A_j|^q 1_{B(x_j,d)}$, so that $\Vert F_A\Vert _{L^q(\mathbb {R}^3)}^q = \frac{4\pi }{3} \sum |A_j|^q d^3$, and

$$\begin{aligned} \sum _i |M_i|^q \leqq C' \left( \frac{a}{d}\right) ^{\frac{3q}{p}} \sum _i |A_i|^q \leqq \mathcal {C} \phi ^{\frac{q}{p}} \sum _i |A_i|^q. \end{aligned}$$

To bound $N_i$, we notice that for all $x \in B_i$, the support of $x' \rightarrow \chi _d(x_i-x') - \chi _d(x-x')$ is included in

$$\begin{aligned} \Big ( B(x_i, 2d) \cup B(x, 2d) \Bigr ) \setminus \Big ( B(x,d) \cap B(x_i,d) \Bigr ) \subset B(x, 2d+a) \setminus B(x,d-a) \end{aligned}$$

(remark that by definition of $\phi$, a is less than d for $\phi$ small enough). We get

$$\begin{aligned} |N_i|^q \leqq \frac{1}{d^{3q}}a^{\frac{3q}{p}} \Vert \big | 1_{B(0, 2d+a) \setminus B(0,d-a)} \mathbf {K} \big | \star \big | F_A \big | \Vert _{L^q(B_i)}^q, \end{aligned}$$

so that

$$\begin{aligned} \sum _i |N_i|^q&\leqq \frac{C}{d^{3q}}a^{\frac{3q}{p}} \Vert \big | 1_{B(0, 2d+a) \setminus B(0,d-a)}|x|^{-3} \big | \star \big | F_A \big | \Vert _{L^q(\mathbb {R}^3)}^q \\&\leqq \frac{C'}{d^{3q}}a^{\frac{3q}{p}} \big | \ln \big (\frac{2d+a}{d-a}\big ) \big |^q \Vert F_A\Vert _{L^q(\mathbb {R}^3)}^q \\&\leqq C'' \left( \frac{a}{d}\right) ^{\frac{3q}{p}} \sum _i |A_i|^q \leqq \mathcal {C} \phi ^{\frac{q}{p}} \sum _i |A_i|^q, \end{aligned}$$

using that, for $\phi \ll 1$, $a \ll d$ and $\big |\ln \big (\frac{2d+a}{d-a}\big )\big |$ is bounded by an absolute constant.