2.1 Case with no diffusiophoresis

2.1.1 Case of salt

For an initially round patch $(\langle x^{2}\rangle _{s}(0)=\langle y^{2}\rangle _{s}(0)=\ell _{0}^{2},\langle xy\rangle _{s}(0))=0$ , the solution is

(2.2) $$\begin{eqnarray}\displaystyle \langle x^{2}\rangle _{s}(t) & = & \displaystyle \left(\ell _{0}^{2}+\frac{D_{s}}{\unicode[STIX]{x1D70E}}\right)\exp (2\unicode[STIX]{x1D70E}t)-\frac{D_{s}}{\unicode[STIX]{x1D70E}},\end{eqnarray}$$

(2.3) $$\begin{eqnarray}\displaystyle \langle y^{2}\rangle _{s}(t) & = & \displaystyle \left(\ell _{0}^{2}-\frac{D_{s}}{\unicode[STIX]{x1D70E}}\right)\exp (-2\unicode[STIX]{x1D70E}t)+\frac{D_{s}}{\unicode[STIX]{x1D70E}},\end{eqnarray}$$

(2.4) $$\begin{eqnarray}\displaystyle \langle xy\rangle _{s}(t) & = & \displaystyle 0.\end{eqnarray}$$

A large patch such that $\ell _{0}^{2}\gg D_{s}/\unicode[STIX]{x1D70E}$ (satisfied whenever $Pe_{s}\gg 1$ ), is exponentially stretched in the dilating direction (since $\langle x^{2}\rangle _{s}(t)\approx \ell _{0}^{2}\exp (2\unicode[STIX]{x1D70E}t)$ ), and exponentially compressed towards the Batchelor scale $\ell _{s}=\sqrt{D_{s}/\unicode[STIX]{x1D70E}}$ in the compressing direction (following $\langle y^{2}\rangle _{s}(t)\approx \ell _{0}^{2}\exp (-2\unicode[STIX]{x1D70E}t)+D_{s}/\unicode[STIX]{x1D70E}$ ).

Neglecting diffusion at short time, it is possible to estimate the time needed for the patch to reach the Batchelor scale under exponential contraction: $\unicode[STIX]{x1D70F}_{s}=\ln (\ell _{0}/\ell _{s})/\unicode[STIX]{x1D70E}$ . This expression can be expressed using the Péclet number $Pe_{s}=\ell _{0}^{2}/\ell _{s}^{2}$ as

(2.5) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}_{s}={\textstyle \frac{1}{2}}\ln Pe_{s}. & & \displaystyle\end{eqnarray}$$

Of course all the previous reasoning is valid for the mixing of colloids in the absence of diffusiophoresis. By replacing $s$ by $c$ in the expressions, one finds that the Péclet number $Pe_{c}$ is linked to the Batchelor scale $\ell _{c}$ by the relation

(2.6) $$\begin{eqnarray}\displaystyle Pe_{c}=\ell _{0}^{2}/\ell _{c}^{2}. & & \displaystyle\end{eqnarray}$$

This can be also written as

(2.7) $$\begin{eqnarray}\displaystyle \frac{\ell _{c}}{\ell _{0}}=\frac{1}{\sqrt{Pe_{c}}}, & & \displaystyle\end{eqnarray}$$

which shows that the Batchelor scale of the colloids is smaller than that of the salt as one usually has $Pe_{c}\gg Pe_{s}$ .

2.1.2 Mixing time

It is interesting to note that the time $\unicode[STIX]{x1D70F}_{c}$ , needed for the patch of colloids to be compressed towards the Batchelor scale, is directly connected to the mixing time, $T_{mix}$ , here defined as the time needed for the concentration, $c(t)$ , to decrease by 50 %. Indeed, using (2.2) and (2.3) one can estimate the concentration of the colloids patch:

(2.8) $$\begin{eqnarray}\displaystyle \frac{c(t)}{c(0)}=\frac{\ell _{0}^{2}}{\sqrt{\langle x^{2}\rangle _{c}(t)\langle y^{2}\rangle _{c}(t)}}\approx \frac{1}{\sqrt{1+1/Pe_{c}\exp (2\unicode[STIX]{x1D70E}t)}}. & & \displaystyle\end{eqnarray}$$

At time $t=T_{mix,c}$ , $c$ is half of its initial value so that one gets

(2.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}T_{mix,c}\approx \ln \sqrt{3Pe_{c}}=\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}_{c}+\frac{\ln 3}{2}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}_{c}$ is the time to reach the Batchelor scale (2.5). The two times follow a similar (logarithmic) scaling as functions of the Péclet number, and are found to be linked by an affine transformation whose coefficients will depend on the precise definition of the mixing time.

2.2 Batchelor scale with diffusiophoresis

When salt and colloids are mixed together, salt is advected by the linear velocity $\boldsymbol{v}=(\unicode[STIX]{x1D70E}x,-\unicode[STIX]{x1D70E}y)$ while the colloids velocity field is $\boldsymbol{v}+\boldsymbol{v}_{dp}$ . As the salt concentration remains Gaussian at all times with $\langle xy\rangle _{s}=0$ , the diffusiophoretic velocity field is $\boldsymbol{v}_{dp}=(-D_{dp}x/\langle x^{2}\rangle _{s}(t),-D_{dp}y/\langle y^{2}\rangle _{s}(t))$ , which is also a linear flow. The colloids concentration field therefore remains Gaussian at all time, its shape being given by the second-order moments $\langle x^{2}\rangle _{c}(t)$ , $\langle y^{2}\rangle _{c}(t)$ , and $\langle xy\rangle _{c}(t)$ . In the case of an initially round patch of radius $\ell _{0}$ , one still has $\langle xy\rangle _{c}(t)=0$ and the equations for $\langle x^{2}\rangle _{c}(t)$ and $\langle y^{2}\rangle _{c}(t)$ read (Raynal et al. Reference Raynal, Bourgoin, Cottin-Bizonne, Ybert and Volk2018)

(2.10) $$\begin{eqnarray}\displaystyle \frac{\text{d}\langle x^{2}\rangle _{c}}{\text{d}t} & = & \displaystyle -2D_{dp}\frac{\langle x^{2}\rangle _{c}}{\langle x^{2}\rangle _{s}}+2\unicode[STIX]{x1D70E}\langle x^{2}\rangle _{c}+2D_{c},\end{eqnarray}$$

(2.11) $$\begin{eqnarray}\displaystyle \frac{\text{d}\langle y^{2}\rangle _{c}}{\text{d}t} & = & \displaystyle -2D_{dp}\frac{\langle y^{2}\rangle _{c}}{\langle y^{2}\rangle _{s}}-2\unicode[STIX]{x1D70E}\langle y^{2}\rangle _{c}+2D_{c}.\end{eqnarray}$$

This system has a complicated analytical solution given in Raynal et al. (Reference Raynal, Bourgoin, Cottin-Bizonne, Ybert and Volk2018). However the behaviour of its solution may be obtained in the limit of small colloids diffusion coefficient $D_{c}/D_{s}\ll 1$ for which salt is mixed in a much shorter time than the colloids so that one has $\langle y^{2}\rangle _{s}\simeq \ell _{s}^{2}$ for $t>T_{mix,s}$ and

(2.12) $$\begin{eqnarray}\displaystyle \frac{\text{d}\langle x^{2}\rangle _{c}}{\text{d}t} & \simeq & \displaystyle 2\unicode[STIX]{x1D70E}\langle x^{2}\rangle _{c},\end{eqnarray}$$

(2.13) $$\begin{eqnarray}\displaystyle \frac{\text{d}\langle y^{2}\rangle _{c}}{\text{d}t} & \simeq & \displaystyle -2\left(\unicode[STIX]{x1D70E}+\frac{D_{dp}}{\ell _{s}^{2}}\right)\langle y^{2}\rangle _{c}+2D_{c}.\end{eqnarray}$$

Equation (2.12) shows that the large scale $\langle x^{2}\rangle _{c}$ is barely affected by diffusion or diffusiophoresis, a fact that can be easily understood as $x$ is the dilating direction, with exponential stretching by the flow. In the $y$ -direction, using $\ell _{s}^{2}=D_{s}/\unicode[STIX]{x1D70E}$ , we see that the colloids concentration field is now compressed towards the modified Batchelor scale

(2.14) $$\begin{eqnarray}\displaystyle \ell _{c,diff}^{2}=\frac{D_{c}}{\unicode[STIX]{x1D70E}(1+D_{dp}/D_{s})}. & & \displaystyle\end{eqnarray}$$

2.3 Effective Péclet number

Because we want to quantify mixing through an effective Péclet number, and because mixing quantities are closely related to the Batchelor scale, we propose to use (2.6) and define the effective Péclet number $Pe_{eff}$ using the modified Batchelor scale as

(2.15) $$\begin{eqnarray}\displaystyle Pe_{eff}=\ell _{0}^{2}/\ell _{c,diff}^{2}. & & \displaystyle\end{eqnarray}$$

Using (2.14), its expression reads

(2.16) $$\begin{eqnarray}\displaystyle Pe_{eff}=Pe_{c}\left(1+\frac{D_{dp}}{D_{s}}\right). & & \displaystyle\end{eqnarray}$$

We have tested this prediction by computing the mixing time of the colloids, defined as the time needed for $c(t)/c(0)=\ell _{0}^{2}/\sqrt{\langle x^{2}\rangle _{c}(t)\langle y^{2}\rangle _{c}(t)}$ to decrease by 50 %. The evolution of $T_{mix,c}$ is displayed in figure 1 as a function of $Pe_{c}=\unicode[STIX]{x1D70E}\ell _{0}^{2}/D_{c}$ (a), and as a function of the effective Péclet number $Pe_{eff}=Pe_{c}(1+D_{dp}/D_{s})$ (b), for a wide range of parameters. In this case, the mixing time was computed without approximations by using the complicated analytical solution given in Raynal et al. (Reference Raynal, Bourgoin, Cottin-Bizonne, Ybert and Volk2018) where $D_{c}$ , $D_{s}$ and $D_{dp}>0$ were varied over several orders of magnitude. It is remarkable to observe the almost perfect rescaling of $T_{mix,c}$ as a function of $Pe_{eff}$ spanning over height decades.

In the case of non-Gaussian patches, because patches tend to relax towards a Gaussian shape after a transient under the combined action of diffusion and stretching by the flow (Villermaux Reference Villermaux2019), the scaling given in (2.16) should remain the same.

Note that in the case of a simple shear, also studied in Raynal et al. (Reference Raynal, Bourgoin, Cottin-Bizonne, Ybert and Volk2018), the concentration field does not reach a state with a constant Batchelor scale at large time, so that it is not possible to derive an effective Péclet number for any linear flow with this method.

Figure 1.

$\unicode[STIX]{x1D70E}T_{mix}$ for different $D_{dp}>0$ , $D_{s}$ and $\unicode[STIX]{x1D70E}$ , as a function of the Péclet number $Pe$ (a), and of the effective Péclet number $Pe_{eff}$ (b). Black solid line: no salt; — ⋅ — : $D_{s}=1360~\unicode[STIX]{x03BC}\text{m}^{2}~\text{s}^{-1}$ , $D_{dp}=290~\unicode[STIX]{x03BC}\text{m}^{2}~\text{s}^{-1}$ ; $\ast$ : $D_{s}=1360~\unicode[STIX]{x03BC}\text{m}^{2}~\text{s}^{-1}$ , $D_{dp}=1000~\unicode[STIX]{x03BC}\text{m}^{2}~\text{s}^{-1}$ ; ▵: $D_{s}=1360~\unicode[STIX]{x03BC}\text{m}^{2}~\text{s}^{-1}$ , $D_{dp}=10^{4}~\unicode[STIX]{x03BC}\text{m}^{2}~\text{s}^{-1}$ ; $\times$ : $D_{s}=1360~\unicode[STIX]{x03BC}\text{m}^{2}~\text{s}^{-1}$ , $D_{dp}=10^{5}~\unicode[STIX]{x03BC}\text{m}^{2}~\text{s}^{-1}$ ; ○: $D_{s}=10~\unicode[STIX]{x03BC}\text{m}^{2}~\text{s}^{-1}$ , $D_{dp}=10^{4}~\unicode[STIX]{x03BC}\text{m}^{2}~\text{s}^{-1}$ ; $+$ : $D_{s}=10~\unicode[STIX]{x03BC}\text{m}^{2}~\text{s}^{-1}$ , $D_{dp}=10^{5}~\unicode[STIX]{x03BC}\text{m}^{2}~\text{s}^{-1}$ ; - - -: time needed to reach the Batchelor scale; note that this latter follows the same scaling as the mixing time.

3.1 Estimation of the Batchelor scale without diffusiophoresis

In order to derive the Batchelor scale, it is useful to investigate how small scales are produced, which can be done by looking at the equation for the scalar gradient $\boldsymbol{G}=\unicode[STIX]{x1D6FB}C$ . In the absence of diffusiophoresis it reads

(3.1) $$\begin{eqnarray}\displaystyle \underbrace{{\textstyle \frac{1}{2}}D_{t}G^{2}}_{(0)}=\underbrace{D_{c}\;G_{i}\unicode[STIX]{x2202}_{j}^{2}G_{i}}_{(a)}\underbrace{-G_{i}G_{j}\unicode[STIX]{x2202}_{i}v_{j}}_{(b)}, & & \displaystyle\end{eqnarray}$$

where we have used the convention of summation over repeated indices, and $D_{t}=\unicode[STIX]{x2202}_{t}+v_{k}\unicode[STIX]{x2202}_{k}$ stands for the material derivative. In the case of chaotic advection by a large-scale velocity field, small scales are produced by stretching (term $(b)$ ) until they become so small that the Batchelor scale is reached so that a balance between dissipation and diffusion takes place. When mixing is efficient enough, which is the case of global chaos, a quasi-static situation takes place where the left-hand side of (3.1) is negligible compared to the two other terms so that the balance reads (Raynal & Gence Reference Raynal and Gence1997):

(3.2) $$\begin{eqnarray}\displaystyle \underbrace{D_{c}\;G_{i}\unicode[STIX]{x2202}_{j}^{2}G_{i}}_{(a)}\approx \underbrace{G_{i}G_{j}\unicode[STIX]{x2202}_{i}v_{j}}_{(-b)}. & & \displaystyle\end{eqnarray}$$

Given the characteristic length scale $L=\ell _{0}$ and velocity scale $V$ of the velocity field, to get an order of magnitude of the stretching rate $\unicode[STIX]{x2202}_{i}v_{j}\sim V/L$ , one can use this last equation to derive the Batchelor scale for both species:

(3.3) $$\begin{eqnarray}\displaystyle \ell _{c} & {\sim} & \displaystyle \frac{L}{\sqrt{Pe_{c}}}\quad \text{(colloids, no diffusiophoresis)},\end{eqnarray}$$

(3.4) $$\begin{eqnarray}\displaystyle \ell _{s} & {\sim} & \displaystyle \frac{L}{\sqrt{Pe_{s}}}\quad \text{(salt)},\end{eqnarray}$$

with $Pe_{c}=VL/D_{c}$ and $Pe_{s}=VL/D_{s}$ . Using these relations, the time needed to mix a patch of scalar of size $L$ will follow the same scaling relation as the time $\unicode[STIX]{x1D70F}_{B}=\ln Pe/\unicode[STIX]{x1D6EC}$ , needed to reach the Batchelor scale, where $\unicode[STIX]{x1D6EC}\propto V/L$ is the most negative Lyapunov exponent of the flow (Raynal & Gence Reference Raynal and Gence1997; Bakunin Reference Bakunin2011). In the case of joint mixing with diffusiophoresis, the Batchelor scale ( $\ell _{c,diff}$ ) will have a different expression so that the relation (3.3) is not expected to hold anymore. However we shall follow the path of the previous section to derive an effective Péclet number based on the modified Batchelor scale through the relation

(3.5) $$\begin{eqnarray}\displaystyle \ell _{c,diff}\sim L/\sqrt{Pe_{eff}} & & \displaystyle\end{eqnarray}$$

so as to rescale all measurements of the mixing time as a function of $Pe_{eff}$ .

3.2 Batchelor scale in the case of diffusiophoresis

In absence of diffusiophoresis, we found the Batchelor scale to be $\ell _{c}=L/\sqrt{Pe_{c}}=\sqrt{D_{c}/\unicode[STIX]{x1D70E}_{v}}$ with $\unicode[STIX]{x1D70E}_{v}=V/L$ . This expression being the same as the one found for the pure strain flow of § 2.1, it may be anticipated that the effective Péclet number for the mixing of colloids is given by the relation $Pe_{eff}=Pe_{c}(1+D_{dp}/D_{s})$ . However as we shall see in the following, the simple result of § 2.1 will not hold in the case of chaotic advection. In order to derive a correct expression for the Batchelor scale with diffusiophoresis, we must again look carefully at the equation for $\unicode[STIX]{x1D6FB}C$ in the presence of the velocity drift $\boldsymbol{v}_{dp}=D_{dp}\unicode[STIX]{x1D6FB}\ln S$ . Taking the gradient of (1.2), we obtain

(3.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}G_{i}+v_{j}\unicode[STIX]{x2202}_{j}G_{i}+v_{dp\,j}\unicode[STIX]{x2202}_{j}G_{i}+G_{i}\unicode[STIX]{x2202}_{j}v_{dp\,j}+G_{j}\unicode[STIX]{x2202}_{i}v_{j}+G_{j}\unicode[STIX]{x2202}_{i}v_{dp\,j}+C\unicode[STIX]{x2202}_{i}\unicode[STIX]{x2202}_{j}v_{dp\,j}=D_{c}\,\unicode[STIX]{x2202}_{j}^{2}G_{i}, & & \displaystyle\end{eqnarray}$$

which we shall multiply by $G_{i}$ and sum over all components to get

(3.7) $$\begin{eqnarray}\displaystyle \underbrace{\frac{1}{2}D_{t}G^{2}}_{(0)}=\underbrace{D_{c}\,G_{i}\unicode[STIX]{x2202}_{j}^{2}G_{i}}_{(a)}\underbrace{-G_{i}G_{j}\unicode[STIX]{x2202}_{i}v_{j}}_{(b)}\underbrace{-v_{dp\,j}G_{i}\unicode[STIX]{x2202}_{j}G_{i}}_{(c)}\underbrace{-G^{2}\unicode[STIX]{x2202}_{j}v_{dp\,j}}_{(d)}\underbrace{-G_{i}G_{j}\unicode[STIX]{x2202}_{i}v_{dp\,j}}_{(e)}\underbrace{-CG_{i}\unicode[STIX]{x2202}_{i}\unicode[STIX]{x2202}_{j}v_{dp\,j}}_{(f)}.\quad & & \displaystyle\end{eqnarray}$$

As opposed to (3.1), equation (3.7) contains four additional terms involving the diffusiophoretic drift $\boldsymbol{v}_{dp}=D_{dp}\ln S$ . In order to get an order of magnitude of each term, one needs to estimate the magnitude of $\boldsymbol{v}_{dp}$ . This can be done in the case $D_{c}\ll D_{s}$ for which the salt patch is compressed towards $\ell _{s}$ long before diffusion affects the colloids concentration field. The magnitude of diffusiophoretic drift is then

(3.8) $$\begin{eqnarray}\displaystyle V_{dp}\sim \frac{D_{dp}}{\ell _{s}}=\frac{D_{dp}}{L}\sqrt{Pe_{s}}, & & \displaystyle\end{eqnarray}$$

in agreement with the numerical simulations of Volk et al. (Reference Volk, Mauger, Bourgoin, Cottin-Bizonne, Ybert and Raynal2014).

Estimating the different terms of (3.7) in the quasistatic regime, when the colloids patch has been compressed towards its Batchelor scale $\ell _{c,diff}$ so that $G\sim C/\ell _{c,diff}$ , we get

(3.9) $$\begin{eqnarray}\displaystyle (a) & {\sim} & \displaystyle \frac{G^{2}D_{c}}{\ell _{c,diff}^{2}}=\frac{G^{2}D_{c}}{L^{2}}Pe_{eff},\end{eqnarray}$$

(3.10) $$\begin{eqnarray}\displaystyle (b) & {\sim} & \displaystyle \frac{G^{2}V}{L}=\frac{G^{2}D_{c}}{L^{2}}Pe_{c},\end{eqnarray}$$

(3.11) $$\begin{eqnarray}\displaystyle (c) & {\sim} & \displaystyle \frac{G^{2}}{\ell _{c,diff}}\frac{D_{dp}}{\ell _{s}}=\frac{G^{2}D_{c}}{L^{2}}\frac{D_{dp}}{\sqrt{D_{c}D_{s}}}\sqrt{Pe_{c}\,Pe_{eff}},\end{eqnarray}$$

(3.12) $$\begin{eqnarray}\displaystyle (d) & {\sim} & \displaystyle \frac{G^{2}D_{dp}}{\ell _{s}^{2}}\ll (c),\end{eqnarray}$$

(3.13) $$\begin{eqnarray}\displaystyle (e) & {\sim} & \displaystyle \frac{G^{2}D_{dp}}{\ell _{s}^{2}}\ll (c),\end{eqnarray}$$

(3.14) $$\begin{eqnarray}\displaystyle (f) & {\sim} & \displaystyle G^{2}\ell _{c,diff}\frac{D_{dp}}{\ell _{s}^{3}}\ll (c).\end{eqnarray}$$

As one has $\ell _{c,diff}\ll \ell _{s}$ , the last three terms are always much smaller than $(c)$ so that they will be neglected in the analysis.

3.3 Effective Péclet number in the salt-attracting case ( $D_{dp}>0$ )

This configuration, which was not addressed in the case of chaotic advection, is similar to the case of pure deformation addressed in § 2. As colloids and salt are released together, the velocity drift is opposed to molecular diffusion so that the mixing time of colloids increases compared to the case with no diffusiophoresis, leading to

(3.15) $$\begin{eqnarray}\displaystyle Pe_{eff}^{attract}\gg Pe_{c}. & & \displaystyle\end{eqnarray}$$

In the quasistatic regime, where production and dissipation balance, the dominant terms are then $(a)$ and $(c)$ so that filaments are produced by diffusiophoresis and dissipated by diffusion, as was the case in the analytical example. Equating $(a)$ and $(c)$ , we get

(3.16) $$\begin{eqnarray}\displaystyle Pe_{eff}^{attract}\sim \frac{D_{dp}^{2}}{D_{c}D_{s}}Pe. & & \displaystyle\end{eqnarray}$$

Note that, because of hypothesis (3.15), the analysis can only be true if one has $D_{dp}^{2}\gg D_{c}\,D_{s}$ , which is satisfied in experiments as $D_{dp}\sim D_{s}$ and $D_{c}\ll D_{s}$ (Abécassis et al. Reference Abécassis, Cottin-Bizonne, Ybert, Ajdari and Bocquet2009; Deseigne et al. Reference Deseigne, Cottin-Bizonne, Stroock, Bocquet and Ybert2014).

The effective Péclet number found here is completely different from that of (2.16) (linear strain). This is not surprising: whilst both flows display exponential stretching, in the chaotic advection case, as explained, a quasi-stationary state is reached where gradients of concentration are created by the flow and dissipated by diffusion at the same rate. This is completely different in the case of the simple strain: because this linear flow is barely mixing, when the Batchelor scale is reached, all terms in the equation of gradients (3.1) decay at the same exponential rate, so that the left-hand-side (term (0)) is far from negligible compared to the two others.

3.4 Effective Péclet number in the salt-repelling case ( $D_{dp}<0$ )

This second configuration was first addressed in Deseigne et al. (Reference Deseigne, Cottin-Bizonne, Stroock, Bocquet and Ybert2014) where the authors proposed an expression $Pe_{eff}^{repell}\propto VL/D_{eff}$ , with $D_{eff}\propto D_{dp}^{2}/D_{s}$ , based on the Ranz model of mixing (Ranz Reference Ranz1979). In this configuration, diffusiophoresis acts as an enhanced diffusion so that we will assume

(3.17) $$\begin{eqnarray}\displaystyle Pe_{eff}^{repell}\ll Pe, & & \displaystyle\end{eqnarray}$$

and the quasistatic equilibrium is now given by $(b)\sim (c)$ , with gradients produced by the flow and smoothed by diffusiophoresis. Under this assumption we obtain the same result as proposed in Deseigne et al. (Reference Deseigne, Cottin-Bizonne, Stroock, Bocquet and Ybert2014):

(3.18) $$\begin{eqnarray}\displaystyle Pe_{eff}^{repell}\sim \frac{D_{c}D_{s}}{D_{dp}^{2}}\,Pe_{c}, & & \displaystyle\end{eqnarray}$$

which, using hypothesis (3.17), is found to hold under the same hypothesis $D_{dp}^{2}\gg D_{c}\,D_{s}$ as in the salt-attracting case.

3.5 Comparison with numerical simulations

We have tested the two scaling relations in numerical simulations of chaotic mixing by a sine flow with random phase, which ensures that global chaos is achieved (Pierrehumbert Reference Pierrehumbert1994, Reference Pierrehumbert2000) so that the mixing time is expected to scale linearly with the logarithm of the Péclet number. The flow is a modified version of the one used in Volk et al. (Reference Volk, Mauger, Bourgoin, Cottin-Bizonne, Ybert and Raynal2014): it is composed of two sub-cycles of duration $T/2$ for which the velocity field is $\boldsymbol{v}(\boldsymbol{r},t)=(f(t)\sin (y+\unicode[STIX]{x1D719}_{n}),0)$ for $nT\leqslant t<(n+1/2)T$ and $\boldsymbol{v}(\boldsymbol{r},t)=(0,f(t)\sin (x+\unicode[STIX]{x1D713}_{n}))$ for $(n+1/2)T\leqslant t<(n+1)T$ , where $(\unicode[STIX]{x1D719}_{n},\unicode[STIX]{x1D713}_{n})_{n\geqslant 1}\in [0,2\unicode[STIX]{x03C0}]^{2}$ are sequences of random numbers and $f(t)=2\sin ^{2}(2\unicode[STIX]{x03C0}t/T)$ . Because $\int _{0}^{T/2}f(t)\,\text{d}t=1$ , this flow corresponds to the same iterated map as the one with $f(t)=1$ without temporal discontinuities.

The equations were solved in a square periodic domain of length $L=2\unicode[STIX]{x03C0}$ , with $T=1.6\,\unicode[STIX]{x03C0}$ and identical concentration profiles $C(x,y,t=0)=S(x,y,t=0)=1+\sin x$ , with the same code as used in Volk et al. (Reference Volk, Mauger, Bourgoin, Cottin-Bizonne, Ybert and Raynal2014). In order to test the scaling relations, we performed a set of simulations while keeping the same sequence of random phases, varying $D_{c}\in [2\times 10^{-5},8\times 10^{-4}]$ , $D_{s}=(0.25\times 10^{-2},0.5\times 10^{-2},10^{-2})$ , and $D_{dp}=(\pm 10^{-3},\pm 2\times 10^{-3},\pm 4\times 10^{-3})$ . This corresponds to Péclet numbers $Pe_{c}\in [7.8\times 10^{3}{-}3\times 10^{5}]$ for the colloids, $Pe_{s}\in [600{-}2500]$ for the salt.

The evolution of the mixing time of the colloids, $T_{mix,c}$ , as a function of $Pe_{c}$ is displayed in figure 2(a) for all simulations with $D_{dp}^{2}/D_{c}D_{s}>1$ , and different symbols for the salt-attracting case (circles) and the salt-reppelling case (triangles). The solid line is a fit for the non-diffusiophoretic cases (salt or colloids without salt), defined as

(3.19) $$\begin{eqnarray}\displaystyle T_{mix}=3.2\log (Pe/120) & & \displaystyle\end{eqnarray}$$

for this particular flow.

Figure 2.

(a) Mixing time of the colloids, $T_{mix,c}$ , as a function of the Péclet number for all numerical simulations with $D_{dp}^{2}/(D_{c}Ds)>1$ ; the evolution of the flow is kept identical, varying $D_{c}$ , $D_{dp}$ and $D_{s}$ . (b)  $T_{mix,c}$ as a function of the effective Péclet number. $+$ : Mixing of colloids without diffusiophoresis ( $D_{dp}=0$ );  $\times$ : mixing of salt; ●: ‘salt-attracting’ case ( $D_{dp}>0$ ); ▴: ‘salt-repelling’ case ( $D_{dp}<0$ ). The solid line is a curve of expression $T_{mix,c}=3.2\ln (Pe/120)$ .

As shown in figure 2(b), all values of the mixing time are found to follow well the curve obtained without diffusiophoresis when they are plotted as a function of the effective Péclet numbers $Pe_{eff}^{attract}=(D_{dp}^{2}/D_{c}D_{s})Pe$ for ‘salt-attracting’ case, and $Pe_{eff}^{repell}=(D_{c}D_{s}/D_{dp}^{2})Pe$ for the ‘salt-repelling’ case; note also that the effective Péclet number varies here over five orders of magnitude.

The validity of our analysis can be further checked by focusing on the salt-repelling case. In that case the relation (3.18) is indeed equivalent to having an effective diffusivity

(3.20) $$\begin{eqnarray}\displaystyle D_{eff}^{repell}\sim \frac{D_{dp}^{2}}{D_{s}}, & & \displaystyle\end{eqnarray}$$

or an effective Péclet number independent of $D_{c}$ , such that the mixing time becomes independent of $Pe_{c}$ . Figure 3 displays $T_{mix,c}$ , as a function of $Pe_{c}$ in the salt-repelling case ( $D_{dp}<0$ ), linking the data points corresponding to the same values of $D_{dp}$ and $D_{s}$ . As predicted, the different curves seem to exhibit a plateau when $Pe_{c}$ is large enough so that $D_{dp}^{2}/D_{c}D_{s}\geqslant 10$ (filled symbol).

Figure 3.

Mixing time $T_{mix}$ as a function of the Péclet number for different values of $D_{dp}$ and $D_{s}$ in the salt-repelling case ( $D_{dp}<0$ ); ▫: $D_{dp}=10^{-3}$ ; ○: $D_{dp}=2\times 10^{-3}$ ; ▿: $D_{dp}=4\times 10^{-3}$ ; ——: $D_{dp}/D_{s}=0.1$ ; $\cdots \cdots$ : $D_{dp}/D_{s}=0.2$ ; — ⋅ —: $D_{dp}/D_{s}=0.4$ ; – –: $D_{dp}/D_{s}=0.8$ ; — ⋅ ⋅ —: $D_{dp}/D_{s}=1.6$ The full symbols are those for which $D_{dp}^{2}/(D_{c}Ds)\geqslant 10$ ; open symbols: $D_{dp}^{2}/(D_{c}Ds)<10$ .