## 1 Introduction  
Microfluidic systems constitute a modern instrument to intensify processes, both for a laboratory and on a large scale, due to their special properties [1]. Microreactors provide a drastically shorter mixing time, which can contribute to conducting chemical and biochemical reactions more effectively in comparison with conventional reactors. High surface-to-volume ratios lead to an intensification of mass transport processes, which is especially important for mass transport limited reactions [2; 3; 4]. As a result, conversion and yield in microreactors are both higher than in conventional reactors [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13]. Due to low thermal inertia and high thermal-transfer efficiencies, it is possible to control optimal temperature precisely. Continuous processes in microreactors lead to a significant shortening of reaction duration [14; 15]. The relationship between the reaction rate and the rate of mixing is very important for the reactor performance of a chemical reactor because the reaction rate depends not only on reaction conditions, but also on mixing efficiency [16; 17; 18; 19]. In the case of bimolecular chemical reactions, mixing plays an important role because intimate contact between the components must occur to ensure chemical transformations [20; 21]. There is a large number of microreactor systems that use different mixing principles. They can be classified in two main groups [16]: active microreactors and passive microreactors. Active microreactors use an external energy source to build time-dependentperturbations, which enhance the mixing process, i.e., in oscillatory baffled reactors a piston oscillates the flow so that around each reactor baffle eddies are created leading to turbulent mixing of solution [22; 23]. Passive microreactors provide a high interfacial area between the species due to their specially designed geometry [24; 25; 26; 27; 28]. Passive microreactors can be classified into two subgroups according to the mixing process: molecular diffusion or chaotic advection [17]. Despite these different classifications, both maximize the interfacial area for molecular diffusion as the final step of the mixing process. In general, the time for diffusional transport, the so-called micromixing time, can be described in laminar flow as a quadratic function of the characteristic mixing length L [29]:  
\[\tau_{\rm micro}=\rm A\frac{L^{2}}{D} \tag{1}\]  
where D denotes the diffusion coefficient and A is the shape factor, which depends on the form of the channel cross section. If there are no hydrodynamic instabilities in the flow, the characteristic mixing length is typically equal to half of the microreactor channel diameter, which normally lies in the range from tens of micrometers to several millimeters. However, diffusion is rather a slow process with mixing times of up to several minutes. Special microreactor geometries (interdigital [24; 30], split-and-recombine [25; 31], focusing [26; 32], etc.) were designed to focus, stretch or recombine the flow and, as a result, decrease the characteristic mixing length and improve the mixing process. Microreactors based on chaotic advection intensity the mixing process due to the generation of vortices. Guided flow streaming leads to hydrodynamic instabilities and can significantly improve the transverse mass transfer. Microreactors based on chaotic advection may have different geometries: T- and Y-shaped [33; 34; 35; 36], herringbone [27], chicane or serpentine [37; 38; 39], with embedded obstructions [40], etc. The most important feature of these microreactors is the effect of volume flow rate on the intensity of the vortex structures which appear, [39; 41; 42] as well as on reactor performance [2; 43; 38]. Modern microreactors can provide almost instant mixing in comparison with the reaction time-scale, even in the case of fast reactions [16; 38; 44]. However, the mixing time, and therefore reactor performance, depends greatly on the volume flow rate.