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Tutorial Review of Mixing in a Rotating Soft Microchannel under Electrical Double Layer Effect: A Variational Calculus Approach

We study the effect of the grafted polyelectrolyte layer on the flow dynamics, and its consequences on underlying mixing in the rotating microfluidic channel. For this analysis, the method used by Sadeghi et al. (J. Fluid Mech., vol. 887, 2020, pp. A13; Phys. Rev. Fluids., vol. 4 (6), 2019, 063701-23), is modified by incorporating the non-linear effect stemming from the polyelectrolyte layer induced electrostatics to solve the coupled system of equations, integrated with the non-homogeneous boundary conditions. This method is used to obtain the velocity distribution in the asymptotic limit of geostrophic plug flow under the framework of variational calculus approach. We analyze the mixing dynamics from the perspective of both qualitative assessment and quantitative evaluation. For the qualitative estimation, we focus on the Poincar\'e map analysis, while the entropy of mixing approach is used for the mixing quantification. Results show that the grafted polyelectrolyte layer in contact with the ionic solution leads to the development of an electrical double layer, which upon interacting with the external electric field, strengthens the electroosmotic pumping in the fluidic channel. Such polyelectrolyte layer modulated strong electroosmotic pumping together with its intrinsic feature of offering a frictional drag to the underlying transport helps to modulate the primary as well as the secondary flows in the channel under the influence of rotational forces. With an alteration in the electroosmotic pumping and frictional drag force, tuneable through the thickness of the grafted polyelectrolyte layer, we obtain different types of secondary flow vortex configurations viz., a standard double-vortex, dumbbell-shaped vortex and the transition state between the formers. A significant change in the structure and strengths of these vortices modulates the chaotic mixing in the present configuration


Introduction
The rotational fluidics in micro-electro-mechanical-systems (MEMS) and CD-based labon-a-chip (LOC) platforms offers several advantageous features and have made significant advances in the field over the past few decades (Ducrée et al. 2006;Duffy et al. 1999;Stone et al. 2004). Of these features, the periodic helical motion of the secondary flow vortices set by the Coriolis force and the lateral wall confinement renders an intriguing nature to the mixing in the rotational microfluidics (Ng & Qi 2015). It may be mentioned here that the underlying flow dynamics in the miniaturized assays integrated on the rotating platforms is encountered with a few serious challenges. First, for example, the increasing magnitude of the Coriolis force adversely affects the axial momentum and triggers a rigid body type motion with increasing magnitude of the rotational speed beyond no-flow limits (Stewartson 1957). Second, ensuring a considerable amount of mixing efficiency in rotating devices/systems leads to the intervention of the geometrical and physical features of the centrifuges. Alongside above all issues, the direct exposure of fluid samples, typically bio-samples, to the binding surfaces of rotating micro-totalanalysis-systems (μTAS), leads to a significant deterioration of samples' quality as well.
One of the promising solutions to overcome these challenges could be the application of a grafted soft polyelectrolyte layer 1 (PEL) on the walls of rotating microchannels. It may be mentioned in this context here that with the advent of nanotechnology, several methods have shown their potential applicability in grafting PEL of desirable thickness at the wall of microfluidic channels in a synergetic way (Das et al. 2015). Quite notably, such technology at the interfacial scale provides tremendous benefits in different applications like biological analysis, immunoassays, the biochemistry analysis of biofluids, to name a few (Das et al. 2015). The pivotal objective behind the use of grafted PEL at the walls of the narrow fluidic assays has been the exploration of the enriched physicochemical interaction of the PEL macromolecules with the electrolyte ions, and its affluent consequences on the underlying flow dynamics (Keh & Ding 2003;Ohshima & Kondo 1990). Important to mention, the imposition of soft PEL at the walls of the fluidic pathways has demonstrated distinguishable outcomes on account of the notable fluidic functionalities, attributable primarily to the complex interactions between the polymeric 1 Microfluidic channels with built-in soft polymeric layers are also known as soft microchannels. Polymeric layer following the chemical interaction with the electrolyte solution forms polyelectrolyte layer or PEL in short. 5 2015). The grafted polyelectrolyte layer modulated electrostatics under the application of the electric field brings about a rich flow dynamics in the rotating fluidic environment, attributed primarily to the complex coupling of different forces that are in use. It is anticipated that the stronger electroosmotic effect, together with the frictional drag offered by the PEL, can modulate the rotational flow dynamics non-trivially, leading to an enhancement in the mixing performance following the formation of dumbbell-shaped vortices in the field. Nevertheless, no analysis has been reported to date considering the aforementioned effects in tandem in the rotational microfluidic platform.
Here we analyze the effect of grafted PEL on the flow dynamics and its consequences on the underlying mixing in a rotating rectangular microfluidic channel under the influence of electrical forcing. Our analysis primarily focuses on the low Rossby number flow and develops a theoretical framework towards modelling the electrohydrodynamics in a rotating microfluidic channel having grafted PEL at the walls. The solution to the set of governing transport equations using the boundary conditions specific to the problem under present consideration involves added complexities, and it is, therefore, necessary to pay attention to several involved issues that are in demand. These critical issues include the non-homogeneous boundary conditions imposed by the PEL electrostatics, the Darcy frictional drag at the PEL and EL(electrolyte layer) interface, and the coupled system of equations owing to the presence of generated pressure gradients in the lateral and transverse directions under the effect of lateral confinement (Ng & Qi 2015). A recent study has shown the efficacy of the variational calculus method to tackle the issues, as mentioned above, quite effectively (Sadeghi 2018;Sadeghi et al. 2019Sadeghi et al. , 2020. From the results of our study, we also establish that the approximate solutions provided by the variational calculus method are consistent with the imposed boundary conditions at the walls and PEL-EL interface and, at times, including the effect of lateral wall confinement accurately. To the best of our knowledge, the present study, for the first time, focuses on the variational calculus method for solving the underlying electrohydrodynamics in a rotating microfluidic channel having built-in soft layers. On solving the Eulerian velocity fields, we further use the Lagrangian approach to understand the mixing in the present rotating fluidic system. In the Lagrangian approach, the Poincaré maps offer qualitative information about the mixing phenomenon, while the entropy of the mixing provides quantification of mixing performance (Phelan et al. 2008;Stone & Stone 2005). By establishing the accuracy of the present theoretical approach, we discuss some new physical insights obtained 6 from this study in the forthcoming sections. We first discuss the flow dynamics thoroughly and then discuss the mixing performance of the proposed micromixing assay. We believe that the findings of the present endeavor will provide an essential basis for the efficient design of the rotating microfluidic platform towards meeting the demand of desired fluidic functionalities, including enhanced mixing without compromising the net throughput.  . It assists the pressure-driven flow in the duct, which has a rectangular cross-section of height ' 2H ' and width ' 2W ' as shown in figure 1(a). The pressuredriven flow is maintained by the axial gradient of the gyrostatic pressure p , which is the net effect of applied axial pressure gradient and the centrifugal acceleration (Kaushik et al. 2017b(Kaushik et al. , 2017a. The axial flow developed from these primary actuation parameters is translationally symmetric along the duct, while the secondary flow with the vortical flow structure is symmetric about the 7 transverse centerline of the channel (Hart 1971;Johnston et al. 1972;Lezius & Johnston 1976).
In the rotating frame of reference, the vortical flow takes place due to the Coriolis force. If the channel is observed from the inlet, the Coriolis force sweeps the fluid towards the right wall of the channel, from where it turns back to the horizontal walls, and then to the left wall. This typical flow trajectory gives rise to a double-vortex configuration for higher rotational speed (Kheshgi & Scriven 1985;Speziale 1982), causing periodic helical motion of the fluid particles as shown in figure 1(a). The channel is considered to be sufficiently long, such that the flow structure at the intermediate cross-section ( yz plane -a region of our interest in this study) remains almost unaffected by the entry and exit effects of the channel. The flow in the channel is assumed to be free from convective inertia for a very small Reynolds number   Re 1 , typically considered in the microfluidic analysis (Ng & Qi 2015). Note that due to low   Re 1 , the underlying transport phenomenon falls in the regime of low Rossby number flow   Ro 1  as well. We may mention here that the Rossby number is the ratio of inertia force to the Coriolis force (Ng & Qi 2015). This order analysis is suggestive of the dominating effect of the Coriolis force on the underlying transport of the problem under present consideration.

PEL Electrostatics
As mentioned, the grafted polyelectrolyte layer following the electrostatic interaction with the ionic solution leads to the development of the electrical double layer. Since the electroosmotic force contributes to the flow actuation mechanism in the present configuration, we here focus on the description of the potential distribution both in the electrolyte layer and polyelectrolyte layer by appealing to the Poisson equation (Chanda et al. 2014 k is the universal Boltzmann constant and T is the absolute temperature of the fluid (Masliyah & Bhattacharjee 2006;R. J. Hunter 1981). Note that the validity leading to the applicability of the Boltzmann charge distribution in the context of the present analysis is detailed in the supporting information part of this paper (section S1). Now, by using this charge distribution given in equation (2.2) (Ng & Qi 2015). If the effect of gravity is ignored, the governing equations for both the electrolyte layer and polyelectrolyte layer in the rotating frame of reference can be written as (Ng & Qi 2015): Note that in equations (2.5) and (2.6), is the velocity vector in the rotating frame, is the deformation rate tensor,  is the viscosity,  is the density of electrolyte, p is the gyrostatic pressure field, and c  signifies Darcy's frictional drag parameter. The term c  u appearing in equation (2.5.b) represents the frictional drag force offered by the polyelectrolyte layer to the transport of electrolyte. It may be mentioned here that this frictional resistance depends on the grafting density and the thickness of the polyelectrolyte layer (Chanda et al. 2014;Das et al. 2015).
We further non-dimensionalize governing equations (2.5) and (2.6) using following reference and dimensionless parameters: For brevity in the presentation, henceforth, we drop the overbars from the dimensionless variables and use a symbol G for px    .

Euler-Lagrange form of governing equations and boundary conditions
We use the Ritz technique, which is consistent with the variational calculus method, to solve the transport equations of the present flow configuration [(2.4), (2.10)-(2.12)]. Following this method, we first compose the total functional using the Euler-Lagrange forms of these equations. For this task, we multiply these equations with their corresponding test functions, integrate over the control volume, and then add them together to get the total functional of the system (Reddy & Rasmussen 1982;Rektorys 1977). The minimization of this functional with respect to the Ritz coefficients (introduced in subsection 3.3) gives the approximate solution to the problem. The derivation of the functional over the total domain (cross-section of the channel), instead of solving differential equations [(2.4), (2.10)-(2.12)] for every sub-domain if any in the flow configuration, sets uniformity in the solution over the entire cross-section of the channel (Sadeghi 2018;Sadeghi et al. 2020). This advantageous feature prevails even with the consideration of the polyelectrolyte layer, which forms a separate sub-domain, in the underlying analysis [(2.10)-(2.11)] (Sadeghi 2018;Sadeghi et al. 2019Sadeghi et al. , 2020. In such a case, a single functional is derived for the entire cross-section, and the Darcy drag   2  u and the term 2 which are needed to be solved over the complete channel cross-section, can be written as: Below, we write the boundary conditions that accompany the set of governing equations The first and second boundary conditions in (3.4) depict the no-flux boundary condition at the walls and the center (i.e., at y and z-axis), respectively.
For the velocity distribution The first boundary condition (3.5.a) depicts the no-slip boundary condition at the walls of the channel for all velocity components u , v and w . The second boundary condition (3.5.b) is considered for the symmetric nature of the profiles of both u and v velocity components at y and 13 z -axis. The last boundary condition (3.5.c) stands for the anti-symmetric nature of w velocity at y and z -axis.  Gaikwad et al. 2020;Zaccone et al. 2009

Formulation of functional
In the above equations (3.7), the integrand of each integral represents the energy or work associated with the corresponding variable. The integrand of first integral [Eq. (3.7.a)] represents the electrical energy in the domain b  , whereas the integrands of equations (3.7.b)-(3.7.d) represent the kinetic energy associated with the velocities u , v and w (Reddy 1986;Reddy & Rasmussen 1982). In equation (3.7.e), the integrand represents the flow work done by the pressure to conserve the mass of the system (Reddy 1986;Reddy & Rasmussen 1982). Now, we perform the integrations in equations (3.7) and add them together to get the functional  as,

Extremization of functional: Ritz method
Now, we obtain the solution to the problem using the Ritz method in which the variables u , v , w , p and  are approximated by the corresponding basis functions. Note that the selection of suitable basis functions is integral as well as an essential part of this method. We stick to the following basics for selecting these functions in the present endeavour, i.e., the basis functions should satisfy the boundary conditions (3.4)-(3.5) as well as the symmetric or asymmetric nature of the variables (Ng & Qi 2015). In this study, the u and v velocity components, as well as the potential  has symmetric nature about y and z-axis. In contrast, the velocity w has an antisymmetric nature about y and z-axis. Also, the basis function chosen for pressure should maintain orthogonality with functions that have already been selected for v and w velocity components.
Accounting all these aspects, we consider the following test functions for the functional  as: After minimizing the functional (3.8), we get, In equation (3.9.e), the basis function of pressure fi p has an additional term 2  The distribution of electrostatic potential  , velocity field u , and pressure p in yz  plane is obtained by solving equations (3.11) with the test functions (3.9). For a concise presentation, we do not outline the intermediate steps here. Instead, the detailed solution procedure to solve equations (3.11) is given in the supporting information part of the paper (section S2).

The physical significance of model parameters
In rotational microflows, the thickness of the Ekman layer varies between  (Duffy et al. 1999 (Gaikwad et al. 2018;Kaushik et al. 2017a).
To estimate the range of the DH parameter ( or p  ), we resort to the electrolyte concentration and the thermodynamic state properties of the system. Under the standard laboratory conditions, the thickness of the electrical double layer varies in the range ~ 10-300 nm, if the concentration of the symmetric electrolyte (such as KCl, NaCl) is varied between 1-0.001mM (Masliyah & Bhattacharjee 2006). Considering this range, for the typical length scale of the microchannel (10-10 2 m  as mentioned before), the value of the dimensionless DH parameter may fall in the range 1 to 100. It is important to mention here that the aforementioned values of the basic and derived parameters are typical to the microfluidic setup and commonly used in the literature as well (Das et al. 2015;Gaikwad et al. 2018;Kaushik et al. 2017a).

Model benchmarking
Before going to discuss the results at hand, we make an effort to benchmark our theoretical model in this section. In doing so, we take advantage of full-scale numerical simulations as elaborated next. Besides, we also discuss the consistency of the velocity profiles of the present problem from different perspectives, as reported in the literature. In the present set up, the flow is driven by the combined influences of the gyrostatic pressure gradient and the electroosmotic force. In the asymptotic limit of geostrophic plug flow 20   Ro~0 , as depicted in figure 4(a), a plug-like axial velocity profile appears with a geostrophic core at the center, Ekman layer at the upper wall (i.e., at 1 z  ) and Stewartson layer at the left wall of the channel (i.e., at 1 y  ). As seen in figure 4(a), the magnitude of axial velocity   u increases from zero to its peak within both the Ekman layer and the Stewartson layer, while it assumes a nearly constant trend in the geostrophic core. To be precise, this typical flow behavior is in complete agreement with the well-established Taylor-Proudman theorem, i.e., Pedlosky 1987). Analogous to this theorem and upon satisfying the constraint of mass conservation principle  

Effect of PEL on rotational electrohydrodynamics
It may be mentioned here that, in rotational microfluidics, the Coriolis force adversely affects the deliverance of the prime objective of the micro total analysis systems   TAS  , i.e., augmented mixing without compromising the net throughput (Ng & Qi 2015). Although assisted by the electroosmotic effect, yet the rotational induced forces bring in a restriction on the underlying mixing performance as well as on the net throughput beyond certain rotational speeds.
Therefore, ensuring these objectives with equal priority demands a severe challenge in the design of state of the art Lab-on-CD based microfluidic devices. To this end, the grafted PEL at the walls of the microfluidic channel has shown tremendous potential, attributed primarily to their inherent physicochemical characteristics to control the underlying transport. Considering this feature of PEL, we discuss, in this section, the flow dynamics in a rotating microfluidic channel, as influenced by the grafted PEL at the walls. Note that for the subsequent discussion, we consider a We observe in figure 5 that the primary flow and the intensity of depression in the geostrophic core is comparatively higher for lower values of  (=1) and  (=12), and higher value of d (=0.3); although the rotational speed is remaining unaltered. This variation in the intensity of depression in the geostrophic core is attributed to the effect of Coriolis force   Re  u , which not only varies with the rotational speed Re  but also depends on the magnitude of flow velocity u (Kheshgi & Scriven 1985;Ng & Qi 2015;Speziale 1982). In the present study, the magnitude of flow velocity largely depends on the electroosmotic effect and Darcy's frictional drag in PEL, thus is governed by the magnitude of pertinent parameters viz., ,  , and d . Note that  = 1 signifies relatively lower drag force in the PEL region, whereas, for  =12 (figure 6(a)) and d = 0.3 (figure 6(b)), the electroosmotic effect becomes stronger. As shown in figure 6(a)-(c), for  =12 and d = 0.3, the stronger ionic interactions between PEL and electrolyte ions leads to an increasing magnitude of electrostatic potential    , which, in turn, enhances the electroosmotic effect. It is also observed in figure 6(a)-(c) that the thickness of EDL obtained for  =12 and d = 0.3 is higher than their respective counterparts, signifying a thicker influencing regime of the electroosmotic effect in the channel for the said values. A relatively higher magnitude of potential together with its influence up to a larger distance for d = 0.3, as seen in figure 6(b), is suggestive of achieving higher EDL thickness for the said value. Therefore, the magnitude of primary flow velocity is expected to be higher for lower  , lower  , and higher  figure 6(c), we can observe that despite having a higher magnitude of  (as observed in figure 6(a)), the magnitude of the electroosmotic force is smaller for    .
On the contrary, for    (see figure 6(c)), the electroosmotic force becomes higher, albeit the magnitude of  is seen to smaller (see figure 6(a)). In this variation, as per the definition of the electroosmotic force   2  , the higher counter-ion density of the electrolyte for    gives rise to the higher magnitude of the electroosmotic force. However, due to thinner EDL for    as shown in figure 6(a), the influence of electroosmotic force being acted in the domain is comparatively smaller than that for 12   (see figure 6(c)). In other words, the thicker influencing regime of the electroosmotic effect corresponding to the thicker EDLs for    plays a dominant role here. On account of this, a minute variation in the electroosmotic force is obtained in figure   6(c) for a change in  from 12 to 50. Consequently, as observed in figure 5

Formation of standard double-vortex configuration
For a comparatively smaller strength of the Coriolis force, the motion of the fluid as discussed above (in the context of figure 7(a)) remains undisturbed. In effect, the weak secondary flow corresponding to the smaller strength of the Coriolis force makes a parallel arrangement of the streamlines at the middle section (geostrophic core) and the horizontal walls (Ekman layer) of the channel (see figure 7(a)-(b1)). Such vortex geometry is commonly observed in the paradigm of rotational microfluidics and defined as the standard double-vortex since its shape resembles what has been observed in the literature for both low Rossby number flows as well as low rotational Reynolds number flows (for weak Coriolis force) (Kheshgi & Scriven 1985;Nandakumar et al. 1991;Ng & Qi 2015). Note that such a small magnitude of Coriolis force Re u   is attributable to the weak strength of the primary flow (as discussed previously). As a result, we observe in figure 7(b1) that the higher PEL drag      or higher DH parameter of EL      or thin PEL   0.05 d  , leads to the development of a standard double-vortex. Quite notably, combined effects of the weak primary flow strength (refer to figure 5(a)-(c) for primary flow) and the weak Coriolis force result in the standard double-vortex structure, as discussed above.

Formation of dumbbell-shaped vortex
For the stronger Coriolis force, the magnitude of secondary flow increases through the Ekman layer (Ng & Qi 2015). It may be mentioned here that the stronger Coriolis force arises due Therefore, the higher the strength of these forces, the higher will be the strength of dumbbellshaped vortices appearing in the secondary flow structure.   (refer to figure 6(c) and section 6.1 for a detailed discussion). This complex variation in the electroosmotic force and its effective influencing regime near the walls (in figure 6(c)) with a change in  gives rise to a minimal change in the primary flow, as observed in figure 5(b). It is because of this reason the difference in the velocity profiles as well as its consequences to establish a difference in the thickness of the Ekman layer for the chosen values of  becomes very small, as witnessed in figure 5(b). Notably, this minute variation in the Ekman layer thickness with  , as observed in figure 5(b), is further reflected by a small change in the vortex structure in figure   8(c)-(d). It may be mentioned here that the influence of the Coriolis force on the underlying transport gives rise to this correlative effect between the primary flow (in figure 5(b)) and the secondary flow (in figure 8(c)-(d)). Note that the relatively thicker influencing regime of the electroosmotic force for    assists in the formation of the dumbbell-shaped vortices with comparatively higher magnitude, as seen in figure 8(c).

Influence of PEL thickness
Next, in figure 9

Effect of PEL modulated electrohydrodynamics on the mixing: analysis, results, and discussion
In line with the prime objective of the present endeavour, we investigate in this section the mixing dynamics in the soft microchannel embedded in the rotating platform. We look at the underlying mixing from the perspective of both qualitative and quantitative assessments. To perform the qualitative analysis, we focus on the Poincaré maps at different axial locations of the channel. On the other hand, to quantify the mixing strength, we analyze the variation in the mixing index obtained by using the concept of entropy of mixing. Both these methods are based on the Lagrangian approach, in which the passive tracer particles of multispecies are traced in the 3D computational domain with the help of Eulerian velocity fields. We now systematically discuss the procedure and results obtained from these methods as follows.

Poincaré maps
To obtain the qualitative information of the mixing dynamics, we focus on the Poincaré maps obtained at different axial locations of the channel. The traces of the passive tracer particles in 3D computational domain produce the Poincaré maps. These traces can be calculated by using the following Lagrangian equations, which rely on the Eulerian velocity fields   ,, u v w obtained from the previous calculations (section 2 and 3).
It can be seen that the equations (7.1) are integrable with respect to time. In contrast, the Poincaré maps show the y and z locations of the particles in the cross-section of the channel at the desired axial (along x -coordinate for this analysis) location. This implies that the secondary flow velocities should be integrated with respect to the axial coordinate x . In doing so, we divide equations (7.1.b) and (7.1.c) by (7.1.a).
The magnitudes of the velocities u , v and w at the updated locations ( y and z ) are obtained by using the function-griddedInterpolant(cubic) in Matlab 2019a. Unless mentioned specifically, the step size in the axial direction, i.e., x  is taken as 10 -3 , and the total length of the channel, to be consistent with our previous assumption made in Section 4, is considered as 10 3 . In In figure 10(a), we observe that the tracer particles (Pa,1 and Pa,2), trapped in the dumbbellshaped vortices (depicted in figures 8 and 9), follow a periodic helical motion, primarily attributed to the periodic motion of the fluid particles therein. Also, the trajectory of a blue particle Pa,1(≡(0,0,0.7)) released from the upper half of the channel exactly forms a mirror image of the trajectory of red particle Pa,2(≡(0,0,-0.7)) released from the lower half. Note that the pathlines of particles Pa,1 and Pa,2 depicted in figure 10(a) show similar morphology with the corresponding streamline contours shown in figures 8 and 9. By depicting the traces of particles Pa,1 and Pa,2 in figure 10(a), we intend to convey that in the present flow configuration, the stream of the particles (represented by blue particle Pa,1) released from the upper half of the channel (about the line 0 z  and parallel to y -axis) will mix insignificantly with the particles (represented by red particle Pa,2) being released in the lower half. Considering this aspect, we inject the particles of the different species through the left half and right half (two symmetry halves about the line 0 y  and parallel to z -axis) of the channel, as shown in figure 10(b). The colors of the particles, red for the right half and blue for the left half (if observed from the inlet), distinguish the species considered for the calculation.  Having outlined the solution procedure at the beginning of this section, we now discuss the axial evolution of the Poincaré maps depicted in figure 11 for = 10, = 0.2, = 0.5, Re = 150, = 12, and = 5. We observe in figure 11 that, beyond the location 350 x  , the change in the Poincaré maps is insignificant except at the geostrophic core of the channel. This observation is suggestive of the efficient mixing of species, albeit qualitative, beyond 350 x  and except the core region. As shown in figure 12, the attributable factor behind this observed discrepancy at the core region is the delay in attaining the periodic behaviour of the fluid particles therein (precisely, the periodic motion is getting delayed). The location of the blue particle Pb,1(≡(0,0,0.5)) and the red particle Pb,2 (≡0,0,0.05) shown in figure 12, represents the particles being released from the non-geostrophic and geostrophic region of the channel, respectively. Each belongs to different species (red/blue). In the non-geostrophic region, the periodicity in the motion is quickly obtained almost after 50 x  , visualized from the trajectory of particles in the dumbbell-shape for 50 x  (see figures 11 and 12(blue particle Pb,1)). However, the trajectories of the particles at the core region (red particle Pb,2 in figure 12) attain well-defined periodicity at a sufficiently larger distance from the inlet ( 500 x  ), as observed in figures 11 and 12 (see for red particle Pb,2). The periodicity of this motion depends on the strength of secondary flow developed in the flow field. However, due to the very small magnitude of v and w velocities at the geostrophic core (as shown by the surface plots in figure 4(b) and 4(c)), the strength of secondary flow and its effect in developing the periodic fluid motion becomes minuscule in the interior. Consequently, the primary flow ( uvelocity) imposes the dominating translational motion instead of a periodic motion to the fluid particles at the geostrophic core, as observed in figure 12. This flow feature results in qualitatively poor mixing at the geostrophic core as compared to that at the non-geostrophic region. This 35 observation also implies that for the given set of governing parameters, the particles mostly passing through the core region encounter a relatively larger mixing length 3 . For example, to achieve approximately complete mixing, the particles (red particle Pb,2) being released from the core region are expected to travel a considerable distance in the channel. This aspect is visualized from the depicted variations in both the figures 11 and 12: in figure 11, the stream of particular species at the core region remains poorly mixed even up to axial distance 500 x  ; and in figure 12, the time taken by red particle Pb,2 to complete the particular number of circulations (say, two in this case) is higher than that of a blue particle Pb,1.
Influence of Physico-chemical parameters of PEL on mixing In the context of this discussion, it is important to mention here that in figure 13,  (refer to figure 9(b)-(c)). In contrast, the strength of the vortices for 0.3 d  is higher than that of 0.1 d  as seen in figure 9(b)-(c). On account of this competition between the strength and the size of the vortices, we obtain almost an equal mixing performance for both the values of d (=0.1, 0.3), as witnessed in figure 13. Also, it can be added here that the mixing performance obtained in a rigid channel can be higher than that of the channel having thicker PELs grafted on its wall. The vortex structure obtained for the case of minimal PEL thickness (say 0.05 d  ) has a very higher recirculation area (refer to figure 9(a)) as compared to that of the higher PEL thickness (cf. figure 9(c)). However, a smaller PEL thickness ( 0.05 d  ) has some adverse consequences on the underlying mixing, as discussed from the quantitative perspective in the next Section 7.2. Also, in section 7.2, we will highlight the effect of a higher frictional drag, mainly stemming from the grafted PEL, on the mixing performance since it has similar adverse consequences.
Also, we observe in figure 13 that the lobes of the dumbbell-shaped vortices trap the particles of both the species. This phenomenon adversely affects the mixing performance following the event discussed as follows. The particles from these lobes do not enter into the chaotic regime; instead, keep on moving following the undisturbed trajectories. Such trajectories are shown in figure 14(a)-(b). In figure 14(a)-(b), the plots demarcate the isolated motion of the particles Pc,1(≡(0,-0.5,0.75)) and Pc,2(≡(0,0.5,0.75)) located inside the dumbbell lobes, from a particle Pc,3(≡(0,0,0.5)) moving through the remaining region of the dumbbell-shaped vortex. It can be seen in figure 14 that the particles Pc,1 and Pc,2 from the dumbbell lobes never interact with a particle Pc,3 from the remaining dumbbell-shaped region. Throughout the length of the channel, the number density of particular species residing inside the dumbbell lobes (represented by particles Pc,1 and Pc,2) remains unaltered. As a result, the entropy of such discrete sections (lobes) remains constant, attributed primarily to non-intrusion of other species' particles into the lobes. This is a disadvantage of the rotational micromixers, even in the presence of the PEL modulated electrostatics. However, for 0.1 d  and 20   , the density of trapped particles inside the lobes is negligible due to the standard double-vortex configuration (refer to figure 13).
The insights gained from figure 13 will be of help for the design of the proposed rotating microfluidic devices, ensuring efficient mixing. To this end, the parameters modulating the PEL electrostatics   d  , and the underlying hydrodynamics    seem to have huge consequences on the underlying mixing within the limit of asymptotic geostrophic plug flow. Having discussed the qualitative analysis of the mixing, we now focus on the quantification of the mixing dynamics in the following section.

The entropy of mixing and mixing index
To quantify the mixing dynamics following the Lagrangian approach, we analyze the variation in the mixing index, obtained by using the concept of entropy of mixing. In this method, In line with our observation and subsequent discussion in section 7.1, here in figure 15 , as shown in figure   16(a) and 16(c) respectively. Figure 16 depicts  . As a result, for these relevant cases, the present soft narrow fluidic configuration will again imitate the similar characteristics of the rigid narrow fluidic channel, thus rendering an unacceptable outcome.
However, for lower   12   , the higher electroosmotic effect offers higher mixing performance as well as enhances the net axial throughput in the channel (see figure 16(b)). Thus, taking a note on this aspect, the judicial selection of the physicochemical structure of PEL seems to be imperative for the optimum mixing performance without negotiating the throughput in the proposed mixer substantially.

Summary
We theoretically investigate the flow dynamics and its consequences to the underlying chaotic mixing phenomenon in a soft rotating microchannel. We employ the variational calculus method for the flow field analysis, while we take the help of the Lagrangian approach to predict the mixing both qualitatively as well as quantitatively. The results presented in this study, which we believe as the first to be obtained by the use of the variational calculus method, are in excellent qualitative agreement with the experimental observations (Hart 1971). For qualitative analysis of the mixing, we perform the Poincaré maps, whereas, for quantification, we estimate the variation of mixing index by the use of entropy of mixing.
For a fixed rotational speed of the channel   Re  , the Coriolis force, which also leads to the inviscid geostrophic core formation, becomes the sole function of the flow velocity. As a consequence, by increasing the magnitude of primary flow through strong electroosmotic pumping and a weak frictional drag, we achieve a considerable intensity of the geostrophic depression in the interior of the channel. . Of these, the dumbbell-shaped vortices formed due to higher strength of Ekman pumping observed for strong electroosmotic effect in case of 12   , bring in more chaotic nature to the secondary flow, which in turn, establish an excellent mixing performance. Whereas, in contrast, by increasing the recirculation zone of the standard double-vortex configuration, the higher frictional drag for    provides a good quality of mixing. Notably, the large recirculation zone of standard double-vortex configuration, which also appears in case of very small PEL thickness 0.05 d  , plays an essential role to reduce the mixing length substantially. This inference underlines that the microchannel with built-in polymeric layers (supposedly provides higher drag in the PEL) or rigid microchannel (equivalence in very small PEL thickness) imbedded in CD platforms contradicts the prime objectives of achieving the maximum axial throughput and mixing performance simultaneously with equal priority. However, as shown in this analysis, this is not the case with variation in the DH parameter of EL.
The results of the present study also indicate that the tracer particles residing at the geostrophic core impede the process of mixing. The primary flow velocity at the geostrophic core, which substantially dominates its effect by imparting translational motion to this stream of particles, is the key cause for this retardation of the mixing process. More interestingly, it is also observed that the particles trapped inside the lobes of dumbbell-shaped vortices never interact with the particles of species co-injected at the inlet. For both the cases mentioned above, the information entropy at the geostrophic core and the lobes of the dumbbell-shaped vortices remain almost unaltered and minimum over the total length of the channel, thus augmenting the mixing length 45 unreasonably. This aspect remains a challenge for further studies in this research paradigm.
With concern to this study, it seems reasonable to conjecture that the influence of the convective inertia on the flow dynamics and its consequences on the underlying mixing phenomenon is expected to bring significant weightage to the studied problem. For a sufficiently high Reynolds number, the standard double-vortex configuration or dumbbell-shaped vortices bifurcate into a non-trivial four-vortex configuration, thus expected to affect the mixing performance. In such cases, the elevated roll-cell instability is another fundamental concern (Kheshgi & Scriven 1985;Speziale 1982) to be accounted for on the underlying analysis. In a similar line, the effect of change in conformational entropy on the rotational soft microfluidics, primarily attributable to intra-molecular interaction inside the polyelectrolyte layer or an interaction between the polyelectrolyte and foreign bio-fluidic or chemical molecule is not thoroughly understood. In addition to that, the experimental investigation, which would mimic the present study, is worth pursuing in the future endeavour essentially for potential improvements of the rotating microfluidic devices/systems, largely used in biomedical/biochemical applications.