A theoretical analysis of rotating electromagnetohydrodynamic and electroosmotic transport of couple stress fluid through a microchannel

The main motivation behind this research work is the electroosmotic flow (EOF), which characterizes the movement of charged particles in a fluid when subjected to an external electric field, which offers precise control over fluid movement without mechanical components. The study of bio‐fluids in rotating microfluidic platforms is becoming much more significant. Hence, it is important to discuss the non‐Newtonian fluids in such environments. This paper analytically investigates the hydrodynamic characteristics of combined electromagnetohydrodynamic (EMHD) and EOF transport of non‐Newtonian fluid through a rotating microchannel with Navier‐slip boundary conditions at the walls. The couple stress fluid model is considered a non‐Newtonian fluid in the flow domain. The linearized Poisson‐Boltzmann equation is considered inside the electric double layer (EDL). The modified Stokes equation is governed by a combined imposed magnetic and EOF field. Closed form expressions are obtained for electrical potential, flow velocity, and volume flow rate in the channel using analytical methods. The dependence of rotating electromagnetic flow velocity and flow rates on appropriate parameters is explained. It is found that the rotational flow velocity displays an opposite pattern when the couple stress parameter and electric field intensity parameters are varied due to the influence of channel rotation. It is also found that the secondary flow rate ratio is amplified with the Hartmann number and slip length parameter, due to the stronger impact of the non‐Newtonian behaviour than the Newtonian situation on the volume flow rate at a relatively higher rotational value. The findings of this study may aid in the design and analysis of practical thermal micro/nano‐equipment for transporting biofluids.

Electromagnetohydrodynamics plays a vital role in biomedical and industrial engineering [11][12][13].The electromagnetohydrodynamic (EMHD) flow has received a lot of attention with regard to improving flow control.In this analysis, a working fluid is subjected to magnetic and electric fields.It is vital to mention that the Lorentz force (combination of electric and magnetic forces), which drives the EMHD mechanism, has garnered a lot of attention because of its benefits, which include transport of high flow rates, biochemical fields, fabrication process, and controlling the electrolyte flows [14][15][16][17].The combined impact of Lorentz and electroosmotic forces have been investigated recently.Numerous theoretical and practical research on EMHD flows has been conducted over the past decade, focusing on diverse magnetic-field effects in small-scale confinements due to an externally applied electric field [18][19][20][21].Gorthi et al. [22] analysed the combined effect of the magnetic and electric field through a microchannel by taking the immiscible conductive fluids.Their work mainly focused on the implication of magnetic strength and the electric field.They highlighted the effect of a transverse magnetic field on the interface of the two-layer system.However, in the present analysis, we included rotational effects as an additional/complementary source to drive the flow in the channel.Mondal et al. [23] presented theoretical and numerical solutions for heat transfer and entropy generation of EMHD flow of Newtonian fluid through a vertical microchannel with slip boundary conditions at the walls.They observed that the temperature gradient significantly minimizes the entropy generation rate in the microfluidic channel.The same work can be extended by considering the rotational effects without thermofluidic analysis.Recent research by An et al. [24] investigated a fractional viscoelastic fluid in a narrow channel under the presence of electric and magnetic fields under slippage at channel walls.They showcased an excellent relationship between flow velocity and physical parameters, which are not reported in the literature.However, they have not considered rotational effects in their analysis.Furthermore, several researchers in the past investigated the flow transport through narrow confinement by taking Navier slip boundary conditions [25][26][27].
In modern medical diagnostic devices, rotating microfluidic platforms are widespread.These devices are commonly employed to transport rheological fluids in an angular motion for biochemical testing [28][29][30].The centrifugal force and Coriolis force are two major forces that influence rotation.Especially, from the literature, it is perceived that the Coriolis forces are important in improving/generating secondary flow and, as a result, mixing in the flow process [31][32][33].Chang and Wang [34] examined the steady EOF of Newtonian fluid between two parallel pates for the first time.They presented the analytical solutions for axial and transverse rotating flow velocities.They showed the significance of spiral plots on the flow dynamics under the influence of rotational effects.An extension of same work was presented by Mondal and Wongwises [35] which explores the rotating electrokinetic flow of nanofluids in a narrow fluidic channel under the influence of Lorentz force.They noticed that the flow reversal occurs due to the higher strength of the Coriolis force.We considered slip effects as an additional source to analyse the flow behaviour in our analysis.Also, several researchers have devoted significant effort to examining rotational microfluidic transport in the last few decades because of its enormous practical significance and wide range of applications in the biochemical analysis [36][37][38][39].Most of the above studies have conducted the rotating EOF through narrow fluidic channels without considering the Navier slip boundary conditions at channel walls.The main purpose of this study is to analyse the combined EMHD and EOF through rotating a microchannel by taking two different slip lengths at both walls.
Because of an extremely non-linear relationship between shear stress and rate of strain and different rheological properties, non-Newtonian fluids are much more complicated and always more difficult to handle than Newtonian fluids.Due to their potential applications in a wide range of typical industrial and mechanical issues, non-Newtonian fluids are of interest to many researchers [40,41] than the flow of nanofluids through different geometrical shapes as such flows are important in flow dynamics for several reasons [42][43][44].Non-Newtonian fluid models have recently become highly relevant in the design and manufacture of small-scale electrokinetic devices.Stokes [45] pioneered the notion of polar effects in continuous theory and subsequently developed the couple stress fluid (CSF) model, a non-Newtonian model.The most extensively accepted model for describing couple stress in a liquid medium is the Stokes couple stress hypothesis.Two tensors are present in the Stokes model's constitutive equations.The first is the non-symmetric stress tensor, and the second is the couple stress tensor, which has a size-dependent effect.The CSF model has recently gained a lot of interest as a way to characterise the rheological behaviour of flow dynamics in various geometrical contexts [46][47][48][49][50].
In the current research study, our primary emphasis is directed towards a comprehensive examination of diverse aspects pertaining to the theoretical analysis of the electroosmotic and EMHD flow of CSF in a rotating microchannel.These aspects have not been thoroughly investigated in existing literature.The study of the EOF of non-Newtonian fluids, especially in transporting biological fluids, holds significant potential for improving diagnostic instruments, drug administration mechanisms, synthetic organs, and environmental pollutant detectors.The findings are verified by comparing the current results to those in the literature.Further, this research has provided information on the dependence of rotating EMHD flow velocity on the couple stress parameter and other active physical factors.The complex interplay between the effect of rotational forcing and electromagnetic force improves the performance of Lab-On-a-Chip (LOC) systems or thermofluidic devices.

MATHEMATICAL FORMULATION
We consider the combined electroosmotic and magnetic actuated flow of CSF through a rotating microfluidic channel.The physical dimensions of the rotating microchannel are represented in Figure 1.It is presumed that the length of the microchannel is much larger than the width ( ≫ ), and the width is much larger than the height ( ≫ 2).This assumption leads to the flow analysis being one-dimensional.The entire microchannel is rotated at a constant angular velocity  = (0, 0, Ω) anti-clockwise about  ′ -axis.Further, we have assumed an imposed electric field  = (  , 0, 0) and magnetic field  = (0, 0,  0 ) along axial and vertical directions, respectively.The combined force of electroosmotic and magnetic effects initially drives the flow along the axial direction.The presence of these combined forces in the channel, as well as the Coriolis force due to the microchannel's rotation, causes a secondary flow in the  ′ -direction.

Electrostatics in the rotating microfluidic channel
The potential distribution inside EDL can be described by Poisson-Boltzmann equation when the electric potential is much lesser than thermal potential (in this case Debye-Hückel approximation is applicable) under symmetric ions and is given by [14,16,17]: where,   is the net charge density,   is the absolute temperature,  is the elementary/free charge,  is the dielectric constant,  0 is the bulk concentration,   is the Boltzmann constant,  0 is the valence.
The suitable boundary conditions for the present investigation are considered as: Hence the dimensionless form of Equation ( 1) can be expressed in the following way: where  = ,  2 = 2 0  2  2 0 ∕     is the dimensionless Debye-Hückel parameter/electroosmotic parameter.Next, the dimensionless boundary conditions can be represented as: Therefore, the net charge distribution inside EDL can be shown in the following way by substituting the solution of Equation (2) in Equation (1):

The velocity distribution in the flow field
The flow field equations for the present investigation are given by [34,37,51], Continuity equation and Momentum equation where  is the fluid density,  is the velocity field,  is the body couple, (, ) are viscosity coefficients,  is the couple stress viscosity coefficient,  =  − 1 2 |Ω × | is the modified pressure which arises due to centrifugal force,  is the fluid pressure,  is the position vector, and  is the body force term which includes electrical body force and magnetic body force and can be expressed as  =   +  .Here,   = (    , 0, 0) and   =  ×  =(( +  × )) × ,  is the electrical conductivity of the fluid and  =  ( +  × ) is the current density vector.
In the present study, we propose the following additional assumptions • The flow is hydrodynamically fully developed, steady, laminar and incompressible.
• The aspect ratio of the channel is 2 ≪  ≪  which essentially explains the creeping flow transport and hence the entire convective term vanishes.• The velocity of the fluid is  =( ′ ( ′ ),  ′ ( ′ ), 0).• We neglect the body couple effect from this analysis.• In the microfluidic transport, the induced magnetic Reynolds number (R  ), which is the relative measure of magnetic advection to magnetic diffusion, is assumed to be negligible (R  ≪ 1).As a result, the induced magnetic field is assumed to be very small in comparison to the applied magnetic field in this research.• The applied electric and magnetic fields are assumed to drive the flow in the channel, and the Coriolis force causes the flow in the transverse direction due to the rotation effect and helps in mixing.In this study, we ignore the externally applied pressure gradient (modified pressure).
Based on the assumptions mentioned above, the following are the set of modified equations: that is The boundary conditions for the flow distribution in the rotating microfluidic channel are given by [52,53] ( where  ′ 1 and  ′ 2 -slip coefficients at the channel boundaries.Next, introducing the non-dimensional variables for the above flow field Equations ( 7)- (10) . Thus, the dimensionless form of Equations ( 7) and ( 8) can be written as: where  1 and  2 are the dimensionless slip length parameters at upper and lower walls respectively,  = is the electric field intensity parameter.
To obtain the velocity distribution in the channel, we define a complex function as follows: () = () +  () for combining Equations ( 11) and ( 12).The complex differential equation and boundary conditions using complex function () can be written as: and Therefore, the velocity of the flow through a rotating microfluidic channel can be obtained by solving Equation ( 13) and using boundary conditions mentioned in Equation ( 14).The analytical expression is given by: where  =  2 ,  = (2 +  2 ),  =   2 cosh() , and  = −  .Further, the expressions of constants  1 ,  2 ,  3 , and  4 are presented in the Appendix.
Next, integrating Equation ( 15), we obtain the dimensionless volume flow rate in the channel as follows: where   and   are the axial and transverse rotating electromagnetic volume flow rates in the channel, respectively.

MODEL VALIDATION
The validation of the current results is described in this section.It is essential to verify the accuracy of the present analytical rotating flow velocity.To verify our results, we first compare our analytical solution with Chang and Wang's [34] results without the magnetic field, then with Mondal and Wongwises's [35] results with magnetic field as shown in Figure 2. It is worth noting here that the fluid behaviour tends to the Newtonian case when the couple stress parameter  is large.For this comparison, we used  = 150 to describe Newtonian fluid behaviour.Without considering the magnetic effect, we compared our flow velocity with Chang and Wang's [34] results, especially when the rotational parameter  = 1 while other parameters considered as follows:  = 10, Ha =  =  1 =  2 = 0, and  = 150.The present results are then validated with Mondal and Wongwises [35] results by taking magnetic field, particularly at  = 1, as shown in Figure 2. Other parameters are fixed for this validation as:  = 10,  =  1 =  2 = 0,  = 1, and  = 150.It is clearly observed from Figure 2 that the present results exactly match with the literature results.Hence our analytical solution is feasible and accurate.

Effect of couple stress and transverse electric field parameters on rotating flow velocity in the narrow-fluidic channel
In the ensuing subsections, the hydrodynamic characteristics of the flow field in a rotating microfluidic channel will be discussed.The impact of  on  and − velocity profiles at low ( = 0.1) and relatively high ( = 1) values of  are elucidated in Figure 3a,c, respectively.Also, the effect of  on the  and − velocity profiles at low ( = 0.1) and relatively high ( = 1) values of  are shown in Figure 3b,d, respectively.It can be observed from Figure 3a that the − velocity profiles are enhancing consistently, at the same moment, the − velocity profiles are decreasing with respect to  at smaller value of ( = 0.1).It is important to describe here that the large value of  = √  2  tends to the Newtonian fluid case (couple stress fluid/non-Newtonian fluid → Newtonian fluid for higher value of  (say  = 150)).In this investigation, we consider smaller values for  = 5, 15, and 30 to describe the non-Newtonian behaviour in the flow domain.Further, the resistance of fluid movement or the effect of couple stresses in the fluid medium is primarily described by the couple stress coefficient .The increase in non-Newtonian parameter  represents the less inter molecular force between the mobile ions, which is responsible to augment the axial or − velocity due to combined electromagnetic force and Coriolis force, which occurs due to the channel's rotation.Next, from Figure 3c we observe that the axial/primary flow velocity first increase in EDL, that is, near the channel boundary/walls, and then opposite trend can be observed towards the centreline of the channel with increase in  at high value of  (such as  = 1).The reason behind this is the presence of electromagnetic body force inside EDL due to the combined applied electric and magnetic force, which acts as a driving force to transport the electrolyte solution and also, higher viscosity in the flow domain enhances flow resistance due to the fluid molecules and mobile ions which reduce axial flow velocity towards the centreline of the channel.This phenomenon may be useful in many applications of rotational microfluidics, especially in CD-based LOC platforms.Furthermore, it can be seen from Figure 3c that the transverse/secondary velocity decreases with increasing values of .This is because of the higher couple stresses effect, which always retards the flow movement.Moreover, the magnitude of the − velocity is more for higher value of  = 1 than smaller value of  = 0.1, because of the strong Coriolis force effect.
In this study, we consider  = 0 as a special case to describe the flow behaviour when there is no flow aiding force.The axial velocity is increased near the channel boundary and then subsequently decreases towards the core region of the channel when  = 0.1 and  = 1 (see Figure 3b,d).Furthermore, it is noticed that the magnitude of the  velocity higher when  = 0.This is attributed to the flow opposing magnetic force (the term  2  which is present in Equation ( 13)) and which dominates the flow driving magnetic force (the term   which is present in Equation ( 13)).Also, it can be noticed from Figure 3b,d that the traverse velocity or  velocity increases for increasing strength of electric field parameter.This may be due to the strong driving magnetic force effect in the channel.We further notice that for both  = 0.1 and  = 1 rotational speeds, the rotational flow velocity exhibits an opposite pattern when the couple stress parameter  and electric field parameter  are varied.

Centreline and boundary flow velocity distribution in the channel
The effect of  on centreline rotating velocity versus  and the impact of  on centreline rotating velocity versus  are depicted in Figure 4. First, we discuss the variation of centreline rotating velocity for different values of  when  = 0.1.It is seen that the axial centreline velocity along  direction decreases with  for small range of  versus  ∶ 0 − 1.5.This result may be useful in dissolving air/gas bubbles under the effect of Joule heating.Further, as  increases the impact of distribution of  on the axial flow velocity becomes insignificant.It is because of the higher resistive Lorentz force due to an applied magnetic field reduces the magnitude of the axial centreline velocity.Next, we explain the distribution of secondary centreline velocity with  for different values of  when  = 0.1.The magnitude of secondary centreline velocity (( = 0)) first increases with  up to  ≈ 1, and then decreases when  is increased further (see

Figure 4a
).It is identified from Figure 4a that for higher values of , the effect of  is negligible on both centreline velocities (( = 0) & ( = 0)).The physical reason behind this is that for a higher value of  the fluid rotates almost like a rigid-body along with the microchannel.Secondly, we discuss the distribution of centreline velocity versus  for different values of  when  = 5.It can be seen from the Figure 4b that the centreline velocity decreases consistently in the -direction as  increases.The reason behind this decreasing nature is that the energy transfer from axial momentum to transverse momentum because of the Coriolis force effect.However, when  = 0.1, the axial centreline velocity shows decreasing and increasing nature with respect to .As already seen from Figure 4a, for smaller values of  the magnitude of ( = 0) is almost similar in the case of  = 5 (see the transverse centreline velocity in Figure 4b) as well.Finally, the effect of  on rotating centreline velocity with the change in  1 is presented in Figure 4c.It is observed that the axial centreline velocity enhances gradually with increasing values of  1 .This is because of the decrease in flow resistance near the upper wall of the microchannel.Due to the combined effect of the applied electric and magnetic fields, electromagnetic forcing results in flow in the axial direction.In this analysis, the flow domain experiences a Coriolis force as a result of the microchannel rotating about the -axis.This Coriolis force alters the flow in the primary/axial direction and creates flow in the secondary direction.It is important to note that,  = 0 represents there is no rotational effect on the flow domain.We notice from Figure 4c that ( = 0) shows an insignificant nature (( = 0) = 0) when  = 0. Furthermore, it can be noticed from Figure 4c  the flow dynamics in a rotating microfluidic channel strongly depend on the Coriolis force, which generates the secondary flow along the transverse direction.

Effect of Hartmann number on volume flow rate ratio transport in the channel
The volumetric transport of the fluid over the channel is a crucial factor in rotating narrow fluidic pumping.To understand the deeper analysis of volume flow rate in the channel, we have defined the special parameter flow rate ratio (  ) in this investigation, which is represented as where   represents the relative increment of volume flow rate of CSF or non-Newtonian fluid ( = 30) to the Newtonian fluid ( = 150),    is the flow rate ratio in the axial or − direction and    is the flow rate ratio in the transverse or − direction.The impact of  on   with  1 for small ( = 0.1) and relatively high ( = 1) values of rotational parameter () is presented in Figure 5.We observe that the axial flow rate ratio decreases with increasing slip length  1 at the upper wall when  = 0.1 (see Figure 5a).This observation can be explained by the fact that the consistent increment in the volume flow rate for the Newtonian case (| =150 ) with respect to  1 .Furthermore, the value of    decreases with respect to , this observation is the result of the Lorentz force caused by the imposed magnetic field.In the present formulation, we consider the imposed magnetic field in the opposite direction of the flow, and the Lorentz force eventually resists the advancement of the flow rate in the channel.Figure 5b shows the transverse flow rate ratio (   ) evolution with  1 for different values of  when  = 0.1.It can be noticed that the value of    first decreases when  1 is increased from 0 to 0.04 and then increases for other values of  1 .This decrement of    is a result of strong resistiveness between the fluid particles due to the Lorentz force acting in the flow domain for a relatively higher values of  even though the value of  is small.Next, we describe the variation of axial and transverse flow rate ratio for different magnitudes of  when  = 1 in Figure 5c,d, respectively.It is important to mention here that the physical behaviour/nature of Figure 5c is almost same as Figure 5a except the magnitude.The effect of  (i.e., for a small  = 0.1 and relatively high  = 1 value of rotational parameter) on the axial flow rate ratio    is insignificant, this may be due to the weak flow rate transportation due to the Coriolis force in the flow domain.Furthermore, this phenomenon may be useful to enhance the mixing characteristics in the narrow-fluidic channels.At relatively high  = 1, the influence of  on secondary flow rate ratio versus  1 is depicted in Figure 5d.It can be seen from Figure 5d that the value of    first decreases for a small range of  1 that is,  1 ∈ (0, 0.01) and then subsequently increases for the remaining values of  1 .We also observe that the effect of  on − direction flow rate ratio is more significant, as    showing opposite behaviour for relatively higher value of  = 1 (see Figure 5d) than the small value of  = 0.1 (see Figure 5b).The reason behind this observation is that the combined electromagnetic and Coriolis force enhances the secondary flow rate ratio in the flow regime due to the applied electric and magnetic fields, also rotation of the microfluidic device as well.Finally, we would like to mention in this section is that for relatively high value of  the flow rate ratio along the secondary direction increases with respect to  for higher values of  1 , this is mainly because of the sharp increment in flow rate for the non-Newtonian case (| =30 ) than the Newtonian case (| =150 ).

Effect of rotational parameter on volume flow rate ratio transport in the channel
In this section, we discuss the distribution of   versus  1 for different values of  at small ( = 0.1) and higher ( = 2) values of Hartmann number.In Figure 6a, the effect of  on the − direction flow rate ratio (   ) with  1 when  = 0.1 is depicted.We observe from Figure 6a that the rotational parameter  has a significant effect on    .As  increases, the axial flow rate ratio increases constantly when  = 0.1, this is due the strong Coriolis force caused by rotation in the flow field than the viscous force.Note that for a small magnitude of Coriolis force is responsible for the weak strength of the primary/axial flow.Figure 6a indicates that the primary flow rate ratio decreases with increasing values of slip length  1 , the reason behind this observation is that the increase in flow resistance near the boundary of the microchannel.Furthermore, the value of    decreases with a steeper rate at higher slip length than that a smaller slip length values for a low magnetic field  = 0.1.The variation of secondary flow rate ratio with  1 at  = 0.1, 0.5, 1 when  = 0.1 is displayed in Figure 6b.Generally, in rotating environment the secondary flow velocity is generated or developed due to the Coriolis force effect.One may observe that the transverse flow rate ratio decreases with the increase of  as identified from Figure 6b.This is due to the adverse Coriolis force effect in the fluid flow domain and also the higher viscosity near channel walls increases the flow resistance because of the interaction of the fluid particles and mobile ions.This outcome may be useful for the design of Lab-on-a-Disc (LD) type devices.In addition, we notice that the dominancy of the enhancement in the flow rate for the Newtonian fluid (| =150 ) with  decreases the transverse flow rate ratio    .Likewise, we observe from the same figure that for relatively higher value of  ( = 1), the profile of    attains parabolic like profile because of strong Centrifugal effect.Next, we continue our discussion to understand the effect of  on    with  1 when  = 2 is depicted in Figure 6c.It is observed from Figure 6c that the axial flow rate ratio decreases with growing values of .It should be mentioned here that, based on the present formulation the dimensionless Lorentz force can be represented as   =   −  2 .For higher value of  ( > 1), the linear term ( ) is weaker than the quadratic term ( 2 ), resulting in a negative Lorentz force that reduces the axial flow rate ratio    .We also observe from Figure 6a,c that the magnitude of axial flow rate ratio for slip case ( 1 ≠ 0) is smaller than that for the no-slip case ( 1 = 0).Finally, In Figure 6d, it can be noticed that the pattern/nature of    for higher value of  (such as Ha = 2) is almost similar for lower value of  (see Figure 6b).This observation confirms that the Newtonian case (| =150 ) dominates the non-Newtonian case (| =30 ) in terms of decreasing the value of secondary flow rate ratio    , even when the magnetic field is increased ( = 2).

Angle flow transport in the channel
After discussion of volume flow rate and flow rate ratio, it is important to discuss the angle flow rate in a rotating microfluidic-channel. Before going to explain the details of the angle flow rate, we define the angle flow rate as  =  −1 (     ).The influence of electroosmotic parameter  on angle flow rate  with  when  = 0.1 and  = 5 are illustrated in Figure 7a,b, respectively.It is observed from Figure 7a that for smaller (< 1), the angle flow rate  is insignificant.Moreover, at higher values of  (1 ≤  ≤ 10), the magnitude of  decreases simultaneously with increasing values of .This may be due to increasing the density of mobile/free ions in EDL leads to decrease the angle flow rate in the channel.It is also can be seen from Figure 7a that, for small value of (= 0.1) flow aiding/driving term ( ) tends to quite weaker than the flow opposing term ( 2 ) as the value of Hartmann number is constant.Further, we found an interesting result at a higher magnitude of the electric field intensity parameter ( = 5) on angle flow rate, which shows the opposite behaviour when considering the small magnitude of ( = 0.1).A monotonic increment in magnitude of angle flow rate  with both  and  is noticed (see Figure 7b).This is expounded by the fact that the increase in  represents the decrease in Debye length and the resistance due to electrostatic force decreases.Therefore, the angle flow rate  enhances with .It is interesting to note that the angle flow rate value is higher for  = 5 compared to the low magnitude of the electric field intensity parameter ( = 0.1) (see Figures 7a,b).Hence, the strong effect or higher magnitude of electric field parameter enhances the angle flow rate in the flow domain.

CONCLUSIONS
The hydrodynamics of fully developed electromagnetic flow of CSF in a rotating narrow fluidic channel has been described.The primitive transport equations are developed initially and subsequently solved analytically to obtain electrical potential, rotating flow velocity (both axial and transverse velocities), and net volume flow rate in the flow domain using Navier-slip boundary conditions.The influence of the couple stress viscosity parameter and electric field intensity parameter on rotating flow velocity is of prime concern in the present analysis.It is seen that axial velocity increases with increasing numeric values of couple stress parameter within EDL due to the relative strength of electromagnetic force and rotational effect, also the magnitude of transverse velocity increases consistently for increasing couple stress parameter values.It is also pointed out that the driving magnetic force plays an important role to enhance the secondary flow velocity with a relatively high rotational effect.It is further observed that the effect of the Hartmann number is insignificant on both centreline velocities at higher rotating speeds because the fluid rotates almost like a rigid body along the centreline of the microchannel for higher values of the rotational parameter.Furthermore, we found that the magnitude of transverse flow rates typically increases for rotational parameter values more than the Hartmann number values.We also found that the magnitude of the axial flow rate ratio is much smaller for non-zero values of slip length than for the no-slip case.We further found that the electric field intensity parameter has a significant impact on angle flow rate in the flow domain.
Further, the combined effect of an applied electromagnetic force and Coriolis force (the strength of rotation) can be used to control the flow rates in the channel.Our findings suggest that a combination of applied electromagnetic force and Coriolis force (due to rotation strength) can effectively regulate flow rates within the channel.Furthermore, the results of this research may provide the valuable insights for researchers and engineers in developing innovative microfluidic technologies which offering a comprehensive understanding of the hydrodynamics in rotating microfluidic channels.We believe that the findings of this study will be of considerable interest because of their importance for the design of various cooling systems/devices and when dealing with bio-fluids such as blood, mucus, etc. Future studies may consider the steric effects and high zeta potential at the walls and will be communicated imminently.

A C K N O W L E D G E M E N T S
The authors thank the potential reviewers for their insightful comments, which helped me to improve the quality of the current work.

F
I G U R E 1 3D flow domain represents the rotating electroosmotic and EMHD flow of couple stress fluid in a narrow fluidic channel.
Figure 4a).It is identified from Figure4athat for higher values of , the effect of  is negligible on both centreline velocities (( = 0) & ( = 0)).The physical reason behind this is that for a higher value of  the fluid rotates almost like a rigid-body along with the microchannel.Secondly, we discuss the distribution of centreline velocity versus  for different values of  when  = 5.It can be seen from the Figure4bthat the centreline velocity decreases consistently in the -direction as  increases.The reason behind this decreasing nature is that the energy transfer from axial momentum to transverse momentum because of the Coriolis force effect.However, when  = 0.1, the axial centreline velocity shows decreasing and increasing nature with respect to .As already seen from Figure4a, for smaller values of  the magnitude of ( = 0) is almost similar in the case of  = 5 (see the transverse centreline velocity in Figure4b) as well.Finally, the effect of  on rotating centreline velocity with the change in  1 is presented in Figure4c.It is observed that the axial centreline velocity enhances gradually with increasing values of  1 .This is because of the decrease in flow resistance near the upper wall of the microchannel.Due to the combined effect of the applied electric and magnetic fields, electromagnetic forcing results in flow in the axial direction.In this analysis, the flow domain experiences a Coriolis force as a result of the microchannel rotating about the -axis.This Coriolis force alters the flow in the primary/axial direction and creates flow in the secondary direction.It is important to note that,  = 0 represents there is no rotational effect on the flow domain.We notice from Figure4cthat ( = 0) shows an insignificant nature (( = 0) = 0) when  = 0. Furthermore, it can be noticed from Figure4cthat for non-zero values of  the transverse centreline velocity decreases.It is worth adding that