Magnetic micro‐fluidics in 3D microchannel at the micro‐scale: Unlocking nano‐porous electrode potential for lithium‐ion micro‐batteries

Enhancing the nanosized‐electrolyte's characteristics in Lithium‐driven micro‐batteries (LIMBs) is indispensable to improve the overall efficiency, security, and lifespan of these energy devices, designing nano‐sized electrolyte with a wide electrochemical stability window while keeping them compatible with electrode materials is one of the improvement goals. Battery technologies must go through this optimization process in order to be used practically. A sensing mechanism to keep an eye on the health of Li‐ion energy devices through the magnetization. Magnetic micro‐fluidic patterns that change could be a sign of battery deterioration or other problems with performance. Li‐ion battery health is one application of magnetic sensing that you can do with magnetic sensing. Battery health variations and other performance problems can be found using magnetic mass transport patterns. Present study examines the effects of magnetic field on Eyring–Powel mass transport in nano‐porous channels over a stretching sheet. The principal equations exhibiting the phenomenon are transformed into non‐linear differential equation by second‐order approximation by using a similarity transformation. Furthermore, a semi‐analytic technique named optimal homotopy asymptotic method (OHAM) is used to solve the transformed Eyring–Powell model. The numerical results demonstrated the impact of variations in velocity, skin‐friction coefficient and Sherwood number for the proposed scheme.


| INTRODUCTION
Lithium-ion Microbatteries (LiMBs) designate groundbreaking frontiers in energy storage technology, indicating a new era in miniaturized power sources.These micro-sized and diminutive yet powerful energy devices have harvested immense attention for their potential to revolutionize various fields, from wearable electronics to biomedical implants, owing to their high energy density, long cycle life, and compact form factor.][3][4][5] Lithium-driven micro-batteries (LIMBs) and powerdevices are modern-age energy devices that can be used in micro devices like micro-robotics, micro-machines, IOT and bio-medical devices.In recent years, investigators have gradually been attentive on discovering the role that magnetism plays within LIMBs and power-devices, yielding numerous extraordinary and evocative developments.Magnetic fields display assorted helpful characteristics for LIMBs, Li-S, and Li-O 2 batteries.The core properties involve five magnetic field-prompted mechanisms: magnetic force, magnetization influence, magnetohydrodynamic effect, spin impact, and NMR effect.These things profit LIBs better the dissemination rate of Li ions, offer electrode surface augmentations, and more-jointly forward-moving energy density and steadiness.It is basically understood that beyond magnetization's innate qualities and magnetic power, the magnetic realm interacts with the electric realm in a contactless energy share process.Greater energy density results from operating the battery at a higher voltage, which is made possible by a wider electrochemical window.The blend of the magnetic and electric realms produces a Lorentz energy, which applies to charged fragments in a magnetic realm.This vitality prompts the fragments to quicken the electrochemical procedure and change their earliest heading of travel.A sensing mechanism to keep an eye on the health of Li-ion energy devices through the magnetization.Magnetic mass transport patterns that change could be a sign of battery deterioration or other problems with performance.Li-ion battery health is one application of magnetic sensing that you can do with magnetic sensing.To get componentexplicit flags, atomic attractive reverberation (NMR) utilizes a magnetic realm to energize proton groups.[8][9][10][11] In the field of fluid dynamics and heat transfer, the Eyring-Powell model offers a unified explanation regarding flow properties exhibited by non-Newtonian materials, hence differing in a big way from their corresponding Newtonian counterparts.The Eyring-Powell model explains the amount of force needed to break some bonds between two atoms in a solution-to move them around as well as their speed starting from the moment when bond rupture occurs.It differs markedly from most earlier theories for such phenomena because it introduces important nonlinear effects in rheological studies of non-Newtonian materials.
The Eyring-Powell model can be used to describe the flow or diffusion of species in a nano-porous medium in the context of mass transportation.Micro-fluidic investigations in porous stretching sheet (SS) for widening surfaces has been discussed practically in many engineering applications, particularly in contemporary batteries.Because of its various experimental practices in industrial applications particularly; paper sheet making, drawing of artificial layer, counterfeit filaments, metal ejection and turning.In the midst to make various types of sheets, the liquefy complications initially starts from an opening and a short period later connected with achieving the desired stability.At the same time, structural attributes of a conclusive thing completely depend upon the expanding and cooling rates.The investigation of magnetic mass transport can have a positive impact on the preparation and customization of electrolytes, leading to innovative electrolytes that enhance specific aspects of battery performance.By optimizing magnetic field parameters, scientists can control conditions that maximize the benefits of mass transport, which is crucial for the practical implementation of battery technologies.Additionally, magnetic sensing can be utilized to monitor the condition of Li-ion batteries, as changes in magnetic mass transport patterns may indicate battery degradation or other performance-related issues.Magnetic sensing can be employed to monitor the state of Li-ion batteries, permitting for the recognition of changes in battery health or performance issues through the analysis of magnetic mass transport patterns.Consequently, the progression of laminar flow (viscous flow) over SS has created many related issues as discussed in References 12-14, for each consolidating an imaginative impact but then giving an accurate arrangement.Studies are altogether limited to a progression of Newtonian fluids.Currently, it has been comprehensively predictable that in industrial practices and business, non-Newtonian fluids are supplementary fitting than Newtonian fluid.][17] Harish et al. 18,19 discussed mass and heat transport (HMT) of non-Newtonian fluid stretched on SS with attractive field impact.Hussain et al. 20 discovered a progression of Walter's-B liquid preceding an extending chamber.Hayat et al. researched the progression of third graduate liquid because of an exponentially stretching sheet in the ebb and flow of alluring field. 21Harish et al. broke down the impact of an attractive and warmth move Jeffrey fluid through stretching layer. 22Ashraf et al. worked on the Eyring-Powell fluid for boundary layer flow over micro-porous stretching sheet using OHAM. 23hmed et al. measured the progression of law of intensity liquid over a flimsy extending layer with an attractive impact.A non-Newtonian liquids well known model named as Eyring-Powell fluid on low and high shear rates describes mathematically the flow behavior of the fluids. 24Specifically, it tends to be used to express the advancements in some industrialized supplies like ethylene glycol and powdered graphite.Paanigrahi et al. deliberated the SS of a fluid model over non-straight extending layer with thermal diffusion. 24Hayat et al. 25 worked on 2-D flow of Eyring-Powell fluid and found out that Brownian motion and thermophoretic phenomenon enhances temperature.][28][29][30] The main focus of the present study is to examine the effects of magnetic field on Eyring-Powell fluid for mass transport over a micro-porous stretching sheet.The governing equations are transformed into non-linear differential equation by second-order approximation using similarity transformations.A powerful semi analytic technique named optimal homotopy asymptotic method (OHAM) is used for solving the transformed Eyring-Powell model.As OHAM solves the dimensionless semi analytic issues illuminated in References 31-34.It is one of the very ground-breaking semi analytic technique in unraveling different sorts of non-straight conditions for instance, homogeneous, non-homogeneous, fixed, and decoupled.

| MATHEMATICAL FORMULATION
Let us study 2-dimensional stream of an incompressible fluid of Eyring-Powell stretching layer flow in 3D microchannel of micro-porous electrodes that ensues together with the liquid stream in the half space  > 0 and plane  ¼ 0. Also, the mass exchange results are considered.The speed    ð Þ and the fixation C   ð Þ of the stretching out layer are identified with separation  from beginning stage 0, while Here, it can be noted that the expanding speed at stretching layer is where  > 0 is constant.A magnetic connected field of solidarity β 0 is experienced opposite to the stream heading.Furthermore, we supposed that the induced magnetic field is irrelevant brought about by magnetic analogue of the Reynolds number.Over the prior presumptions that the limit layer stream administered by underneath articulations 31,32 : The conditions for the related boundaries are: Using the given transformation, we get: Using the above equation which is same as the equation of continuity as: Simplify the equation of boundary as we get the required equation: The physical amounts of Sherwood number sh and coefficient of skin-friction C f which is well-characterized as: The Reynolds number can be defined as Re The equation of zeroth-order is: With initial boundary conditions: And the solution of zeroth-order is given as: The equation of first-order is: With initial boundary conditions: And the solution of first-order is given as: The equation of second-order is: With initial boundary conditions: And the solution of second-order is given as: The final solution of the equations is given as: Auxiliary constants C 1 and C 2 mentioned in References 32-34 can be computed for different values of γ and are presented in Table 1.
The equation of zeroth-order is: With initial boundary conditions: And the solution of zeroth-order is given as: The equation of first-order is: With initial boundary conditions: And the solution of first-order is given as: The equation of second-order is: With initial boundary conditions: And the solution of second-order is given as: The final solution of equations is given as: Auxiliary constants C 1 and C 2 mentioned in References 32-34 can be calculated for different Sc values and are given in Table 2.

| RESULTS AND DISCUSSION
In this section, the required model with boundary conditions of 35 has been broken down semi-scientifically by utilizing OHAM.The subtleties of the strategy is discussed in References 31,34,36-38.The impact of Hartmann number, Sherwood number and skin grating coefficient are investigated as seen in Figures 2-6.Graphical outcomes exhibiting the impact of appropriate velocity parameters, Sherwood number and coefficient of skin-friction are presented.
Figure 2 shows the result of f 0 ξ ð Þ against ξ for different values of γ = 0, γ = 0.5, γ = 1.0 and γ = 1.5, where β = 0.2,  = 1,  = 0.6 and  = 2.0.It is clear that f 0 ξ ð Þ is ascending with increasing for varying values of γ.In short, the chart shows that as ξ ξ changes, f 0 (ξ) f 0 (ξ) also changes for various values of γ, as it affects the properties of nano-scale electrolytes in LIMBs.This connection can be utilized to improve how efficient and effective they are when it comes to storing energy.A decreasing trend of f' (ξ) with increasing ξ suggests that the electrolyte shows better flow dynamics as ξ increases.More efficient ion transport is then possible which is very important concerning the usage of microbatteries based on lithium.The impact of k: Changing values of k and how they affect f' (ξ) show that one can control the performance of the electrolyte if they adjust their k.
The graph depicts that f'(xi) rises with k for different values of ξ suggesting that both ξ and k are very important in defining the behavior of nanoscale electrolytes in LIMBs.These observations could be further elaborated upon if one wishes so as to improve the battery properties for purposes such as increasing efficiency or lifespan in lithium-operated micro-batteries.
Figure 4   These trends that has emerged from looking into the values f(ξ 0 ) (ξ) is that, as γ increases, so does mine turn out to be suggesting that there could be more effective ion transport processes in the electrolyte with still higher values of γ.Compatibility and Stability: In order for there to be material electrodes that are compatible with their ion carrying atmosphere it is best that we have stable ionic liquids so as to maintain broad electrochemical potential windows.Figure 6 shows the result of Àϕ ξ ð Þ plotted beside ξ for varying values of  = 1.5,  = 1.7,  = 2.0 and  = 2.3 where β = 0.2,  = 1,  = 0.6 and ξ = 0.It is clearly shows in above figure that Àϕ ξ ð Þ is ascending with increasing for various values of .
The graph illustrates an upward trend as C f √Re x ascends with increasing γ for all k considered values.The relevance of γ and k to nanosized electrolytes in lithiumdriven micro-batteries' efficiency, flow dynamics, and overall performance has been shown by this connection.Altering these factors can enhance LIMBs' form as well as use.
A higher value of γ in C f √Re x suggest fluid flow efficacy in micro-batteries might be improved.Surface characteristics: Elevated surface actions and declining resistance due to increased γ and k related to C f √Re x .Although higher values of Cf-Rex increase c. Efficiency: C f √Re x rising with increasing γ is indicative of increased efficiency in the micro-batteries' fluid flow.The parameter C f √Re x is correlated with friction factor and Reynolds number, it suggests improved surface interaction and reduced resistance with increasing γ and k.Optimization of performance: Increased fluid pressure has a beneficial effect on the electrolyte's ion transport and diffusion rates.This may result in faster rates of charging and discharging, which would make the battery more appropriate for uses requiring rapid energy release.
The findings of the analysis of MHD Eyring-Powel mass transport over a micro-porous SS have been discoursed through different types of diagrams, that is, Figures 2-6.In this regard, a powerful semi-analytic technique OHAM is used to unraveling different sorts of non-straight conditions for instance, homogeneous, nonhomogeneous, fixed, and decoupled.The results reveal the impact of variations in the velocity, Sherwood number and skin-friction coefficient are presented.It is clearly shows in Figures 2-6 that the results of f 0 ξ ð Þ is ascending with increasing for various values of γ, f 0 ξ ð Þ is ascending with increasing for different values of , C f ffiffiffiffiffiffiffi Re x p is ascending with increasing for different values of , ϕ ξ ð Þ is ascending with increasing for varying values of  and Àϕ ξ ð Þ is ascending with increasing for changing values of , respectively.The use of the Eyring-Powell model stands out as a great instrument in the research involving heat transfer and general liquid dynamics that is not the viscous when referred to Non-Newtonian fluids since it is in this equation that people get to see how such liquids rheologically behave quite well which makes it possible for you to come up with industrial installation as well as application more efficiently.The basic reason for this is that through using the model, they are able to get an insight into how the behavior of these kinds of fluids is exhibited in real life situations which is an added advantage in coming up with more efficient processes.Also, when dealing with non-Newtonian fluids, engineers and researchers must understand and use Eyring-Powell model in order to increase the effectiveness and efficiency achieved in their works. 39,40

| EXPLORING THE PRACTICAL PURPOSES OF MICRO-FLUIDIC PARAMETERS IN LITHIUM-ION BASED BATTERY-SYSTEM VIA COMPUTER CODE
The investigations that is emblem of micro-machines and fluid mechanics is always fascinating because of new discoveries.Some parameters of fluids like fluid pressure, viscosity, and velocity all have a significant impact on the behavior and performance of electrolytes in lithium-ion batteries.It's arguable how changing these settings may affect battery performance and efficiency.During charge and discharge cycles, improved mass transport within the electrolyte facilitates the passage of ions between electrodes.This enhances battery performance, particularly with regard to the effectiveness of charging and discharging.However, decreasing fluid velocity can impede ion movement and diminish mass transfer, which can have an impact on the battery's overall efficiency.Controlled lower rates, however, could be preferable for some battery types or uses.Viscosity: Greater viscosity can impact the velocity of electrolyte flow by limiting it and raising internal resistance.A higher viscosity can have an impact on the pace of ion diffusion by limiting the flow of electrolyte and raising internal resistance.Battery performance may suffer as a result, particularly in terms of power density.On the other hand, reduced viscosity promotes more even ion mobility in the electrolyte, which can accelerate the charging and discharging of batteries.On the other hand, issues like diminished structural stability and electrolyte leakage may arise if the viscosity is too low.[43]

| CONCLUSIONS
In this study, a powerful semi-analytic procedure named as optimal homotopy asymptotic method (OHAM) has been incorporated to solve the transformed Eyring-Powell model.Resulting outcomes in Figures 2-6 can be outlined from the present study as fluid speed ascends with an expanding estimation of β though inverse impact is found for the varying situation of γ and K, Fluid speed is expanded for the non-Newtonian as contrasted and the comparing Newtonian fluid for the impact of , a skin rubbing coefficient abatement with an expansion in estimations of γ, and Sherwood number increments with expanding estimations of γ just as Sc.To summarize, the Eyring-Powell model for stretching sheets in lithium-ion batteries provides a computational and mathematical framework to investigate the response of a micro-porous stretching sheet to mass transport phenomena caused by a magnetic field.The model is based on the Eyring-Powell rheological characteristics.
Because of this, the battery is more suitable for arrangements that necessitate rapid energy proclamation.On the other hand, low fluid pressure can decrease mass transfer and impede ion mobility, which can have an impact on the battery as a whole.Increased fluid pressure has a beneficial effect on the electrolyte's ion transport and diffusion rates.This may result in faster rates of charging and discharging, which would make the battery more appropriate for uses requiring rapid energy release.Ion mobility and mass transfer may be slowed down by decreasing fluid pressure.Specific necessities and essential possessions of ionic-electrolytes for LIMB propositions have been widely deliberated.
The study of multiphase flow in 3D microchannel of magnetic crystalline electrolyte within micro-porous electrodes of Lithium-ion micro-batteries can benefit to prepare and tailor the properties of electrolytes by enhancing the magnetic properties of electrolytes, scientists may lead to ingenuity of electrolytes that improve specific aspects of battery performance.Scientists could explore optimizing magnetic field parameters (strength, direction, frequency) to control conditions that maximize the positive impact on mass transport.This optimization procedure is necessary for practical implementation of battery technologies.The practical application of battery technologies requires this optimization process.Miniaturized Li-ion battery health can be tracked using the magnetic effect as a sensing mechanism.4][45] The magnetic effect can be used as a sensing mechanism to monitor the condition of Li-ion batteries.Changes in magnetic mass transport patterns may indicate battery degradation or other performance-related issues.Electrolyte flow in 3D microchannel of magnetic crystalline electrolyte within micro-porous electrodes of Lithium-ion micro-batteries to use magnetic sensing is to monitor the health of your Li-ion battery.By optimizing the magnetic characteristics of electrolytes, researchers can enhance their preparation and customization capabilities.This would result in the development of innovative electrolytes that enhance particular aspects of battery performance.To maximize the beneficial effects on mass transportation, scientists could investigate optimizing the magnetic field's strength, direction, and frequency.Multiphase fluidics in microchannel of magnetic crystalline electrolyte within micro-porous electrodes of lithium-ion micro batteries is significant investigation theme that will benefit advancement in the field of transportable electronics and beyond by growing the effectiveness, constancy, and general presentation of micro battery technology.
Model for expression of multiphase flow in 3D microchannel of magnetic crystalline electrolyte within micro-porous electrodes.(B) Physical model for multiphase flow in 3D microchannel of micro-porous electrodes of lithium-ion micro-batteries.

Figure 5
shows the result of ϕ ξ ð Þ against ξ for various values of  = 1.5,  = 1.7,  = 2.0 and  = 2.3 where β = 0.2,  = 1,  = 0.6 and γ = 0.5.It is clearly shows in above figure that ϕ ξ ð Þ is ascending with increasing values of .Decreased fluid pressure has a link to Ion mobility and mass transfer may be slowed down by decreasing fluid pressure.

1
Auxiliary constants at different values of γ.
T A B L E 2 Auxiliary constant at different Sc values.