Electro-osmotic optimized flow of Prandtl nanofluid in vertical wavy channel with nonlinear thermal radiation and slip effects

The simulations have been performed for the nonlinear radiative flow of Prandtl nanofluid following the peristaltic pumping in a wavy channel. The applications of entropy generation for the electrokinetic pumping phenomenon are also focused as a novelty. The complex wavy channel induced the flow of Prandtl nanofluid. Moreover, the formulated problem is solved by using the convective thermal and concentration boundary conditions. The Keller Box numerical procedure is adopted as a tool for the simulation task. The results are also verified by implementing the built-in numerical technique bvp4c. The comparison tasked against obtained numerical measurement has been done with already reported results with excellent manner. The physical characteristics based on the flow parameters for velocity, heat transfer phenomenon, concentration field, and entropy generation pattern is visualized graphically. It has been observed that the presence of thermal slip and concentration enhanced the heat transfer rate and concentration profile, respectively. The skin friction coefficient declines with electro-osmotic force and slip parameter. The increasing variation in Nusselt number is observed for electro-osmotic parameter for both linear and non-linear radiative phenomenon.


Introduction
Peristalsis is a topic of interest for researchers from past three decades because of its applications in physiological and industrial processes. The design of dialysis and heart-lungs machines involves peristaltic mechanism. Peristaltic pumps play an important role in agricultural processes like pushing of chemicals/water from tank/ river into the unfertile field. Important industrial applications of peristalsis include the roller and finger pumps to transport fluids. Transporting fluids through this process is helpful in avoiding pollution in industrial expenditures/wastages/consumption. Choudhari et al. 1 assumed the peristaltic motion of Bingham material due to the elastic tube with porous walls and distinct flow characteristics. Prasad et al. 2 identified the role of slip features for Casson fluid flow due to the peristaltic movement in inclined channel. The peristaltic investigation for non-Newtonian magnetized material in view of electroosmosis aspect was visualized by Tanveer et al. 3 Divya et al. 4 observed the hemodynamics applications regarding the peristalsis flow of Jeffrey fluid and additional under the consideration of distinct flow properties. Saunier and Yagoubi 5 presented the novel contribution on the peristaltic flow with the surface impact and compressive stresses. The Powell-Eyring nanofluid thermal observation based on the peristaltic phenomenon has been focused by Ahmed et al. 6 Khazayinejad et al. 7 reported a radiative analysis for the biological material subject to the peristaltic pumping and porous space. Parveen et al. 8 observed the thermophysical classification of nanofluid with peristaltic activity and slippage features. Abo-Elkhair et al. 9 focused on the peristaltic determination of hybrid nano-materials in view of uniform Reynolds number assumptions. Javid et al. 10 determined the cilia transport problem for the peristaltic mechanism and electrokinetic features.
The electroosmotic flow is the flow induced by the application of an electric field across the channel in the presence of electric double layer at channel walls. The electroosmotic flow is most significant in small channels because characteristic length of the channel is much greater than electric double layer (EDL) in small channels and in all heterogeneous fluids based system electric double layers exists. Electroosmotic flow plays a vital role in microfluidic devices such as, micro-reactors, biosensors and DNA analysis, etc. Ramesh and Prakash 11 explained the inspiration of electroosmosis phenomenon in microfluidic vessel with inspection of heat transfer. Sharma et al. 12 discussed the diffusive convective flow regarding the electroosmosis enrollment due to nanofluid flow. Lodhi and Ramesh 13 implemented the Darcy theory for a electroosmosis problem subject to the viscoelastic fluid. Noreen et al. 14 observed the electroosmotic activity against the bio-fluid flow via microchannel. The research communicated by Lv et al. 15 explored the electroosmosis significances for the viscoelastic material in a metallic cylinder. Noreen and Waheed 16 determined the electroosmotic evaluation for the microtube flow with porous activity. Ghosh et al. 17 indentified the insight significances of electrophoretic due to electrical layer. Zhou et al. 18 prescribed the physiological applications for the electroosmosis pattern along with the joule heating effects. The Prandtl nanofluid determination with electroosmosis assumptions was noticed by Abbasi et al. 19 Enhancing heat transfer in a channel is important for designing more compact heat exchangers, which are used in a variety of engineering applications including electronic device cooling, air-conditioning equipment, and ocean thermal energy conversion technologies, among others. For researchers and engineers, the advancement of these technologies is a major concern. However, in engineering and technology departments, poor thermal conductivity is the main impediment to heat transfer of these possible fluids. Choi 20 generated the idea of nanofluid and introduced us to its high thermal conductivity compared to other fluids. Saleem et al. 21 addressed the hybrid nanofluid thermal performances for the peristaltic flow subject to the curved tube. Narla and Tripathi 22 inspected the optimized thermal analysis regarding the peristaltic phenomenon in curved channel. Akram et al. 23 determined the heat transfer rate based on the utilization of hybrid nanofluid following the peristaltic pattern and electroosmotic approach. Akhtar et al. 24 determined the heat transfer investigation for the peristaltic transport of nanofluid in elliptic duct. Song et al. 25 investigated the ciliated motion of nanofluid with double diffusion convection with implementation of Hall features. Ge-JiLe et al. 26 observed the nanofluid thermal properties for the creeping flow with claim of biotechnology applications. Mahanthesh 27 numerically investigated the flow and heat transport of nanoliquid with aggregation kinematics of nanoparticles. In an other study Mahanthesh 28 highlighted the influence of quadratic thermal radiation on the heat transfer characteristics. Mahanthesh et al. 29 discussed the impacts of non-linear thermal radiation, Hall current and heat source on the dynamics of dusty nano fluid.
The creation of entropy is linked to the irreversibility of thermodynamic processes; as irreversibility grew, the system's useful work dropped and its efficiency fell. As a result, researchers aim to come up with a realistic way to reduce the rate of entropy formation in order to improve a system's performance. For the first time Bejan 30 discovered that the rate of entropy generation can be used to calculate irreversibility. A rich research on the optimized flow of various materials has been presented by research. [31][32][33][34][35][36][37][38][39] This research inspired the nonlinear radiative flow of Prandtl Eyring nanofluid presence of entropy generation phenomenon. The peristaltic motion with applications of electroosmotic pumping regarding the Prandtl Eyring nanofluid has been considered in the wavy channel. The mixed convection and chemical reaction consequences are also taken into consideration. Additionally, the problem is further extended by utilizing the electric field and joule heating effects. The whole analysis is performed with implementation of thermal and concentration slip constraints. The solution procedure has been followed by employing implicit finite difference method. After carefully observing the scientific research, it is claimed that such novel thermal investigation for the electroosmotic phenomenon and entropy generation features is not focused by researchers yet. The electroosmotic phenomenon presents applications in capillary electro-chromatography, medicine, microfluidic devices, capillary electrophoresis, microchips, geomechanics, petroleum engineering, etc. Moreover, the motivations for considerations the entropy generation effects are justified due to its significances in the energy storage devices, cooling systems, extrusion processes, solar collectors, exchangers turbo apparatus, combustion, heat, etc.

Mathematical modeling for electroosmotic pumping of Prandtl Eyring nanofluid
We consider the two-dimensional flow of Prandtl Eyring nanofluid in a symmetric vertical channel by due to peristaltic wave. The applications of electroosmotic phenomenon have been accounted. Moreover, the energy equation is modified in view of joule heating and thermal radiation. For chemical reactive fluid, the chemical reaction features are utilized in the concentration equation. The flow geometry is presented in Figure 1.
Furthermore, the left wall ÀH and right wall H may be defined as Where, a i (i = 1 À 3) are non-similar wave amplitudes, l is wave length, t is time, A is half width of the channel, j is axial co-ordinate, c k (k = 1 À 3) are parameters related to wave amplitude, and c is speed of complex peristaltic wave. For the mathematical formulation of the problem under consideration the basic conservassion laws of mass, momentum, energy, and concentration along with Nernst-Planck and Poisson equations are used which expressed as.

Continuity equation
The equation describes that rate at which the fluid enters the channel is equal to the rate at which liquid leaves the channel and fluid mass in dynamic system remain conserved.

Momentum equations
The momentum equations actually describing the Newton's second law of motion. The left hand side is describing the inertial forces while the terms on right hand side are describing pressure gradient forces, viscous forces, and body forces.
Energy equation By employing the first law of thermodynamics, the emergy blance equation is

Concentration equation
The concentration equation in presence of chemical reaction in differential form is 16 : In above equations, D tc is Dufour diffusivity, D ct is Soret diffusivity, r f is density of fluid, C f is heat capacity of nanofluid, K f is thermal conductivity, D s is Dufour diffusivity, d=dt is material time derivative, T is temperature having specific value T 0 at both left and right walls, q r is radiative thermal heat flux, C is concentration of mass, and k c is chemical reaction constant. B represents the body forces and t =À PI + S is stress tensor and for Prandtl fluid in which P = tr A 1 ð Þ 2 , A 1 = rV + (rV ) T , A and C 1 denotes the material parameters. The rheological equations of mathematical model under consideration are: In the presence of electro-static and mixed convection the equation of momentum for the Prandtl Eyring fluid is 16,19 : In above equations U , V are velocity components, h h are components of shear stress for Prandtl fluid. The expressions for energy and concentration equations is 19 : The right hand side of above equation is material derivative of heat transfer, the first term on right hand side is due to thermal conductivity, second term is joule heating, third term is due to non-linear Radiative heat flux fourth term is mass convention and last term is viscous dissipation. In view of Roseland approximation heat flux can be written as ∂ h , in which s Ã is the Stefan-Boltzmann constant and k Ã is mean absorption coefficient.
Electric field The electric potential function u can be calculated from Poisson equation with e 1 (dielectric constant) r e (electrolyte electric density) which are presented as: z, n À , and n + be charge balance, electric charge magnitude, negative ions and positive ions, respectively. Following Nernst-Planck expression: with chemical species diffusivity (D), average electrolyte solution temperature T , and Boltzmann constant K B :

Boundary conditions
The flow constraints for model are: with heat coefficient (K h ) and mass coefficients (K m ), thermal conductivity coefficients (h 1 ) and mass diffusivity coefficients (h 2 ).

Linear transformations and dimensionless formulation
Transforming equations from laboratory frame to moving frame we use following linear transformation and dimension less variables for dimension less formulation In which Re is Reynolds number, U 1 is Helmholtz-Smoluchowski velocity, a and b be materials constants, Sc be Schmidt number, N ct is Soret parameter, N tc is Dufour number, Gr is thermal Grashof number, Gc is the concentration Grashof number, Pr Prandtl constant, g 1 is chemical reaction parameter, Ec is Eckert number, b 1 is velocity slip parameter, S Joule heating constant, Bi t is thermal Biot number, and Bi m is concentration Biot number. After using the stream function c defined by u = ∂c=∂h and v =À ∂c=∂j continuity equation (1) satisfied while flow model retained as: Sc Incorporating the dimensionless expressions and following constraints of Re, Pe, d ( 1, Poisson equation (14) transformed as: in which k = eza ffiffiffiffiffiffiffiffiffiffi ffi is the electro-osmotic parameter. The dimensionless form of Nernst-Planck equation is: The solution of the equation (24) subject to the conditions n 6 = 1 at u = 0 and ∂n 6 ∂y = 0 at ∂u ∂y = 0 is obtained in the form Imposing the boundary condition u = 0 at y = h and ∂u ∂y = 0 at y = 0. For analytical simulations of equation (23) in view of n 6 , one get Boundary conditions are: Where h h ð Þ= 1 + a 1 sin a 1 j ð Þ+ a 2 sin a 2 j ð Þ+ a 3 sin a 3 j ð Þ: The reporesnetation of skin fraction, Nusselt number and Sherwood number is: Nu = ∂h ∂j ∂u ∂h Abbasi et al.

Special cases
Results for simple peristaltic flow can be obtained by a 2 = a 3 = 0:0: If U 1 = 0, the results are in absence of electric field. Results for no-slip flow case can be obtained by Results for viscous fluid case can be recovered by neglecting Prandtl Eyring parameters. That is a = b = 0:0:

Results and discussion
The physical interpretations are presented in this section.

Axial velocity
The physical attribution and mechanism of axial velocity u y ð Þ has been testified for electro-osmotic constant k ð Þ, rheological fluid constants (a, b), electro-osmotic velocity     u(h) due to U 1 are noticed. In core regime of surface, the velocity declined with electro-osmotic constant. However, the vicinity regime surface of boundary, the increasing aspect of velocity is noted. The appearance of electro-osmotic force is attributing the role of resistive force. Moreover, a reverse velocity change with U 1 is resulted for Q = 5: Figures 7 and 8 focused the influence of thermal Grashof constant Gr and mass concentration Grashof number Gc. The leading change in velocity rate in left channel surface is resulted while velocity reduces in all remaining zone with Grash of number. Here, the absence of buoyancy forces is denoted for . The trend of velocity change for enhancing mass concentration Grashof number Gc is shown contrary behavior as compared to Gr: and Bi m = 3:0: Figure 9 reports the nature of temperature profile for Biot number Bi t . The observations are visualized for linear and non-linear radiation impact. The improving change in Biot number Bi t result decrement in u(h). Owing to temperature jumps, the lower heat transfer is obtained. Figure 10 reports the dynamic of u h ð Þ for Dufour constant Nt c . The increasing attention in temperature due to Nt c has been observed. By definition, the Dufour constants explores the ratio between temperature and concertation differences. Figure 11 claimed the change in u(h) for Eckert number Ec: The declining temperature rate in left half and right half is observed, respectively. Physically, the increment claimed in Eckert number improves the internal kinetic energy due to dissipation factor and subsequently temperature raise. Figure 12 conveyed the contribution of radiation parameter Rn, the temperature profile decreases by strengthen the radiation process continuously and the fall in temperature occur due to the Radiative process the fall in internal kinetic energy of the fluid and as a result fall    in temperature is noticed. The observations for u(h) and Gr are defined in Figure 13. When Gr increases, lower heat transfer rate is noted. The association of electric force in flow regime present reduced temperature change due to Grashof number. For O = 1:2 that is for non-linear radiation the fall in temperature has maximum magnitude.

Concentration profile
The graphical predictions for Bi m , g 1 , Sc, and N ct on f(h) profile when other involved parameters are fixed with a 1 = 0:1, a 2 = 0:2, a 3 = 0:3, a 1 = 1:0, a 2 = 2:0, a 3 = 3:0, Q =À 1:0, a = 1:0, b=0:1, k= 0:5, U 1 = 2:0, Pr = 1:0, Sc = 0:7, N ct =0:5, N tc =0:5, Br=0:5, g = 1:0, Gr = 0:5, Gc = 0:5, b 1 = 0:1, Bi t =3:0, Rn=0:5, O= 1:4 and Bi m = 3:0: The concentration profile observations Joule heating constant is explored via Figure 14. The declining concentration change due to Bi m is noted for S = 6 2: The profile of concentration due chemical reaction parameter g 1 is noted in Figure 15. The lower concentration rate due to g 1 is resulted. The physical influence of Sc and N ct on concentration field in discussed in Figures 16 and 17, respectively. With change in Sc and N ct , concentration profile rises.       Figure  19 illustrates the association of Dufour number N tc on Nusselt and Sherwood number. The boosting profile of Nu and Sh due to N tc is observed. However, no significant change is noted due to joule heating constant.    show the impact of g on Sherwood number. The rising behavior of Sherwood number with larger g is founded.

Wall shear force
The impact of wall shear force defined via relations (24) is visualized for electro-osmotic velocity (U 1 ) in Table 1. For assisting flow case (U 1 =À 2:0), the wall shear force fluctuated when electro-osmotic constant gets vary. The deacceleration in wall shear force due to electro-osmotic force has been noted. Here, for U 1 = 0, the role of electro-osmotic constant dismiss which is due to electric double layer. The higher wall shear force magnitude for slip appearance and electro-osmotic is noted. The lower wall shear force is noted for assisting flow U 1 =À 2 ð Þ , opposing flow (U 1 = 2) and without electric force (U 1 = 0): For no-slip appearance, the wall shear force declined. The rise in thermal radiation declines the entropy generation number along the cross-section with large magnitude for O = 1:2 but opposite results are noted for O = 1:0. In Figure 22

Bejan number
The irreversibility in the present study is defined by Bejan number Be in the present study is plotted for various values of Radiation parameter Rn. The analysis is carried out for both linear radiation phenomenon O = 1:0 and non-linear radiation process O = 1:4: In Figure 23(a) the effects of Rn of Bejan number is plotted. The Bejan number Be declines for thermal radiation parameter Rn for linear Radiative flow while rises for non-linear Radiative flow. The rise in non-linear Radiative case is due to rise in temperature ratio which cases the irreversibility. The Bejan number for enhancing viscous dissipation is plotted in Figure 23

Verification of results
The verification and confirmation of simulation for obtained data is checked first by making comparison of present results with the work of and Ramesh and Prakash. 11 Figure 24 is plotted to show the accuracy of technique for viscous fluid and in absence of body forces. Moreover, Figure 25 is plotted to show the velocity profile for Prandtl Eyring nanofluid. The velocity profile shows a good agreement by Keller box scheme and bvp4c solver for flow parameters.

Concluding Remarks
The entropy generation flow of Prandtl nanofluid with applications of electro-osmotic is numerically investigated. The considered flow is induced due to wavy channel. The simplification in the governing model has been done by following the Debye-Huckel linearization approach. The finite difference numerical analysis is reported. The major outcomes are: The axial velocity change in core channel surface decline with electro-osmotic constant. The increasing change in velocity rate in left channel surface has been obtained for Grashof number.    The temperature rate enahcned with Dufour constant however, the impact of Eckret number is reverse. When chemical reaction parameter increases, the lower concetration rate is noted. The entropy generation number rises for the increasing values of Brinkman number. The magnitude of entropy generation is larger for nonlinear radiative phenomeon as compared to linear radiative framework. The Bejan number enhanced with joule heating parameter.
With enhancing Dufour number, the local Nusselt number also inreases.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: velocity slip parameter c k (k = 1 À 3) wave amplitude l wave length