Peristaltic motion of MHD nanofluid in an asymmetric microchannel with Joule heating, wall flexibility and different zeta potential
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Abstract
A mathematical scrutiny is introduced for the flow of magnetohydrodynamic nanofluid through an asymmetric microfluidic channel under an applied axial electric field. The impacts of wall flexibility, Joule heating and upper/lower wall zeta potentials are considered. Electric potential expressions can be modeled in terms of an ionic Nernst–Planck equation, Poisson–Boltzmann equation and Debye length approximation. Appropriate boundary conditions have been utilized to get the results of highly nonlinear coupled PDEs numerically. The impact of physical factors on the characteristics of flow, pumping, trapping and heat transfer has been pointed out. The outcomes may well assist in designing the organonachip like gadgets.
Keywords
Peristalsis Electroosmosis Magnetohydrodynamic (MHD) nanofluid TrappingNomenclature
 \(a_{1}\), \(a_{2}\)
Amplitude of upper and lower wall [L]
 \(B_{0}\)
Magnetic field [A/L]
 \(B_{r}\)
Brinkman number [–]
 c
Wave speed [L/T]
 C
Dimensional concentration field
 \(C _{0}\), \(C_{1}\)
Concentrations of fluid at upper and lower walls
 \(c_{f}\)
Specific heat [ML^{2}/T^{2}K]
 \(c_{p}\)
Heat capacity of fluid [ML^{2}/T^{2}K]
 \(d_{1}\), \(d_{2}\)
Constant heights [L]
 \(D_{T}\)
Coefficient of thermophoresis diffusion
 \(D_{B}\)
Coefficient of Brownian motion
 e
Electron charge [C]
 \(E_{0}\)
Electric field [M/T^{3}A]
 \(E_{c}\)
Eckert number [–]
 F
Flow rate [L^{3}/T]
 \(\tilde{h}_{1}\), \(\tilde{h}_{2}\)
Dimensional upper and lower walls [L]
 \(h_{1}\), \(h_{2}\)
Nondimensional upper and lower walls [–]
 \(H_{r}\)
Hartmann number [–]
 \(K_{B}\)
Boltzmann constant
 \(k_{f}\)
Thermal conductivity of fluid [ML/T^{3}K]
 m
Electroosmotic parameter
 \(n^{\pm}\)
Positive, negative ions
 \(n_{0}\)
Average number of \(n^{+}\) or \(n^{}\) ions
 \(N_{ b}\)
Brownian motion parameter
 \(N_{ t}\)
Thermophoresis parameter
 p̃
Pressure field [ML/T^{2}]
 p
Pressure field [–]
 \(P_{r}\)
Prandtl number [–]
 \(R_{e}\)
Reynolds number [–]
 \(S_{c}\)
Schmidt number [–]
 t̃
Dimensional time [T]
 t
Dimensionless time [–]
 T̃
Temperature field [K]
 \(T_{0}\), \(T_{1}\)
Temperatures of fluid at upper and lower walls [K]
 \(U_{\mathrm{HS}}\)
Helmholtz–Smoluchowski velocity
 ũ, ṽ
Dimensional velocity components in stationary frame [L/T]
 u, v
Nondimensional velocity components in wave frame [–]
 x̃, ỹ
Dimensional coordinates in stationary frame [L]
 x, y
Nondimensional coordinates in wave frame [–]
 \(z_{v}\)
Valence of ions
 α
Wave number [–]
 β
Mobility of the medium
 \(\gamma _{1}\)
Ratio of the effective heat capacity of the nanoparticle to the heat capacity of the fluid [–]
 \(\gamma _{3}\)
Joule heating parameter [–]
 ∈_{0}
Permittivity of free space
 ∈
Relative permittivity of the medium [–]
 \(\tilde{\zeta_{1}}\), \(\tilde{\zeta_{2}}\)
Dimensional upper and lower wall zeta potential
 \(\zeta_{1}\), \(\zeta_{2}\)
Nondimensional upper and lower wall zeta potential
 \(\varTheta ( y,t )\)
Temperature field [K]
 λ
Wave length of peristaltic wave [L]
 \(_{\lambda D}\)
Debye length
 \(\mu_{f}\)
Dynamic viscosity of fluid [M/LT]
 \(\rho_{f}\)
Density of fluid [M/L^{3}]
 \(\rho_{e}\)
Net ionic charge density [M/L^{3}]
 \((\rho c)_{p}\), \((\rho c)_{f}\)
Heat capacity of the nanoparticles and the base fluid [ML^{2}/T^{2}K]
 \(\sigma _{e}\)
Electrical conductivity [T^{3}I^{2}/L^{3}M]
 ϕ̃
Dimensional electric potential distribution [ML^{2}/T^{2}I]
 ϕ
Nondimensional electric potential distribution [–]
 φ
Phase difference [L]
 Ψ
Dimensional Stream function [L^{2}/T]
 Ω
Nondimensional concentration field [–]
MSC
76W05 76E25 35Q351 Introduction
Current past reveals that the extra consideration has been given to area of research entitled “microfluidics” so that advances in microfabrication technologies are accessible. In a microfluidic system, electroosmotic flow (EOF) is significant. Electroosmosis is basically an electrokinetic mechanism, in which we examine the ionic development of fluids affected by electric fields. Due to this process, a Stern layer (a charged surface with a high concentration of counter ions) is created. Resulting Stern layer with diffusing layer forms an Electric Double Layer (EDL). The potential (interfacial) between diffuse double layer and Stern layer is called zeta potential, a prominent aspect of many electrokinetic mechanisms. Electrokinetic transportation has become a vivid area of modern fluid mechanics. The combined impacts of electrokinetic and peristaltic phenomena are critical in controlling biological transport mechanisms. Electrokinetics contains electrophoresis, electroosmosis, diffusiophoresis and several other phenomena. Many microfluidic apparatus such as Lung chips, proteomic chips, labonachip (LOC), portable blood analyzers, microperistaltic pumps, organonachip, microelectromechanical systems (MEMS), microperistaltic pumps, DNA and bioMEMS, as well as microscopic full analysis systems are built upon the ideology of EDL and electroosmosis. Moreover, microfluidic apparatus is also associated with MEMS, automation and parallelization, costeffectiveness analysis, integration, miniaturization, separation, study of biological/chemical factors and high efficiency progression. Bandopadhyay et al. [1] examined the peristaltic modulation of electroosmotic flow in the microfluidic channel for the viscous fluid. Shit et al. [2] analyzed the rotation of EOF in a nonuniform microfluidic channel through slip velocity. Tripathi et al. [3] discussed the impacts of electroosmosis and peristalsis, for unsteady viscous flow. Ranjit et al. [4] worked on the electromagnetohydrodynamic flow via peristaltically induced microchannel along the effects of Joule heating and wall slip. Furthermore, Tripathi et al. [5] scrutinized the mathematical model on electroosmosis in peristaltic biorheological flow through an asymmetric microfluidic channel. Jhorar et al. [6] proposed the peristaltic modulation of electroosmosis in an asymmetric microfluidic channel for viscous fluid. Ranjit et al. [7] explained the effect of zeta potential and Joule heating through porous microvessel on peristaltic blood flow. In addition, Prakash et al. [8] investigated the EOF Williamson ionic nanoliquids in a tapered microfluidic channel under the effects of peristalsis and thermal radiation. Tripathi et al. [9] considered the electroosmosis of microvascular blood flow.
In the existing literature, traditional liquids such as water and natural oil fail to accomplish the current demands for improving thermal conductivities. Currently, nanofluid research is a major topic of research because it enhances the thermal conductivity of conventional liquids. Nanofluidic flow problem has numerous uses in biomedical engineering such as the delivery of a drug by using nanoparticles, heat exchanges and tumor cure. However, researchers have paid great attention to the all abovementioned phenomenon. Das et al. [10] found that the effect of electrical and thermal conductivities of the wall in a vertical channel for nanofluid flow. Hassan et al. [11] explored the properties of the wall in a porous channel, for the peristaltic flow of MHD nanofluid. Alghamdi et al. [12] demonstrated the smooth solutions for threedimensional HallMHD equations through regularity criteria. Ahmed et al. [13] discussed the generalized timeconvection of nonlocal nanofluids in a vertical channel. Pramuanjaroenkij et al. [14] studied the enhancement of heat transfer for the hybrid thermal conductivity model of nanofluid, numerically. Moreover, Arabpour et al. [15] analyzed the influence of slip boundary conditions on the flow of doublelayer microchannel nanofluid. Akbarzadeh et al. [16] investigated the first two laws of thermodynamics for nanofluid flow with porous inserts and corrugated walls in a heat exchanger tube. In addition, Mosayebidorcheh et al. [17, 18] explained the peristaltic flow of nanofluid and heat transfer through asymmetric straight and divergent wall channels. Rahman [19] assumed the expansion/contraction of MHD nanofluid through permeable walls. Prakash et al. [20] demonstrated the effect of thermal radiation on electroosmosis modulation and peristaltic transport of ionic nanoliquid in biological microfluidic channel.
Peristaltic flow is a flow by wave propagation along the flexible walls of channel. Peristalsis is an inbuilt feature of many biomedical and biological systems. Physiologically, it plays a crucial role in several situations, for example, function of ureter, mixing of food and chyme transport in the tract of gastrointestine, transportation of oocytes in the fallopian tube of females and the transmission of sperms in the male reproductive tract. Moreover, it is also useful in transport of cilia and bile duct, movement of lymph in lymphatic vessels and vascular movement of blood vessels, roller pump design (for pumping fluids without contact of pumping machinery) in peristaltic and acupressure pumps for cardiopulmonary and dialysis machinery. The updated version of hose pumps are operated by the peristaltic principle. Peristalsis is particularly advantageous for the transportation of slurries and chemicals that are corrosive in nature; therefore, preventing the rotation of pump drive and damage to moving parts. On a LOC device, it is generally essential to deliver a small amount of biological fluid by peristalsis (in a smaller level) than in a typical LOC system. Thus, contamination of the sample is prevented. Such uses have unlocked a new approach for doctors and mathematicians to maneuver their gadgets to scrutinize better results. Gala et al. [21] explained the regularity criterion for Boussinesq equations with respect to zero thermal conductivity. Also, Gala et al. [22] described the weak solutions for quasigeostrophic equations through uniqueness criteria in Orlicz–Morrey spaces. Nanofluid transport in an asymmetric peristaltic flow was incorporated by Noreen [23]. Latha et al. [24] used the impacts of heat dissipation on the Jeffery and Newtonian fluid of peristaltic flow in an asymmetric channel. Besides, Latha et al. [25] also worked on the asymmetric channel with partial slip conditions for peristaltic transport of couple stress fluid. Noreen [26] determined the magnetothermal hydrodynamic peristaltic transport for Eyring–Powell nanofluids through an asymmetric conduit. Furthermore, Abd Elmaboud et al. [27] developed the peristaltic transport for couple stress fluid through the rotating channel. Bhatti et al. [28] developed the peristaltic impulsion of solid (magnetic) particles in biological fluids, thermally. The electromagnetic transport for twolayer immiscible liquids incorporated in [29] by Elmaboud et al. Moreover, Saravana et al. [30] depicted the effect of heat transfer and flexible walls on the peristaltic flow of a Rabinowitsch fluid through an inclined channel.
The literature review showed that most of earlier studies dealt with electrokinetic or peristaltic pumping to drive fluid flow. The combined outcomes of peristalsis and electrokinetic phenomena can be critical for improving/controlling the mechanism of peristaltic transport. Inspired by the extensive uses of electroosmosis, peristalsis and nanofluids in current biomedical engineering/industry, some mathematical models of fully developed flows driven by the combined outcomes of electroosmotic and peristaltic pumping have been examined for the Newtonian fluid model and nanofluid. However, MHD nanofluids for electroosmotic peristaltic transport for Burgino model have not been taken into account. To fill this research gap, we present a new mathematical model to study the electroosmotic peristaltic pumping analysis of MHD nanofluid in an asymmetric microchannel. Joule heating, viscous dissipation effect and zeta potential of different values are likewise taken in this model. First, the relevant equations for EOF model along the axial electric field have been modeled and then solved for long wavelength and low Reynolds number. Afterwards, the resulting equations are solved numerically by utilizing the Mathematica software. Consequences of pertinent factors on the characteristics of flow, pumping, trapping, and heat transfer have been pointed out.
2 Formulation and solution
2.1 Distribution of potential
2.2 Analysis of flow
3 Solution methodology
4 Graphical analysis
4.1 Characteristics of flow
This subsection explains the detailed analysis of velocity distribution. Figures 2(a)–(e) have been plotted to observe the changes in velocity profile, across the microfluidic channel under the influence of Hartmann number \(H_{r}\), electroosmotic parameter m, the mobility of the medium β, zeta potentials \(\zeta_{1}\) and \(\zeta_{2}\). Figure 2(a) demonstrates that the velocity (axial) decreases in the central region of the channel as \(H_{r}\) increases, whilst the reverse trend is viewed near the channel walls because decrease in axial velocity subject to increase in magnetic field strength. Since magnetic field and axial velocity are perpendicular to each other, they produce a Lorentz force, which has a propensity to slow the movement of the fluid. Thus, the (axial) velocity has a rapid acceleration effect for \(H_{r}\) in the middle area of the channel and reduces close to the channel wall. Figure 2(b) shows an increased behavior for the subregion \(0.1\leq y\leq0.1\). Since m is the fraction of the conduit height to the \(\lambda_{D}\), it signifies that the increase of \(\lambda_{D}\) leads to a decrease in EDL, so that a large amount of fluid rapidly flows in the central region. Figure 2(c) portrays that the (axial) velocity increases as the mobility of the medium in the middle region of the conduit increases, while decreasing in the left region of the conduit, because β is directly dependent on the Helmholtz–Smoluchowski velocity \(U_{\mathrm{HS}}\). This physically can be interpreted as follows: the velocity of fluid reduces with increasing thickness of EDL and the flow of fluid reduces in the presence of EDL. Figure 2(d) shows the effect of (axial) velocity distribution with respect to \(\zeta_{1} \) on the upper microchannel wall. It is also observed that the change in \(\zeta_{1}\) significantly enhances the axial velocity distribution of the EOF. As the zeta potential behavior of the upper wall increases, the velocity has an increased effect on the lower wall, while the opposite behavior is observed at the upper wall of the channel. In Fig. 2(e), a parallel behavior of velocity distribution is observed due to the zeta potential \(\zeta_{2}\). For different values of \(\zeta_{1}\) and \(\zeta_{2}\), the intersection summit is not completely in the middle of the channel walls. Clearly, a higher value of the zeta potential produces a strong field of EDL and thus a decrease in the fluid velocity. This zeta potential phenomenon produces different rates of flow at different locations in the microchannel, thus changing the momentum flux. The presence of zeta potential in an EOF is a key phenomenon in controlling fluid flow in the microchannel. We examined that our results are consistent with previous studies without zeta potential [11].
4.2 Characteristics of pumping
4.3 Characteristics of trapping
4.4 Heat and concentration characteristics
5 Summary and conclusions

The (axial) velocity increases in the middle section of the channel, with a reduction in the vicinity due to electroosmotic parameter and mobility of the medium.

The magnitude of axial pressure gradient firstly decreases then increases with the increase of electroosmotic parameter, Hartmann number, mobility of the medium and different zeta potentials.

Since a higher potential is applied at the upper wall than at the lower wall, streamline circulates significantly close to that wall where potential is high.

The construction of trapped bolus depends strongly on the electroosmotic parameter, Hartmann number, mobility of the medium and high zeta potentials.

The heat transfer rate impacts the energy dissipation caused by the existence of Joule heating impact.
Notes
Acknowledgements
The corresponding author would like to thank Ton Duc Thang University, Ho Chi Minh City, Vietnam for the financial support.
Availability of data and materials
Not applicable.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Funding
This research received no external funding. The APC was given by Ton Duc Thang University, Ho Chi Minh City, Vietnam. However, no grant number is available from source.
Competing interests
The authors declare that they have no competing interests.
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