Joule heating and buoyancy effects in electro-osmotic peristaltic transport of aqueous nanofluids through a microchannel with complex wave propagation

Highlights

• Electroosmotic flow of aqueous nanofluids through a microchannel is investigated.

• Joule heating and buoyancy effects on transport characteristics are also computed.

• This model can be utilized in engineering the bio-mimetic pumping systems.

• Volumetric flow rate accelerates with reducing the electric double layer thickness.

• Pressure difference increases with adding the electric field in flow direction.

Abstract

Electro-osmotic peristaltic transport of aqueous nanofluids in a two-dimensional micro-channel is examined analytically. Such flows arise in bio-mimetic pumping systems at the very small scale of interest in physiological treatment e.g. occular drug delivery systems. Complex waveforms are imposed at the walls to mimic sophisticated peristaltic wave propagation scenarios. Nano-particles are assumed to be in local thermal equilibrium. Joule electro-thermal heating is included. The dimensional conservation equations are linearized and transformed from the wave to the fixed (laboratory) frame under lubrication theory approximations. The emerging dimensionless model features a number of important thermo-physical, electrical and nanoscale parameter, namely thermal and solutal (basic density) Grashof numbers, nanoscale Brownian motion parameter, thermophoresis parameter, Helmholtz-Smoluchowski velocity (maximum electro-osmotic velocity), Debye electrokinetic length and Joule heating to surface heat flux ratio. Closed-form solutions are derived for the nano-particle volume fraction, temperature, axial velocity, averaged volumetric flow rate, pressure difference across one wavelength, skin friction (wall shear stress function), Nusselt number (wall heat transfer rate) and stream function distribution in the wave frame. The influence of selected parameters on these flow variables is studied with the aid of graphs. Bolus formation is also visualized and streamline distributions are observed to be strongly influenced and asymmetric in nature.

Introduction

Nanofluids constitute suspensions of nanometer-sized particles in base fluids. In the medical context, many different nanoparticles have been explored in clinical applications, ranging from single-walled (SWCNT) and multi-walled carbon nanotubes, to metal oxides (gold, silver, titanium, copper etc.) and fullerene. The bio-compatibility [1] of nanofluids has led to their significant deployment in diverse areas of biomedical technologies including enzymes [2], anti-bacterial wound treatment [3], solid lipid and dendrimer nanofluids in ophthalmic [4], [5], [6], hypothermia regulation [7], radiofrequency ablation in cancer care [8] and orthopedic lubrication [9]. The effectiveness of nano-particle doping of fluids was first demonstrated by Choi [10] wherein it was shown that thermal conductivity characteristics of base fluids (e.g. silicon oil, ethylene glycol etc) and other features may be enhanced with metallic nano-particles. Following experimental investigations, two major theoretical approaches to simulating volume-averaged properties of nanofluids have emerged. The Buongiorno MIT model [11] emphasizes the contribution of Brownian diffusion and thermophoresis for heat transfer enhancement applications. The Tiwari-Das formulation [12] features a nano-particle volume fraction and allows the simulation of different types of nano-particles. In the former [11] a separate mass (species) diffusion equation in addition to momentum and heat conservation equations is required whereas in the latter [12] only momentum and energy conservations equations are considered. Both models have been deployed extensively in medical engineering for formulating boundary value problems and circumvent the need for conducting very costly numerical simulations of nano-particle interactions which require the use of direct numerical simulation, Lattice Boltzmann methods etc. Latiff et al. [13] used Maple symbolic software to study the transient nano-bio-polymeric flow from an extending/contracting sheet. Tan et al. [14] used both an immersed finite element method and Brownian adhesion dynamics algorithm to simulate the interaction of nano-particles with deforming red blood cells (RBCs) in drug delivery. Gentile et al. [15] studied analytically the longitudinal transport of nanoparticles in intra-vascular blood vessels using the Taylor-Aris dispersion model and Casson viscoplastic theory for blood. Bég et al. [16] used a Nakamura finite difference algorithm to study the bioconvection flow in nanofluids through deformable channels as a model of nanotechnological microbial fuel cells. Tan et al. [17] studied fluid-structure interaction aspects of nano-particle diffusion in vascular networks with a combined particulate and continuum model, also addressing particulate ligand–receptor binding kinetics. Bég et al. [18] presented a computational simulations to study multiple aluminum oxide nano-particle transport in cylindrical vessels with a single-phase model and three different two-phase models (volume of fluid (VOF), mixture and Eulerian), observing that two-phase models correlate more closely with experimental measurements. Tan et al. [19] applied a Lattice Boltzmann-immersed boundary method to simulate nano-particle dispersion in blood vessels, confirming that dispersion rate is strongly influenced by local disturbances in the flow due to RBC motion and deformation.

Peristalsis is a significant mechanism encountered in many complex biological transport processes. It utilizes deformability of the conveying vessel to generate contracting and expanding waves which propel contents very efficiently over large distances and through tortuous paths. An excellent appraisal of the fluid dynamics of peristalsis has been presented by Jaffrin and Shapiro [20] for two-dimensional Newtonian flows. Applications of peristalsis arise in arthropumps [21] (which combine peristaltic and pulsatile i.e. periodic flow), piezoelectric actuated micro-pumps [22], swallowing mechanisms [23], waste management pumping systems [24] and also in geophysical (coastal) processes [25]. Extensive analytical investigations of peristaltic pumping flows have been communicated over a number of decades. Wilson and Pattin [26] studied peristalsis in two-dimensional conduits, observing that a lateral bending wave propagating along the walls of the channel generates a mean flow. Gupta and Shehadri [27] considered peristaltic wave propagation in viscous incompressible flow in non-uniform conduits. Ishikawa et al. [28] studied numerically the microbial flora transport in peristaltic flow in gastrointestinal tract, noting that viscous effects and wave amplitude significantly influence bacterial population and also concentration distributions of oxygen and nutrient. These studies did not consider nanofluid pumping by peristalsis. Bég and Tripathi [29] probably presented the first mathematical study of peristaltic transport of nanofluids in two-dimensional channels. They explored in detail the influence of cross-diffusion (Soret and Dufour) effects and also Brownian motion and thermophoresis on pressure difference distributions and streamline evolution. More recently Akbar et al. [30] used Mathematica symbolic software and a Chebyschev spectral collocation method to study the effects of various nanoparticle geometries (bricks, cylinders and platelets) on magnetized peristaltic nanofluid dynamics with heat transfer in a vertical channel. They noted that temperature is enhanced significantly for brick-shaped nanoparticles.

Electroosmotic flow (EOF) refers to the electrically-driven transport of a fluid relative to the stationary charged surfaces which bound it, for example micro-channel walls. It has substantial benefits in microfluidic pumping and allows very effective regulation of micro-channel flow fields via electric fields since it does not require the customary moving components featured in conventional micro-pumps. Electro-osmotic pumping has been observed to produce a continuous pulse-free flow. These pumps are also much more amenable to fabrication at the microscale and are increasingly being deployed in biomedical systems including pharmacological delivery, plasma separation, electro-osmotically actuated bio-microfluidic systems etc. Essentially flow actuation is achieved by applying an electric field to an electrolyte in contact with a surface. The contact of the surface with the electrolyte results in a net charge density in the solution. The viscous drag causes the liquid to flow tangentially to the surface and produces a consistent net migration of ions. Many theoretical and computational studies electro-osmotic flow with and without heat transfer have been reported. Babaie et al. [31] studied numerically with a finite difference method the fully developed electroosmotic flow of power-law fluids via a slit microchannel with a constant wall heat flux boundary condition, noting that both zeta potential and non-Newtonian behavior strongly modify heat transfer rate and volumetric flow rate at low values of the dimensionless Debye-Hückel parameter. Hu et al. [32] presented both particle-based numerical and current-monitoring laboratory results for electro-osmotic flow through microchannels with 3D prismatic elements. Masilamani et al. [33] used Lattice Boltzmann and finite difference algorithms to analyse the non-Newtonian electro-osmotic flow in micro-channels, noting the significant modification in flow patterns with rheological effects. Sadeghi et al. [34] used a power-series analytical method to derive solutions for the fully developed electroosmotic slip flow in hydrophobic microducts of general cross section under the Debye–Hückel approximation, considering many different microgeometries (e.g. trapezoidal, double-trapezoidal, isosceles triangular, rhombic, elliptical, semi-elliptical, rectangular etc), They showed that flow rate increases in a linear fashion with slip length for thinner electric double layers (EDLs) Liao et al. [35] employed a finite element Galerkin algorithm to compute the mixed electro-osmotic/pressure-driven flows in triangle microchannels by solving the Poisson and Naiver–Stokes equations, highlighting that electrolytic solution mass flux is enhanced with positive pressure gradient and Debye length ratio. Marcos et al. [36] used a control volume integration method to simulate the steady state developing electro-osmotic flow in closed-end cylindrical micro-channels computing in detail the influence of electric field strength and channel geometry on pressure and velocity fields. These studies generally neglected the contribution of Joule heating (dissipation). This phenomenon relates to the heat generated from the electrical current arising from the flowing liquid with net free charges, which is common for the electro-osmotic flows at the micro/nanoscale. It is therefore a significant effect to consider in biomicrofluidics systems simulations since it may impact considerably on both pumping rates and wall heat transfer rates since it produces temperature gradients in cross-stream and axial directions which can alter the applied electric potential field and the flow field. Inclusion of Joule heating in mathematical EOF models therefore circumvents over-prediction of micro-pump characteristics. Bosse and Arce [37] investigated the influence of Joule dissipation on solute dispersion in a free convection electrophoretic cell for the batch mode of operation. Xuan and Li [38] investigated analytically the impact of Joule heating effects on the transport of heat, electricity, momentum and mass species in capillary-based electrophoresis, showing that the thermal end effect induces significant depletion in temperature close to capillary ends, and that in these zones higher electric field strengths are necessary to ensure current continuity. Jing et al. [39] studied theoretically the Joule heating and viscous dissipation effects in steady, laminar, hydrodynamically and thermally fully developed pressure-driven flow in a microchannel with surface charge-dependent slip. They found that Joule heating and viscous dissipation demonstrate a non-monotonic variation with the continually increasing zeta potential and that owing to deceleration in the flow there is an associated decrease in Nusselt number with zeta potential. Further studies have been communicated by Yavari et al. [40] who considered non-uniform Joule heating and variable thermophysical property effects for EO dynamics in microtubes noting that a reduction in electrical resistivity of the fluid by increasing temperature elevates the total energy generation due to the Joule heating and manifests in a decrease in Nusselt number. Several studies have also examined the combined peristaltic pumping of electro-osmotic flow and heat transfer in micro-channels with Joule heating effects. Recently Sutradhar et al. [41] presented perturbation solutions for electro-magnetic peristaltic transport of Casson blood in a permeable microvessel and observed that temperature is amplified with greater Joule heating. Nanofluid peristaltic pumping with Joule heating was analyzed by Hayat et al. [42] who also considered viscoelastic characteristics, wall slip and radiative heat transfer effects. Very recently Tripathi et al. [43] presented analytical solutions for peristaltic transport of electro-osmotic nanofluids in finite micro-channels with Joule heating effects. This model however did not consider Brownian motion and thermophoresis effects as reflected in the Buongiorno formulation [11] and was also restricted to axi-symmetric pumping (the same peristaltic waves imposed at both micro-channel walls). In the present article, a generalized nanofluid model is employed to study more comprehensively the electro-osmotic flow and heat transfer in two-dimensional micro-channels with Joule heating. Furthermore complex waveforms are imposed at the micro-channel walls to consider asymmetric peristaltic pumping. The computations are relevant to more realistic designs for ocular electro-osmotic pumps in drug delivery systems.