Effect of electro-osmosis and mixed convection on nano-bio-fluid with non-spherical particles in a curved channel

This paper resigns to study effects of electro-kinetic force due to presence of electrical charge layer on the walls of the channel. The nano-bio-fluid fills the void between two concentric curved plates. The flow is induced due to peristaltic wave on flexible walls. The effects of mixed convection along with heat transfer are accounted. Furthermore, the focus is on effects of shapes of non-spherical nanoparticles in nano-bio-fluid and its effects on the flow. Nanofluids are important in treatment of cancer and other diseases in tissues which are normally not reachable by normal drug procedures. The problem is modeled for four types of non-spherical nanoparticles of alumina in aqueous base fluid. Numerical solution is obtained using Mathematica. Some important results are displaced through graphs. Empirical observations display that a significantly greater velocity for nanofluid with blade shape particles is offered followed by brick shaped particles. Numerical experiment also deems a rise in heat transfer due to presence of blade shapes particles.


Introduction
Peristaltic flows are being profusely discussed in several industrial and physiological submissions. Some of these comprise of urine transport in ureters, swallowing of food, chyme movement in the GI tract, in roller and finger pumps, etc. In all above mentioned applications, biological fluids are transported by inducing a wave on the surface of channel or tube due to contraction and relaxation of muscles along the length of tube or channel, etc. known as peristaltic wave. Study of peristaltic transport along with nanofluids and heat transfer has enriched this area a great deal. Nanotechnology has huge impact in technology world due to its extraordinary physical and chemical features. Nanofluids uses a base fluid with low thermal efficiency like water or ethylene glycol with some metallic nano-sized particles which spread uniformly throughout the fluid and consequently increase the thermal conductivity of fluid. Its applicability in phenomena like targeted drug transport, microelectronics, chiller, nuclear reactor coolant, refrigerator, fuel cells, hybrid-powered engines, space technology, pharmaceutical processes grinding and so on, brought about much interest of scientist and engineers community.
Choi [1] was the first who introduced the term nanofluid. Its essential features are improvement of thermal conductivity in non-conducting or fluid with low thermal conductance. A comprehensive analysis of nanofluids was argued by Buongiorno [2]. Buongiorno argued fluids such as water (poor conductor of heat), can efficiently be used as good conductor when transformed to nanofluid, i.e. addition of metallic solids in these fluids helps to improve conductivity drastically. Many authors in recent years have taken in account the analysis of nanofluid in channel driven by peristalsis wave. Akbar et al. [3] investigate the flow of nanofluid in an asymmetric channel due to peristaltic pumping in the presence of induced magnetic field. Aly and Ebaid [4] find the exact solution for the nanofluids flowing through channel with asymmetric wave on channel wall also the slip effects are encounter for the velocity, temperature and concentration. Further, Ebaid and Aly [5] solved analytically and found exact solution for peristaltic flow in with a channel flexible wall and slip conditions. The application of the work is related to cancer treatment. The effects of hall current on flow of nanofluid through a porous channel are observed by Nowar [6]. A brief study on carbon nanofluid with heat transfer Park et al. and alumina nanoparticles, ethanol and isopropanol blend with diesel by Shaafi and Velraj [7,8]. Akram and Nadeem [9] investigate apt magnetic field on hyperbolic tangent nanofluid flowing in a channel with peristaltic transport. Akbar and Nadeem [10] show peristaltic flow of a Phan-Thien-Tanner nanofluid in a diverging tube. The flow was induced due to asymmetric peristaltic wave on the wall. Reddy and Reddy [11] study Joule heating effect due to transverse magnetic on peristaltic transport of a nanofluid in a channel with compliant walls. Akbar et al. [12] discussed interaction of nanoparticles on peristaltic flow of water base nanofluid in the presence of heat source and wall heat flux. Zamzamian et al. [13] investigated the effect of nanofluid on the efficiency of flatplate solar collectors. Copper oxide nanoparticle made by flame spray pyrolysis for photo electrochemical water splitting e part I. CuO nanoparticle preparation. Bég and Tripathi [14] use Mathematica to simulate bio-Nanoengineering model which involves peristalsis transport along with double-diffusive convection of nanofluids. Nourani et al. [15] presented the thermal behavior of paraffin-nano-Al 2 O 3 stabilized by sodium stearoyl lactylate as a stable phase change material with high thermal conductivity. They also discussed Soret and Dufour effects. Akbar [16] considered Cu-water nanofluid and saw the effects of endoscopy. Homayonifar et al. [17] devoted the Numerical simulation of Nano-carbon deposition in the thermal decomposition of methane. Nadeem et al. [18] show the flow of nanofluids in eccentric cylinders along with heat and mass transfer. Narla et al. [19] use Non-Newtonian Jeffery base fluid flowing in a curved channel. Nadeem and Maraj [20] simulate peristaltic flow of nanofluid in a curved channel with compliant walls.
The Literature mentioned shows the importance of nano bioscience in biomedical engineering, cancer treatment and targeted drugging, etc. One thing all the mentioned studies lack is the structure of nano-size particles used such as shapes and sizes. Some experimental and theoretical studies [21][22][23][24][25] warrant that the shape of particle tends to mark clear variation in the flow properties. Lack of such study indulges us to work and analyze this aspect of nanofluid. In this study nanoparticles of platelet, blade, cylinder and brick shapes are accounted. Mathematical equations for the flow situation and shape are modeled in a curved channel with compliant walls. The flow is induced by peristalsis wave with mixed convection and electro-osmosis effects. Using the non-dimensional parameters and assumptions of the long wavelength and low Reynolds number equations are simplified. The problem is solved numerically using 6th order RK based shooting method. The discussion of the physical phenomena is done using graphical observation.

Formulation of problem 2.1 Properties and shape of nanoparticles
The flow of non-spherical alumina (Al 2 O 3 ) nanoparticles of platelet, blade, cylinder and brick shape are accounted here. It is established that all the thermal and physical quantities, i.e. density r nf , k nf is thermal conductivity of nanofluid, thermal diffusibility a nf , specific heat Cp nf , thermal expansion coefficient b nf depends directly on concentration x of nanoparticles and defined as here the subscript f represents properties of base fluid, s defines the properties of solid nanoparticles, whereas, nf is for nanofluid.
To discuss the effects of shape of nanoparticles, the model for thermal conductivity [26], used is here C Shape k and C Surface k are the parameters displaying the effects of thermal conductivity due to shape of particle and resistance due to surface respectively. The shape parameter C Shape k is described by Hamilton-Crosser equation renovated by Yu and Choi [27]. The thermal conductivity model of non-spherical nano size particles are defined as [27,28] is X here is defined as, here c is the sphericity of the particles, k pi is defined as thermal conductivity along the axes of nano size particles. k pi along with volume ratio and depolarization factor can be defined as [28,29]. As the nanoparticles flows through the fluid the interaction generates interfacial resistance called Kapitza resistance R k , as interface hinders the heat flow and reduces the thermal conductivity. This gives rise to negative thermal conductivity coefficient due to surface resistance defined as, where f is the shape factor defined as normalized surface area per unit volume of the nanoparticles used and l k is Kapitza length, defined as The effective viscosity of nanofluid as defined in [27][28][29], due to different shapes of the particles is here A 1 andA 2 are viscosity enhancement coefficients. Table 1 described shapes, sphericity c, shape factor f in 1/nm, Kapitza length l k in nm and A 1 andA 2 . The models are verified experimentally by Timofeeva et al. [30].

Electro-osmosis effects
The Poisson equation in the presence of electric Debye layer (EDL) for electric potential distribution can be written as wherer e is the density of the total ionic charge, e is the permittivity, r e = e(n + À n À )z, n + is the number of cations and n À is the number of anions. The Boltzmann distribution in the absence of EDL overlap, can be defined as where n 0 describes the concentration of ions at bulk, which does not depend upon surface electrochemistry, e, z, k B and T are the electron charge, charge balance, Boltzmann constant and average temperature of the electrolytic solution.
Let us define the normalized electro-osmotic potential F and zeta potentialz of the medium through a medium, i.e. F 0 ¼F=z After applying the approximation of Linearized Debye-Huckel, we have Along with their imposed boundary conditions ∂F 0 ∂y y¼0 ¼ 0 and F 0 y¼h ¼ 1: The potential can be obtained in the following form as where m is electro-osmotic parameter which is defined as where l d is the characteristic thickness of EDL or Debye length and E is the electro-kinetic body force.

Geometry and flow problem
Alumina-water incompressible nanofluid is assumed to flow through a curved channel with semi widthã. The radius of curvature is taken asR Ã with center O as displayed in Figure 1. A peristaltic wave with long wavelength l and small amplitudeb is transmitted along the wall which induces the flow. It is assumed that the speed of wave is c and the wall is described using equation HðX; tÞ ¼ã þbsin 2p l ðX À ctÞ ; Two-dimensional velocity profile in axial and radial direction is assumed. The inertial effects are negligible. Additionally, upper and lower walls of the channel are kept at a constant temperature with lower wall at higher temperature.
The governing continuity, momentum and heat equations with effects of nanoparticles for curved channel are described as [ where p andT~are fluid pressure and temperature profile, r e E is electro-osmosis term due to charged particles forming a layer on the wall of channel and (rb) nf g(T À T 0 ) is due to mixed convection. The values for k nf , a nf , m nf and r nf are obtained using equations (1)-(5) and equation (10). Last term in equation (21) represents viscous dissipation. The problem in fixed ðR; XÞ frame is shifted to wave ðr; xÞ frame which is moving with the wave speed c. In wave frame velocity transformed to ðṽ;ũÞ, the following translational transformations are applied r ¼ R; x ¼ X À ct;ṽ ¼Ṽ ;ũ ¼Ũ À c: ð22Þ Then using the non-dimensional parameters, where Re, Gr specify the Reynolds number and Grashof number, d, k define wave number and curvature coefficient. Using long wavelength and creeping flow assumptions equations (18)- (21) finally take the form We assumed the flexible walls with effects of mass per unit area, viscous damping and flexural rigidity on the plate [32][33]. The pressure gradient takes the form The problem formulation completes with non-dimensional boundary conditions for velocity and temperature profile, uðÀhÞ ¼ À1; uðhÞ ¼ À1; uðÀhÞ ¼ 1; uðhÞ ¼ 0:

Graphical results and discussion
The problem is solved numerically using implicit RK with shooting scheme in Mathematica. The variation of sundry parameters on velocity and temperature profiles is observed using graphical utility.

Velocity analysis
The behaviour of velocity u against r for variation in the curvature k, electro-osmostic parameter or Debye length m and Grashoff Number Gr. Different colours are assigned to each shape of nanoparticle to clearly distinguish between    Figure 2. It is depicted that greater curvature causes in slow down of fluid in the earlier part of the flow, i.e. the peak velocity is achieved quicker for lower curvature values. It is also observed a significantly greater velocity for nanofluid with blade shape particles as it offers minimum resistance due to flater surface, followed by brick shaped particles. Whereas, cylinder and platelets have comparable velocities, with platelets the velocity is slightly higher. Effects of electro-osmosis parameter (m = 1, 3, 5) on velocity profile are shown in Figure 3 Influence of m (1 m 5) on velocity u (0 u 60) lies in greater range as compared with the previous profile of velocity u (0 u 30) against k (5 k 100), hence, flow velocity has marked an increase as m increases and the effect is consistent with all particles. It is due to the fact that value of Debye length helps by reduction of resistive force and flow is assisted by electric field generated, the results are consistent with experimental findings [34]. Also, electrokinetic effects are similar if the effects of Helmhotz-Smoluchowski velocity U HS are observed. The effects of mixed convection are observed in Figure 4 by changing Gr, between 0.1 and 0.5. It is noticed that the increase in Grashoff number increases the velocity of nanofluids.

Variations in temperature
To discuss the effects of important parameters on temperature, we have drawn Figures 5-7. Figure 5 is drawn to discuss the variation on u for k = 5, 10, 100. It shows that temperature increases with the increase the value of k. Also, the nanofluid with best thermal performance is with blade shape particles whereas, brick, cylinder and platelet follow, respectively. Figure 6 describes the effects of m = 1.0, 3.0, 5.0 on u profile. It depicts that temperature increases with the increase of the value of m. Figure 7 is drawn to discuss the effects of Grashoff number on temperature profile. The slight temperature shows the enhancement when Gr is increased.

Concluding remarks
This communication is a design to discuss the importance of shapes of nanoparticles in nanofluid when transporting inside a mechanical system. Four non-spherical shapes, i.e. cylinder, blade, brick and platelet shape of nanoparticles are accounted. Water-alumina nanofluid is assumed to be transported through a curved channel. The flow is induced due to peristaltic wave on the compliant wall. The effects of mixed convection and electro-osmosis due to charged particles on the wall are observed. Nanoparticle and flow   situation are modeled in form of partial differential equation with assumption of laminar and incompressible fluid along with, long wavelength of peristaltic wave on the wall. The following fallouts are concluded.
Due to low resistance of surface of blade shape particles the velocity of the fluid is maximum followed by brick, cylinder and platelet shaped particles. Again, best thermal performance is displayed by blade shape particles also, with brick, cylinder and platelet to follow respectively. These results are consistent with variation of all the displayed parameters.
The peak value of the velocity shifted to left when an increase in curvature is observed but the change is not drastic, whereas, efficiency in thermal performance is observed with greater curvature.
Velocity and temperature show a significant increase in magnitude with the increase of Debye length, i.e. electroosmosis reduces the resistance and assists the flow.
With the increase in Grashoff number velocity and temperature increase.