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

N. Ijaz; Ahmed Zeeshan; S.U. Rehman

doi:10.1051/meca/2017040

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Volume 19 / No 1 (2018)

Mechanics & Industry, 19 1 (2018) 108

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Issue		Mechanics & Industry Volume 19, Number 1, 2018


Article Number		108
Number of page(s)		8
DOI		https://doi.org/10.1051/meca/2017040
Published online		31 August 2018

Mechanics & Industry 19, 108 (2018)

Regular Article

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

N. Ijaz¹, Ahmed Zeeshan¹^* and S.U. Rehman²

¹ Department of Mathematics and Statistics, FBAS, International Islamic University, Islamabad, Pakistan
² Department of Computer Science, University of Engineering and Technology, Taxilla, Chakwal Sub Campus, Chakwal, Pakistan

^* e-mail: ahmad.zeeshan@iiu.edu.pk

Received: 1 June 2017
Accepted: 14 September 2017

Abstract

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.

Key words: Nano-bio-fluids / non-spherical nanoparticles / electro-osmosis / mixed convection / curved channel

© AFM, EDP Sciences 2018

1 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 flat-plate 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-Nano-engineering model which involves peristalsis transport along with double-diffusive convection of nanofluids. Nourani et al. [15] presented the thermal behavior of paraffin-nano-Al₂O₃ 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 [21–25] 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.

2 Formulation of problem

2.1 Properties and shape of nanoparticles

The flow of non-spherical alumina (Al₂O₃) nanoparticles of platelet, blade, cylinder and brick shape are accounted here. It is established that all the thermal and physical quantities, i.e. density ρ_nf, k_nf is thermal conductivity of nanofluid, thermal diffusibility α_nf, specific heat Cp_nf, thermal expansion coefficient β_nf depends directly on concentration χ of nanoparticles and defined as $See Formula in PDF$ (1) $See Formula in PDF$ (2) $See Formula in PDF$ (3) $See Formula in PDF$ (4) 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 $k_{n f} k_{f} = 1 + C_{k}^{S h a p e} + C_{k}^{S u r f a c e},$ (5) here $C_{k}^{S h a p e}$ and $C_{k}^{S u r f a c e}$ are the parameters displaying the effects of thermal conductivity due to shape of particle and resistance due to surface respectively. The shape parameter $C_{k}^{S h a p e}$ 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 $C_{k}^{s h a p e} = (1 + \frac{m χ X}{1 - χ_{e} X}) k_{f},$ (6)X here is defined as, $X = \frac{1}{3} \sum_{i = a, b, c} \frac{k_{p i} - k_{f}}{k_{p i} - (3 / ψ - 1) k_{f}},$ (7)here ψ 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, $C_{k}^{S u r f a c e} = - f \cdot l_{k},$ (8)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 $l_{k} = R_{k} k_{f} .$ (9)

The effective viscosity of nanofluid as defined in [27–29], due to different shapes of the particles is $μ_{n f} = μ_{f} (1 + A_{1} χ + A_{2} χ^{2}),$ (10) here A₁andA₂ are viscosity enhancement coefficients.

Table 1 described shapes, sphericity ψ, shape factor f in 1/nm, Kapitza length l_k in nm and A₁andA₂. The models are verified experimentally by Timofeeva et al. [30].

Table 1

Contribution of particles shape effect and surface resistance to thermal conductivity of nanoparticle suspensions.

2.2 Electro-osmosis effects

The Poisson equation in the presence of electric Debye layer (EDL) for electric potential distribution can be written as $\nabla^{2} \tilde{Φ} = - \frac{\tilde{ρ_{e}}}{ϵ},$ (11) where ${\tilde{ρ}}_{e}$ is the density of the total ionic charge, ϵ is the permittivity, ρ_e = e(n⁺ − n⁻)z, n⁺ is the number of cations and n⁻ is the number of anions. The Boltzmann distribution in the absence of EDL overlap, can be defined as $n^{\pm} = n_{0} \exp [\pm \frac{\tilde{Φ} e z}{k_{B} T}],$ (12)where n₀ describes the concentration of ions at bulk, which does not depend upon surface electrochemistry, e, z, k_Band T are the electron charge, charge balance, Boltzmann constant and average temperature of the electrolytic solution.

Let us define the normalized electro-osmotic potential $\tilde{Φ}$ and zeta potential $\tilde{ζ}$ of the medium through a medium, i.e. $Φ^{'} = \tilde{Φ} / \tilde{ζ}$

After applying the approximation of Linearized Debye-Huckel, we have $\sinh \frac{\tilde{Φ} e z}{k_{B} T} \approx \frac{\tilde{Φ} e z}{k_{B} T} .$ (13)

Along with their imposed boundary conditions $\frac{\partial Φ^{'}}{\partial y}_{y = 0} = 0 and Φ^{'}_{y = h} = 1 .$ (14)

The potential can be obtained in the following form as $Φ^{'} = \frac{\cosh m y}{\cosh m h},$ (15) where m is electro-osmotic parameter which is defined as $m = \sqrt{\frac{2 n_{0}}{ϵ k_{B} T}} a e z = \frac{a}{λ_{d}}, λ_{d} \propto \frac{1}{m},$ (16)where λ_d is the characteristic thickness of EDL or Debye length and $\overline{E}$ is the electro-kinetic body force.

2.5 Geometry and flow problem

Alumina-water incompressible nanofluid is assumed to flow through a curved channel with semi width $\tilde{a}$ . The radius of curvature is taken as ${\tilde{R}}^{*}$ with center O as displayed in Figure 1. A peristaltic wave with long wavelength λ and small amplitude $\tilde{b}$ 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 $\overline{H} (\overline{X}, \overline{t}) = \tilde{a} + \tilde{b} \sin [\frac{2 π}{λ} (\overline{X} - c \overline{t})],$ (17)

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 [31] $\frac{\partial}{\partial \overline{R}} {(\overline{R} + {\tilde{R}}^{*}) \tilde{V}} + {\tilde{R}}^{*} \frac{\partial \tilde{U}}{\partial \overline{X}} = 0,$ (18) $ρ_{n f} (\frac{\partial \tilde{V}}{\partial \overline{t}} + \tilde{V} \frac{\partial \tilde{V}}{\partial \overline{R}} + \frac{{\tilde{R}}^{*} (\tilde{U})}{(\overline{R} + {\tilde{R}}^{*})} \frac{\partial \tilde{V}}{\partial \overline{X}} - \frac{{(\tilde{U})}^{2}}{(\overline{R} + {\tilde{R}}^{*})}) = - \frac{\partial \overline{p}}{\partial \overline{R}} μ_{n f} (\frac{1}{(\overline{R} + {\tilde{R}}^{*})} \frac{\partial}{\partial \overline{R}} ((\overline{R} + {\tilde{R}}^{*}) \frac{\partial \tilde{V}}{\partial \overline{R}}) + {(\frac{{\tilde{R}}^{*}}{\overline{R} + {\tilde{R}}^{*}})}^{2} \frac{\partial^{2} \tilde{V}}{\partial {\overline{X}}^{2}} - \frac{\tilde{V}}{{(\overline{R} + {\tilde{R}}^{*})}^{2}} - \frac{2 {\tilde{R}}^{*}}{{(\overline{R} + {\tilde{R}}^{*})}^{2}} \frac{\partial \tilde{U}}{\partial \overline{X}}),$ (19) $ρ_{n f} (\frac{\partial \overline{U}}{\partial \overline{t}} + \tilde{V} \frac{\partial U}{\partial \bar{R}} + \frac{{\tilde{R}}^{*} \tilde{U}}{(\overline{R} + {\tilde{R}}^{*})} \frac{\partial \tilde{U}}{\partial \overline{X}} - \frac{\tilde{U} \tilde{V}}{(\overline{R} + {\tilde{R}}^{*})}) = \frac{{\tilde{R}}^{*}}{(\overline{R} + {\tilde{R}}^{*})} \frac{\partial \overline{p}}{\partial \overline{X}} + {\overline{p}}_{e} \overline{E} + μ_{n f} (\frac{1}{(\overline{R} + {\tilde{R}}^{*})} \frac{\partial}{\partial \overline{R}} ((\overline{R} + {\tilde{R}}^{*}) \frac{\partial \tilde{U}}{\partial \overline{R}}) + {(\frac{{\tilde{R}}^{*}}{\overline{R} + {\tilde{R}}^{*}})}^{2} \frac{\partial^{2} \tilde{U}}{\partial {\overline{X}}^{2}} - \frac{\tilde{U}}{{(\overline{R} + {\tilde{R}}^{*})}^{2}} - \frac{2 {\tilde{R}}^{*}}{{(\overline{R} + {\tilde{R}}^{*})}^{2}} \frac{\partial \tilde{V}}{\partial \overline{X}}) + {(ρ β)}_{n f} g (T - T_{0}),$ (20) $(\frac{\partial \tilde{T}}{\partial \overline{t}} + \tilde{V} \frac{\partial \tilde{T}}{\partial \overline{R}} + \frac{{\tilde{R}}^{*} \tilde{U}}{(\overline{R} + {\tilde{R}}^{*})} \frac{\partial \tilde{T}}{\partial \overline{X}}) = α_{n f} (\frac{1}{(\overline{R} + {\tilde{R}}^{*})} \frac{\partial \tilde{T}}{\partial \overline{R}} + \frac{\partial^{2} \tilde{T}}{\partial {\overline{R}}^{2}} + {(\frac{{\tilde{R}}^{*}}{\overline{R} + {\tilde{R}}^{*}})}^{2} \frac{\partial^{2} \tilde{T}}{\partial {\overline{X}}^{2}}) - \frac{μ_{n f}}{{(ρ C_{p})}_{n f}} [2 {(\frac{\partial \tilde{V}}{\partial \overline{R}})}^{2} + (\frac{{\tilde{R}}^{*}}{\overline{R} + {\tilde{R}}^{*}} \frac{\partial \tilde{V}}{\partial \tilde{X}} - \frac{\tilde{U}}{\overline{R} + {\tilde{R}}^{*}}) (\frac{\partial \tilde{U}}{\partial \tilde{X}} - \frac{\tilde{U}}{\overline{R} + {\tilde{R}}^{*}} + \frac{{\tilde{R}}^{*}}{\overline{R} + {\tilde{R}}^{*}} \frac{\partial \tilde{V}}{\partial \tilde{X}}) + \frac{\partial \tilde{U}}{\partial \tilde{R}} (\frac{\partial \tilde{U}}{\partial \tilde{R}} - \frac{\tilde{U}}{\overline{R} + {\tilde{R}}^{*}} + \frac{{\tilde{R}}^{*}}{\overline{R} + {\tilde{R}}^{*}} \frac{\partial \tilde{V}}{\partial \tilde{X}}) + 2 {(\frac{{\tilde{R}}^{*}}{\overline{R} + {\tilde{R}}^{*}} \frac{\partial \tilde{U}}{\partial \tilde{X}} + \frac{\tilde{V}}{\overline{R} + {\tilde{R}}^{*}})}^{2}],$ (21) where $\bar{p} and \tilde{T}$ are fluid pressure and temperature profile, ${\overline{ρ}}_{e} \overline{E}$ is electro-osmosis term due to charged particles forming a layer on the wall of channel and (ρβ)_nfg(T − T₀) is due to mixed convection. The values for k_nf, α_nf, μ_nf and ρ_nf are obtained using equations (1)–(5) and equation (10). Last term in equation (21) represents viscous dissipation.

The problem in fixed $(\overline{R}, \overline{X})$ frame is shifted to wave $(\overline{r}, \overline{x})$ frame which is moving with the wave speed c. In wave frame velocity transformed to $(\tilde{v}, \tilde{u})$ , the following translational transformations are applied $\begin{array}{l} \overline{r} = \overline{R}, \\ \overline{x} = \overline{X} - c \overline{t}, \\ \tilde{v} = \tilde{V}, \\ \tilde{u} = \tilde{U} - c . \end{array}$ (22)

Then using the non-dimensional parameters, $\tilde{x} = \frac{\overline{x}}{λ}, \tilde{r} = \frac{\overline{r}}{\tilde{a}}, v = \frac{\tilde{v}}{c}, u = \frac{\tilde{u}}{c}, \tilde{p} = \frac{{\tilde{a}}^{2}}{λ μ c} p, δ = \frac{\tilde{a}}{λ}, \tilde{k} = \frac{{\tilde{R}}^{*}}{\tilde{a}}, t = \frac{c \overline{t}}{λ}, R e = \frac{ρ a c δ}{μ}, \tilde{θ} = \frac{\tilde{T} - \tilde{T_{1}}}{\tilde{T_{0}} - \tilde{T_{1}}}, \overline{E} = m^{2} U_{H S} φ, B r = \frac{μ_{f} c^{2}}{k_{f} (\tilde{T_{0}} - \tilde{T_{1}})}, G r = \frac{ρ_{f} g α a^{2}}{μ_{f} c} (T_{1} - T_{0}),$ (23) where Re, Gr specify the Reynolds number and Grashof number, δ, k define wave number and curvature coefficient. Using long wavelength and creeping flow assumptions equations (18)–(21) finally take the form $\frac{μ_{n f}}{μ {(\tilde{r} + \tilde{k})}^{2}} \frac{\partial}{\partial \tilde{r}} ({(\tilde{r} + \tilde{k})}^{2} \frac{\partial u}{\partial \tilde{r}}) + \frac{μ_{n f}}{μ {(\tilde{r} + \tilde{k})}^{2}} \frac{\partial}{\partial \tilde{r}} ({(\tilde{r} + \tilde{k})}^{2} \frac{u}{\tilde{r} + \tilde{k}}) = \frac{\tilde{k}}{\tilde{r} + \tilde{k}} \frac{\partial p}{\partial \tilde{x}} - G r \frac{{(ρ β)}_{n f}}{ρ_{f}} \tilde{θ} - m^{2} U_{H S} φ,$ (24) $\frac{1}{{(\tilde{r} + \tilde{k})}^{2}} ((\tilde{r} + \tilde{k}) \frac{\partial \tilde{θ}}{\partial \tilde{r}} + {(\tilde{r} + \tilde{k})}^{2} \frac{\partial^{2} \tilde{θ}}{\partial {\tilde{r}}^{2}}) + \frac{B r k_{f}}{k_{n f} {(1 - ϕ)}^{2.5}} (\frac{\partial^{2} u}{\partial {\tilde{r}}^{2}} + \frac{1}{{(\tilde{r} + \tilde{k})}^{2}} {(1 + u)}^{2} + \frac{2}{{(\tilde{r} + \tilde{k})}^{2}} \frac{\partial u}{\partial \tilde{r}} (1 + u)) = 0 .$ (25)

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 $\frac{\partial p}{\partial x} = E_{1} \frac{\partial^{2} η}{\partial t^{2} \partial x} + E_{2} \frac{\partial^{2} η}{\partial t \partial x} + E_{3} \frac{\partial^{5} η}{\partial x^{5}}$ (26) at z = h(x, t) = 1 + η(x, t) and η(x, t) = ϵSin2π(x − t), in which $E_{1} = \frac{\overline{m} a^{3} c}{λ^{3} μ}, E_{2} = \frac{D a^{3}}{λ^{2} μ}, E_{3} = \frac{B a^{3}}{c λ^{5} μ} .$

The problem formulation completes with non-dimensional boundary conditions for velocity and temperature profile, $u (- h) = - 1, u (h) = - 1, θ (- h) = 1, θ (h) = 0 .$ (27)

Fig. 1

Schematic diagram of the problem.

3 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. Figures 2–4 describe the variations in velocity, while Figures 5–7 exhibit the effects of parameters on temperature profiles.

Fig. 2

Variation in velocity profile for k = 5 (solid line),k = 10 (dotted), k = 100 (long dash). Purple colour indicates particle of blade shape, blue colour lines are for brick shaped particles, red lines are for platelets shaped particles and cyan colour shows behavior of cylindrical particles.

Fig. 3

Variation in velocity profile for m = 1 (solid line),m = 3 (dotted),m = 5 (long dash). Purple colour indicates particle of blade shape, blue colour lines are for brick shaped particles, red lines are for platelets shaped particles and cyan colour shows behavior of cylindrical particles.

Fig. 4

Variation in velocity profile for G_r = 0.1 (solid line), G_r = 0.3 (dotted),G_r = 0.5 (long dash). Purple colour indicates particle of blade shape, blue colour lines are for brick shaped particles, red lines are for platelets shaped particles and cyan colour shows behavior of cylindrical particles.

Fig. 5

Variation in temperature profile for k = 5 (solid line),k = 10 (dotted),k = 100 (long dash). Purple colour indicates particle of blade shape, blue colour lines are for brick shaped particles, red lines are for platelets shaped particles and cyan colour shows behavior of cylindrical particles.

Fig. 6

Variation in temperature profile for m = 1 (solid line), m = 3 (dotted), m = 5 (long dash). Purple colour indicates particle of blade shape, blue colour lines are for brick shaped particles, red lines are for platelets shaped particles and cyan colour shows behavior of cylindrical particles.

Fig. 7

Variation in temperature profile for G_r = 0.1 (solid line),G_r = 0.3 (dotted), G_r = 0.5 (long dash). Purple colour indicates particle of blade shape, blue colour lines are for brick shaped particles, red lines are for platelets shaped particles and cyan colour shows behavior of cylindrical particles.

3.1 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 particles. Red colour indicates the platelet particles, purple colour shows the blade, cyan colour for the cylinder and blue colour fixed for the brick. Also, taking є = 0.15, m = 1, U_HS = 1, E₁ = 0.1, E₂ = 0.4, E₃ = 0.3 are assumed as default values.

Now, the variation in velocity for different values of curvature (k = 5, 10 100) is shown in 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 the velocity has 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, electro-kinetic 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.

3.2 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 θ 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 θ 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.

4 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. electro-osmosis reduces the resistance and assists the flow.

With the increase in Grashoff number velocity and temperature increase.

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S. Nadeem, A. Riaz, R. Ellahi, N.S. Akbar, Effects of heat and mass transfer on peristaltic flow of a nanofluid between eccentric cylinders, Appl. Nanosci. (2013), doi:10.1007/s13204-013-0225-x [Google Scholar]
V.K. Narla, K.M. Prasad, J.V. Ramanamurthy, Peristaltic transport of Jeffrey nanofluid in curved channels, Proc. Eng. 127 (2015) 869–876 [CrossRef] [Google Scholar]
S. Nadeem, E.N. Maraj, The mathematical analysis for peristaltic flow of nanofluid in a curved channel with compliant walls, Appl. Nanosci. (2012), doi:10.1007/s13204-012-0165-x [Google Scholar]
E. Kaloudis, E. Papanicolaou, V. Belessiotis, Numerical simulations of a parabolic trough solar collector with nanofluid using a two-phase model, Renew. Energy 97 (2016) 218–229 [CrossRef] [Google Scholar]
G. Aaiza, I. Khan, S. Shafie, Energy transfer in mixed convection MHD flow of nanofluid containing different shapes of nanoparticles in a channel filled with saturated porous medium, Nanoscale Res. Lett. 10 (2015) 490 [CrossRef] [PubMed] [Google Scholar]
F.N. Sayed, V. Polshettiwar, Facile and sustainable synthesis of shaped iron oxide nanoparticles: effect of iron precursor salts on the shapes of iron oxides, Sci. Rep. 5 (2015) 9733 [CrossRef] [PubMed] [Google Scholar]
M.M. Bhatti, A. Zeeshan, R. Ellahi, Electromagnetohydrodynamic (EMHD) peristaltic flow of solid particles in a third-grade fluid with heat transfer, Mech. Ind. 18 (2017) 314 [CrossRef] [Google Scholar]
R. Ellahi, M. Hassan, A. Zeeshan, Shape effect of nanosize particles in Cu-H₂O nanofluid on entropy generation, Int. J. Heat Mass Trans. 81 (2015) 449 [CrossRef] [Google Scholar]
E.V. Timofeeva, J.L. Routbort, D. Singh, Particle shape effects on thermophysical properties of alumina nanofluids, J. Appl. Phys. 106 (2009) 014304 [CrossRef] [Google Scholar]
W. Yu, S.U.S. Choi, The role of interfacial layers in the enhanced thermal conductivity of nanofluids: a renovated Hamilton-Crosser model, J. Nanoparticle Res. 6 (2004) 355–361 [CrossRef] [Google Scholar]
R. Ellahi, M. Hassan, A. Zeeshan, Shape effects of spherical and nonspherical nanoparticles in mixed convection flow over a vertical stretching permeable sheet, Mech. Adv. Mater. Struct. (2016) 1232454, DOI:10.1080/15376494 [Google Scholar]
A. Zeeshan, M. Hassan, R. Ellahi, M. Nawaz, Shape effect of nanosize particles in unsteady mixed convection flow of nanofluid over disk with entropy generation, Proc. Inst. Mech. Eng. E: J. Process Mech. Eng. (2016) 0954408916646139 [Google Scholar]
E.V. Timofeeva, J.L. Routbort, D. Singh, Particle shape effects on thermophysical properties of alumina nanofluids, J. Appl. Phys. 106 (2009) 014304 [CrossRef] [Google Scholar]
S. Nadeem, I. Shahzadi, Mathematical analysis for peristaltic flow of two phase nanofluid in a curved channel, Commun. Theor. Phys. 64 (2015) 547 [CrossRef] [Google Scholar]
M.M. Bhatti, A. Zeeshan, R. Ellahi, N. Ijaz, Heat and mass transfer of two-phase flow with electric double layer effects induced due to peristaltic propulsion in the presence of transverse magnetic field, J. Mol. Liq. 230 (2017) 237–246 [CrossRef] [Google Scholar]
S. Nadeem, A. Riaz, R. Ellahi, Peristaltic flow of a Jeffrey fluid in a rectangular duct having compliant walls, Chem. Ind. Chem. Eng. Q 19 (2013) 399–409 [CrossRef] [Google Scholar]
P. Goswami, J. Chakraborty, A. Bandopadhyay, S. Chakraborty, Electrokinetically modulated peristaltic transport of power-law fluids, Microvasc. Res. 103 (2016) 41–54 [CrossRef] [Google Scholar]
B.C. Liechty, B.W. Webb, R.D. Maynes, Convective heat transfer characteristics of electro-osmotically generated flow in microtubes at high wall potential, Int. J. Heat Mass Trans. 48 (2005) 2360–2371 [CrossRef] [Google Scholar]
R. Ellahi, M. Hassan, A. Zeeshan, Aggregation effects on water base Al₂O₃-nanofluid over permeable wedge in mixed convection, J. Chem. Eng. 11 (2016) 197–186 [Google Scholar]

Cite this article as: N. Ijaz, A. Zeeshan, S.U. Rehman, Effect of electro-osmosis and mixed convection on nano-bio-fluid with non-spherical particles in a curved channel, Mechanics & Industry 19, 108 (2018)

All Tables

Table 1

Contribution of particles shape effect and surface resistance to thermal conductivity of nanoparticle suspensions.

In the text

All Figures

	Fig. 1 Schematic diagram of the problem.
In the text

	Fig. 2 Variation in velocity profile for k = 5 (solid line),k = 10 (dotted), k = 100 (long dash). Purple colour indicates particle of blade shape, blue colour lines are for brick shaped particles, red lines are for platelets shaped particles and cyan colour shows behavior of cylindrical particles.
In the text

	Fig. 3 Variation in velocity profile for m = 1 (solid line),m = 3 (dotted),m = 5 (long dash). Purple colour indicates particle of blade shape, blue colour lines are for brick shaped particles, red lines are for platelets shaped particles and cyan colour shows behavior of cylindrical particles.
In the text

	Fig. 4 Variation in velocity profile for G_r = 0.1 (solid line), G_r = 0.3 (dotted),G_r = 0.5 (long dash). Purple colour indicates particle of blade shape, blue colour lines are for brick shaped particles, red lines are for platelets shaped particles and cyan colour shows behavior of cylindrical particles.
In the text

	Fig. 5 Variation in temperature profile for k = 5 (solid line),k = 10 (dotted),k = 100 (long dash). Purple colour indicates particle of blade shape, blue colour lines are for brick shaped particles, red lines are for platelets shaped particles and cyan colour shows behavior of cylindrical particles.
In the text

	Fig. 6 Variation in temperature profile for m = 1 (solid line), m = 3 (dotted), m = 5 (long dash). Purple colour indicates particle of blade shape, blue colour lines are for brick shaped particles, red lines are for platelets shaped particles and cyan colour shows behavior of cylindrical particles.
In the text

	Fig. 7 Variation in temperature profile for G_r = 0.1 (solid line),G_r = 0.3 (dotted), G_r = 0.5 (long dash). Purple colour indicates particle of blade shape, blue colour lines are for brick shaped particles, red lines are for platelets shaped particles and cyan colour shows behavior of cylindrical particles.
In the text

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