Research Article  Open Access
Numerical Research on Convective Heat Transfer and Resistance Characteristics of Turbulent Duct Flow Containing NanorodBased Nanofluids
Abstract
A coupled numerical model for nanorodbased suspension flow is constructed, and the convective heat transfer and resistance characteristics of the nanofluid duct flow are investigated. The numerical results are verified by experimental results and theoretical models. Most of nanorods are located randomly in the bulk fluid, while particles near the wall aligned with the flow direction. Friction factor of nanofluids with nanorods increases with higher particle volume concentration or aspect ratio, but the increment reduces when the Reynolds number gets larger. The relative Nusselt number is obtained to characterize the intensity of convective heat transfer. The results show that the Nusselt number of nanofluids increases when the particle volume concentration or aspect ratio becomes larger. Compared to increasing the aspect ratio of nanorods, increasing the particle volume concentration would be more effective on enhancing the convective heat transfer intensity in industrial applications although it will cause a slight increase of resistance.
1. Introduction
Nanofluids, which are fluids with nanosized particles suspended, are considered to be nextgeneration mediums for advanced heat transfer or cooling technology [1]. Since Choi put forward the concept firstly [2], the study of nanofluids has been intensively performed and reported in worldwide. Abnormal heat transfer characteristics were reported by experiments including the effective heat conductivity, natural and forced convection heat transfer, and boiling heat transfer of nanofluids [3–5]. Factors that can influence thermophysical properties have been discussed in many literatures, such as nanoparticle size and shape, material of particle and base fluid, particle loading, temperature, and so on, but quantitative understanding is still a lack on the topic [1, 3]. Another hot topic is the flow resistance characteristic of nanofluids [6, 7]. The pumping power needed for the bulk fluid did not increase dramatically as the particle loading grows in experiments [5], whereas the viscosity of nanofluids shows a marked increase after adding nanoparticles into base fluid [1, 6]. The rheology of nanofluids also leads to the complexity on characterizing physical properties of bulk flow [8, 9].
Although many researchers have reviewed lots of measured experimental data and demonstrated that particle shape plays an important role on convective heat transfer characteristic of nanofluids [10, 11], few models can be found in literatures proposed for nanofluid flow containing nonspherical nanoparticles. Nanorods, as well as nanotubes, nanofibers, and nanobelts are used more and more for nanofluid preparation because of larger aspect ratio and surface area compared with spherical nanoparticles. Such kind of onedimensional nanomaterials are available for heat transfer application with the development of nanoscale synthesis technology [12]. Xie et al. [13] first found the thermal conductivity of nanofluids with elongated nanoparticles is superior to that of spherical nanoparticles. Murshed et al. [14] measured enhanced thermal conductivity of waterbased nanofluid containing TiO_{2} nanorods with the aspect ratio of 4; they showed that the results give 12% higher than that predicted by classical Hamilton–Crosser model. Then, two models derived considering interfacial layer existence between particle and base fluid for the effective thermal conductivities of nanofluids with spherical and cylindrical nanoparticles [15]. Yang and Han [16] investigated the thermal conductivity of nanofluids dispersed by Bi_{2}Te_{3} nanorod in perfluoronhexane and oil. They found the thermal conductivities of nanofluids decrease with increasing temperature due to nanorod aspect ratio. The results measured by Zhang et al. [17, 18] employing carbon nanotubes (CNTs) agreed well with the predictions of the unitcell model proposed by Yamada and Ota [19] by considering shape factor for cylindrical particles. By using the excluded volume concept, Koo et al. [20] constructed a model which can reproduce the nonlinear increase of the thermal conductivity for carbon nanotube and nanofiber suspensions. Timofeeva et al. [21] studied the particle shape effect on thermophysical properties of aluminaEG/H_{2}O suspensions and concluded that elongated particles and agglomerates resulted in higher viscosity at the same volume fraction due to structural limitation of rotational enhancements in effective thermal conductivities diminished by interfacial effects. Molecular dynamics (MD) simulation based on the model developed by Ghosh et al. [22] have shown that cylindrical nanoparticles pick up thermal energy much faster than spherical nanoparticles during a collision with a blockshaped heat source [23]. Comparative experimental results reported by Yu et al. [10] indicated that convective heat transfer characteristic of nanofluids containing nonspherical nanoparticles is affected significantly by the shearinduced alignment and orientational motion of particles. By considering the thickness and thermal conductivity of the interfacial nanolayer, Jiang et al. [24] proposed a model which provides good predictions for the effective thermal conductivity of CNTbased nanofluids. Other existing models are summarized by Yang et al. [11], and three similar theoretical models are proposed for nanofluids with finite cylindrical particles by anisotropy analysis [25–27]. All of the three models give minimal relative errors to experimental results compared to other models. Lin et al. [7, 28] took numerical simulation on the phaolefinsbased nanofluids fluid flow containing cylindrical Al_{2}O_{3} nanoparticles in laminar and turbulent pipes, respectively. The results show that the friction factor of flow decreases when the Reynolds number and particle aspect ratio become larger. The Nusselt number of convective heat transfer increases with the increase of Reynolds number, particle aspect ratio, and volume concentration. By comparing the defined performance evaluation criterion (PEC), nanofluids containing rodlike nanoparticles with large aspect ratio and a suitable particle volume concentration are more effective for convective heat transfer process at higher Reynolds number. A coupled numerical model is constructed by Yuan et al. [29] to simulate convective heat transfer and resistance characteristics of TiO_{2}/water nanofluids with cylindrical particles in laminar channel flow. The results show that the Nusselt number of nanofluid flow related to the Reynolds number, axial length, Prandtl number, and particle volume concentration and a fitted formula is proposed to predict the Nusselt number of nanofluid flow containing cylindrical nanoparticles.
As summarized above, high potential of rodlike nanoparticles in enhancing thermal conductivity of nanofluids has got more attention by researchers. Related reports focus mostly on viscosity and thermal conductivity properties, while progress made on convective heat transfer characteristics is still highly needed. In addition, lack of numerical modelling on nanofluid flow brings an obstacle to industrial applications. The objective of this work is to develop a coupled model for nanorodbased suspension flow and investigate the convective heat transfer and resistance characteristics of the nanofluid duct flow numerically. The orientation and movement of nanorod are considered, and key factors that can influence the friction factor and Nusselt number of the fluid flow are discussed in sequence.
2. Model for NanorodBased Nanofluids
2.1. Governing Equations
The duct flow for nanofluid is considered as impressible, and the governing equations include the continuity equation, modified Navier–Stokes equation with the additional term of nanorods, and the energy conservation equation [28, 30, 31]:where , p, ρ_{nf}, and T are the nanofluid velocity, pressure, density, and temperature, respectively; μ is the dynamic viscosity of the pure fluid; μ_{a} is the apparent viscosity; and are the mean second and fourthorder tensors of particle orientation, respectively; is the rateofstrain tensor; and C_{nf} is the thermal diffusivity coefficient of the nanofluid, (k is the turbulent kinetic energy, ε is the turbulent dissipation rate, C_{μ} = 0.09, and turbulent Prandtl number, Pr_{T} = 0.9) is the eddy thermal diffusivity coefficient.
Substituting instantaneous velocity, pressure, temperature, rateofstrain tensor, and tensors of particle orientation which consist of mean and fluctuation part in (1–3) and averaging, we havewhere U_{i}, P, and are the mean nanofluid velocity, pressure, and temperature, respectively, is the fluctuation flow velocity, and is the mean rateofstrain tensor.
In (5), the nanofluid density iswhere the subscripts “f” and “p” stand for pure fluid and nanoparticles, respectively, and ϕ_{e} is the effective particle volume concentration, which is used for replacing the nominal particle volume fraction ϕ, in consideration of the particle aggregation:where s and s_{e} are the effective size of aggregates and primary particle size, respectively, and f_{i} is the fractal index and varies from 1.5 to 2.45 for those of rodlike nanoparticles. Yu et al. [10] gives the value as: s/s_{e} = 1.48 and f_{i} = 1.95 according to experimental data.
Batchelor’s theory was extended by Mackaplow and Shaqfeh [32] to account for twobody interactions and gave the apparent viscosity in (5):where n and l are the number density and half length of nanorods, respectively, μ is the dynamic viscosity of the pure fluid, and r is the particle aspect ratio, which is the ratio of length of diameter.
The thermal diffusivity coefficient in (6) is [33, 34]where the heat capacitance of the nanofluid is given by
In (10), the classical Hamilton–Crosser model is employed for predicting thermal conductivity of nanorodbased nanofluid as follows:in which k_{f} and k_{p} are the thermal conductivity of pure fluid and nanoparticles, respectively, and K is the shape factor given by K = 3/γ, where γ is the particle sphericity, defined as the ratio of the surface area of a sphere with the same volume as that of the particle and the surface area of the particle. The model predicts well the thermal conductivity of nanofluid with ZnO nanoparticles with experimental data by Ferrouillat et al. [35] in rodlike shape.
The Reynolds stress tensor in (5) iswhere eddy viscosity . For solving (5), the equation and equation for turbulent flow are given bywhere C_{1} = 1.44, C_{2} = 1.92, σ_{k} = 1.0, and σ_{ε} = 1.3.
2.2. Nanoparticle Orientation Distribution
The mean second and fourthorder tensors of particle orientation in (5) describe the orientation distribution of nanoparticle population in flow field; they can be calculated by Advani and Tucker [36] aswhere p_{i} is a unit vector parallel to the particle’s axis and is the probability density function for particle orientation at any position with p being the orientation vector. The governing equation of is the modified Fokker–Planck equation considering the rotary effect caused by Brownian movement [37, 38]:where D_{rB} is the Brownian rotary diffusion coefficient, is the gradient operator projected onto the surface of the unit sphere, and is the particle angular velocity. Cintra and Tucker [39] expressed aswhere is the vorticity tensor, . The last term in (18) is introduced to model the behavior at higher concentrations, in which D_{rI} is a rotary diffusion coefficient resulting from particleparticle interactions. Folgar and Tucker [40] suggested that D_{rI} is isotropic and given by , where the interaction coefficient C_{I} is fixed to 0.01.
For cylindrical particles, the Brownian rotary diffusion coefficient in (17) is
With aspect ratio r, the rotational friction coefficients around long axis and short axis are [41, 42].with
Substituting instantaneous and which consist of mean and fluctuation part in (17) and averaging, we have the mean equation of probability density functions for the particle orientation:where and are the dispersion coefficients of linear and angular displacement in which ν is the kinetic viscosity of the pure fluid [43], which appear based on the dispersion mechanism:
Integral on the angle after multiplying component of unit vector p_{i}, (22) is transformed to the equation about :
The mean fourthorder particle orientation tensor needs to be closured by model. Considering the shearingstretch feature of duct flow, the orthotropic closure model [44] is fit for (24):in which det(a_{ij}) is the determinant of secondorder tensor a_{2}.
2.3. General Dynamics Equation for RodLike Nanoparticles
The particle volume concentration ϕ in (7), (11), and (12) is considered distribution inhomogeneity in the flow. In order to obtain the distribution of ϕ in the flow, it is necessary to solve the general dynamics equation for nanorods:
Based on the Reynolds average, the above equation is transformed towhere is the mean particle volume distribution function based on the volume and the last term on the lefthand side represents the change in resulting from turbulent diffusion and is usually assumed to be [45]:where ε_{i} is the eddy diffusion coefficient which is a function of position and , where v_{t} and Sc_{t} are the turbulent viscosity and the Schmidt number, respectively.
D_{tB} in (26) is the Brownian translational diffusion coefficient for rodlike nanoparticles:and the translational friction coefficients parallel and perpendicular to the long axis are [41, 42]with
Substituting (29) in (27), we have averaged general dynamics equation for nanoparticles. With the moment transformation which involves multiplying the equation by v^{k} and then integrating over the entire size regime, the equation for rodlike nanoparticles finally becomeswhere the zero moment and the firstorder momentdescribe the total particle number density and volume of nanoparticles at a given point.
3. Numerical Method
Schematic of a turbulent duct flow containing nanorodbased nanofluid is shown in Figure 1. The length and diameter of the channel are L and D, respectively.
The boundary conditions are given as follows: inlet: U_{x} = U_{0}, U_{y} = 0, T = T_{in}, m_{1} = m_{10}; outlet: , U_{y} = 0, , ; and wall: U_{x} = U_{y} = 0, m_{1} = 0, . Notably, numerical simulations performed here are forced convection process with uniform wall temperature, other than those with uniform wall heat flux.
The ratio of channel length to height is L/D = 125. The nanofluid is a mixture of water and rigid ZnO nanorods. Thermophysical properties of water and ZnO are presented in Table 1. The temperatures of inlet nanofluid and wall are T_{in} = 293 K and T_{w} = 323 K, respectively. Reynolds number of the fluid flow is defined as Re = u_{0}D/ν_{nf} ranging from 2500 to 15,000. The nominal particle volume fraction ϕ for nanofluids is 0.4%, 0.93%, and 1.3%, and aspect ratio r is 8, 12 and 16 for comparing with experimental results in Ferrouillat et al. [35].

The SIMPLEC algorithm is employed for solving coupled continuity equation and moment equation, where the convection item and diffusion item are discreted using QUICK scheme and secondorder center difference, respectively. No slip condition is set for wall boundary condition.
4. Results and Discussion
4.1. Calculation Verification
To validate the model and code for nanofluid flow, classical models and experimental results are used for comparing the numerical results of friction factor and the Nusselt number of duct fluid flow in Figures 2 and 3. The definition of f and Nu is as follows:where ∆p is the pressure drop, is the average velocity of the fluid in the flow direction, and h is the heat transfer coefficient.
Figure 2 shows the comparison of friction factor of nanofluids varying with the Reynolds number. The solid points in figure are measured by Ferrouillat et al. [35], and the solid curve is the Blasius equation which is f = 0.316Re^{−0.25} for turbulent duct flow. The numerical results are drawn by hollow points and are obtained by setting the nanorod aspect ratio as 8, because the aspect ratio of nanoparticles in reference experiments are about 6∼10. From the figure, the numerical results show agreement with the experimental results and the curve of Blasius equation. In turbulence regime, the friction factor of nanofluid flow drops slowly when the Reynolds number grows.
Comparison of calculated Nusselt number of nanofluids with Dittus–Boelter relation is shown in Figure 3. The classical Dittus–Boelter relation is commonly used for predicting the Nusselt number in fully developed turbulent duct flow, which is Nu = 0.023Re^{0.8}Pr^{0.4}. The difference of Nusselt number predicted by current model and DB relation is within about 10%, and the difference declines in high Nu region.
4.2. Orientation Distribution of Nanorods
The orientation of nanorods in fluid makes an effect on nanofluid fluid flow. The orientation distribution of nanorods population can be described by the mean secondorder orientation tensor of particle. The components of in z direction are shown in Figure 4, in which the volume concentration and aspect ratio are set as ϕ = 1.4% and r = 16, respectively. It can be seen from the figure that the distributions of components of a_{ii} coincide with each other, and the median distribution of which along the axial direction is 1/π. The value of near the wall is larger that that of central area, which means that the nanorods align with the flow direction due to the existence of wall. The orientation state of nanorods can hardly keep in yz component because is always zero in z direction. But is not zero near the wall, which illustrates the wall effect on orientation state of nanorods.
4.3. Friction Factors
The friction factor and pressure drop of the duct flow are of importance to the practical applications. Figure 5 shows the friction factors of nanofluids at different volume concentrations as a function of the Reynolds number. The aspect ratio of nanorods in experiments and simulations are set as r = 8. All the friction factors of nanofluid flow are higher than those of base fluid and decrease when the Reynolds number grows higher. The friction factor increases slightly with the increase of particle volume concentration, and the difference diminishes when the Reynolds number becomes larger.
In Figure 6, the friction factor as a function of the Reynolds number by different aspect ratio is given, and the volume concentration ϕ is 0.93%. As can be seen in figure, the shape of nanoparticles takes effect on friction factor of nanofluid fluid flow. Nanorods with larger aspect ratio bring more flow resistance when the Reynolds number is the same. When the Reynolds number gets larger, the friction factor of nanofluid approaches to that of base fluid, which means the additional pumping power for nanofluid is not significant in high Reynolds number region.
4.4. Convective Heat Transfer
The Nusselt number describes the ratio of convective heat transfer across the wall. The relative Nusselt number is defined here as the ratio of the Nusselt number of nanofluids to that of base fluid, that is, Nu_{nf}/Nu_{f}. Figures 7 and 8 show the calculated and experimental results of Nu_{nf}/Nu_{f} as a function of the Reynolds number with different particle aspect ratios or different volume concentrations. The solid points give the measured Nu with particle volume concentration as 0.93% and reveal that the Nusselt number of nanofluid is smaller than that of base fluid when Re is less than 7000. The overall experimental data indicate that Nu_{nf}/Nu_{f} grows as the Reynolds number gets larger except the point when Re is about 2100. The calculated Nu_{nf}/Nu_{f} by model in this paper is larger than that of experiments when the volume concentration is set as 0.93%. In addition, all the numerical results are greater than 1, which means the convective heat transfer of nanofluid is better than base fluid at the same Reynolds number. The Nu_{nf}/Nu_{f} is monotone increasing as a function of the Reynolds number, no matter the particle volume concentration or aspect ratio varies in figures. In Figure 7, the relative Nusselt number increases as the particle volume concentration gets larger. Similarly, in Figure 8, the relative Nusselt number increases as the particle aspect ratio gets larger. Both the particle loading and particle shape influence the convective heat transfer characteristic obviously. Compared to increasing the aspect ratio of nanorods, increasing the particle volume concentration would be more effective on enhancing the convective heat transfer intensity in industrial applications although it will cause a slight increase of resistance.
5. Conclusions
The convective heat transfer and resistance characteristics of the nanofluid duct flow are investigated based on a coupled numerical model for nanorodbased suspension flow. The numerical results are verified by experimental results and theoretical models. Most of nanorods are located randomly in the bulk fluid, while particles near the wall aligned with the flow direction. Friction factor of nanofluids with nanorods increases with higher particle volume concentration or aspect ratio, but the increment reduces when the Reynolds number getting larger. The relative Nusselt number is obtained to characterize the intensity of convective heat transfer. The results show that the Nusselt number of nanofluids increases when the particle volume concentration or aspect ratio becomes larger. Compared to increasing the aspect ratio of nanorods, increasing the particle volume concentration would be more effective on enhancing the convective heat transfer intensity in industrial applications although it will cause a slight increase of resistance.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (no. 11632016) and Jiangsu Key Laboratory of Green Process Equipment (GPE201705).
References
 S. S. Murshed and P. Estellé, “State of the art review on viscosity of nanofluids,” Renewable and Sustainable Energy Reviews, vol. 76, pp. 1134–1152, 2017. View at: Publisher Site  Google Scholar
 S. Choi and J. Eastman, “Enhancing thermal conductivity of fluids with nanoparticles,” ASMEPublicationsFed, vol. 231, pp. 99–106, 1995. View at: Google Scholar
 L. Yang and K. Du, “A comprehensive review on heat transfer characteristics of TiO_{2} nanofluids,” International Journal of Heat and Mass Transfer, vol. 108, pp. 11–31, 2017. View at: Publisher Site  Google Scholar
 M. Bahiraei, “Particle migration in nanofluids: a critical review,” International Journal of Thermal Sciences, vol. 109, pp. 90–113, 2016. View at: Publisher Site  Google Scholar
 M. Islam, B. Shabani, G. Rosengarten, and J. Andrews, “The potential of using nanofluids in PEM fuel cell cooling systems: a review,” Renewable and Sustainable Energy Reviews, vol. 48, pp. 523–539, 2015. View at: Publisher Site  Google Scholar
 R. Saxena, D. Gangacharyulu, and V. K. Bulasara, “Heat transfer and pressure drop characteristics of dilute alumina–water nanofluids in a pipe at different power inputs,” Heat Transfer Engineering, vol. 37, no. 18, pp. 1554–1565, 2016. View at: Publisher Site  Google Scholar
 J. Lin, Y. Xia, and X. Ku, “Friction factor and heat transfer of nanofluids containing cylindrical nanoparticles in laminar pipe flow,” Journal of Applied Physics, vol. 116, no. 13, p. 133513, 2014. View at: Publisher Site  Google Scholar
 S. Kim, C. Kim, W.H. Lee, and S.R. Park, “Rheological properties of alumina nanofluids and their implication to the heat transfer enhancement mechanism,” Journal of Applied Physics, vol. 110, no. 3, p. 034316, 2011. View at: Publisher Site  Google Scholar
 H. Chen, Y. Ding, and C. Tan, “Rheological behaviour of nanofluids,” New Journal of Physics, vol. 9, no. 10, p. 367, 2007. View at: Publisher Site  Google Scholar
 L. Yu, D. Liu, and F. Botz, “Laminar convective heat transfer of aluminapolyalphaolefin nanofluids containing spherical and nonspherical nanoparticles,” Experimental Thermal and Fluid Science, vol. 37, pp. 72–83, 2012. View at: Publisher Site  Google Scholar
 L. Yang, X. Chen, M. Xu, and K. Du, “Roles of surfactants and particle shape in the enhanced thermal conductivity of TiO_{2} nanofluids,” AIP Advances, vol. 6, no. 9, p. 095104, 2016. View at: Publisher Site  Google Scholar
 G. Cao and Y. Wang, Nanostructures and Nanomaterials: Synthesis, Properties and Applications, World Scientific, London, UK, 2011.
 H. Xie, J. Wang, T. Xi, and Y. Liu, “Thermal conductivity of suspensions containing nanosized SiC particles,” International Journal of Thermophysics, vol. 23, no. 2, pp. 571–580, 2002. View at: Google Scholar
 S. Murshed, K. Leong, and C. Yang, “Enhanced thermal conductivity of TiO_{2}—water based nanofluids,” International Journal of Thermal Sciences, vol. 44, no. 4, pp. 367–373, 2005. View at: Publisher Site  Google Scholar
 S. Murshed, K. Leong, and C. Yang, “Investigations of thermal conductivity and viscosity of nanofluids,” International Journal of Thermal Sciences, vol. 47, no. 5, pp. 560–568, 2008. View at: Publisher Site  Google Scholar
 B. Yang and Z. Han, “Temperaturedependent thermal conductivity of nanorodbased nanofluids,” Applied Physics Letters, vol. 89, no. 8, p. 083111, 2006. View at: Publisher Site  Google Scholar
 X. Zhang, H. Gu, and M. Fujii, “Effective thermal conductivity and thermal diffusivity of nanofluids containing spherical and cylindrical nanoparticles,” Journal of Applied Physics, vol. 100, no. 4, p. 044325, 2006. View at: Publisher Site  Google Scholar
 X. Zhang, H. Gu, and M. Fujii, “Effective thermal conductivity and thermal diffusivity of nanofluids containing spherical and cylindrical nanoparticles,” Experimental Thermal and Fluid Science, vol. 31, no. 6, pp. 593–599, 2007. View at: Publisher Site  Google Scholar
 E. Yamada and T. Ota, “Effective thermal conductivity of dispersed materials,” Heat and Mass Transfer, vol. 13, no. 1, pp. 27–37, 1980. View at: Publisher Site  Google Scholar
 J. Koo, Y. Kang, and C. Kleinstreuer, “A nonlinear effective thermal conductivity model for carbon nanotube and nanofiber suspensions,” Nanotechnology, vol. 19, no. 37, p. 375705, 2008. View at: Publisher Site  Google Scholar
 E. V. Timofeeva, J. L. Routbort, and D. Singh, “Particle shape effects on thermophysical properties of alumina nanofluids,” Journal of Applied Physics, vol. 106, no. 1, p. 014304, 2009. View at: Publisher Site  Google Scholar
 M. M. Ghosh, S. Roy, S. K. Pabi, and S. Ghosh, “A molecular dynamicsstochastic model for thermal conductivity of nanofluids and its experimental validation,” Journal of Nanoscience and Nanotechnology, vol. 11, no. 3, pp. 2196–2207, 2011. View at: Publisher Site  Google Scholar
 M. Ghosh, S. Ghosh, and S. Pabi, “Effects of particle shape and fluid temperature on heattransfer characteristics of nanofluids,” Journal of Materials Engineering and Performance, vol. 22, no. 6, pp. 1525–1529, 2013. View at: Publisher Site  Google Scholar
 H. Jiang, Q. Xu, C. Huang, and L. Shi, “The role of interfacial nanolayer in the enhanced thermal conductivity of carbon nanotubebased nanofluids,” Applied Physics A, vol. 118, no. 1, pp. 197–205, 2015. View at: Publisher Site  Google Scholar
 L. Yang, X. Xu, W. Jiang, and K. Du, “A new thermal conductivity model for nanorodbased nanofluids,” Applied Thermal Engineering, vol. 114, pp. 287–299, 2017. View at: Publisher Site  Google Scholar
 L. Yang, K. Du, and X. Zhang, “A theoretical investigation of thermal conductivity of nanofluids with particles in cylindrical shape by anisotropy analysis,” Powder Technology, vol. 314, p. 328, 2017. View at: Publisher Site  Google Scholar
 L. Yang and X. Xu, “A renovated Hamilton–Crosser model for the effective thermal conductivity of CNTs nanofluids,” International Communications in Heat and Mass Transfer, vol. 81, pp. 42–50, 2017. View at: Publisher Site  Google Scholar
 J.Z. Lin, Y. Xia, and X.K. Ku, “Flow and heat transfer characteristics of nanofluids containing rodlike particles in a turbulent pipe flow,” International Journal of Heat and Mass Transfer, vol. 93, pp. 57–66, 2016. View at: Publisher Site  Google Scholar
 F. Yuan, J. Lin, and X. Ku, “Convective heat transfer and resistance characteristics of nanofluids with cylindrical particles,” Heat Transfer Engineering, vol. 39, no. 6, pp. 1–10, 2018. View at: Google Scholar
 J. Buongiorno, “Convective transport in nanofluids,” Journal of Heat Transfer, vol. 128, no. 3, pp. 240–250, 2006. View at: Publisher Site  Google Scholar
 G. Batchelor, “Slenderbody theory for particles of arbitrary crosssection in stokes flow,” Journal of Fluid Mechanics, vol. 44, no. 3, pp. 419–440, 1970. View at: Publisher Site  Google Scholar
 M. B. Mackaplow and E. S. Shaqfeh, “A numerical study of the rheological properties of suspensions of rigid, nonBrownian fibres,” Journal of Fluid Mechanics, vol. 329, no. 1, pp. 155–186, 1996. View at: Publisher Site  Google Scholar
 H. Brinkman, “The viscosity of concentrated suspensions and solutions,” Journal of Chemical Physics, vol. 20, no. 4, p. 571, 1952. View at: Publisher Site  Google Scholar
 G. Batchelor, “The effect of Brownian motion on the bulk stress in a suspension of spherical particles,” Journal of Fluid Mechanics, vol. 83, no. 1, pp. 97–117, 1977. View at: Publisher Site  Google Scholar
 S. Ferrouillat, A. Bontemps, O. Poncelet, O. Soriano, and J.A. Gruss, “Influence of nanoparticle shape factor on convective heat transfer and energetic performance of waterbased SiO_{2} and ZnO nanofluids,” Applied thermal engineering, vol. 51, no. 1, pp. 839–851, 2013. View at: Publisher Site  Google Scholar
 S. G. Advani and C. L. Tucker, “The use of tensors to describe and predict fiber orientation in short fiber composites,” Journal of Rheology, vol. 31, no. 8, pp. 751–784, 1987. View at: Publisher Site  Google Scholar
 J.Z. Lin and Z.J. You, “Stability analysis in spatial mode for channel flow of fiber suspensions,” Applied Mathematics and MechanicsEnglish EditionShanghai, vol. 24, no. 8, pp. 871–879, 2003. View at: Google Scholar
 Z. J. You and J. Z. Lin, “Stability in the circular pipe flow of fiber suspensions,” Journal of Hydrodynamics, vol. 15, no. 2, pp. 12–18, 2003. View at: Google Scholar
 J. S. Cintra and C. L. Tucker, “Orthotropic closure approximations for flowinduced fiber orientation,” Journal of Rheology, vol. 39, no. 6, pp. 1095–1122, 1995. View at: Publisher Site  Google Scholar
 F. Folgar and C. L. Tucker, “Orientation behavior of fibers in concentrated suspensions,” Journal of Reinforced Plastics and Composites, vol. 3, no. 2, pp. 98–119, 1984. View at: Publisher Site  Google Scholar
 J. G. de la Torre and V. A. Bloomfield, “Hydrodynamic properties of complex, rigid, biological macromolecules: theory and applications,” Quarterly Reviews of Biophysics, vol. 14, no. 1, p. 81, 1981. View at: Publisher Site  Google Scholar
 G. Li and J. X. Tang, “Diffusion of actin filaments within a thin layer between two walls,” Physical Review E, vol. 69, no. 6, p. 061921, 2004. View at: Publisher Site  Google Scholar
 J. A. Olson, “The motion of fibres in turbulent flow, stochastic simulation of isotropic homogeneous turbulence,” International Journal of Multiphase Flow, vol. 27, no. 12, pp. 2083–2103, 2001. View at: Publisher Site  Google Scholar
 Z. You and J. Lin, “Effects of tensor closure models and 3D orientation on the stability of fiber suspensions in a channel flow,” Applied Mathematics and Mechanics, vol. 26, no. 3, pp. 307–312, 2005. View at: Google Scholar
 S. K. Friedlander, Smoke, Dust, and Haze: Fundamentals of Aerosol Dynamics, Oxford University Press, New York, NY, USA, 2000.
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Copyright © 2018 Fangyang Yuan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.