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Convective heat transfer is present in many engineering processes such as thermal powerplants, heat exchangers, environmental comfort, energy storage systems and electronic cooling. Fluids such as oil,water and ethylene glycol mixture are poor heat transfer fluids, sincethe thermal conductivity of these fluids, which is generally small, play important role on theheat transfer coefficient between the heat transfer medium andthe heat transfer surface. The termnanofluid refersto a liquid containing (e.g. water, ethylene glycol, engine oils, etc.)a suspension of solid particles (nanoparticles) in conventional heat transfer fluids. It seems that Choi (1995) is the first who used the term nanofluids to refer to the fluid with suspended nanoparticles. Nanoparticles can be metallic particles such as those of $\mathrm{Cu},\mathrm{Ag},\mathrm{Au}$, etc. or metallic oxides or non-metallic oxide particles: $\mathrm{CuO},\mathrm{Al}_{2}\mathrm{O}_{3},\mathrm{TiO},\mathrm{SiO}$, etc. having dimensions in the range 1 to 100 nm . Significant features of nanofluids over base fluids include enhanced thermal conductivity, greater viscosity, and enhanced value of critical heat flux. Of these the most talked about is the enhanced thermal conductivity, a phenomenon which was first reported by Masuda et al. (1993), Ding et al. (2007) reported the use of nanofluids in a wide variety of industries ranging from transportation, heating, ventilation, and air conditioning(HVAC), and energy production and supply to electronics, textiles, paper production, etc. It is worth to mention that there are several patents which use nanofluids in heat transfer problemssuch as heat pumps and solar energy collectors (see,Laforgia et al. 2013 and Olson, 2013).Choi et al. (2001) showed that the addition of a small amount (less than $1\%$ by volume) of nanoparticles to conventional heat transfer liquids increased the thermal conductivity of the fluid up to approximately two times. Therefore, the effective thermal conductivityof nanofluids is expected to enhance heat transfer compared withconventional heat transfer liquids (Masuda et al., 1993). All of these industries deal with heat transfer in some or the other way, and thus have a strong need for improved heat transfer mediums. This could possiblybe nanofluids, because of some potential benefits over normal fluids- large surface area provided by nanoparticles for heat exchange, reduced pumping power due to enhanced heat transfer, minimal clogging, innovation of miniaturized systems leading to savings of energy and cost. Choi’s (1995) results have been supported by other researchersfrom time to time. Eastman et al. (2001) reported an increase of $40\%$ in the effectivethermal conductivity of ethyleneglycol with $0.3\%$ volume of copper nanoparticles of 10 nm diameter. Further $10-30\%$ increase of the effective thermal conductivity in alumina/water nanofluids with $1-4\%$ of alumina was reported by Das et al. (2003). Buongiorno and Hu(2005)suggested the possibility of using nanofluids inadvanced nuclear
systems.Another recent application of the nanofluid flow is in thedelivery of nano-drug as suggested by Kleinstreuer et al. (2008).

A comprehensive survey of convectivetransport in nanofluids wasmade by Buongiorno(2006),Choi (2009), Kakaç and Pramuanjaroenkij(2009), Das and Choi (2009),Fan and Wang (2011), etc.We have alsotomentionthe valuable published books by Das et al. (2007) and Schaefer (2009).Buongiorno(2006) haspointed out that the nanoparticle absolute velocity can be viewed asthe sum of the base fluid velocity and a relative velocity (that hecalls the slip velocity). He considered in turn seven slip mechanisms:inertia, Brownian diffusion, thermophoresis, diffusiophoresis,Magnuseffect, fluid drainage, and gravity settling (Nield and Kuznetsov, 2009).Numerous models and methods have beenproposed by different authors to study convective flows of nanofluidsand we mention here the papers by Khanafer et al. (2003), Tiwari andDas (2007), Oztop and Abu-Nada (2008),Rohni et al. (2009), Mansur et al. (2009), Nield and Kuznetsov(2010), Kuznetsov andNield (2010), Tham et al. (2012),Aminossadati and Ghasemi (2012),etc.

The problem of viscous boundary-layer flowon a full and truncated cone is a classical problem, and it has been considered by manyresearchers, for example Chiou and Na (1980), Kumari et al. (1989), Chamkha (2001), Molla et al. (2009), Mahdy et al. (2010), Chamkhaand Rashad (2012), Chamkha et al. (2012), etc. Engineering applications of this particular geometry (truncated cone) are in the field of heat exchangers, cooling of electronic devices, etc. (seeShinmura, 1996; Hamilton et al. 1999; McCutcheon et al., 2005 and Nakamura et al., 2010).

Differentfrom the previous investigations, following the nanofluid equationsmodel proposed by Tiwari and Das (2007), we consider in thispaper the development of the steady mixed convection boundary-layer flowon a vertical impermeablefrustum of a cone in a nanofluid. The problem is formulated so that we can consider three different types of naoparticles, namely Cu (cooper), $\mathrm{Al}_{2}\mathrm{O}_{3}$ (alumina) and $\mathrm{TiO}_{2}$ (titania), and water as a base fluid. However, in order to save space, we have considered here only the case of Cu nanoparticles. The motivation for this study is that nanotechnologyhas been widely used in industry because materials withsizes of nanometers possess unique physical properties. The dimensionless governing partial differential equations have been solved numerically and the results for the particular case of the full cone have been compared with results reported by Kumari et al. (1989). The results are in very good agreement.

Consider the steady mixed boundary layer flow near a vertical truncated cone (with half angle $\phi$ ) as shown in Fig. 1 in a water-based nanofluid. The origin O of the coordinate system is placed at the vertex of the full cone, where $x$ is the co-ordinatemeasured from the origin Oalong the surface of the full cone and $y$ is the coordinate normal to the surface of the truncated cone, respectively. The distance of the leading edge of the truncated cone measured from the origin O is denoted as $x_{0}$. The surface of the truncated cone is held at a constant temperature $T_{w}$, while the temperature of the ambient nanofluid is $T_{\infty}$ with $T_{w}>T_{\infty}$ for a heated truncated cone (assisting flow) and $T_{w}<T_{\infty}$ for a cooled truncated cone (opposing flow), respectively.The velocity far away from the truncated cone in the vertical direction is $U_{\infty}$ and the viscous dissipation is assumed to be negligible.

Under the above assumptions along with the assumptions of Boussinesq and boundary layer approximations, the basic equations governing the steady mixed convection boundary layer flow of a
nanofluidnear a vertical frustum of a cone can be written as in Chamkha and Rashad (2012) or Pop and Ingham (2001) combined with the mathematical nanofluid model given by Tiwari and Dass(2007):

along with the boundary conditions

where $u$ and $v$ are the velocity components along $x$ and $y$ axes, respectively, $T$ is the temperature of the nanofluid, $g$ is the gravity acceleration, $U_{\infty}$ is the constant velocity of the outer (inviscid) flow, $r=x\sin\phi,\rho_{nf}$ is the effective density, $\mu_{\mathrm{nf}}$ is the effective dynamic viscosity, $(\rho\beta)_{nf}$ is the thermal expansion coefficient and $\alpha_{nf}$ is the thermal diffusivity of the nanofluid, which are given as (see Khanafer et al., 2003, and Oztop and Abu-Nada, 2008),

Here, $\varphi$ is the solid volume fraction, $\mu_{f}$ is the dynamic viscosity of the basic fluid, $\rho_{f}$ and $\rho_{s}$ are the densities of pure fluid and nanoparticles, respectively, $\left(\rho C_{p}\right)_{\mathrm{nf}}$ is the heat capacity of the nanofluidand $k_{\mathrm{nf}}$ is the thermal conductivity of the nanofluidgiven by

where $\left(\rho C_{p}\right)_{f}$ and $\left(\rho C_{p}\right)_{s}$ are the specific heat parameters of the base fluid and nanoparticles and $k_{f}$ and $k_{s}$ are the thermal conductivities of the base fluid and nanoparticles. The viscosity of the nanofluid $\mu_{nf}$ can be approximated as the viscosity of a base fluid $\mu_{f}$ containing dilute suspension of fine spherical particles and is given byBrinkman (1952). The effective thermal conductivity of the nanofluid
$k_{nf}$ is approximated by the Maxwell-Garnett’s model, which is found to be appropriate for studying heat transfer enhancement using nanofluids (Khanafer et al., 2003;Maïga et al., 2004, etc.).

Further, we introduce the following non-similarity variables

where $\bar{x}=x-x_{0},v_{f}$ is the kinematic viscosity of the based fluid, $\operatorname{Re}_{x}=U_{\infty}\bar{x}/v_{f}$ is the local Reynold number and $\psi$ is the stream function, which is defined as

Substituting (7) and (8) into Eqs. (2) and (3), we obtain the following partial differential equations of parabolic type

and the boundary conditions (6) become

where $\lambda$ is the constant mixed convection parameter, which is given by

with $Gr_{x}=g\beta_{f}\left(T_{w}-T_{\infty}\right)x^{3}\sin\phi/\nu_{f}^{2}$ being the local Grashof number. It should be stated that $\lambda>0$ ( $T_{w}>T_{\infty}$ ) corresponds to a heated truncated cone (assisting flow), $\lambda<0$ ( $T_{w}<T_{\infty}$ ) corresponds to a
cooled truncated cone (opposing flow) and $\lambda=0$ ( $T_{w}=T_{\infty}$ ) corresponds to the forced convection flow along the truncated cone, respectively. Using (7) and (8), the velocity components $u$ and $v$ are given by

The quantities of physical interest are also the skin friction coefficient $C_{f}$ and the local Nusselt number $Nu_{x}$ which are defined as

where $\tau_{w}$ is the wall skin friction and $q_{w}$ is the heat flux from the surface of the truncated cone, which are given by

Using relations (7) in (15) and (14), we obtain

The following two particular cases are also of interest:

In this case, Eqs. (9) and (10) reduce to the following ordinary differential equations

subject to

where primes denote differentiation with respect to $\eta$.
ii) Case of $\xi$ large ( $\xi\gg 1$ )

In this case Eqs. (9) and (10) reduce to

subjected to the boundary conditions (19). Also, the skin friction coefficient $C_{f}$ and the local Nusselt number $Nu_{x}$ become

and

In this case, we limit the analysis to the case when $\xi\gg 1$, the other case $\xi=0$ (full cone) being similar. To discuss the behaviour of the solution of the boundary value problems (17-19) for $\lambda\gg 1$, we introduce the following new variables

Substituting (24) into Eqs. (17) and (18), we obtain

and the boundary conditions (19) become

whereprimes denote now the differentiation with respect to $z$.The boundary conditions (27) suggest an expansion for $F$ and $\theta$ of the following form

where ( $F_{0},\theta_{0}$ ) and ( $F_{1},\theta_{1}$ ) are given by

Thus, from (22), we have

for $\lambda\gg 1$. It should be mentioned that Eqs. (29) describe the problem of free convection from a full vertical cone in a nanofluid.

As we have mentioned before, the problem is formulated so that we can consider different types of nanoparticles (e.g. $\mathrm{Cu},\mathrm{Al}_{2}\mathrm{O}_{3},\mathrm{TiO}_{2}$, etc.) and water as a base fluid. However, in order to save space, we have considered here only the case of Cu nanoparticles. The thermophysical properties of the base fluidand nanoparticles are listed in Table 1.Following Aminossadati and Ghasemi(2012), we considered the range of nanoparticle fraction parameter $\phi$ as $0\leq\phi\leq 0.05$ and the Prandtl number of the base fluid (water) is kept constant at $\operatorname{Pr}=6.7$. Itcan be stated that the present study reduces to that of a viscous (Newtonian) fluid when $\varphi=0$.

The non-linear partial differential equations (9) and (10) with the boundary conditions (11) have beensolved numerically for several values of the governing parameters. The derivatives with respect to $\xi$ were discretized using the first order upwind finitedifferences and the resulting ordinary differential equations with respect to $\eta$ were solved using bvp4croutine from Matlab. For marching in $\xi$ direction a step $\Delta\xi=0.01$ has been used, while the absolute error tolerancesin bvp4c was 1e-9. Also the sets (17$19)$, $(20,21)$ and $(29,30)$ of ordinary differential equations were solved numerically using the same Matlab routine (bvp4c). We believe this is a better numerical method than the popular Keller box method used for solving such problems, especially in obtaining the multiple (dual) solutions. Therefore, in order to validate the present numerical method, wehave compared our results with those reported byKumari et al. (1989)fortwo values of Prwhen $\xi=0$ (full cone) $\lambda=0$ (forced convection)and $\phi=0$ (Newtonian fluid)as shown in Table 2. The comparisons arefound to be in very good agreement. However, in order to compare the results, we have to take in Kumari et al. (1989), $\xi=1$ and to use the transformation $\eta=\sqrt{3}\bar{\eta},f(\eta)=\frac{1}{\sqrt{3}}\mathrm{~F}(\bar{\eta})$ and $-\theta(\eta)=\Theta(\bar{\eta})$.

Figures 2and 5 display the variation of the reduced skin friction coefficient $f^{\prime\prime}(0)$ and the reduced Nusselt number $-\theta^{\prime}(0)$ for $\xi=0$ (full cone) and $\xi\gg 1$ when $\lambda<0$ (opposing flow) and $\phi=0$ (regular fluid), 0.03 and 0.05 . It was found that there are regions of uniquesolutions for $\lambda>0$ (assisting flow), dual (upper and lower branch) solutions exist for $\lambda<0$ (opposing flow). This happens because the forced flow and the flow due the buoyancy are in opposite directions. The solution for each value of $\phi$ exists up to a critical value of $\lambda$ ( $\lambda_{c}<0$ say) (opposing flow). Beyond this value we are unable to get the solution using the boundary layer approximations and the full Navier-Stokes equation has to be used. The upper curve for aparticular value of $\phi$ ends at $\lambda_{c}(<0)$ and the lower branchsolution continues further and terminates at a certain value of $\lambda<0$. Itshould be remarked that the computations have been performed increasing $\lambda$ until the point where the solution does not converge ( $\lambda=\lambda_{c}<0$ ), and the calculations were terminated at that point. In both cases of $\xi=0$ and $\xi\gg 1$, the values of $\lambda_{c}$ decrease with the increases of the nanoparticle concentration parameter $\phi$. The values of $\lambda_{c}<0$ are given in Table 3 for $\phi=0,0.03,0.05$, and $\xi=0,5$ and $\xi\gg 1$.

Figures 3and 6 present thevelocity and temperature (upper and lower branch) profilesfor $\xi=0$ (full cone) and $\xi\gg 1$ when $\lambda=-0.2$ and $\varphi=0,0.03$ and 0.05 , which support the existence of the dual nature of the solutions presented inFigs. 2 and 5.The velocity and thermalboundary layer thicknessesincrease
with the increases of the nanoparticle volume fraction parameter $\phi$. This is due to the increase of the viscosity and thermal conductivity of the nanofluid with the increasing of the nanoparticles volume fraction, see Eqs. (5) and (6).A reverse flow is noticed from Figs. 3a and 6a in the case of the lower branch solution.It is worth mentioning that the lower branch solutions are unstable and the boundary layer thicknesses of these solutions are higher as for the upper branch solutions. This can be shown by performing a stability analysis, but this is out of the scope of the present paper. Such an analysis has been done by Weidman et al. (2006), Rosca and Pop (2013) and Trîmbi\textcommabelowta\textcommabelows et al. (2013).

Furthermore, Figs. $4(\xi=0)$ and $7(\xi\gg 1)$ illustrate thevelocity and temperature profilesfor $\lambda>0$ (assisting flow) and $\phi=0,0.03,0.05$. It easily to notice thatnear the surface of the truncated cone there is an overshoot of the velocity profile for large values of $\lambda(>0)$. This is in agreement with the physical phenomenon that for large values of $\lambda(>0)$ the convective heat transfer dominates the conductive heat transferand buoyancy is much larger than convective flow. In addition, it is seen that the velocity profiles increase with $\lambda$, while the temperature profiles decrease with $\lambda(>0)$, which is in agreement with Fig. 8.

Figure 8 displays the asymptotic behaviour of the reduced skin friction coefficient $f^{\prime\prime}(0)$ and reducedNusselt number $-\theta^{\prime}(0)$ for $\lambda\gg 1$ when $\xi\gg 1$ and $\phi=0,0.03$ and 0.05 . It is further observed that there is a very good agreement between the numerical solution of Eqs. (20) and (21) and the asymptotic solution of Eqs. (29) and (30). Reduced skin friction and Nusselt number decrease with $\phi$ and increase with $\lambda$. The values of the coefficients $F_{0}{}^{\prime\prime}(0),F_{1}{}^{\prime\prime}(0),\theta_{0}{}^{\prime}(0)$ and $\theta_{1}{}^{\prime}(0)$ involved in Eq. (31) are given in Table 4.

Finally, Fig. 9 presents the variation of velocity profiles and temperature profiles with $\xi$ between the two limit cases: full cone ( $\xi=0$ ) and $\xi$ large ( $\xi\gg 1$ ). One can see that both velocity and temperature profiles decrease with the increases of $\xi$ and as a consequence the skin friction coefficient is lower and Nusselt number is higher for larger values of $\xi$ (truncated cone), which is important in some practical applications.

The steady mixed convection boundary layer flow of a nanofluidover a vertical frustum of a cone has been theoretically studied.A frequently used nanofluid model based on formulas for thermal
conductivity and dynamic viscosity is considered.The working fluid ischosen as water with the Prandtl number of $\operatorname{Pr}=6.7$. The effects of theparticle volume fraction parameter $\phi$, the mixed convection parameter $\lambda$ and the dimensionless coordinate $\xi$ on the flow and heat transfer characteristics are determined for the Cu nanoparticles. From this investigation, some conclusions were summarized as follows:
a. The reduced skin friction coefficient $f^{\prime\prime}(\xi,0)$ increases due to increase in the parameter $\lambda$ while it decreases due toincrease in particle volume fraction parameter $\phi$ when $\lambda>0$.
b. The reduced local Nusselt number $-\theta^{\prime}(\xi,0)$ is also influenced by the particle volume fraction parameter $\phi$ and the mixed convection parameter $\lambda$. Thus, it increases with the increase of $\lambda$ and decreases with the increase of the volume fraction parameter $\phi$ for positive $\lambda$. The increasing volume fraction of Cu nanoparticle increases the thermal conductivity which results in increasing of the thermal boundary layer.
c. The nanofluidtemperature increases due to increasing in theparticle volume fraction parameter $\phi$, while it decreases due to increase in the mixed convection parameter $\lambda$ when assisting flow is considered.
d. Dual solutions were found in the opposing flow case in the region $\lambda\in\left[\lambda_{c},0\right]$. When $\lambda<\lambda_{c}$ it is not possible to find a solution if the boundary approximation model is used.
e. It was shown that the numerical method based on bvp4c from Matlab works very efficient for this kind of problems.

The work of the first author was supportedby the Sectorial Operational Programme for Human Resources Development 2007-2013, co-financed by theEuropean Social Fund, under the project number POSDRU/88/1.5/S/60185 withthe title "Modern Doctoral Studies: Internalization andInterdisciplinarity". The work of the corresponding author was supported from the grant PN-II-RU-TE-2011-3-0013, UEFISCDI, Romania.

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