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We study the unsteady convective flow of two waterbased nanofluids containing Copper and Titanium oxide along a vertical stretching or shrinking cone with viscous dissipation and internal heat generation. The problem is transformed to twodimensional flow over a cone using Mangler's transformation. The coupled nonlinear conservation equations are solved numerically using the spectral local linearization method. We present an analysis of how some physical parameters affect the flow structure, the heat and mass transfer rates and the fluid properties. The accuracy of the results is determined by comparison with previously published studies, for some limiting cases.
Chemical Reaction, Nanofluid Flow, Stretching or Shrinking Cone, Spectral Local Linearization Method.
Common fluids such as water, ethylene glycol and oil have low heat transfer characteristic owing to their poor thermal conductivities. It is now understood that the thermophysical properties of these fluids can be significantly enhanced by suspending nanosized metallic particles such as Aluminum, titanium, Gold, Copper, Iron or their oxides, resulting in what is commonly called a nanofluid, see Choi and Eastman [1].
During the last several years many authors have studied the boundary layer flow of nanofluid fluids through different geometries and with different conditions. Examples include Kameswaran et al. [2] who studied hydromagnetic nanofluid flow due to a stretching or shrinking sheet and Kameswaran et al. [3] who found solutions for the equations for the stagnationpoint flow of a nanofluid over a stretching surface. The steady nanofluid boundary layer flow along a vertical cone in a porous medium was investigated by Fauzi et al. [4]. Boutra et al. [5] studied free convection enhancement within a nanofluid’ filled enclosure with square heaters and Ambethkar and Kumar [6] examined solutions of 2D unsteady incompressible flow with heat transfer in a driven square cavity sing streamfunctionvorticity formulation. Cheng [7] discussed natural convection in boundary layer flow
Figure 1. Geometry and the coordinate system
We consider a twodimensional unsteady boundary layer flow of an incompressible viscous nanofluid along a vertical stretching or shrinking cone embedded in a porous medium. The coordinate system and the physical model are shown in Fig. 1. The cone stretches or shrinks with velocity U_{0} = u_{0}x^{m/}^{3} where m is an exponent. Mangler’s transformation is used (see Schlichting, [24]) to reduce the axisymmetric system to a twodimensional problem. Then equations can be written as,
$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ (1)
$\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}=\frac{1}{\rho_{n f}} \frac{\partial p}{\partial x}+$$v_{n f}\left(\frac{\partial^{2} u}{\partial y^{2}}\right)+\frac{(\rho \beta)_{n f}}{\rho_{n f}}\left(TT_{\infty}\right) g \cos \Omega\frac{\sigma B_{0}^{2}}{\rho_{n f}} u$ (2)
$\frac{\partial T}{\partial t}+u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}=$$\alpha_{n f} \frac{\partial^{2} T}{\partial y^{2}}+\frac{Q}{\left(\rho c_{p}\right)_{n f}}\left(TT_{\infty}\right)+\frac{\mu_{n f}}{\left(\rho c_{p}\right)_{n f}}\left(\frac{\partial u}{\partial y}\right)^{2}$ (3)
$\frac{\partial C}{\partial t}+u \frac{\partial C}{\partial x}+v \frac{\partial C}{\partial y}=D_{\mathrm{m}} \frac{\partial^{2} C}{\partial y^{2}}R\left(CC_{\infty}\right)$
The boudary conditions are given by
u = U_{0} = u_{0}x^{m/}^{3}, v = 0, T = T_{w}, C = C_{w}, at y = 0 and t >0,
u →U_{∞} = u_{∞}x^{m/}^{3}, T→ T_{∞}, C→ C_{∞}, as y→∞ and t > 0. (5)
subject to the initial conditions
u = v = 0, T = T_{∞}, C = C∞ for t < 0.
The parameters are the effective dynamic viscosity μ_{nf} , the kinematic viscosity ν_{nf }, the thermal diffusivity α_{nf }, the heat capacity (ρc_{p})_{nf }, the density ρ_{nf }, the thermal expansion coefficient (ρβ)_{nf} and the thermal conductivity k_{nf} of nanofluid, which are given by (see Oztop and AbuNada [25]),
$\mu_{n f}=\frac{\mu_{f}}{(1\phi)^{2.5}} v_{n f}=\frac{\mu_{n f}}{\rho_{n f}} \quad \alpha_{n f}=\frac{k_{n f}}{\left(\rho c_{p}\right)_{n f}}$
$\left(\rho c_{p}\right)_{n f}=(1\phi)\left(\rho c_{p}\right)_{f}+\phi\left(\rho c_{p}\right)_{s}, ? \rho_{n f}=(1\phi) \rho_{f}+\phi \rho_{s}$
$(\rho \beta)_{n f}=(1\phi)(\rho \beta)_{f}+\phi(\rho \beta)_{s}, \frac{k_{n f}}{k_{f}}=\frac{\left(k_{s}+2 k_{f}\right)2 \phi\left(k_{f}k_{s}\right)}{\left(k_{s}+2 k_{f}\right)+\phi\left(k_{f}k_{s}\right)}$ (6)
We introduce the following transformations,
$\eta=\sqrt{\frac{U_{\infty}}{v_{f} x \xi}} y f \eta \xi=\frac{\psi}{\sqrt{v_{f} U_{\infty} x \xi}}$
$\xi=1e^{\tau}, z=\frac{U_{\infty}}{x} t \theta \eta \xi=\frac{TT_{\infty}}{T_{\mathrm{w}}T_{\infty}}$
$\Phi \eta \xi=\frac{CC_{\infty}}{C_{\mathrm{w}}C_{\infty}} T_{\mathrm{w}}=T_{\infty}+T_{0} x^{\frac{(2 m3)}{3}}$
$C_{\mathrm{w}}=C_{\infty}+C_{0} x^{\frac{(2 m3)}{3}}$ (7)
And the stream function ψ is chosen such that
u = ∂ψ/∂y, v =  ∂ψ/∂x. (8)
Substituting the transformations (7) into Eqs. (1)  (4), the (1) is automatically satisfied and Eqs. (2)(4) reduce to
$f^{\prime \prime \prime}+\phi_{1}\left\{\frac{1}{2}(1\xi) \eta f^{\prime \prime}+f_{1} f^{\prime \prime}+\xi\left[\left(\frac{m}{3}\right)\left(1\left(f^{\prime}\right)^{2}\right)+\frac{M n^{2}}{\phi_{2}}\left(1f^{\prime}\right)+\frac{\phi_{3}}{\phi_{2}} \lambda \theta\right]\right\}$
$=\phi_{1} \xi(1\xi)\left\{f_{2} f^{\prime \prime} \frac{\partial f}{\partial \xi}+\left(1f_{2} f^{\prime}\right) \frac{\partial f^{\prime}}{\partial \xi}\right\}$ (9)
$\theta^{\prime \prime}+\frac{k_{f}}{k_{n f}} \operatorname{Pr} \phi_{4}\left\{\frac{1}{2}\xi \eta \theta^{\prime}+f_{1} f \theta^{\prime}\xi\left[\left(\frac{2 m3}{3}\right) f^{\prime} \theta\frac{\delta}{\phi_{4}} \theta\right]+\frac{E c}{\phi_{5}} f^{\prime \prime 2}\right\}$
$=\frac{k_{f}}{k_{n f}} \operatorname{Pr} \phi_{4} \xi(1\xi)\left\{f_{2} \theta^{\prime} \frac{\partial f}{\partial \xi}+\left(1f_{2} f^{\prime}\right) \frac{\partial \theta}{\partial \xi}\right\}$ (10)
$\Phi^{\prime \prime}+S c\left\{\frac{1}{2}\xi \eta \Phi^{\prime}+f_{1} f \Phi^{\prime}\xi\left[\left(\frac{2 m3}{3}\right) f^{\prime} \Phi+\gamma \Phi\right]\right\}$
$=S c \xi(1\xi)\left\{f_{2} \Phi^{\prime} \frac{\partial f}{\partial \xi}+\left(1f_{2} f^{\prime}\right) \frac{\partial \Phi}{\partial \xi}\right\}$ (11)
The boundary conditions in Eq. (5) are transformed to
f = 0, f′= ε, θ = Φ = 1 at η = 0, 1≥ ξ ≥ 0,
f′= 1, θ = Φ = 0 as η → ∞, 1 ≥ ξ ≥ 0, (12)
where
$f_{1}=\xi\left(\frac{m+3}{6}\right)\left(\frac{m3}{6}\right)(1\xi) \log (1\xi), 2 f_{2}=\left(\frac{m3}{3}\right) \log (1\xi)$$\phi_{1}=\phi^{2.5}\left[\phi+\phi\left(\frac{\rho_{s}}{\rho_{f}}\right)\right] \quad \phi_{2}=\left[\phi+\phi \frac{\rho_{s}}{\rho_{f}}\right] \phi_{3}=\left[\phi+\phi \frac{(\rho \beta)_{s}}{(\rho \beta)_{f}}\right]$$\phi_{4}=\left[1\phi+\phi \frac{\left(\rho c_{p}\right)_{s}}{\left(\rho c_{p}\right)_{f}}\right], ? \phi_{5}=(1\phi)^{2.5}\left[1\phi+\phi \frac{\left(\rho c_{p}\right)_{s}}{\left(\rho c_{p}\right)_{f}}\right]$ (13)
In the above equations, the prime denotes differentiation with respect to η. The parameters are defined as
$M n=\sqrt{\frac{\sigma B_{0}^{2} x}{\rho_{f} U_{\infty}}}, ? \lambda=\frac{G r_{x}}{R e_{x}^{2}} G r_{x}=\frac{g \beta_{f}\left(T_{\mathrm{w}}T_{\infty}\right) x^{3} \cos \Omega}{v_{f}^{2}}$$R e_{x}=\frac{U_{\infty} x}{v_{f}}, \operatorname{Pr}=\frac{v_{f}\left(\rho c_{p}\right)_{f}}{k_{f}} \delta=\frac{Q x}{\left(\rho c_{p}\right)_{f} U_{\infty}}$$E c=\frac{U_{\infty}^{2}}{\left(c_{p}\right)_{f}\left(T_{w}T_{\infty}\right)}, ? S c=\frac{v_{f}}{D_{m}} \gamma=\frac{R x}{U_{\infty}} \varepsilon=\frac{u_{0}}{u_{\infty}}$ (14)
where Rex is the local Reynolds number and Grx is the local Grashof number (see Gangadhar et al. [26], Mahdy [27]). It must be noted that λ > 0 corresponds to the case of buoyancy assisting the flow while λ < 0 corresponds to buoyancy opposing the flow and λ = 0 suggests pure forced convection. When ξ = 0 and ϕ = 0 (regular fluid), Eq. (9) reduces to the ordinary differential equation,
f′′′+1/2 η f′′= 0, (15)
with boundary conditions (when ε = 0) are
f(0, 0) = 0, f′(0, 0) = 0, f′(1, 0) = 1
In studies of this nature, we also often interested in the skin friction coefficient Cfx , the Nusselt number Nux and the Sherwood number Shx. These defined as
C_{fx} = 2τ_{w}/ρ_{f}U^{2}_{∞}, Nu_{x }= xq_{w} /k_{f} (T_{w}  T_{∞}),
Sh_{x} = xq_{m}/ D_{m}(C_{w}  C_{∞}) . (16)
where τ_{w} is the shear stress at the cone surface, q_{w} and q_{m} are the heat and mass flux from the cone surface, respectively,
$\tau_{w}=\mu_{n f}\left(\frac{\partial u}{\partial y}\right)_{y=0}, ? q_{w}=k_{n f}\left(\frac{\partial T}{\partial y}\right)_{y=0} q_{m}=D_{m}\left(\frac{\partial C}{\partial y}\right)_{y=0}$ (17)
and substituting (6) into (16) and (17), we get
$\frac{1}{2} \sqrt{R e_{x}} C_{f_{x}}=\frac{(1\phi)^{2.5}}{\sqrt{\xi}} f^{\prime \prime}(0, \xi)$
$R e_{x}^{1 / 2} N u_{x}=\frac{1}{\sqrt{\xi}} \frac{k_{n f}}{k_{f}} \theta^{\prime}(0, \xi), ?$
$R e_{x}^{1 / 2} S h_{x}=\frac{1}{\sqrt{\xi}} \Phi^{\prime}(0, \xi)$ (18)
We use the spectral local linearization method (SLLM) to solve the system of equations (9)(11). Eqs. (9)  (11) are linearized using the GaussSeidel approach (see Motsa [28]). The principle of the SLLM algorithm is to linearize and decouple the system of equations. Nonetheless, this method has only been used in a limited number of studies, hence its general validation in complex systems remains to be made. The detail derivation of SLLM algorithm is described in [28].
Here, the computational domain in the ηdirection is chosen so that L = 30. This value was found to give accurate results for all selected physical parameters. Increasing η does not change the results to a significant extent. The number of collocation points used in the spectral method discretization is N_{x} = 100 in all cases. We note that the computation of some quantity, say F^{n}^{+1 }_{r}_{+1}, at each time step is achieved by iterating using the local linearization method using a known value at the previous time step n as the initial approximation. The calculations were carried out until the desired tolerance level ϵ = 10^{−}^{6} was attained. The tolerance level is the maximum value of the infinity norm of the difference between the values of the calculated quantities, that is to ensure the accuracy of the results, a sufficiently small step size Δξ was used. The step size was chosen to be small enough such that further reduction did not change the results.
The nanofluid velocity profiles f′ (η, ξ) for different values of the nanoparticle volume fraction ϕ and stretching or shrinking parameter ε are given in Fig. 2. Fig. 2(a) shows that the nanofluid velocity decreases when the nanoparticle volume fraction increases. Fig. 2(b) shows that the TiO_{2}water nanofluid has marginally higher values increasing stretching parameter values ε > 0 while opposite trend is observed for a shrinking parameter ε < 0. The stretching or shrinking is due to the impulsive force that acts in xdirection (+ve or ve). The findings in the case of Cuwater nanofluid are similar to the result obtained by Grosan and Pop [29].
The unsteady boundary layer flow of two water based nanofluids along a vertical stretching or shrinking cone was studied. The flow was subject to viscous dissipation, internal heat generation and a chemical reaction. The nonsimilar partial differential equations were solved using the spectral local linearization method. We have investigated the effects of the nanoparticle volume fraction (ϕ), magnetic field parameter (Mn), buoyancy parameter (λ), stretching or shrinking parameter (ε), heat generation parameter (δ), Eckert number (Ec), chemical reaction parameter (γ) on the nanofluid velocity, temperature and concentration profiles as well as the skin friction coefficient, heat and mass transfer coefficients. We have used the values m = 4, Pr = 6.7 and Sc = 1 unless otherwise stated. We have considered Copper (Cu) and Titanium oxide (TiO_{2}) nanoparticles with water as the base fluid. We note that ϵ < 0 for a shrinking cone and ϵ > 0 indicates that the cone is stretching. The thermophysical properties of the base fluid and the nanoparticles are listed in Table 1. To determine the accuracy of the numerical method, solutions for some special cases are presented in Tables 2 and 3. The results are in excellent agreement with the work of Kameswaran et al. [3], Wang [22] and Jafar et al. [23].
Table 1. Thermophysical properties of the base fluid and the nanoparticles Oztop and AbuNada [25]
Physical properties 
Cp (J/kgK) 
Ρ (Kg/m^{3}) 
K (W/mK) 
β ×10^{5} (K^{1}) 
Pure water (H_{2}O) 
4179 
997.1 
0.613 
21 
Copper (Cu) 
385 
8933 
401 
1.67 
Titanium Oxide (TiO_{2}) 
686.2 
4250 
8.9538 
0.9 
Table 2. Comparison of the skin friction coefficient f’’(0, 1) and heat transfer rate θ′(0, 1), for various values of stretching or shrinking parameter ε when Mn = λ = δ = Ec = 0,m = 3, Pr = 1 and ϕ = 0.

Suali et al. [30] 
Present results 
SLL Method 

ϵ 
f’’(0, 1) 
ө^{’}(0, 1) 
f’’(0, 1) 
ө^{’}(0, 1) 
4 7.086378 2.116738 7.086378 2.116738 3 4.276545 1.870671 4.276542 1.870671 0.2 1.051130 0.913303 1.051130 0.913303 0.1 1.146561 0.863452 1.146561 0.863452 0.2 1.373886 0.501448 1.373886 0.698748 0.5 1.495672 0.501448 1.495670 0.501447 1.15 1.082232 0.2979953 1.082491 0.297346 
Table 3. Comparison of the skin friction coefficient f’’(0, 1) for various values of stretching or shrinking parameter ε when Mn = λ = 0,m = 3 and ϕ = 0.

Wang [22] 
Jafar et al. [23] 
Kameswaran et al. [3] 
Present results 
ϵ 
f’’(0, 1) 
f’’(0, 1) 
f’’(0, 1) 
f’’(0, 1) 
0.0 1.232588 1.2326 1.232588 1.232588 0.1 1.14656 1.1466 1.146561 1.146561 0.2 1.05113 1.0511 1.051130 1.051130 0.5 0.71330 0.7133 0.713295 0.713295 1.0 0.00000 0.0000 0.000000 0.000000 2.0 1.88731 1.8873 1.887307 1.887307 5.0 10.26475 10.2648 10.264749 10.264751 
Figure 2. a) Effect of various nanoparticle volume fraction (ϕ). (b) Effect of stretching or shrinking parameter (ε) on velocity profiles
Figs. 3 and 4 illustrate the effects of nanoparticle volume fraction (ϕ), stretching or shrinking parameter (ε), heat generation parameter (δ) and Eckert number (Ec) on the temperature profiles θ(ξ,η) for both Cuwater and TiO_{2}water nanofluids. The temperature profiles increase with increases in ϕ, δ and Ec. We note that an increase in the nanoparticle volume fraction increases the thermal conductivity of the nanofluid significantly, and that internal heat generation increases the temperature of the nanofluid. An increase in the Eckert number increases dissipation due to fluid viscosity or frictional heating. The surface gets cooler when the dissipation increases and as a result there is a transfer of heat from the surface to the nanofluid which causes the temperature to increase. On the other hand, the Cuwater nanofluid temperature decreases with increasing stretching parameter ε > 0 and increases with shrinking parameter ε < 0.
Figure 3. (a) Effect of nanoparticle volume fraction (ϕ). (b) Effect of stretching or shrinking parameter (ε) on temperature profiles.
Figure 4. (a) Effect of heat generation parameter (δ). (b) Effect of Eckert number (Ec) on temperature profiles.
Fig.5 shows that the concentration profiles increase with nanoparticle volume fraction but reduce with stretching. It is observed from Fig. 6 that the concentration profiles decrease with Mn, λ and γ for both nanofluids. It is observed in Fig.7(a) that the Cuwater nanofluid has higher skin friction coefficient values compared to the TiO_{2}water nanofluid for the increasing values of ξ. Fig.7(b) shows that the value of the skin friction coefficient of a Cuwater nanofluid are higher than for TiO_{2} water nanofluid when ϵ > 0 but the opposite trend is observed for ϵ < 0. These results show that the skin friction coefficient decreases with increasing nanoparticle volume friction and stretching or shrinking parameters.
Figure 5. (a) Effect of nanoparticle volume fraction (ϕ). (b) Effect of stretching or shrinking parameter (ε) on concentration profiles
Figure 6. a) Effect of magnetic field parameter (Mn), (b) Effect of buoyancy parameter (λ), (c) Effect of chemical reaction parameter (γ) on concentration profiles.
Figure 7. (a) Effect of nanoparticle volume fraction (ϕ). (b) Effect of stretching or shrinking parameter (ε) on skin friction coefficients.
Fig. 8(a) shows that the heat transfer rate for a Cuwater nanofluid is smaller compared to the TiO_{2}water nanofluid at the surface of the cone as the nanoparticle volume friction increases. We observe from Figs. 8 (a) and (b) that the heat transfer rate decreases with the increasing ϕ but the opposite trend is observed for ϵ. The Cuwater gives a smaller heat transfer rate in comparison to the TiO_{2}water nanofluid for increasing δ. Again, we note that δ, Ec reduce the heat transfer rate in both Figs. 9 (a) and (b).
Figure 8. (a) Effect of nanoparticle volume fraction (ϕ). (b) Effect of stretching or shrinking parameter (ε) on heat transfer rate
Figure 9. (a) Effect of heat generation parameter (δ). (b) Effect of Eckert number (Ec) on heat transfer rate.
Figure 10. (a) Effect of nanoparticle volume fraction (ϕ). (b) Effect of stretching or shrinking parameter (ε) on mass transfer rate
The impact of increasing the nanoparticle volume fraction, stretching or shrinking parameters, magnetic field, buoyancy force parameter and chemical reaction parameters on the mass transfer rate at the cone surface is shown in Fig. 10. Fig. 10 (a) shows that the Cuwater nanofluid takes higher values than the TiO_{2}water nanofluid for the increasing value of ϕ. The mass transfer rate reduces with increasing ϕ for both nano fluids but the opposite trend is observed in Fig. 10 (b).
Figure 11. (a) Effect of magnetic field parameter (Mn), (b) Effect of buoyancy parameter (λ), (c) Effect of chemical reaction parameter (γ) on mass transfer rate.
Fig. 11 shows that the mass transfer rate increases with increasing Mn, λ and γ for both nanofluids. The effects of magnetic field parameter Mn and buoyancy force parameter λ on the velocity and the skin friction coefficient are shown in Figs. 12 and 13 respectively. We note that the Cuwater nanofluid assumes higher velocity and skin friction coefficient than the TiO_{2}water nanofluid for the increasing values of Mn, λ.
Figure 12. a) Effect of magnetic field parameter (Mn), (b) Effect of buoyancy parameter (λ) on velocity profiles.
Figure 13. (a) Effect of magnetic field parameter (Mn), (b) Effect of buoyancy parameter (λ) on skin friction coefficients.
The influence of the magnetic field and buoyancy force parameters on the nanofluid temperature and the local Nusselt number are captured in Figs. 14 and 15, respectively. It is noted from Fig. 15 that a Cuwater nanofluid has smaller values of the local Nusselt number than a TiO_{2}water nanofluid for the increasing value of Mn, λ. We further observe that the local Nusselt number decreases with increases in Mn and λ.
Figure 14. (a) Effect of magnetic field parameter (Mn), (b) Effect of buoyancy parameter (λ) on temperature profiles
Figure 15. (a) Effect of magnetic field parameter (Mn), (b) Effect of buoyancy parameter (λ) on heat transfer rate.
Fig. 16 shows the effect of the Prandtl number on the temperature profiles. The temperature profiles and the thermal boundary layer thickness quickly decreases with increasing Prandtl numbers. The Prandtl number is a means to increase fluid viscosity resulting in a reduction in the flow velocity and temperature. Here, the thermal boundary layer thickness decreases with increasing Prandtl number, which is consistent with the findings of various researchers. Fig 17 shows the streamlines for different value of ϵ when the other values are fixed.
Figure 16. Effect of (Pr) on temperature profiles
Figure 17. Streamlines for (a) ϵ =  1.2, (b) ϵ = 1.0, (c) ϵ = 2.0 when the other parameters are fixed.
The unsteady boundary layer flow of a viscous, incompressible fluid along a vertical stretching or shrinking cone was investigated. The effects of viscous dissipation, internal heat generation and a chemically reactive species have been taken into account for Cuwater and TiO2water nanofluids.
It was found that the viscous dissipation has the effect of increasing the nanofluid temperature within the boundary layer region while the rate of heat transfer from the surface decreases with an increase in viscous dissipation. The internal heat generation has the tendency to increase the nanofluid temperature and reduce the rate of heat transfer at the surface of the cone. The nanoparticle concentration decreases while the wall mass transfer rate increases with the increase in the strength of a chemical reaction.
All the authors are thankful to University of KwaZuluNatal, South Africa and Amity University, Kolkata, India for the necessary support.
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