Impulsive nanofluid flow along a vertical stretching cone

Impulsive nanofluid flow along a vertical stretching cone

Sami M.S. Ahamed Sabyasachi Mondal*Precious Sibanda

School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal Private Bag X01 Scottsville 3209, South Africa

Department of Mathematics, Amity University, Kolkata, Newtown-700135, West Bengal, India

Corresponding Author Email: 
sabya.mondal.2007@gmail.com
Page: 
1005-1014
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DOI: 
https://doi.org/10.18280/ijht.350437
Received: 
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Accepted: 
| | Citation

OPEN ACCESS

Abstract: 

We study the unsteady convective flow of two water-based 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 two-dimensional 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.

Keywords: 

 Chemical Reaction, Nanofluid Flow, Stretching or Shrinking Cone, Spectral Local Linearization Method.

1. Introduction

 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 thermo-physical properties of these fluids can be significantly enhanced by suspending nano-sized 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 stagnation-point 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 2-D unsteady incompressible flow with heat transfer in a driven square cavity sing streamfunction-vorticity formulation. Cheng [7] discussed natural convection in boundary layer flow

2. Mathematical Formulation

Figure 1. Geometry and the coordinate system

We consider a two-dimensional 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 U0 = u0xm/3 where m is an exponent. Mangler’s transformation is used (see Schlichting, [24]) to reduce the axisymmetric system to a two-dimensional 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(T-T_{\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(T-T_{\infty}\right)+\frac{\mu_{n f}}{\left(\rho c_{p}\right)_{n f}}\left(\frac{\partial u}{\partial y}\right)^{2}$     (3)

$\frac{\partial C}{\partial t}+u \frac{\partial C}{\partial x}+v \frac{\partial C}{\partial y}=D_{\mathrm{m}} \frac{\partial^{2} C}{\partial y^{2}}-R\left(C-C_{\infty}\right)$

The boudary conditions are given by

u = U0 = u0xm/3, v = 0, T = Tw, C = Cw, at y = 0 and t >0,

u →U = uxm/3, T→ T, C→ C, as y→ and t > 0.        (5)

subject to the initial conditions

u = v = 0, T = T, C = Cfor t < 0.

The parameters are the effective dynamic viscosity μnf , the kinematic viscosity νnf , the thermal diffusivity αnf , the heat capacity (ρcp)nf , the density ρnf , the thermal expansion coefficient (ρβ)nf and the thermal conductivity knf of nanofluid, which are given by (see Oztop and Abu-Nada [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,

$\eta=\sqrt{\frac{U_{\infty}}{v_{f} x \xi}} y f \eta \xi=\frac{\psi}{\sqrt{v_{f} U_{\infty} x \xi}}$

$\xi=1-e^{-\tau}, z=\frac{U_{\infty}}{x} t \theta \eta \xi=\frac{T-T_{\infty}}{T_{\mathrm{w}}-T_{\infty}}$

$\Phi \eta \xi=\frac{C-C_{\infty}}{C_{\mathrm{w}}-C_{\infty}} T_{\mathrm{w}}=T_{\infty}+T_{0} x^{\frac{(2 m-3)}{3}}$

$C_{\mathrm{w}}=C_{\infty}+C_{0} x^{\frac{(2 m-3)}{3}}$     (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

$f^{\prime \prime \prime}+\phi_{1}\left\{\frac{1}{2}(1-\xi) \eta f^{\prime \prime}+f_{1} f^{\prime \prime}+\xi\left[\left(\frac{m}{3}\right)\left(1-\left(f^{\prime}\right)^{2}\right)+\frac{M n^{2}}{\phi_{2}}\left(1-f^{\prime}\right)+\frac{\phi_{3}}{\phi_{2}} \lambda \theta\right]\right\}$

$=\phi_{1} \xi(1-\xi)\left\{f_{2} f^{\prime \prime} \frac{\partial f}{\partial \xi}+\left(1-f_{2} f^{\prime}\right) \frac{\partial f^{\prime}}{\partial \xi}\right\}$     (9)

$\theta^{\prime \prime}+\frac{k_{f}}{k_{n f}} \operatorname{Pr} \phi_{4}\left\{\frac{1}{2}-\xi \eta \theta^{\prime}+f_{1} f \theta^{\prime}-\xi\left[\left(\frac{2 m-3}{3}\right) f^{\prime} \theta-\frac{\delta}{\phi_{4}} \theta\right]+\frac{E c}{\phi_{5}} f^{\prime \prime 2}\right\}$

$=\frac{k_{f}}{k_{n f}} \operatorname{Pr} \phi_{4} \xi(1-\xi)\left\{f_{2} \theta^{\prime} \frac{\partial f}{\partial \xi}+\left(1-f_{2} f^{\prime}\right) \frac{\partial \theta}{\partial \xi}\right\}$     (10)

$\Phi^{\prime \prime}+S c\left\{\frac{1}{2}-\xi \eta \Phi^{\prime}+f_{1} f \Phi^{\prime}-\xi\left[\left(\frac{2 m-3}{3}\right) f^{\prime} \Phi+\gamma \Phi\right]\right\}$

$=S c \xi(1-\xi)\left\{f_{2} \Phi^{\prime} \frac{\partial f}{\partial \xi}+\left(1-f_{2} f^{\prime}\right) \frac{\partial \Phi}{\partial \xi}\right\}$     (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{m-3}{6}\right)(1-\xi) \log (1-\xi), 2 f_{2}=\left(\frac{m-3}{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

Cfx = 2τw/ρfU2, Nux = xqw /kf (Tw - T),

Shx = xqm/ Dm(Cw - C) .     (16)

where τw is the shear stress at the cone surface, qw and qm 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)

3. Solution Method

We use the spectral local linearization method (SLLM) to solve the system of equations (9)-(11). Eqs. (9) - (11) are linearized using the Gauss-Seidel 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 η-direction 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 Nx = 100 in all cases. We note that the computation of some quantity, say Fn+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 = 106 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.

4. Results and Discussion

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 TiO2-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 x-direction (+ve or -ve). The findings in the case of Cu-water 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 non-similar 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 (TiO2) 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 Abu-Nada [25]

Physical properties

Cp

(J/kgK)

Ρ

(Kg/m3)

K

(W/mK)

β ×105

(K-1)

Pure water (H2O)

4179

997.1

0.613

21

Copper (Cu)

385

8933

401

1.67

Titanium Oxide (TiO2)

686.2

4250

8.9538

0.9

Table 2. Comparison of the skin friction coefficient f’’(0, 1) and heat transfer rate (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)

(0, 1)

f’’(0, 1)

(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 2a) 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 Cu-water and TiO2-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 Cu-water 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 Cu-water nanofluid has higher skin friction coefficient values compared to the TiO2-water nanofluid for the increasing values of ξ. Fig.7(b) shows that the value of the skin friction coefficient of a Cu-water nanofluid are higher than for TiO2- 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 Cu-water nanofluid is smaller compared to the TiO2-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 Cu-water gives a smaller heat transfer rate in comparison to the TiO2-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 Cu-water nanofluid takes higher values than the TiO2-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 Cu-water nanofluid assumes higher velocity and skin friction coefficient than the TiO2-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 Cu-water nanofluid has smaller values of the local Nusselt number than a TiO2-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.

5. Conclusions

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 Cu-water and TiO2-water 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.

Acknowledgements

All the authors are thankful to University of KwaZulu-Natal, South Africa and Amity University, Kolkata, India for the necessary support.

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