Flow and heat transfer of an Oldroyd-B nanofluid thin film over an unsteady stretching sheet

doi:10.1016/j.molliq.2016.04.108

Journal of Molecular Liquids

Volume 220, August 2016, Pages 665-670

https://doi.org/10.1016/j.molliq.2016.04.108 Get rights and content

Highlights

•
We establish the unsteady boundary layer governing equations of Oldroyd-B fluid.
•
We obtain the similarity solutions of mathematical formulation.
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The solid volume fraction promotes heat transfer of the liquid film remarkably.
•
The Ag-PVA nanofluid has better enhancement on heat transfer than Cu-PVA nanofluid.
•
The Ag-PVA nanofluid has a lower skin friction coefficient than Cu-PVA nanofluid.

Abstract

The unsteady boundary layer flow and heat transfer of an incompressible Oldroyd-B nanofluid thin film due to a stretching sheet have been investigated analytically. Two different types of nanoparticles Cu and Ag are considered with poly vinyl alcohol (PVA)–water used as a base fluid. The velocity and temperature of the stretching sheet are assumed to vary both along the sheet and with time. The unsteady boundary layer governing equations are firstly established and then reduced to the coupled 4-order nonlinear ordinary differential equations by similarity transformation. The analytical solutions are obtained using the homotopy analysis method (HAM), and show good agreement compared with previous results. The influences of various relevant parameters such as unsteady parameter, volume fraction of Cu/Ag and Prandtl number on the flow field are elucidated through tables and graphs.

Introduction

In the recent years, researches on the flow and heat transfer of viscoelastic fluid over a stretching sheet have got more and more attentions for its application in polymer processing engineering, steel fiber coating engineering, chemical equipment processing and foodstuff processing, etc. The viscoelastic fluid has elastic and viscous simultaneously. As a non-Newtonian fluid, it cannot be described by Newtonian constitutive relation, so researchers have proposed various constitutive models to predict different rheological features, including the Maxwell model, Oldroyd-B model, and Burgers' model [1], [2], [3], [4], [5], [6], [7], [8].

Due to its wide applications in industries, the flow characteristic of viscoelastic fluids over continuously moving surfaces has been extensively investigated. Shehzad et al. [9] presented the boundary layer flow of a Maxwell fluid with variable thermal conductivity due to a stretching surface. Kumari and Nath [10] explored flow of Maxwell fluid over an exponentially stretching vertical surface with magnetic field and viscous dissipation. Hayat et al. [11] proposed mass transfer effects in three-dimensional flow of Maxwell fluid over a stretching surface. Sajid et al. [12] studied heat transfer characteristics of an Oldroyd-B fluid in the presence of a constant applied magnetic field. Hayat et al. [13] analyzed the three-dimensional flow and heat transfer of an Oldroyd-B fluid over a bidirectional stretching surface. Other researches on viscoelastic fluids over a stretching surface can be found in Refs. [14], [15], [16], [17], [18].

On the other hand, as an important component of viscoelastic fluid, the polymer has the advantages of light weight, high corrosion resistance, low cost, insulation and so on, but has relatively poor thermal conductivity. To achieve its heat transfer efficiency, the metal or metallic oxide nanoparticles are mixed in polymer fluid. In fact, investigations about the heat transfer mechanics of the Newtonian fluid mixed into nanoparticles can be found in a large amount of literatures [19], [20], [21], [22], [23], [24]. Pop et al. [25] studied flow and heat transfer of a nano-liquid over an unsteady stretching surface. On the basis of Pop's work, Ganji et al. [26] analyzed magnetic field and thermal radiation effect on the flow of a nanofluid. Until recently, the flow characteristics of viscoelastic nanofluid begin to be studied. Haq et al. [27] investigated MHD boundary layer flow of a Maxwell fluid in the presence of nanoparticles using numerical method. Ramesh and Gireesha [28] obtained the numerical solutions of a Maxwell nanofluid over a stretching surface with the effect of heat source/sink. Hayat et al. [29] considered the heat and mass transfer of three-dimensional Maxwell nanofluid flow under the Brownian motion and the thermophoresis effects.

All of the above studies mainly focus on infinite fluid. In fact, the flow and heat transfer of finite film is more common in practical application such as injection molding and tape casting etc. Wang [30] first studied liquid film of Newtonian fluid on an unsteady stretching surface. Noor et al. [31] explored thermocapillarity and magnetic field effects in a thin liquid film. More researches about thin film can be found in Refs. [32], [33], [34], [35], [36]. But, they are mainly focused on the Newtonian fluid field.

To the best of our knowledge, no one has ever attempted to study the flow of an Oldroyd-B nanofluid film over an unsteady stretching sheet. Compared with Maxwell fluid, the constitutive relation of Oldroyd-B fluid has more applications, but is more complex. Main objective of this study is to analyze the flow and heat transfer characteristics of PVA-water based viscoelastic nanofluids containing two different types of nanoparticles: namely, copper (Cu) and silver (Ag) due to unsteady stretching sheet. Experimental studies show that the PVA-water has viscoelastic properties [37]. The thermal and mechanical stabilities of PVA-water are improved greatly by adding nanoparticles, and the PVA-nanocomposite materials can get a wider range of application [38], [39]. The unsteady boundary layer equations of Oldroyd-B fluid are established firstly. The analytic solutions are presented by using the homotopy analysis method (HAM). Moreover, the effects of relevant parameters on the velocity and temperature fields are shown graphically.

Section snippets

Mathematical analysis

In this paper, we consider volume fraction of Cu/Ag effects on the unsteady Oldroyd-B fluid flow driven by a stretching sheet in PVA-water based nanofluid containing different type of nanoparticles. In this study, the PVA-water with high concentration (10%) is used as a base fluid. Consider the thin elastic sheet that emerges from a narrow slit at origin of the Cartesian coordinate system shown in Fig. 1. The continuous sheet aligned with the x-axis at y = 0 moves in its own plane with a variable

Analytical method

In this section, we solve the problem consisting of Eqs. (16) − (19) by HAM. The functions f (η) and θ (η) can be expressed as follows: $f (η) = \sum_{k = 0}^{+ \infty} a_{k} η^{k}, θ (η) = \sum_{k = 0}^{+ \infty} b_{k} η^{k},$ in which a_k and b_k are the coefficients.

The initial guesses for f (η) and θ (η) of the Eqs. (16), (17) along with boundary condition (18), (19) are $f_{0} (η) = η + (S - 2) η^{2} + (- \frac{2}{3} S + \frac{4}{3}) η^{3} + (\frac{1}{6} S - \frac{1}{3}) η^{4},$ $θ_{0} (η) = 1 .$

The auxiliary linear operators are taken as follows: $L_{f} = \partial^{4} / \partial η^{4}, L_{θ} = \partial^{2} / \partial η^{2},$ with L_f[C₁ + C₂η + C₃η² + C₄η³] = 0 and L_θ[C₁ + C₂η] = 0, here C_i(i = 1, 2, 3, 4) are constants

Results and discussion

The HAM provides a great freedom to select the auxiliary parameters. The auxiliary parameters h_f and h_θ are important to control and adjust the convergence of series solutions. The range of values for h_θ can be obtained as shown in Fig. 2. The graph shows that the range of admissible values of h_θ is − 0.6 ≤ h_θ ≤ − 0.2 using the 10th-order approximation. To construct series solutions, we take the admissible value h_f = h_θ = − 0.35.

The various physical properties of the base fluid and two types of

Conclusions

In this article, the flow and heat transfer of an Oldroyd-B nanofluid in a finite thin film over an unsteady stretching sheet has been analyzed. The reduced momentum and energy equations for the Oldroyd-B fluid have been obtained in the presence of nanoparticles. Analytical solutions are obtained using the homotopy analysis method (HAM) with similarity transformation. We can draw a conclusion that the velocity profile decreases with an increase in nanoparticle volume fraction, while the

Acknowledgements

This work is supported by the National Natural Science Foundation of China (nos. 21206009, 21576023), Funding Project for Academic Human Resources Development in Beijing University of Civil Engineering and Architecture (no. 21221214111). The authors sincerely thank the editor and referees for their helpful comments and suggestions.

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Flow and heat transfer of an Oldroyd-B nanofluid thin film over an unsteady stretching sheet

Highlights

Abstract

Introduction

Section snippets

Mathematical analysis

Analytical method

Results and discussion

Conclusions

Acknowledgements

Comput. Fluids

J. Taiwan Inst. Chem. Eng.

J. Hydrodyn.

Comput. Math Appl.

J. Egypt. Math Soc.

J Egypt Math Soc.

Nonlinear Anal. Real

Int. J. Heat Mass Transf.

Int. J. Therm. Sci.

Alex. Eng. J.

Int. J. Heat Mass Transf.

Appl. Math. Model.

Int. J. Heat Mass Transf.

J. Taiwan Inst. Chem. E

Ain Shams Eng. J.

Int. J. Heat Mass Transf.

J. Non-Newtonian Fluid

Int. J. Heat Mass Transf.

Int. J. Heat Mass Transf.

Powder Technol.