International Journal of Engineering Mathematics

Volume 2015, Article ID 272079, 15 pages

http://dx.doi.org/10.1155/2015/272079

## Homotopy Simulation of Nonlinear Unsteady Rotating Nanofluid Flow from a Spinning Body

^{1}Gort Engovation (Engineering Sciences Research), 11 Rooley Croft, Bradford BD6 1FA, UK^{2}Department of Mathematics, University of Peshawar, Khyber Pakhtunkhwa 25120, Pakistan^{3}Aerospace, Department of Engineering and Mathematics, Sheaf Building, Sheffield Hallam University, Sheffield S1 1WB, UK

Received 10 May 2015; Revised 16 August 2015; Accepted 16 August 2015

Academic Editor: Josè A. Tenereiro Machado

Copyright © 2015 O. Anwar Bég 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.

#### Abstract

The development of new applications of nanofluids in chemical engineering and other technologies has stimulated significant interest in computational simulations. Motivated by coating applications of nanomaterials, we investigate the transient nanofluid flow from a time-dependent spinning sphere using laminar boundary layer theory. The free stream velocity varies continuously with time. The unsteady conservations equations are normalized with appropriate similarity transformations and rendered into a ninth-order system of nonlinear coupled, multidegree ordinary differential equations. The transformed nonlinear boundary value problem is solved using the homotopy analysis method (HAM), a semicomputational procedure achieving fast convergence. Computations are verified with an Adomian decomposition method (ADM). The influence of acceleration parameter, rotational body force parameter, Brownian motion number, thermophoresis number, Lewis number, and Prandtl number on surface shear stress, heat, and mass (nanoparticle volume fraction) transfer rates is evaluated. The influence on boundary layer behavior is also investigated. HAM demonstrates excellent stability and leads to highly accurate solutions.

#### 1. Introduction

Nanofluids continue to stimulate significant interest in modern engineering and medical sciences [1]. These fluids offer significant thermal enhancement characteristics. Nanofluids contain suspended metallic nanoparticles, which increase the thermal conductivity of the base fluid by a substantial amount. Fabrication techniques for nanofluids are also being continuously refined [2]. The heat transfer coefficient of nanofluids increases with volume concentration and they offer other advantages. Nanofluids have been deployed in a tremendous spectrum of applications including sterilization of medical suspensions [3], nanomaterial processing [4, 5], automotive coolants [6], microbial fuel cell technology [7], polymer coating [8], intelligent building design [9], microfluid delivery devices [10], and aerospace tribology [11]. Although initially nanofluid dynamics research concentrated on quantification of fundamental thermophysical properties of nanofluids (including thermal conductivity, density, viscosity, and heat transfer coefficient), over the past few years the focus has become increasingly orientated towards modelling and simulation. In this regard building on mathematical models of nanofluid transport, an extensive range of numerical approaches have been implemented to simulate increasingly more practical problems of nanofluids. The inherent nonlinearity of nanofluid flows which involve momentum, heat, and mass transfer necessitates very robust computational algorithms for their analysis. The approaches range from intensive molecular dynamics methods [12, 13] which can model interfacial tension between nanoparticles, control volume methods [14], genetic algorithms [15], control volume finite elements [16], Nakamura finite difference codes [17], homotopy analysis techniques [18], ANSYS finite element commercial code [19], differential transform methods [20], MAPLE integration quadrature routines [21], Crank-Nicolson finite difference schemes [22], Keller box implicit methods [23], Mathematica integration subroutines [24], Blottner difference methods [25], and dual reciprocity boundary element methods [26]. These studies have generally examined Brownian motion and thermophoresis effects for various nanoparticle suspensions and considered two-dimensional boundary layer, channel, and cavity flows. They have ignored Coriolis body force effects which arise in rotational fluid mechanics. External flows from spinning bodies are significant in many branches of chemical and industrial engineering including electrolysis treatments [27] and polymer deposition on components [28]. In such systems the rotation strongly influences boundary layer growth and structure on the body periphery. This in turn controls heat and mass transfer rates whether flows are laminar, transitional, or indeed fully turbulent [29]. The bodies may be conical, spherical, elliptical, disk-shaped, and indeed concentric and eccentric systems. Many mathematical and numerical studies of such flows have been presented. Faltas and Saad [30] used a collocation method to analyze steady axisymmetric flow between two spinning eccentric spheres with a linear slip of Basset-type boundary condition at both surfaces. Andersson and Rousselet [31] studied partial momentum and thermal slip from a rotating disk with a Runge-Kutta method. Niazmand and Renksizbulut [32] employed a finite volume code to investigate unsteady heat transfer and thermal patterns around a rotating sphere (as a model of a particle) with surface blowing delineating three distinct wake regimes, namely, steady and axisymmetrical, steady but nonsymmetrical, and unsteady with vortex shedding. They found that although rotation strongly induces local modifications in flow patterns, the surface-averaged heat transfer rates were not altered markedly even at large rotational speeds. Roy and Anilkumar [33] used the Keller box method to simulate transient free and forced convection boundary layer flow from a rotating cone for the case when the free stream angular velocity and the angular velocity of the cone vary arbitrarily with the time. Subhashini et al. [34] used the Bellman-Kalaba quasilinearization method to address uniform slot injection/suction and nonuniform total enthalpy wall effects on steady nonsimilar laminar compressible boundary layer flow over a rotating sphere. They showed that greater rotation and total enthalpy at the wall encourages earlier flow separation whereas cooling delays this and furthermore that greater Mach number displaces the point of separation upstream as a result of the adverse pressure gradient.

The focus of the present work is to analyse* rotating nanofluid boundary layer flows at the stagnation point on the external surface of a spinning sphere*. Computational rotating nanofluid dynamics has recently attracted some interest, since the deployment of nanofluids in revolving chemical engineering devices offers significant improvements over existing designs. Rana et al. [35] used a variational finite element algorithm to study unsteady magnetonanofluid transport from a rotating stretching continuous sheet. They showed that greater rotational parameter reduces primary and secondary velocities, temperature, and nanoparticle concentration. They further showed that reduced Nusselt number (wall temperature gradient) was suppressed with both Brownian motion and thermophoresis effects whereas reduced Sherwood number (wall mass transfer gradient) was enhanced. Further studies of swirling nanofluid dynamics have been reported by Nadeem and Saleem [36] for a vertical cone and Malvandi [37] for stagnation point nanofluid flow from a spinning sphere. In the present study we use both a homotopy analysis method (HAM) and an Adomian decomposition method (ADM) to simulate stagnation point nanofluid flow from a rotating sphere. The influence of acceleration parameter, rotational body force parameter, Lewis number, Brownian motion number, and thermophoresis parameter on velocity, temperature, and nanoparticle distributions is examined. This work is relevant to coating applications in the polymeric industry.

#### 2. Mathematical Model

The physical regime under investigation is illustrated in Figure 1, in coordinate system. Transient laminar boundary layer flow of an incompressible Newtonian nanofluid is studied, in the vicinity of the stagnation point region of an isothermal rotating sphere of radius, , rotating with angular velocity, . Soret and Dufour effects are neglected. Following Anilkumar and Roy [38], the free stream and angular velocities depend on time in the form of and ,* where ** and * are arbitrary constants both greater than zero. The flow field is assumed to be axisymmetric and the fluid possesses constant thermophysical properties with the exception of those caused by density changes which generate the buoyancy force, under the Boussinesq approximation. The Buonjiornio nanofluid model is adopted prioritizing Brownian motion and thermophoresis effects [39]. In light of these approximations, the time-dependent conservation equations for mass, momentum, energy, and species (nanoparticle volume fraction) may be presented, as follows, as documented by Malvandi [37]:The appropriate initial and boundary conditions take the formHere , , and * denote* velocity components along the , , and coordinates where these coordinates are orientated, respectively, from the forward stagnation point along the surface (see Figure 1), normal to the surface and in the rotating directions, respectively, is the radial distance from a surface element to the axis of symmetry, is time, is initial condition, is the thermal diffusivity of the nanofluid, is kinematic viscosity of nanofluid, is the thermal conductivity of nanofluid, is the Brownian diffusion coefficient (a measure of the species diffusivity of nanoparticles), is the thermophoretic diffusion coefficient, * defines the* ratio of effective heat capacity of the nanoparticles (e.g., titanium oxide) to the heat capacity of the fluid, * is isothermal specific heat capacity* and * is the nanofluid temperature*, is the free stream temperature (at the edge of the boundary layer), and is the free stream nanoparticle concentration. It is assumed that the base fluid and the nanoparticles are in thermal equilibrium and no slip occurs between them. We note that, in (4), which is a statement of Fick’s law of mass (species) diffusion for nanoparticles, the first term on the left hand side is the* transient concentration gradient*, and the second and third terms are the convective mass transfer terms. The first term on the right hand side denotes the species diffusion and the last term is the relative contribution of thermophoresis to Brownian motion. These effects have also been considered in detail by Dib et al. [40] and Gupta and Saha Ray [41]. The boundary value problem defined in terms of primitive variables may be solved using numerical methods. However to provide a more amenable solution and one which enables the use of scaling parameters it is advantageous to introduce a set of similarity transformation variables. We define the following:Here is the acceleration (unsteadiness) parameter, is the transformed normal coordinate (perpendicular to sphere surface), is the normalized rotation parameter, is sphere radius, is dimensional stream function, is dimensionless stream function, is the dimensionless secondary velocity function, is the dimensionless temperature function, is dimensionless nanoparticle concentration function (volume fraction), Pr is Prandtl number, Le is Lewis number, Nb is Brownian motion parameter, and Nt is the thermophoresis parameter. Introducing these transformations into (1)–(5), the partial differential boundary layer equations contract to the following nonlinear coupled system of self-similar ordinary differential equations; namely,The corresponding transformed boundary conditions are specified as follows:In engineering simulations of nanofluid flows, not only the* velocity, temperature, and nanoparticle volume fraction* distributions but also primary (-)* skin friction ** and secondary* (-)* skin friction coefficients*, , and* local Nusselt number function *are important. We define these asHere is the* local* Reynolds number. The set of ordinary differential equations (8)–(11) are highly nonlinear and purely analytical solutions are difficult if not intractable. An efficient homotopy analysis method (HAM) is therefore adopted. Solutions are validated with the Adomian decomposition method (ADM). Although full solutions were given based on a shooting algorithm in Malvandi [37], unfortunately the exact data needed for a comparison is not available in that work. Therefore another objective of the present study is to provide a dual approach for validated solutions both with HAM and ADM techniques, in order to document correct solutions for other researchers to utilize. This therefore allows future researchers who may wish to extend the model to, for example, magnetohydrodynamics, proper benchmark data with which to validate their own numerical methods.