International Journal of Computational Mathematics

Volume 2014, Article ID 631749, 12 pages

http://dx.doi.org/10.1155/2014/631749

## Computational Modelling of Couette Flow of Nanofluids with Viscous Heating and Convective Cooling

^{1}Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa^{2}Mathematics and Computational Science and Engineering, Nelson Mandela African Institution of Science and Technology (NM-AIST), Arusha, Tanzania

Received 28 April 2014; Accepted 18 November 2014; Published 14 December 2014

Academic Editor: Zhijie Xu

Copyright © 2014 Oluwole Daniel Makinde 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 combined effect of viscous heating and convective cooling on Couette flow and heat transfer characteristics of water base nanofluids containing Copper Oxide (CuO) and Alumina (Al_{2}O_{3}) as nanoparticles is investigated. It is assumed that the nanofluid flows in a channel between two parallel plates with the channel’s upper plate accelerating and exchange heat with the ambient surrounding following the Newton’s law of cooling, while the lower plate is stationary and maintained at a constant temperature. Using appropriate similarity transformation, the governing Navier-Stokes and the energy equations are reduced to a set of nonlinear ordinary differential equations. These equations are solved analytically by regular perturbation method with series improvement technique and numerically by an efficient Runge-Kutta-Fehlberg integration technique coupled with shooting method. The effects of the governing parameters on the dimensionless velocity, temperature, skin friction, pressure drop and Nusselt number are presented graphically, and discussed quantitatively.

#### 1. Introduction

Studies related to laminar flow and heat transfer of a viscous fluid in the space between two parallel plates, one of which is moving relative to the other, have received the attention of several researchers due to their numerous industrial and engineering applications. This type of flow is named in honour of Maurice Marie Alfred Couette, a professor of physics at the French University of Angers in the late 19th century [1]. Couette flow has been used to estimate the drag force in many wall driven applications such as lubrication engineering, power generators and pumps, polymer technology, petroleum industry, and purification of crude oil. Literature survey indicates that interest in the Couette flows has grown during the past decades. Jana and Datta [2] examined the effects of Coriolis force on the Couette flow and heat transfer between two parallel plates in a rotating system. Singh [3] studied unsteady free convection flow of an incompressible viscous fluid between two vertical parallel plates, in which one is fixed and the other is impulsively started in its own plane. Kearsley [4] investigated the problem of steady state Couette flow with viscous heating. Jha [5] numerically examined the effects of magnetic field on Couette flow between two vertical parallel plates. The combined effects of variable viscosity and thermal conductivity on generalized Couette flow and heat transfer in the presence of transversely imposed magnetic field have been studied numerically by Makinde and Onyejekwe [6]. Seth et al. [7] presented a closed form solution for hydromagnetic unsteady Couette flow of a viscous incompressible electrically conducting fluid between two parallel porous plates. Deka and Bhattacharya [8] obtained an exact solution of unsteady free convective Couette flow of a viscous incompressible heat generating/absorbing fluid confined between two vertical plates in a porous medium. Meanwhile, the enhancement of heat transfer in a Couette flow of fluid subjected to a temperature gradient is an important issue that is expected to improve the efficient operation of several engineering and tribological devices. Other relevant applications can be found in engine cooling, solar water heating, cooling of electronics, cooling of transformer oil, improving diesel generator efficiency, cooling of heat exchanging devices, improving heat transfer efficiency of chillers, domestic refrigerator-freezers, and cooling in machining and in nuclear reactor. The common heat transfer fluids such as water, ethylene glycol, and engine oil have limited heat transfer capabilities owing to their low thermal conductivity whereas metals have much higher thermal conductivities than these fluids. With the recent improvements in nanotechnology, the production of particles with sizes on the order of nanometers can be achieved. Consequently, the idea of dispersing these nanoparticles in a base liquid for improving thermal conductivity has been proposed [9]. Such suspension of nanoparticles in a base fluid is called a nanofluid. Due to their small size, nanoparticles fluidize easily inside the base fluid, and as a consequence, clogging of channels and erosion in channel walls are no longer a problem. It is even possible to use nanofluids in microchannels [10]. Niu et al. [11] theoretically studied the slip-flow and heat transfer of a non-Newtonian nanofluid in a microtube by means of theoretical method. In their research, the power-law rheology was adopted to describe the non-Newtonian characteristics of the flow, in which the fluid consistency coefficient and the flow behaviour index depend on the nanoparticle volume fraction. Motsumi and Makinde [12] reported a numerical solution for the effects of thermal radiation and viscous dissipation on boundary layer flow of nanofluids over a permeable moving flat plate. Choi et al. [13] studied the Couette flow of nanofluids composed of negatively charged nanoparticles dispersed in aqueous NaCl solutions. They found that the velocity profile of nanofluids containing charged nanoparticles deviates significantly from the classical linear velocity profile of Couette flow.

In the studies mentioned above, the combined effects of viscous heating and convective cooling on Couette flow of nanofluids have not been discussed while such flows are very important in lubrication technology and tribological problems. Therefore, the objective of the present paper is to analyze the effects of viscous heating and convective cooling on the Couette flow of water base nanofluids containing Copper Oxide (CuO) and Alumina (Al_{2}O_{3}) as nanoparticles. In Sections 2–4, the model nonlinear governing equations together with the analytical and numerical solution are obtained. Pertinent results are presented graphically and discussed quantitatively in Section 5 while the conclusions are drawn in Section 6.

#### 2. Problem Formulation

We consider a two-dimensional steady Couette flow of viscous incompressible water base nanofluids containing Copper Oxide (CuO) and Alumina (Al_{2}O_{3}) as nanoparticles in which an accelerating upper plate drags adjacent fluid along with it and thereby imparts a motion to the rest of the fluid. The lower plate is fixed and kept at a constant temperature while the upper accelerating plate is subjected to a convective heat exchange with the ambient surrounding following Newton’s law of cooling. We choose a Cartesian coordinates system in such a way that the -axis is taken along the channel and the -axis is normal to it as shown in Figure 1.