International Journal of Chemical Engineering

Volume 2018, Article ID 7305973, 25 pages

https://doi.org/10.1155/2018/7305973

## Combined Effects of Thermal Radiation and Nanoparticles on Free Convection Flow and Heat Transfer of Casson Fluid over a Vertical Plate

Department of Mechanical Engineering, University of Lagos, Akoka, Lagos, Nigeria

Correspondence should be addressed to M. G. Sobamowo; moc.liamg@iyinimebgekim

Received 18 February 2018; Revised 27 April 2018; Accepted 9 May 2018; Published 3 September 2018

Academic Editor: Iftekhar A. Karimi

Copyright © 2018 M. G. Sobamowo. 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 influences of thermal radiation and nanoparticles on free convection flow and heat transfer of Casson nanofluids over a vertical plate are investigated. The governing systems of nonlinear partial differential equations of the flow and heat transfer processes are converted to systems of nonlinear ordinary differential equations through similarity transformations. The resulting systems of fully coupled nonlinear ordinary differential equations are solved using the differential transformation method with Padé-approximant technique. The accuracies of the developed analytical methods are verified by comparing their results with the results of past works as presented in the literature. Thereafter, the analytical solutions are used to investigate the effects of thermal radiation, Prandtl number, nanoparticle volume fraction, shape, and type on the flow and heat transfer behaviour of various nanofluids over the flat plate. It is observed that both the velocity and temperature of the nanofluid as well as the viscous and thermal boundary layers increase with increase in the thermal radiation parameter. The velocity of the nanofluid decreases and the temperature of the nanofluid increase, respectively, as the Prandtl number and volume fraction of the nanoparticles in the base fluid increase. The decrease in velocity and increase in temperature are highest in lamina-shaped nanoparticle and followed by platelet-, cylinder-, brick-, and sphere-shaped nanoparticles, respectively. Using a common base fluid to all the nanoparticle types, it is established that the decrease in velocity and increase in temperature are highest in TiO_{2} and followed by CuO, Al_{2}O_{3}, and SWCNT nanoparticles, in that order. It is hoped that the present study will enhance the understanding of free convection boundary layer problems of Casson fluid under the influences of thermal radiation and nanoparticles as applied in various engineering processes.

#### 1. Introduction

The importance and the wide applications of free convection flow and heat transfer in extrusion, melt spinning, glass-fibre production processes, food processing, mechanical forming processes etc. have in recent times aroused various renewed research interests and explorations. In the study of free convection and heat transfer problems, the analysis of incompressible laminar flow of viscous fluid in a steady-state, two-dimensional free convection boundary layer has over the years been a common area of increasing research interests following the experimental investigations of Schmidt and Beckmann [1] and the pioneering theoretical work of Ostrach [2]. In their attempts to study the laminar free convection flow and heat transfer problem in 1953, Ostrach [2] applied method of iterative integration to analyze free convection over a semi-infinite isothermal flat plate. The author obtained the numerical solutions for a wide range of Prandtl numbers from 0.01 to 1000 and validated their numerical results using experimental data of Schmidt and Beckmann [1]. Five years later, Sparrow and Gregg [3] presented a further study on numerical solutions for laminar free convection from a vertical plate with uniform surface heat flux. Considering the fact that the major part of low Prandtl number boundary layer of free convection is inviscid, Lefevre [4] examined laminar free convection of an inviscid flow from a vertical plane surface. In a further work, Sparrow and Gregg [5] developed similar solutions for free convection from a nonisothermal vertical plate. Meanwhile, a study on fluid flow over a heated vertical plate at high Prandtl number was presented by Stewartson and Jones [6]. Due to the disadvantages in the numerical methods in the previous studies [2, 3], Kuiken [7] adopted method of matched asymptotic expansion and established asymptotic solutions for large Prandtl number free convection. In the subsequent year, the same author applied the singular perturbation method and analyzed free convection flow of fluid at low Prandtl numbers [8]. Also, in another work on the asymptotic analysis of the same problem, Eshghy [9] studied free convection boundary layers at large Prandtl number while Roy [10] investigated free convection boundary layer problem for a uniform surface heat flux at high Prandtl number. With the development of asymptotic solution, a combined study of the effects of small and high Prandtl numbers on the viscous fluid flow over a flat vertical plate was submitted by Kuiken and Rotem [11]. In the succeeding year, Na and Habib [12] utilized parameter differentiation method to solve the free convection boundary layer problem. Few years later, Merkin [13] presented the similarity solutions for free convection on a vertical plate while Merkin and Pop [14] used finite difference method to develop numerical solutions for the conjugate free convection problem of boundary-layer flow over a vertical plate. Also, Ali et al. [15] submitted a study on numerical investigation of free convective boundary layer in a viscous fluid.

The various analytical and numerical studies of the past works have shown that the boundary layer problems are very difficult to solve. This is because, besides having very thin regions where there is rapid change of the fluid properties, they are defined on unbounded domains. Although, approximate analytical methods are being used to solve boundary layer problems, they converge very slowly for some boundary layer problems, particularly for those with very large parameters. The numerical methods used in the flow process also encounter some problems in resolving the solutions of the governing equations in the very thin regions and in some cases where singularities or multiple solutions exist. Moreover, in numerical analysis, it is absolutely required that the stability and convergence analysis is carried out so as to avoid divergence or inappropriate results. Such analysis in the mathematical methods increases the computation time and cost. Therefore, in the quest for presenting symbolic solutions to the flow and heat transfer problem using one of the recently developed semianalytical methods, Motsa et al. [16] adopted homotopy analysis of free convection boundary layer flow with heat and mass transfer. In another work, the authors used spectral local linearization approach for solving the natural convection boundary layer flow [17]. Ghotbi et al. [18] investigated the application of homotopy analysis method to natural convection boundary layer flow. Although, homotopy analysis method (HAM) is a reliable and efficient semianalytical technique, it suffers from a number of limiting assumptions such as the requirements that the solution ought to conform to the so-called rule of solution expression and the rule of coefficient ergodicity. Also, the use of HAM in the analysis of linear and nonlinear equations requires the determination of auxiliary parameter which will increase the computational cost and time. Furthermore, the lack of rigorous theories or proper guidance for choosing initial approximation, auxiliary linear operators, auxiliary functions, and auxiliary parameters limits the applications of HAM. Moreover, such method requires high skill in mathematical analysis and the solution comes with large number of terms. Nonetheless, various analyses of nonlinear models and fluid flow problems under the influences of some internal and external factors using different approximate analytical and numerical methods have been presented in the literature [19–47]. Also, the relative simplicity coupled with ease of applications of the differential transformation method (DTM) has made the method to be more effective than most of the other approximate analytical methods. The method was introduced by Ζhou [48] and it has fast gained ground as it appeared in many engineering and scientific research papers. This is because, with the applications of DTM, a closed form series solution or approximate solution can be provided for nonlinear integral and differential equations without linearization, restrictive assumptions, perturbation, and discretization or round-off error. It reduces complexity of expansion of derivatives and the computational difficulties of the other traditional or recently developed methods. Therefore, Lien-Tsai and Cha’o-Kuang [49] applied the differential transformation method to provide approximate analytical solutions to the Blasius equation. Also, Kuo [50] adopted the same method to determine the velocity and temperature profiles of the Blasius equation of forced convection problem for fluid flow passing over a flat plate. An extended work on the applications of differential transformation method to free convection boundary layer problem of two-dimensional steady and incompressible laminar flow passing over a vertical plate was presented by the same author [51]. However, in the later work, the nonlinear coupled boundary value governing equations of the flow and heat transfer processes is reduced to initial value equations by a group of transformations, and the resulting coupled initial value equations are solved by means of the differential transformation method. The reduction or the transformation of the boundary value problems to the initial value problems was carried out due to the fact that the developed systems of nonlinear differential equations contain an unbounded domain of infinite boundary conditions. Moreover, in order to obtain the numerical solutions that are valid over the entire large domain of the problem, Ostrach [2] estimated the values of during the analysis of the developed systems of fully coupled nonlinear ordinary differential equations. Following Ostrach’s approach, most of the subsequent solutions provided in the literature [3, 9, 10, 12, 14, 15, 50, 51] were based on the estimated boundary conditions given by Ostrach [2]. Additionally, the limitations of power series solutions to small domain problems have been well established in the literature. Nevertheless, in some recent studies, the use of power series methods coupled with Padé-approximant technique has shown to be very effective way of developing accurate analytical solutions to nonlinear problems of large or unbounded domain problems of infinite boundary conditions. The application of Padé-approximant technique with power series method increases the rate and radius of convergence of power series solution. Therefore, in a recent work, Rashidi et al. [52] applied differential transformation method coupled with the Padé-approximant technique to develop a novel analytical solution for mixed convection about an inclined flat plate embedded in a porous medium.

Casson fluid is a non-Newtonian fluid that was first introduced by Casson in 1959 [53]. It is a shear thinning liquid which is assumed to have an infinite viscosity at zero rate of shear, a yield stress below which no flow occurs, and a zero viscosity at an infinite rate of shear [54]. If the yield stress is greater than the shear stress, then it acts as a solid, whereas if the yield stress is less than the applied shear stress, then the fluid would start to move. The fluid is based on the structure of liquid phase and interactive behaviour of solid of a two-phase suspension. It is able to capture complex rheological properties of a fluid, unlike other simplified models like the power law [55] and second-, third-, or fourth-grade models [56]. Some examples of Casson fluid are jelly, honey, tomato sauce, and concentrated fruit juices. Human blood is also treated as a Casson fluid in the presence of several substances such as fibrinogen, globulin in aqueous base plasma, protein, and human red blood cells. Concentrated fluids like sauces, honey, juices, blood, and printing inks can be well described using this model. It has various applications in fibrinogen, cancer homeotherapy, protein, and red blood cells, forming a chain-type structure. Due to these applications, many researchers are concentrating on characteristics of Casson fluid. Application of Casson fluid for flow between two rotating cylinders is studied in [57]. The effect of magnetohydrodynamic (MHD) Casson fluid flow in a lateral direction past linear stretching sheet was explained by Nadeem et al. [58].

The role of thermal radiation is very important in some industrial applications, such as glass production and furnace design, and also in space technology applications such as comical flight aerodynamics rocket, space vehicles, propulsion systems, plasma physics, and space craft reentry aerodynamics which operates at high temperatures, in the flow structure of atomic plants, combustion processes, internal combustion engines, ship compressors, and solar radiations. The effect of thermal radiation on magnetohydrodynamic flow was examined by Raptis and Perdikis [59] while Seddeek [60] investigated the impacts of thermal radiation and variable viscosity on magnetohydrodynamics in free convection flow over a semi-infinite flat plate. In another study, Mabood et al. [61] analyzed unsteady stretched flow of Maxwell fluid in the presence of nonlinear thermal radiation and convective condition while Hayat et al. [62] addressed the effects of nonlinear thermal radiation and magnetohydrodynamics on viscoelastic nanofluid flow. Farooq et al. [63] addressed the effects of nonlinear thermal radiation on stagnation point flow. Also, Shehzad et al. [64] presented a study on MHD three-dimensional flow of Jeffrey nanofluid with internal heat generation and thermal radiation.

The previous studies on fluid flow over stretching under investigation are based on viscous fluid flow as shown in the above-reviewed works. To the best of the author’s knowledge, a study on the influence of thermal radiation and nanoparticle shape, size, and type on the free convection boundary layer flow and heat transfer of Casson nanofluids over a vertical plate at low and high Prandtl numbers using the differential transformation method coupled with Padé-approximant technique has not been investigated. Therefore, the present study focuses on the application of differential transformation method coupled with Padé-approximant technique to develop approximate analytical solutions and carry out parametric studies of the effects of thermal radiation and nanoparticles on free convection boundary layer flow and heat transfer of nanofluids of different nanosize particles over a vertical plate at low and high Prandtl numbers. Another novelty of the present study is displayed in the development of approximate analytical solutions for the free convection boundary layer problem without the use of the estimated boundary conditions during the analysis of the problem.

#### 2. Problem Formulation and Mathematical Analysis

Consider a laminar free convection flow of an incompressible Casson nanofluid over a vertical plate parallel to the direction of the generating body force, as shown in Figure 1. The rheological equation for an isotropic and incompressible Casson fluid, reported by Casson [65], isorwhere is the shear stress; is the Casson yield stress; is the dynamic viscosity; is the shear rate; is the component of the deformation rate and is the product of the component of deformation rate with itself; is a critical value of this product based on the non-Newtonian model; is the plastic dynamic viscosity of the non-Newtonian fluid; and is the yield stress of the fluid. The velocity and the temperature are functions of , only.