Mathematical Problems in Engineering

Volume 2017, Article ID 3804751, 12 pages

https://doi.org/10.1155/2017/3804751

## The Influence of Slip Boundary Condition on Casson Nanofluid Flow over a Stretching Sheet in the Presence of Viscous Dissipation and Chemical Reaction

^{1}Department of Mathematics, Deanship of Educational Services, Qassim University, P.O. Box 6595, Buraidah 51452, Saudi Arabia^{2}Department of Mathematics, Faculty of Science, Helwan University, P.O. Box 11795, Ain Helwan, Cairo, Egypt

Correspondence should be addressed to Ahmed A. Afify; moc.oohay@56yfifa

Received 2 March 2017; Revised 19 May 2017; Accepted 30 May 2017; Published 26 July 2017

Academic Editor: Efstratios Tzirtzilakis

Copyright © 2017 Ahmed A. Afify. 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 impacts of multiple slips with viscous dissipation on the boundary layer flow and heat transfer of a non-Newtonian nanofluid over a stretching surface have been investigated numerically. The Casson fluid model is applied to characterize the non-Newtonian fluid behavior. Physical mechanisms responsible for Brownian motion and thermophoresis with chemical reaction are accounted for in the model. The governing nonlinear boundary layer equations through appropriate transformations are reduced into a set of nonlinear ordinary differential equations, which are solved numerically using a shooting method with fourth-order Runge-Kutta integration scheme. Comparisons of the numerical method with the existing results in the literature are made and an excellent agreement is obtained. The heat transfer rate is enhanced with generative chemical reaction and concentration slip parameter, whereas the reverse trend is observed with destructive chemical reaction and thermal slip parameter. It is also noticed that the mass transfer rate is boosted with destructive chemical reaction and thermal slip parameter. Further, the opposite influence is found with generative chemical reaction and concentration slip parameter.

#### 1. Introduction

Nowadays, nanofluids have been utilized as the working fluids instead of the base fluids due to their high thermal conductivity. Choi [1] is the first researcher who established fluids containing a suspension of nanosize particles which are termed the nanofluids. Lee et al. [2] confirmed that the nanofluids possess outstanding heat transfer characteristics compared to those of base fluids. Later on, many investigators [3–7] have indicated that the nanofluids enhanced thermophysical characteristics and heat transfer behavior compared to the base fluids. The thermal conductivity enhanced by 40% when copper nanoparticles with the volume fraction less than 1% are added to the ethylene glycol or oil was studied by Eastman et al. [3]. The enhancement of thermal conductivity of various nanofluids was reviewed by Aybar et al. [4]. They confirmed that the addition of nanoparticles in the fluids increases the thermal conductivity. Wang et al. [5] discussed the viscosity of Al2O3 and CuO nanoparticles dispersed in water, vacuum pump fluid, engine oil, and ethylene glycol. Their results indicated that 30% enhancement with Al2O3/water nanofluid at 3% volume concentration is obtained. Afify and Bazid [6] investigated the impacts of variable viscosity and viscous dissipation on the boundary layer flow and heat transfer along a moving permeable surface immersed in nanofluids. The influences of thermal radiation and particle shape on the Marangoni boundary layer flow and heat transfer of nanofluid driven by an exponential temperature were examined by Lin et al. [7].

Buongiorno, [8] modified the reasons behind the enhancement of heat transfer of nanofluids. Recently, many researchers [9–15] have effectively applied Buongiorno’s model [8]. Nield and Kuznetsov [9] presented the influence of Brownian motion and thermophoresis on natural convection boundary layer flow past a vertical plate suspended in a porous medium. The steady two-dimensional boundary layer flow of a nanofluid past a stretching surface was examined by Khan and Pop [10]. Nield and Kuznetsov [11] discussed the impact of Brownian motion and thermophoresis on the double-diffusive nanofluid convection past a vertical plate embedded in a porous medium. Makinde and Aziz [12] discussed the influence of a convective boundary, Brownian motion, and thermophoresis on the steady two-dimensional boundary layer flow of a nanofluid past a stretching sheet. Rana et al. [13] numerically examined the Brownian motion and thermophoresis effects on the steady mixed convection boundary layer flow of nanofluid past an inclined plate embedded in a porous medium. The influence of variable fluid properties with Brownian motion and thermophoresis on the natural convective boundary layer flow in a nanofluid past a vertical plate was numerically studied by Afify and Bazid [14]. Recently, Makinde et al. [15] numerically examined the effects of variable viscosity, thermal radiation, and magnetic field on convective heat transfer of nanofluid over a stretching surface.

Non-Newtonian nanofluids are widely encountered in many industrial and technological applications, such as the dissolved polymers, biological solutions, paints, asphalts, and glues. The power-law non-Newtonian nanofluid along a vertical plate and a truncated cone saturated in a porous medium were analyzed by Hady et al. [16] and Cheng [17], respectively. Rashad et al. [18] investigated the natural convection of non-Newtonian nanofluid around a vertical permeable cone. The influence of Soret and Dufour on the mixed convective flow of Maxwell nanofluid over a permeable stretched surface was examined by Ramzan et al. [19]. Several authors [20–23] have analytically solved the problem of non-Newtonian nanofluids in various aspects using the homotopy analysis method (HAM). Abou-Zeid et al. [24] examined the impact of viscous dispersion on the mixed convection of gliding motion of bacteria on power-law nanofluids through a non-Darcy porous medium. The influence of nonuniform heat source/sink with the Brownian motion and thermophoresis on non-Newtonian nanofluids over a cone was illustrated by Raju et al. [25].

The Casson fluid model is classified as a subclass of non-Newtonian fluid which has several applications in food processing, metallurgy, drilling operations, and bioengineering operations. The Casson fluid model was discovered by Casson in 1959 for the prediction of the flow behavior of pigment-oil suspensions [26]. Boundary layer flow of Casson fluid and Casson nanofluid flow over different geometries was considered by many authors [27–30]. Mustafa et al. [27] analytically discussed the unsteady boundary layer flow of a Casson fluid over a moving flat plate using the homotopy analysis method (HAM). Nadeem et al. [28] examined the analytical solution of convective boundary conditions for the steady stagnation-point flow of a Casson nanofluid. Makanda et al. [29] studied the numerical solution of MHD Casson fluid flow over an unsteady stretching surface saturated in a porous medium with a chemical reaction effect. Recently, Hayat et al. [30] investigated the combined effects of the variable thermal conductivity and viscous dissipation on the boundary layer flow of a Casson fluid due to a stretching cylinder.

The chemical reaction influence is a significant factor in the study of heat and mass transfer for many branches of science and engineering. A chemical reaction between the base liquid and nanoparticles may regularly occur either throughout a given phase (homogeneous reaction) or in an enclosed region (boundary) of the phase (heterogeneous reaction). Das et al. [31] investigated numerically the effects of chemical reaction and thermal radiation on the heat and mass transfer of an electrically conducting incompressible nanofluid over a heated stretching sheet. The effect of time-dependent chemical reaction on stagnation-point flow and heat transfer of nanofluid over a stretching sheet was presented by Abd El-Aziz [32]. El-Dabe et al. [33] studied the impact of chemical reaction, heat generation, and radiation on the MHD flow of non-Newtonian nanofluid over a stretching sheet embedded in a porous medium. Eid [34] proposed the numerical analysis of MHD mixed convective boundary layer flow of a nanofluid through a porous medium along an exponentially stretching sheet in the presence of chemical reaction and heat generation or absorption effects. Recently, Afify and Elgazery [35] discussed the influences of the convective boundary condition and chemical reaction on the MHD boundary layer flow of a Maxwell nanofluid over a stretching surface.

All these previous studies restricted their discussions on conventional no-slip boundary conditions. However, both velocity slip and temperature jump at the wall have numerous benefits in many practical applications such as micro- and nanoscale devices. In 1823, Navier [36] became the first person to create the slip boundary condition and suggested that the slip velocity is linearly proportional to the shear stress at the wall. Following him, many researchers [37–42] have widely studied the velocity slip and temperature jump at the wall over various geometries placed in viscous fluids and nanofluids. The impacts of slip and convective boundary condition on the unsteady three-dimensional flow of nanofluid over an inclined stretching surface embedded in a porous medium were numerically discussed by Rashad [37]. Afify et al. [38] used Lie symmetry analysis to investigate the effects of slip flow, Newtonian heating, and thermal radiation on MHD flow and heat transfer along the permeable stretching sheet. EL-Kabeir et al. [39] examined the numerical solution for mixed convection boundary layer flow of Casson non-Newtonian fluid about a solid sphere in the presence of thermal and solutal slip conditions. Afify [40] examined the impact of slips and generation/absorption on an unsteady boundary layer flow and heat transfer over a stretching surface immersed in nanofluids. Abolbashari et al. [41] presented an analytical solution to analyze the heat and mass transfer characteristics of Casson nanofluid flow induced by a stretching sheet with velocity slip and convective surface boundary conditions. Recently, Uddin et al. [42] discussed the effects of Navier slip and variable fluid properties on the forced convection of nanofluid and heat transfer over a wedge. The objective of the present paper is to investigate the influences of chemical reaction, viscous dissipation, velocity, and thermal and concentration slip boundary conditions in the presence of nanoparticles attributable to Brownian diffusion and thermophoresis on flow and heat transfer of Casson fluid over a stretching surface. To the best of the author’s knowledge, this work has not been previously studied in the scientific research. Numerical results for the velocity, temperature, and nanoparticle concentration fields are plotted. The friction factor and the heat and mass transfer rates are also tabulated and discussed. The present paper confirmed that the nanoparticles embedded in Casson fluid have many practical applications such as nuclear reactors, microelectronics, chemical production, and biomedical fields.

#### 2. Mathematical Formulation

Consider the steady boundary layer flow of an incompressible Casson nanofluid past a stretching surface. The sheet is stretched with a linear velocity , where is the positive constant. The -axis is directed along the continuous stretching sheet and the -axis is measured normal to the -axis. It is assumed that the flow takes place for . The temperature and the nanoparticle concentration are maintained at prescribed constant values , at the surface and and are the fixed values far away from the surface. It is also assumed that there is a first-order homogeneous chemical reaction of species with reaction rate constant, . The flow configuration is shown in Figure 1. The rheological equation of state for an isotropic and incompressible flow of Casson fluid is given by Eldabe and Salwa [43]:where is the plastic dynamic viscosity of the non-Newtonian fluid, is the yield stress of fluid, is the product of the component of deformation rate and itself, namely, is the component of the deformation rate, and is a critical value of based on non-Newtonian model. Under the boundary layer approximations the governing equations of Casson nanofluid can be expressed as follows (Buongiorno [8] and Haq et al. [44]): subject to the boundary conditions:where and are the velocity components along the - and -axes, respectively, is the density of base fluid, is the kinematic viscosity of the base fluid, is the thermal diffusivity of the base fluid, is the ratio of nanoparticle heat capacity and the base fluid heat capacity, is the Brownian diffusion coefficient, and is the thermophoretic diffusion coefficient. Furthermore, , , and are velocity, thermal, and concentration slip factor. The following nondimensional variables are defined as The continuity equation (2) is satisfied by introducing the stream function such thatIn view of the above-mentioned transformations, (3)–(6) are reduced toHere prime denotes differentiation with respect to , is similarity function, is the dimensionless temperature, is the dimensionless nanoparticle volume fraction, is Prandtl number, is Lewis number, is the thermal slip parameter, is the Casson parameter, is the slip parameter, is the concentration slip parameter, is the Eckert number, is the chemical reaction parameter, is the Brownian motion parameter, and is the thermophoresis parameter, respectively. It should be mentioned here that indicates a destructive chemical reaction while corresponds to a generative chemical reaction. The quantities of physical interest in this problem are the local skin friction coefficient, , the local Nusselt number, , and local Sherwood number, , which are defined as where is the shear stress and and are, respectively, the surface heat and mass flux which are given by the following expressions:The dimensionless forms of skin friction, the local Nusselt number, and the local Sherwood number becomewhere is the local Reynolds number.