1 Introduction

Low thermal conductivity is a primary limitation in the development of energy coefficient heat transfer fluids required in many industrial and commercial applications. To get rid of such issue,a new class of nanofluids has been developed. These fluids are the mixtures of traditional fluids with nanoparticles. It was experimentally demonstrated that the resulting heat transfer nanofluids have significantly higher thermal conductivity than the traditional fluids. The nanofluids are the material through traditional liquids subject to suspended nanoparticles. Metallic or nonmetallic nanometer sized particles are used as the nanoparticles. In particular, the nanofluids have a pivotal role in advanced cooling systems,micro/nano electromechanical devices,and thermal management systems via evaporators,heat exchangers,and industrial cooling applications. Several researchers at present are now engaged to model and simulate the flows of nanofluids. Some recent contributions in this direction may be found in Refs. ^{[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]} and the references therein. Moreover,it is well recognized that several fluids in industry and engineering are non-Newtonian in nature. The well-known Nevier-Stokes equations do not predict the rheological characteristics of non-Newtonian materials. Further,one constitutive relationship cannot describe the flow behaviors of all non-Newtonian fluids. Hence,several models of non-Newtonian fluids were used^{[11, 12, 13, 14, 15]}. The Casson fluid^[16] is one of the non-Newtonian materials predicting the shear thinning liquid (e.g.,blood,Jelly,concentrated fruit juices,etc).

It has been noted that not much has been reported about the flow of non-Newtonian fluids with nanoparticles. Hence,the main aim of the present paper is to address the flow of the Casson fluid in the presence of nanoparticles. The Brownian motion and thermophoretic effects are presented. The flow considered is due to the stretching cylinder. The nonlinear problem is computed and studied.

2 Problem development

The axisymmetric flow of an incompressible Casson fluid with nanoparticles is examined. The behaviors of the Brownian motion and thermophoresis are also considered. The stretchable cylinder is along the horizontal axis,i.e.,the x-axis. Here,the r-axis is taken along the radial direction. The velocity,temperature,and concentration fields through the boundary layer approximations satisfy the following equations:

where u and v represent the velocity components along the x- and r-directions,respectively, U₀ is the reference velocity,β =${{{\mu _B}\sqrt {2{\pi _c}} } \over {{\tau _r}}}$ is the material parameter,in which μ_B is the plastic dynamic viscosity,τ_r is the yield stress of fluid,and πc is the critical value of this product based on the non-Newtonian model,l is the characteristic length,τ = (ρc)_p/(ρc)_f is the ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid with ρ being the density,D_B is the Brownian diffusion coefficient,D_T is the thermophoretic diffusion coefficient,σ is the thermal diffusivity of the fluid,c_p is the specific heat at the constant pressure,C_w is the concentration susceptibility,T_w is the wall temperature,ν is the kinematic viscosity,and k is the variable thermal conductivity of the fluid defined as We introduce the following nondimensional quantities: By Eq. (7),Eq. (1) is identically satisfied,and Eqs. (2)-(5) yield where γ =$\sqrt {{{lv} \over {{a^2}{U_0}}}} $is a curvature parameter,Pr =${\nu \over \sigma }$ is the Prandtl number,N_b=τ${{{D_B}} \over \nu }$ (C_w-C_∞) is the Brownian motion parameter,Nt =${{\tau {D_T}\left( {{T_W} - {T_\infty }} \right)} \over {{T_\infty }\nu }}$is the thermophoretic parameter,Ec = ${{U_0^2{{\left( {x/l} \right)}^2}} \over {{c_p}\left( {{T_W} - {T_\infty }} \right)}}$is the Eckert number,and Le =${\nu \over {{D_B}}}$ is the Lewis number. The local Nusselt number Nu_x and the Sherwood number Sh_x are defined as The dimensionless form of the local Nusselt number Nu_x is The dimensionless form of the Sherwood number Sh_x is given by

3 Homotopy analysis solutions

The initial approximations and auxiliary linear operators are chosen in the forms of

subject to where C_i (i = 1,2,· · · ,7) are the arbitrary constants determined from the boundary conditions. If p ∈ [0,1] denotes an embedding parameter,and h_f,h_θ,and h_φ are the non-zero auxiliary parameters,then the zeroth-order deformation problems are where N_f,N_θ,and N_φ are the nonlinear operators defined by When p = 0 and p = 1, and when p varies from 0 to 1,f(η; p),θ(η; p),and φ(η; p) vary from f₀(η),θ₀(η),and φ₀(η) to f(η),θ(η),and φ(η),respectively. Now,f,θ,and φ in Taylor’s series can be elaborated as follows: where the convergence depends upon h_f,h_θ and h_φ. By proper choice of h_f,h_θ,and h_φ,the series (28)-(30) converge for p = 1. Hence, The mth-order deformation problems are given by The general solutions of Eqs. (35)-(37) are where f_m^*,θ_m^*,and φ_m^* are the special solutions of Eqs. (43)-(45).

We know that the homotopy analysis method (HAM)^{[16, 17, 18, 19, 20]} solution highly depends upon the non-zero auxiliary parameter ħ which helps us to adjust and control the convergence region. Therefore,we draw the ħ-curves for f''(0),θ'(0),and φ'(0). Figures 1-3 depict the ħ-curves for different values of embedded parameters. The suitable range of ħ_f is -1.2 < ħ_f < -0.2,for ħ_θ,it is -0.5 < ħ_θ < -0.8,and for ħ_φ,it is -0.8 < ħφ < -1.2. It is found that the HAM solutions converge entirely when ħ_f = -0.8,ħ_θ = -1.2,and h_φ = -1.2. Table 1 shows the convergence of series solutions. It is evident that the 20th-order estimations are enough for convergent series solutions.

4 Results and discussion

This section discloses the salient features of different physical quantities involved in the temperature and concentration fields. The temperature profile is drawn in Fig. 4 for various values of Casson fluid parameter β. The temperature profile decreases with the increasing η for given values of β. It is shown that the Casson fluid parameter is inversely proportional to the yield stress. The variation of the temperature with η is calculated for different values of Brownian motion parameter N_b in Fig. 5. When N_b increases,both the temperature and the thermal boundary layer thickness decrease. Figure 6 depicts the effect of thermophoretic parameterN_ton the temperature profile. This figure shows thatN_tenhances the temperature field. Figure 7 plots the influence of γ on the temperature field. The effect of γ near the wall is almost negligible,while when η > 1,the temperature increases. Figure 8 explains the variation of variable thermal conductivity parameter ε on the temperature profile. It is noted that θ(η) for the constant thermal conductivity case is less than the variable thermal conductivity. Figure 9 describes the variation of Pr on the temperature field. This figure illustrates that the temperature field decreases when Pr increases. The effect of the Casson fluid parameter β on φ(η) is shown in Fig. 10. This indicates that φ(η) is an increasing function of the Casson fluid parameter β. Figure 11 displays the influence of the Brownian motion parameter N_b on φ(η). The concentration field decreases very slowly when the Brownian motion parameter increases. The enhancement in the thermophoretic parameterN_tleads to an increase in the concentration field. Also,the boundary layer thickness is more pronounced (see Fig. 12). The effect of Le is depicted in Fig. 13. The concentration field and the associated boundary layer thickness are decreasing functions of Le. The Prandtl and Lewis numbers have similar effects on the concentration (see Fig. 14). Figure 15 illustrates the impact of temperature and temperature gradient for different values of Nt. The temperature increases for larger values of Nt,whereas the temperature gradient increases most rapidly near the wall,but when η > 1,it decays very slowly and then vanishes far away. The temperature gradient decreases near the cylinder surface when the fluid thermal conductivity increases (see Fig. 16). Figures 17 and 18 present the effects of N_b and Le on the concentration and the concentration gradient,respectively. For both parameters,the temperature and the temperature gradient decrease.

Table 2 depicts the numerical values of local Nusselt number for different values of β,ε,N_b,γ, Pr,Le,and Nt. The increase in N_b,Pr,γ,and Le enhances the local Nusselt number. It is also observed from this table that for larger values of β,ε,and Nt,the local Nusselt number decreases. The numerical values of local Sherwood number are displayed in Table 3. The local Sherwood number at the wall decreases for larger N_b and Nt. The enhancement in β,ε,Pr,Ec, and Le yields an increase in the local Sherwood number.

5 Conclusions

This article addresses the flow of the Casson fluid with nanoparticles. The resulting problems are computed successfully. It is found that the temperature and the thermal boundary layer thickness are increasing functions of ε and Nt. However,N_b and γ have opposite behavior on the temperature profile. Variable thermal conductivity enhances the fluid temperature,while it decreases θ'(0) from the wall. Both the local Nusselt and local Sherwood numbers increase with Le,whereas N_t has the opposite effect.