Nano uid with nonlinear Rosseland thermal radiation and mixed convection

Two dimensional ow of mixed convection nano uid on horizontal plate with the e ect of nonlinear Rosseland thermal radiation has been investigated. Mathematical model of the problem is based on partial di erential equations and optimal homotopy analysis method (OHAM) is applied to sort out solutions. Moreover, comprehensive study of in uence of emerging parameters is carried out via graphical interpretation and tables.


Introduction
Nanouids have a lot of applications in medical industry, engine cooling, detergency, pharmaceutical processes, heat exchanger and space technology. Nanouids contain nanometer sized particles called nanoparticles. Nanouids consist of base uid which is usually water or oil with nanoparticles like metals, oxides, carbides and carbon. Some commonly used nanouids are T iO2 (Titanium dioxide) in water, CuO (Copper oxide) in water, Al 2 O 3 (Aluminium oxide) in water, ZnO (Zinc oxide) in ethylene glycol. Choi [1] established the concept of nanouids. Nanotechnology gain attention in the heat transfer process due to its characteristics of thermal conductivity.
Email addresses: nazish.iftikhar289@gmail.com (Nazish Iftikhar), Bilalsehole@gmail.com (Muhammad Bilal Riaz), Azharalizafar@gmail.com (Azhar Ali Zafar), syed.husnine@nu.edu.pk (Syed Muhammad Husnine) Velocity along x-axis and y-axis Mixed convection is a phenomenon occurred due to free convection and forced convection. Flow problems having mixed convection has great importance in applied perspective especially in industrial, technical processes. Pal and Mandal studied [2] three types of nanouids along with the thermal radiation and mixed convection. Hayat et al. [3] investigated coupled stress nanouid ow with nonlinear thermal radiation past a stretching surface. Thermal radiation eect in uid ow problems with mixed convection and convective condition are discussed in [4]- [6]. Nonlinear thermal radiation eect has great importance in engineering, nuclear reactors, missiles, and satellites. Hayat et al. [7] studied nonlinear thermal radiation eect in viscoelastic nanouid. Shehzad et al. [8] studied thermophoresis eect and brownian motion in Jerey nanouid with thermal radiation. Pantokratoras [9] investigated natural convection on isothermal plate with the impact of linear or nonlinear Rosseland radiation convection along with radiation parameter. Work has been done in this area by researchers [10]- [14]. Farooq et al. investigated heat transfer phenomena in viscoelastic nanouid with nonlinear radiative eects [15]. Hayat et al. [16] analyze heat transfer in nanouid with nonlinear thermal radiation and inclined magnetic eld. Many researchers pay attention towards nonlinear thermal radiation [17]- [24]. Pantokratoras and Fang [25] studies Blasius ow in the presence of nonlinear Rosseland thermal radiation. Some important phenomenons regarding nonlinear thermal radiations considered by researchers [26]- [29].
In this article, nanouid with nonlinear Rosseland thermal radiation and mixed convection has been investigated. Mathematical model involve partial dierential equations. OHAM is used to investigate solutions. In addition, results are highlighted by tables and graphs.

Mathematical model
Let nanouid ow with nonlinear Rosseland thermal radiation and mixed convection moving in the direction of a horizontal plate with components of velocity u and v. Velocity of the plate is U w (see gure 1). Mathematical model is given below: The boundary conditions are considered as: Introducing similarity transformations: By putting values of u and v in equation (1), it satised. Moreover substituting equations (7) and (8) into equations (2), (3) and (4), we get Dimensionless numbers with parameters are given below: Local Sherwood number is given below and local Reynold is given by Assume initial approximations are Let auxiliary operators are  Convergence of solution can be control in homotopy analysis method by using dierent parameters denoted by c g 0 , c θ 0 and c φ 0 . Values of these parameters can be obtained by minimizing error. BVPh2.0 is applied in order to get minimum error. Three arrays are selected. First array is selected at 2 nd iteration, second array is selected at 4 th iteration and third array is selected at 6 th iteration. Table 2 shows error analysis at 6 th iteration.

Results and discussion
Analysis of graphs for dierent parameters are examined in this section. Figure 2 depicts that there is an increase in velocity as λ increases. Thermal buoyancy force enhances when λ increases due to which velocity is enhanced. Eect of N b on θ(η) and φ(η) presents in gures 3 and 4. By increasing N b, temperature increases on the other hand concentration decreases. Figure 5 demonstrated that an increase in N t, enhances temperature. Thermal conductivity enhances as N t increases due to which temperature increases. Figure  6 displayed that φ(η) increases as N t increases. There is reduction in θ(η) with the increasing value of N r (see gure 7). Physically it is because of production of heat in moving uid which is generated to increase in radiation as a result temperature raises. Figure 8 interprets the inuence of φ(η) for Sc. As Sc increases there is a decrease in concentration. Physically by increasing Schmidt number there is mass diusivity become less and hence φ(η) decreases. Figure 9 shows that temperature decreases as N 1 increases. As θ r increases there is an increase in temperature (see gure 10). Further, Table 2 shows values of parameters which are responsible for convergence. Table 3 depicts error for 6th iteration. Table 4 presents values of Sherwood number corresponding to the parameters.

Conclusion
Nanouid is considered over horizontal moving plate under the inuence of nonlinear Rosseland thermal radiation with mixed convection. Fundamental observations are given below. • N t and N b have the same and opposite eect on φ(η) and θ(η) respectively. • Increasing value of N r accelerates the θ r .
• Enhancement in Sc leads to increase in φ(η).