Boundary-Layer Flow and Heat Transfer over a Rotating Porous Disk in a Non-Newtonian Williamson Nanofluid

Objectives: The present study analyses the effects of nanoparticles over steady, incompressible boundary-layer flow of non-Newtonian Williamson fluid flowing over a rotating porous disk. Methods/statistical analysis: The analysis used nonlinear partial differential equations associated with the non-Newtonian Williamson fluid flow and used an abridged, simplified version of ordinary differential equations under the Boussines q and boundary-layer approximations. The resulting system of nonlinear ordinary differential equations is then solved analytically using the homotopy analysis. Findings: The behavior of three non-dimensional velocity profiles and the non-dimensional temperature profile for important physical parameters like Williamson number, Prandtl number and surface suction/ injection parameters is tabulated and the results are graphed. Application/improvements: Nanoparticles like Copper, Alumina and titania are commonly used with liquid coolants to increase the heat-dissipating capability.


Introduction
The study of fluid flow, particularly fluid flow for non-Newtonian fluids, gained immense popularity in industrial and engineering applications due to their extensive utilization. Common non-Newtonian fluid sare basically pseudoplastic fluids that have various industrial applications; they are used in production of emulsion sheets, polymer sheet extrusion, and are also used for aiding the flow of plasma and blood. In order to study the rheological behaviour non-Newtonian fluids, often the routine system of calculation used, namely, the Navier Stokes equation, alone cannot be used for an in-depth study of the said behaviour or properties; more options are needed for a range of reasons; therefore, various models like Carreau, Power Law, Ellis and Cross Models are commonly used to predict the rheological properties of such fluids.
In 1929, Williamson 1 presented a theory to study the pseudoplatic materials and formulated a constitutive equation to explain the flow characteristics of the pseudoplastic fluids, and the experiments he conducted based on these equations validated the results on the flow characteristics. Recently, these equations have garnered the attention of researchers, as the researchers have come to know that these equations can be put to use to assess/ study the blood flow problems. Many researchers have made valuable contributions in this regard.
Williamson's fluids 2 were investigated using the effects of chemical reaction over steady, viscoelastic Keywords: Boundary-Layer Flow; Heat Transfer; Porous Disk; non-Newtonian; Williamson Nanofluid boundary layer flow. Methods like killer box 3 were used to obtain numerical solutions to be used in assessing the homogeneous/heterogeneous chemical reactions of boundary-layer flow of Williamson fluids flowing over a smooth stretching cylinder. Another study 4 has numerically examined the behavior of variable thermal conductivity and heat generation/absorption on the steady, incompressible boundary-layer flow of Williamson fluids.
One study 5 predicted that in the case of the steadystate solutions of the momentum equation for the fluid flow derived by an infinite disk in radial direction, the centrifugal forces balanced out so the fluid flowed outward in the absence of the pressure gradient. The rotating disk in such cases is expected to be working like a centrifugal fan, and the fluid flow originating from the rotating disk is replaced by an axial one that springs back on the surface of the rotating disk.
The reflection of the non-Newtonian fluid flowing over a rotating disk presented a great challenge to researchers in the field. The non-Newtonian flow of fluid over a rotating disk is a significant problem that needed immediate resolutions because of its extensive use in various fields, including engineering, chemical, and electro-medical processes. Recent research 6-7 on the said problems pointed out a range of outcomes, including the non-Newtonian laminar flow, double diffusion effect on unsteady flow, head transfer over a rotating disk, as well as the boundary-layer stability of non-Newtonian fluid flow over a rotating disk. 8 Rehman et al. 9 investigated the non-Newtonian Casson fluid flow problem over a semi-infinite disk and analyzed the characters of both thermophores and Brownian motion parameters for nanosized particles inside the magnetized Casson liquid. The problem of two turbulent simulations with overlapping small and large Reynolds numbers was reported to be a result of direct numerical simulations performed for the turbulent rotating-disk boundary layer. 10 The study by Khan et al. 11 investigated the problem of stress fluid flow occurring as a result of rotation of a disk.
In many industrial, chemical and power generation processes, massive amount heat is generated. In order to protect the machines operating in such high temperatures, appropriate cooling systems are to be in place to keep the machinery cool and control the heat and keep it at a level within which the machinery can perform unhindered and do not break down. Thus, different coolants like water, ethylene glycol and engine oils of different grades are used.
However, it is a proven fact that the thermal conductivity of liquids is way below that of metallic solids; due to this, all these coolants can never achieve cooling on a level that metallic solids can. In order to achieve the desired goal of reducing and maintaining temperature at an ideal level to keep the machinery functioning, a small amount of metallic nanoparticles are added to these coolants, which greatly increases the heat-dissipating capability of these coolants. The nanoparticles added to these coolants are made of Copper (Cu), Alumina (Al 2 O 3 ) and titania (TiO 2 ), or nano tubes, as described by Tiwari and Das and Lotfi et al. [12][13][14] The aim of the present study was to analyze the features of the boundary-layer flow and heat transfer over a rotating porous disk placed in a Williamson nanofluid. The study aimed analyse the effects of three kinds of nanoparticles: Copper (Cu), Alumina (Al 2 O 3 ) and titania (TiO 2 ). Authors have proposed different models for nano fluids, like those proposed by Buongiorno and others. [15][16][17][18] Oztop and Abu-Nada 19 have analyzed the behaviour of viscous nanofluid flow of the boundary layer's laminar flow and heat transfer over a rotating porous disk. The present study was conducted using the model devised by Tiwari and Das. 13 A set of nonlinear differential equations were incorporated in the homotopy analysis method (HAM) used by the study. The conduct of nanoparticles and suction/injection parameters over the fluid flow and heat transfer are presented and discussed.

Mathematical Formulation
The study analysed the effects of steady, incompressible and axially symmetric flow of Williamson fluid over a rotating porous disk immersed in a nanofluid. Figure 1 shows the schematic diagram of the problem discussed. Let the cylindrical coordinates (r, ϕ, z) be chosen such that the radial component r is along the length of the disk. The porous disk is assumed to be rotating with a constant angular velocity Ω and extends with initial position z = 0, in the space z > 0 of the disk to infinity. The fluid velocity components in the increasing direction of coordinates are taken as u, v, w. The surface constant temperature is assumed to be  w T . The Williamson fluid has a constant pressure P   and a uniform ambient temperature taken as the constant mass flux applied at the disk surface. Under the stated assumptions, the governing system of partial differential equations for the case of steady, incompressible laminar boundary-layer flow of Williamson nanofluid are The related boundary conditions of the problem are where T  and T   denote the temperature of fluid and ambient fluid, respectively. P , Substituting the above transformation in Eqs. (1)-(7) to convert these partial differential equations into a system of ordinary differential equations, we have   with non-dimensional boundary conditions, where   , , are transversal and radial shear stresses at the disk surface, respectively. The surface heat flux q w is given by Using Eqs. (9) and (18) in Eqs. (19) and (20)

HAM Solution
The system of nonlinear ordinary differential Eqs. (13)- (16) subjected to boundary conditions in equations 17,18 has been solved by using the method of homotopy analysis  taking into account the different values of fraction parameter φ and velocity (suction/injection) parameter h 0 . Three types of solid nanoparticles of spherical shape, namely copper (Cu), alumina (Al 2 O 3 )) and titania (TiO 2 ), are considered. To achieve the boundary conditions at infinity, η ∞ the boundarylayer thickness, has been studied for different values of parameters φ and h 0 for the functions H(η), F(η), G(η) and θ(η). To solve the system using the homotopy analysis we made initial estimates for H 0 (η),F 0 (η), G 0 (η) and θ 0 (η) as and the auxiliary operators This follows The first-order homotopy approximations are as follows: The higher-order iterations are obtained through the use of symbolic software MATHEMATICA.   Figure 6 shows the behavior of non-dimensional temperature profile θ(η) for different values of the surface suction/injection velocity h 0 . From the graph, it can be noted that the non-dimensional velocity θ(η) increases correspondiong to an increase in surface suction/injection velocity, h 0 . Figure 7 Shows the behavior of non-dimensional temperature profile θ(η) for different values of φ. From the figure it can be observed that an increase in φ produces a minute increase in the non-dimensional temperature profile θ(η).

Conclusion
The problem of a non-Newtonian boundary-layer flow and heat transfer of Williamson fluid containing nanoparticles and in motion over a rotating porous diskwas analyzed analytically using the method of homotopy analysis. The findings made and our conclusions support the use of Copper(Cu), Alumina (Al 2 O 3 ) and Titania (TiO 2 ) nanoparticles to enhance the heat dissipation capability of coolants are presented as follows.