Multiple physical aspects during the flow and heat transfer analysis of Carreau fluid with nanoparticles

The current work is concerned with the two-dimensional boundary layer flow of a non-Newtonian fluid in the presence of nanoparticles. The heat and mass transfer mechanism for Carreau nanofluid flow due to a radially stretching/shrinking sheet is further investigated in this article. The governing physical situation is modelled in the form of partial differential equations and are simplified to a system of non-linear ordinary differential equations by employing dimensionless variables. Numerical simulations for non-dimensional velocity, temperature and concentration fields has been performed with the assistance of built-in Matlab solver bvp4c routine. One significant computational outcome of this study is the existence of multiple numerical solutions for the flow fields. The impacts of various developing parameters, for instance, Weissenberg number, power-law index, shrinking parameter, suction parameter, Prandtl number, Schmidt number, Brownian motion and thermophoresis parameter on the velocity, temperature and nanoparticles concentration are visualized through tables and graphical experiment. The numerical results demonstrate that the rates of heat and mass transfer are raised by higher Weissenberg number for first solution and an inverse is seen for second solution. Moreover, an increasing trend is seen in nanofluids temperature for both solutions with greater values of thermophoresis parameter. In addition, the numerical results obtained by the applied technique are validated with existing literature and found to be in an excellent agreement.

consideration. A numerical and analytical study for the axisymmetric flow of nanofluid was reported by Mustafa et al. 5 . They see that increment in Schmidt number causes to a thinner nanoparticle boundary layer. Moreover, Hashim et al. 6 numerically investigated the analysis of heat and mass transfer in the Carreau fluid model using Runge-Kutta technique in MATLAB. Ellahi et al. 7 studied the impact of particle shape on Marangoni convection boundary layer flow of nanofluid. They enforced the nanoparticles mass flux and convective boundary conditions in this study. Sheikholeslami 8 presented the effect of variable magnetic field on the flow of Fe 3 O 4 -H 2 O nanofluid in a cavity with circular hot cylinder. Innovative numerical method, namely CVFEM is selected to perform the numerical computations. Recently, the transport of nanofluids are investigated by many researchers, see [9][10][11][12] . It is prominent fact that studies about boundary layer flow on stretching/shrinking sheet have gained a great importance because of its ever-incrementing to do with industry applications in some technology-based cognitive process. Many cases are trans actioned with stretching/shrinking surface appearing in manufacturing of rubber and plastic sheet, polymer-industries, spinning of fibres etc. Scientists are fascinated to develop different methods acting to increment their heat transfer exhibition. The boundary layer flow over a stretching surface was firstly studied by Crane 13 . Later in 2010, Khan and Pop 4 have disciplined the boundary layer rate of flow over a linear stretching sheet. Moreover, Wang 14 was discussed the study for unsteady film solution on the boundary layer flow over a shrinking surface. After that, Rana et al. 15 put into use finite element model for nonlinear stretching sheet to discuss the behaviour of flow and heat transfer. Specifically, flow over a stretching sheet with quadratic 16 , exponential 17 , nonlinear 18 and oscillatory 19 were discussed by different authors. For the case of exponentially stretching sheet, the skin friction at the wall detailed by Elbashbeshy 17 is higher than that processed by Vajravelu 18 for the nonlinear stretching sheet even with u w = cx 5 . Then again, the skin friction at the wall has oscillatory conduct in the position of oscillatory stretching sheet as visualized by Abbas et al. 19 . The classical problem of axisymmetric flow because of radially stretching plat was discussed via Ariel 20 . Later, Sajid et al. 21 who discussed about series solution for axisymmetric flow over a nonlinear stretching sheet. Similarly, Khan and Shehzad 22 are performed the exact solution for steady axisymmetric flow due to nonlinear stretching sheet.
The notable investigation within the sight of the dual solutions for flow and heat transfer characteristics have been presented by several researchers. In this regard, Lio 23 investigated the problem of boundary layer flows caused by a stretching surface. He employed the analytic method known as homotopy analysis technique to get the two branches of solutions. After that, Fang 24 numerically studied the flow and heat transfer features for Newtonian fluid past a stretching sheet. He utilized the Runge-Kutta numerical integration scheme to get the dual solutions for flow fields. Dual solutions for stagnation-point flow in the presence of chemical reaction past a stretching/ shrinking cylinder has been deliberated by Najib et al. 25 . Additionally, the flow and heat transfer characteristics in the presence of nanoparticles over a nonlinearly stretching/shrinking sheet has been examined by Zaimi et al. 26 . They observed that multiple solutions exist for a specific range of shrinking parameter. Recently, Naganthran 27 bestowed a numerical review for the unsteady flow of third grade fluid past a stretching/shrinking sheet. The multiple branches of solution have been derived using bvp4c routine in MATLAB. Recently, Khan et al. 28 also carried out a numerical simulation for slip-flow and heat transfer features of nanofluid past a permeable shrinking sheet in the presence of non-linear thermal radiation.
According to the literature review, a limited amount of research has been linked to evaluating flow and heat transfer characteristics for non-Newtonian fluids due to a radially shrinking sheet in the presence of nanoparticles. However, to the best knowledge of the authors, only few researchers have ever attempted to obtain the multiple solutions for the flow of a non-Newtonian Carreau nanofluids caused by a radially shrinking surface. Additionally, keeping in mind the engineering applications of nanofluids and flow caused by a radially shrinking surface, the main objectives and novelty of this article is: i. To investigate the flow of Carreau nanofluids by employing Buongiorno's model of nanfluid. ii. The mathematical formulation is presented in the company of Brownian motion and thermophoresis. iii. The flow analysis is examined in the neighbourhood of stagnation-point. iv. The impacts of uniform suction are addressed in view of its physical features. v. For numerical treatment, the use of a MATLAB routine bvp4c based on finite difference scheme to acquire the multiple solutions for current governing problem. vi. The set of critical values obtained for different values of controlling parameter are explored graphically.
However, the thermo-physical aspects during the flow of non-Newtonian fluids generated by a stretching surface using nanofluid has various applications in manufacturing processes where the raw material passes through the die for the extrusion in a liquefied state under high temperature with densities gradient. Moreover, the phenomenon of stretching surface into a cooling medium is a mathematical tool for the process of heat treatment in the fields of engineering technology noticed in mechanical, civil, architectural engineering. After modelling the problem, the solution of nonlinear differential equations governing the flow problem has been carried out.

Mathematical Model
Steady incompressible two-dimensional flow of a non-Newtonian Carreau nanofluid driven by a radially stretching/shrinking sheet is considered. Further, in this study heat and mass transfer in the presence of Brownian motion and thermophoresis are investigated. A locally orthogonal set of coordinates (r, z) is chosen in such a way that the origin O is kept in the plane of stretching/shrinking sheet. The velocity of the stretching/shrinking sheet is denoted as u w = ar m in which a and m are constants. The geometry of the physical problem is shown in Fig. 1. The sheet is kept at a constant temperature T w . Here, T ∞ and C ∞ are the ambient temperature and nanoparticle concentration, respectively. Moreover, the free stream velocity is u e (r) = br m for which b is a constant.
The differential equations that model this problem consist of four categories of conservation of mass, momentum, energy and nanoparticle concentration equations, which are expressed as 3,4 : The corresponding boundary conditions for the stretching/shrinking sheet is Here, u and w denotes the velocity components along r− and z− directions, respectively, ν, α, D B , D T , C, T are the kinematic viscosity, thermal diffusivity, Brownian motion coefficient, thermophoresis diffusion coefficient, nanoparticles concentration, fluid temperature. Furthermore, τ refers for the ratio of nanoparticle heat capacity and nanofluid heat capacity, n and Γ denotes the power-law index and the material parameter also known as relaxation time. It is important to note that the Newtonian case is achieved for n = 1 or Γ = 0. The non-dimensional variables for the governing Eqs (1-4) with boundary conditions (5) and (6) are written as follows: e w e 1/2 2 1/2 The velocity components in the perspective of stream function are takes after as: Employing Eqs (7 and 8) into governing Eqs (2-4), we get:  The transformed boundary conditions are: where, primes represent the differentiation with respect to η. The different flow parameters appearing in Eqs (9-13) are characterized by: the Schmidt number.
The parameters of physical and engineering importance are the skin friction, local Nusselt number and local Sherwood number. In mathematical form, these are expressed as: Skin-friction coefficient: The skin friction coefficient (wall shear stress) C f is given as: f f e n z 2 Upon using Eqs (7 and 8), the dimensionless form of skin friction becomes Local Nusselt number: The local Nusselt number (rate of heat transfer) Nu is given as: In view of non-dimensional variables (7), the reduced form of local Nusselt number is Local-Sherwood number: The local Sherwood number (rate of mass transfer) Sh is written as: Making use of Eq (7), local Sherwood number reduces to 1/2 Numerical approach. The set of governing Eqs (9-11) are non-linear in nature and their exact solutions are not feasible. Therefore, the transformed set of ordinary differential Eqs (9-11) alongside the boundary conditions (12) and (13) are numerically integrated via the boundary value problem solver bvp4c in MATLAB. The main theme of this package utilized the finite difference technique. In this method, the system of partially coupled differential equations is altered to a set of first order ordinary differential equations. To do this, let us define the new variables (   with the associated initial conditions as , (0) 1, (0) 1, The above system of seven first order differential Eqs (21-23) with initial conditions (24) and (25) can be solved with the bvp4c function in MATLAB. The maximum residual error is considered here is 10 −5 . In this scheme, the dual solutions are collected by adjusting different initial guesses for y 3 (∞) and y 4 (∞) i.e., f″(0) and θ′(0) according to the different physical parameters. Moreover, the far field conditions (25) has to be satisfied asymptotically by all the profiles. In this analysis, the boundary conditions (25) are implemented for some finite value of similarity variable η denoted by η max . We did our computations for η max = 7, 10 and 15 to achieve asymptotic behaviour of parameters on velocity, temperature and concentration profiles.
Validation of numerical scheme. To establish the accuracy of our computational scheme, a comparison is made for the numeric data of local skin friction coefficient and local Nusselt number in limiting cases with that of Wang and Ng 29 , Rosca et al. 30 , Soid et al. 31 and Wang et al. 32 . These comparisons are depicted in Tables 1 and 2. These tables demonstrate an outstanding agreement between the present numerical results and the existing works. This gives us reliance of our numerical outcomes.

Results and Discussion
In the accompanying segment, our fundamental objective is to comprehend the physics of the numerical model through tables and graphic structures. To dissect the effects the different flow parameters likewise the Weissenberg number We, the suction parameter s, the stretching/shrinking parameter χ, Prandtl number Pr, the Lewis number Le, the Brownian motion Nb and the thermophoresis parameters Nt on skin friction coefficient, local Nusselt number and local Sherwood number. Moreover, we apportioned sensible numerical values to the supervising parameters with a true objective to get a comprehension into the velocity, temperature and concentration profiles. Physically concerned quantities. The impact of sundry supervising flow parameters on skin friction coefficient, the rate of heat transfer and the local Sherwood number are respectively discussed through several graphs. The focus of this dissection is to captured dual solution. solid line represents for the First solution (upper branch solution) and dash lines for the second (lower branch solution). Dual solution exists in some cases. One of them is that when flow flows over a moving surface. As expressed by a few authors for instance by Lio 23 , Fang 24 and Khan et al. 28 who explicated the nature of dual solution occurrence. Furthermore, it uncovers a fascinating actuality that both branches solution arrive at an end at a specific value of suction parameter s and stretching/shrinking parameter χ are known as critical value (critical point) (s c ) and (χ c ) respectively. At that critical point only one solution can be found for both branches. It should be realized that dual solution exists in the range s ≤ s c and χ ≤ χ c in the beyond of critical value i.e., s < s c and χ < χ c , no solution exists. Many researchers [33][34][35][36] performed the stability solution for the both branches. It may be concluded that first branch of the solution is stable and physically reliable while the lower branch is unstable and not physically meaningful. Figures 2 to 4 shows the impact of Weissenberg number We on skin friction coefficient C Re Nusselt number −θ′(0) and Sherwood number −φ′(0) by keeping the other parameters fixed. It is seen that dual solution exists in all the graphs and both solutions are separated by a critical value. For increasing values of Weissenberg number We, the skin friction coefficient increases in upper branch solution and a converse pattern is noted in the lower branch solution, as seen in Fig. 2   enhances. A quite different trend can be noted for concentration profile in Fig. 18. The thermophoretic effect can be discern in Fig. 19. With the development in thermophoresis Nt shows the increment in fluid temperature θ(η) and the thermal boundary layer thickness elevates with advancement of Nt in both solutions. The influence of Prandtl number Pr on temperature profiles is described through Fig. 20. Dual solution occurs with respect to η. With the advancement in Pr, the concentration of nanoparticle declines in both solution in the case of shrinking surface (χ = −2.0). Also, it can be conducted that thickness of concentration boundary layer diminishes in both solution.

Main Findings
This study work presents a numerical-based survey for the subsistence of dual homogeneous attribute solutions for flow on a moving crustal surface in nanofluids. Two-dimensional axisymmetric flow of Carreau fluid model is utilized and bvp4c function in MATLAB is used to gain with effort the dual solutions. In the end, there are some paramount outcomes for this discourse got as: (1) Dual solution exists in the case of moving surface.
(2) At the sheet, the local Nusselt number and local Sherwood number is, respectively greater for higher values of Weissenberg number in the upper solution while decline in the lower solution.