1 Introduction

The Casson fluid is assorted as a non-Newtonian liuid because of its rheological characteristics in relation to the shear stress-strain. Slight values of shear strain make the Casson fluid behave as an elastic solid, whilst above a value of critical stress, it behaves as a Newtonian fluid. The Casson fluid can be depicted as a shear thinning liuid with limitless viscosity at zero shear rate, and zero viscosity at a limitless rate of shear. A number of such common examples of liuids that present Casson fluid characteristics comprise human blood, honey, tomato, sauce, and orange juice. In 1995, Casson established the Casson fluid model for viscoelastic fluid flow. Mustafa et al.^[1] analyzed the analytical solution for flow and heat transfer of Casson fluid stagnation-point towards a stretching flat sheet via the homotopy analysis method (HAM). Nadeem et al.^[2] presented a self-similar solution to get the solution of the governing flow euations of an electrically conducting Casson fluid flow in two lateral directions through a porous linear stretching sheet. Boyd et al.^[3] analyzed the Casson fluid of steady and oscillatory blood flow. Eldabe and Salwa ^[4] addressed the magnetohydrodynamics (MHD) Casson fluid flow between two rotating cylinders.

Nadeem et al.^[5] depicted an enhancement in the heat and mass transfer of a three-dimensional non-Newtonian Casson nanofluid and regulated the hot liuid over the lower surface of the wall by presenting a convective boundary condition. They pointed out that a Newtonian nanofluid induces downgrade skin friction at the wall with respect to the Casson nanofluid, and for a higher Prandtl number, they obtained lower thermal conductivity. Mukhopadhyay et al.^[6] looked at the Casson fluid flow over an unsteady stretching surface by extending the paper of Andersson et al.^[7]. Bhattacharyya^[8] reported the two-dimensional MHD stagnation region flow of an electrically conducting Casson fluid and heat transfer along a stretching surface with the impact of thermal radiation. They depicted that the Casson fluid velocity boundary layer thickness becomes larger than that of Newtonian fluid due to the Casson fluid plasticity. Mahdy and Ahmed^[9] looked at the unsteady mixed convection flow of a non-Newtonian Casson fluid in the stagnation point of a rotating sphere. Mahdy^[10] introduced a numerical solution for heat transfer of a non-Newtonian Casson fluid outside a stretching permeable cylinder through Soret and Dufour impacts.

Speedy progress in industrial and thermal engineering operations reuired more efficient and more compact heat transfer systems. These systems of heat transfer liuids are applied as cooling agent. Anyway such liuids possess low thermal conductivity which badly impacts the heat transfer rate or cooling process. It was actually a defying task for scientists to improve the liuid thermal conductivity. To do this, about a decade ago, the first endeavor was proposed by Choi and Eastman^[11] and Choi et al.^[12] to present the nanofluid. The base fluid with suspended nanoparticles is so-called nanofluid. Not a few liuids possess low capability of thermal conductivity. Particles of nanosized metallic (silver, titanium, and copper) are suspended in the habitual liuids in order to upgrade the property of thermal conductivity. Nanofluid applications appear for improving the capability of heat transfer of transportation, computer microchips, solid state lightening, fuel cells, biomedicine, and microelectronics. Two classes are available to incorporate nanoparticle impact on fluid flow, namely, single-phase nanofluid and two-phase nanofluid models (Sarkar^[13] and Buongiorno^[14]). Further, in the Tiwari and Das model, the effective fluid properties were considered. Such theoretical and experimental contributions on nanofluid heat transfer property have been carried out in Refs.[15]-[38].

Through engineering and industrial operations, heat and mass transfer is caused by buoyancy impacts due to thermal diffusion. Takhar et al.^[39] used the finite difference method to get the solution for the governing boundary layer euations of unsteady free convection flow in the forward stagnation-point region of a sphere, which rotates with time-dependent angular velocity in an ambient fluid. Results illustrated that as the Prandtl number upgrades, heat transfer enhances, whilst the skin friction reduces. Chamkha et al.^[40] examined the problem of unsteady MHD free convection of a rotating fluid through a rotating sphere near the euator. They detected that both the magnetic field and the surface suction reduce the shear stress, whereas they enhance the surface shear stress in the rotational direction. Anilkumar and Roy^[41] introduced a new self-similar solution of unsteady mixed convection rotating fluid flow on a rotating cone with the combined impacts of both mass and thermal diffusion. Heat and mass transfer flow of an electrically conducting fluid in the stagnation point of a rotating sphere with the impact of thermal radiation was addressed by Mahdy and Ahmed^[42].

The investigation of time-dependent flow of Casson nanofluid has not attracted much attention. Hence, the essential aim of the current contribution is to address unsteady MHD Casson single-phase nanofluid flow through variable wall temperature in the stagnation region of a rotating sphere. The model of traditional non-Newtonian Casson nanofluid is revised to incorporate the impacts of thermophoresis and Brownian motion. We provide uantitative and ualitative comparisons with the preceding published data to assess the accuracy of the procedure.

2 Physical model

Let us look at the problem of conjugate MHD forced and natural convection (mixed) of Casson single-phase nanofluid boundary layer flow in the forward stagnation point region of a sphere with variable wall temperature. Water represents a base fluid that contains some types of nanoparticles, e.g., alumina Al₂O₃, silver Ag, copper Cu, and titanate TiO₂. The sphere rotates with a constant angular velocity Ω around an axis parallel to the ambient free stream velocity U(x)=ax, as depicted in Fig. 1. Besides, the fluid has constant physical properties except for the density which creates the buoyancy force, and the viscous dissipation term is ignored.

The axis coordinates x, y, and z are measured from the forward stagnation point along the surface, perpendicular to the surface, and in the rotating direction, respectively. The Casson nanofluid is electrically conducting with existence of a magnetic field B₀ constantly applied in the z-direction. Before time t=0, the sphere is assumed to be motionless in an ambient fluid, and the surface temperature is T_∞. The sphere surface temperature is instantly raised to T_w=T_∞+dxⁿ when the time reaches t=0. In addition, the rheological model euation of state for an isotropic and incompressible flow of a Casson fluid is described as^[2]

in which π=e_ije_ij and e_ij represent the (i, j)th deformation rate components, π_c gives the critical value of this product based on the non-Newtonian model, μ_B means the dynamic viscosity of the non-Newtonian fluid, and P_y gives the yield stress fluid. According to the preceding presumptions and taking into account the Boussines approximation, the unsteady laminar boundary layer euations characterizing the mixed convection Casson single-phase nanofluid flow are^{[24, 47]}

(1)

(2)

(3)

(4)

The initial conditions are described as

(5)

associated with the following compatible boundary conditions:

(6)

whereas the boundary conditions at ambient (i.e., z →∞) are portrayed as

(7)

The components of Casson nanofluid velocity in the x-, y-, and z-directions are symbolized by u, v, and w, respectively, γ refers to the Casson parameter, t is the time, R is the sphere radius, g refers to the gravity acceleration, n represents the sphere surface temperature exponent, μ_nf, ρ_nf, k_nf, σ_nf, and (c_p)_nf refer to the nanofluid viscosity, nanofluid density, nanofluid thermal conductivity, effective electrical conductivity, and specific heat at constant pressure, respectively, and β_nf indicates the expansion effective thermal volumetric coefficient. Now, the effective properties of nanofluid are described by the following formulae^[43-44]:

where φ represents the nanoparticle solid volume fraction (φ=0 points to a regular fluid), ρ_f and ρ_p refer to the density of base fluid and nanoparticles, respectively, (c_p)_f and (c_p)_p mean the specific heat of base fluid and nanoparticles, respectively, σ_f and σ_p indicate the electrical conductivity of base fluid and nanoparticles, respectively, β_f and β_p represent the thermal volumetric coefficient of base fluid and nanoparticles, respectively, μ_f denotes the base fluid viscosity, and k_f and k_p refer to thermal conductivity of base fluid and nanoparticles, respectively. The thermo-physical properties of the nanofluid are depicted in Table 1 as Jawad et al.^[45].

Now, to convert the flow governing euations and their compatible boundary conditions into non-dimensional forms, the following non-similarity transformation are utilized:

(8)

Via the provided non-similarity transformations (8), the remaining flow governing euations (2)-(4) are mutated to a system of ordinary differential euations (ODEs) of the form

(9)

(10)

(11)

associated with the mutated boundary layer conditions

(12)

The resulting primes mean differentiation with respect to the non-similarity variable η, where refers to the Prandtl number, indicates the magnetic field parameter, indicates the rotation parameter, is the mixed convection parameter, is the Reynolds number, and denotes the Grashof number.

Now, the shear stresses in the x- and z-directions are described as

(13)

(14)

The rate of surface heat transfer in terms of the Nusselt number takes the form

(15)

3 Numerical scheme

Now, the highly nonlinear governing euations (9)-(11) associated with boundary conditions (12) are solved numerically via the Keller box procedure. The detail of this techniue is interpreted by Cebeci and Bradshaw^[46]. This method is unconditionally stable and has accuracy of second-order. Four steps are followed to obtain the numerical solutions.

Step1   The resulting governing flow euations are mutated to a system of first-order.

Step2   Then, the derived first-order system is approximated via formula of central difference about the mid-point.

Step3   Now, the deduced algebraic euations are uasi-linearized via Newton's method and insert them in matrix-vector notation.

Step4   In the end, we solve the obtained linear system via the block techniue of tri-diagonal elimination.

Now, the used step size is η=0.01 and the boundary layer thickness is η_∞=7. Additionally, for all the cases, the convergent criteria 10^-6 is considered. The calculations and graphical results are produced through MATLAB software. To assess the validity and accuracy of used code, a comparison is made with previous data from the literature as a limiting case. Figure 2 is provided to assess the accuracy of the used techniue. A comparison with those published by Takhar et al.^[47] is made and found to be in good agreement.

4 Results and discussion

In order to address the impacts of various governing parameters of Casson nanofluid flow behavior thereabout the stagnation point of rotating sphere, numerical computations are pointed out. The ranges of such controlling parameters are the non-Newtonian Casson parameter 0.1≤ γ≤ ∞ (Newtonian fluid), the surface temperature exponent (constant wall temperature) 0≤ n≤ 2, the Prandtl number 0.7≤Pr≤ 10, the rotation parameter 1.0 ≤ λ≤ 50.0, the mixed convection parameter 0.0 ≤ γ^*≤ 5.0, the magnetic field parameter 0.0 ≤ M_g≤ 10, and the nanoparticle volume fraction 0.0 ≤ φ≤ 0.1. Now, with n=1, λ=10, γ^*=1, Pr=6.2, γ=1, M_g=3, and φ=0.05, Fig. 3 portrays the rate of heat transfer expressed in terms of Nusselt number Nu for some different nanoparticles, namely, Cu, Ag, Al₂O₃, and TiO₂. We notice that both Cu and Ag are close with little higher of Cu. TiO₂ gives the minimum value of Nusselt number. Similar to the shear stress -S'(ξ, 0), we observe that Cu is near Ag, while Al₂O₃ is near TiO₂ (see Fig. 4).

Figures 5 and 6 show the impact of power-law index of surface temperature n on the temperature curves and Nusselt number with Cu nanoparticle and for both φ=0.00 (regular fluid) and φ= 0.05. A clear reduction in temperature curves is obtained as n upgrades (see Fig. 5). Figure 6 explains that the Nusselt number becomes larger as n upgrades. From Fig. 7, as the Casson parameter γ goes to ∞, a reduction of the non-Newtonian behavior to Newtonian behavior occurs. Therefore, the velocity boundary layer thickness for the non-Newtonian Casson fluid becomes larger than that for the Newtonian one. This is caused by the plasticity of the fluid that upgrades as the Casson parameter lessens which causes an upgrade in the velocity of the y-direction and the thermal boundary layer thickness. Conseuently, the phenomenon of free convection is downgraded by enhancing γ. In addition, the temperature curves reduce as the Casson parameter goes from non-Newtonian fluid to Newtonian fluid, i.e., with enlarging γ (see Fig. 8).

Besides, the variations of shear stresses in the x- and y-directions with Cu nanoparticle, i.e., F"(ξ, 0) and -S'(ξ, 0) against ξ for some values of Casson parameter γ are illustrated in Figs. 9 and 10. Both F"(ξ, 0) and -S'(ξ, 0) reduce as the Casson parameter upgrades. Per contra, the Nusselt number boosts with increasing γ as portrayed in Fig. 11. The impacts of the nanoparticle volume friction φ on the velocity profiles, the temperature distribution, the rate of heat transfer in terms of Nusselt number and shear stresses for the Cu nanofluid are presented in Figs. 12-16, respectively. As φ upgrades, F'(ξ, η), θ(ξ, η), Nu, F"(ξ, 0), and -S'(ξ, 0) uplift except the velocity in the y-direction reduces. Commonly, the thermal conductivity of the nanofluid has a remarkable effect in such kinds of applications. According to this fact, one can explore the behaviors of F'(ξ, η), θ(ξ, η), Nu, F"(ξ, 0), and -S'(ξ, 0) under the impact of φ. Thus, as φ upgrades, the nanofluid thermal conductivity enhances which yields a good convection. The variation of Nusselt number Nu and shear stresses in the x- and y-directions with Cu nanoparticle, i.e., F"(ξ, 0), -S'(ξ, 0), and the Nusselt number against ξ for some values of mixed convection parameter γ^* are plotted in Figs. 17-19. -θ' (ξ, 0), F"(ξ, 0), and -S'(ξ, 0) increase as the mixed convection parameter γ^* upgrades which is caused by downgrade of viscous and thermal boundary layer thicknesses. This can be expounded as the positive buoyancy force acts as a pressure gradient which accelerates the fluid motion inside the boundary layer. This in turn minimizes the thickness of thermal and momentum boundary layers and conseuently upgrades heat transfer and shear stresses at the surface.

Figures 20-22 exhibit the impact of the rotation parameter λ on the rate of heat transfer expressed in terms of Nusselt number and shear stresses, respectively, with Cu nanoparticles φ=0.05 and Casson fluid γ=1. An enhance in the rotation parameter λ has a strong positive impact on the shear stresses and rate of heat transfer.

Nu, F"(ξ, 0), and -S'(ξ, 0) upgrade by increasing λ. Moreover, the velocity and thermal boundary layer thickness upgrade slightly with upgrading the rotation parameter. The reduction of momentum and thermal boundary layers leads to the increase in the velocity and temperature gradients at the wall. An increment is found in the shear stresses in both the x- and y-directions and the rate of heat transfer and temperature distribution when the magnetic field parameter M_g upgrades (see Figs. 23-26).

Via the physical standpoint, when the magnetic field produces current in the conductive fluid, it creates a resistive-type force (called the Lorentz force) on the fluid in the boundary layer, which makes the motion of the fluid be slow. This downgrade of the velocity distributions improves both the velocity and thermal boundary layer thicknesses which causes an upgrade in the shear stresses in the x- and y-directions and the rate of heat transfer θ'(ξ, 0). The surface shear stresses and rate of heat transfer do not depend on the magnetic field M_g at ξ=0 (at the beginning of the motion). Then, the impact of M_g upgrades with ξ. The temperature distribution reduces when M_g upgrades as portrayed in Fig. 26.

5 Remarks

The aim of our present analysis is to inspect the simultaneous impacts of variable wall temperature and MHD on the time-dependent mixed convection flow of non-Newtonian Casson nanofluid through the stagnation region according to impulsively rotating sphere. The parameters of surface shear stresses and heat transfer rate are improved by the surface temperature exponent parameter, the sphere rotation, and the buoyancy force and by using nanoparticles rather than the regular fluid. As the nanoparticle volume friction upgrades, both F'(ξ, η) and the temperature distribution upgrade, but S(ξ, η) downgrades. The rotation parameter λ and the Casson parameter γ tend to boost the rate of heat transfer θ'(ξ, 0). The nanoparticles can boost the heat transfer capacity of regular fluid when it is added to the base fluid. As the solid volume fraction enhances, the more impact is presented.

Acknowledgements The authors would like to express their gratitude to the reviewers for their constructive and precise suggestions.