On magnetized non-Newtonian rotatory fluid flow field

Abstract

The current pagination is devoted to offering the untapped features of non-Newtonian rotatory magnetized fluid flow over the solid disk. The said non-Newtonian fluid model is a Powell–Eyring fluid model. The flow field is further carried with suspended nanoparticles with velocity slip effects. The strength of the article is a complex mathematical modeling subject to physical effects mentioned before, and the computational results are provided through the self-coded algorithm rather than to move on with the usual built-in scheme. To make the implementation possible, the obtained flow-narrating system is converted into a system having fewer independents. The key involved parameters are Powell–Eyring fluid, magnetic field, velocity slip, thermophoresis, and Brownian motion parameters. The dependent quantities namely axial, tangential velocities, temperature, and concentration are examined against flow-controlling parameters. The obtained outcomes in this direction are offered by means of graphical trends. It is noticed that both axial and radial velocities possess direct relation with Power–Eyring parameter (M). The Powell–Eyring fluid temperature is an increasing function of the thermophoresis parameter. Furthermore, the Powell–Eyring concentration enhances significantly toward the higher values of the Brownian motion parameter.

Keywords

Non-Newtonian fluid solid disk magnetized flow nanoparticles velocity slip

Introduction

Fluid flow realization due to the rotation of disk persuades theoretical and practical significance of engineering and applied sciences. Some important practical fields are the rotor–stator system, gas turbine engineering, air rotational cleaners, medical equipment, chemical engineering, and thermal power–generating systems. Von Karman¹ admitted the fluid flow achieved by a rotating disk and introduced the transformation to transform the partial differential equations (PDEs) into ordinary differential equations (ODEs). Many researchers have investigated his work under different aspects. Shateyi and Makinde² analyzed the hydromagnetic stagnation-point flow over a stretching convectively heated disk. Rashidi et al.³ analyzed the magnetohydrodynamic (MHD) flow of viscous liquid over a rotating disk under entropy-generation effects. Turkyilmazoglu⁴ discussed the rotating disk flow with nanoparticle’s effects. Mustafa⁵ elaborated MHD partial slip nanofluid flow over a disk with MHD. Hatami et al.⁶ used the least square method and examined the incompressible laminar nanoliquid flow generated by contracting and rotating disks. Kumar and Kavitha⁷ examined the three-dimensional (3D) Jeffrey fluid flow between stationary and rotating disks. They noticed that the radial velocity of a fluid increases with the increasing Jeffery fluid parameter, while the axial velocity decreases with the increasing value of Jeffrey fluid parameter. Rehman and colleagues^8–10 investigated the flow over a rotating disk with the impact of MHD and nanoparticles.

The confined flow of non-Newtonian fluids has attracted the research community in the past decades because of its applications such as polymer extrusion, glass-fiber production, and aerodynamic extrusion of plastic sheets, paints, and lubricants. In literature, there is not a single constitutive formula to designate the characteristics of non-Newtonian materials. Various models of non-Newtonian fluids have been proposed regarding the nature of the materials. The Powell–Eyring (PE) fluid model is one model that is quite capable of being used in a chemical engineering system. This model has numerous advantages as compared to other non-Newtonian formulas such as simplicity and physical robustness. In the past, many attempts have been made to characterize the physical attributes of a PE fluid, for instance, Malik et al.¹¹ reveal the variable viscosity shear-thinning fluid flow over a stretchable cylinder. They witnessed that the temperature of the fluid decreases with intensifications in Prandtl number and Reynolds number. Jalil et al.¹² appraised the significances of self-similar solutions for the PE fluid flow in a parallel free stream. Akbar et al.¹³ scrutinized the MHD PE fluid flow over a stretchable sheet. They studied that the opposition of flow increases for the large magnetic field as well as the parameter of PE fluid. Malik et al.¹⁴ considered the MHD flow of PE fluid over a stretchable sheet. They detected that the flow velocity has a reverse relation with Hartmann number and mixed convection parameters. The physical and thermal features of an MHD PE fluid that flows through a stretched surface along with heterogeneous and homogeneous reactions and chemically reactive species is discussed by Khan and colleagues.^15,16 Rehman and colleagues^17,18 consider the stagnation point in the flow of PE fluid through the cylinder with effects of magneto-nanofluid. In addition, they discussed the comparative flow study of the inclined surface with chemical reactions. Khan et al.¹⁹ discussed the flow effects of heterogeneous–homogeneous reactions and MHD. Makinde and colleagues^20,21 analyzed the effect of MHD on stretching convective surface with stagnation point flow and radiative heat.

The nanoliquids have valuable uses such as engine cooling, biomedicine, electronic cooling devices and many more. Choi and colleagues^22,23 first used nanoparticles in the base fluid to improve the thermo-physical properties of a nanofluid with base the fluids. They observed that using the nanoparticles thermal conductivity of fluid is more effectual than base fluids. The thermal conductivity of a nanofluid is experimentally examined and found out that the energy transfer of a nanofluid is larger than that of base fluids.^24,25 Ibrahim²⁶ deliberate the convective viscous nanofluid in the presence of the stagnation point in the flow field. Khan et al.²⁷ deliberated the Williamson MHD nanofluid flow over a plate and cone with chemically reactive species. Salahuddin et al.²⁸ consider the stretching cylinder and resolved numerically the flow problem of MHD tangent hyperbolic nanofluid. Rashad et al.²⁹ examined the Joule heating phenomenon inflow of convective viscous nanofluid through the circular cylinder. They found that the thermal energy increases significantly via the Joule heating phenomenon. Hussain et al.³⁰ observed the flow of nanofluid with combined effects of viscous dissipation and Joule heating over a stretching cylinder. They analyzed that in the case of both viscous dissipation and Joule heating, the fluid temperature significantly increases. Rehman et al.³¹ scrutinized the influences of Joule heating on a MHD PE fluid that flows over a cylindrical surface. Mahanthesh et al.³² designed a 3D flow of a PE nanofluid flow through a stretchable sheet under the impact of Joule heating, applied magnetic field, and thermal radiation.

They examined that the thermal energy of the fluid increases with increasing viscous dissipation. Extensive studies for the nanofluid over different geometries are described in literature.^33–39

The current investigation deals with the flow of a PE nanofluid over a rotating disk with additional effects of MHD and slip condition. The governing nonlinear analysis has been carried out for velocity, temperature, and nanoparticle concentration profiles using Runge–Kutta (RK-5) method. The obtained results are offered with the aid of graphs.

Problem formulation

The non-Newtonian fluid model named Powell–Eyring is considered. The flow is achieved due to the rotation of the solid disk. The flow field has an interaction with the externally applied magnetic field. The nanoparticles are suspended in the PE fluid. The velocity slip assumption is also taken into account. The geometric illustration is shown in Figure 1.

Figure 1.

Geometry of the problem.

The fundamental laws involved in the field of fluid science yields a system of PDEs, that is

u_{r} + \frac{u}{r} + w_{z} = 0

(1)

\begin{matrix} u u_{r} - \frac{v^{2}}{r} + w u_{z} = ν (1 + M) (2 u_{rr} + 2 u_{r} + u_{zr} + w_{rz}) \\ - \frac{1}{ρ β c^{3}} {\begin{matrix} (2 u_{r}^{2} + v_{r}^{2} + \frac{v^{2}}{r^{2}} - \frac{2 v v_{r}}{r} + w_{r}^{2} + u_{z}^{2} + 2 u_{z} w_{r} + 2 \frac{u^{2}}{r^{2}} + v_{z}^{2} + 2 w_{z}) (2 u_{rr}) \\ + (2 u_{r}^{2} + v_{r}^{2} + \frac{v^{2}}{r^{2}} - \frac{2 v v_{r}}{r} + w_{r}^{2} + u_{z}^{2} + 2 u_{z} w_{r} + 2 \frac{u^{2}}{r^{2}} + v_{z}^{2} + 2 w_{z}) (\frac{2}{r} u_{r}) \\ + (\begin{matrix} 4 u_{r} u_{rr} + 2 v_{r} v_{rr} + \frac{2 v_{r} v}{r^{2}} - \frac{2 v^{2}}{r^{3}} - 2 \frac{v v_{rr}}{r} - 2 \frac{v_{r}^{2}}{r} + \frac{2 v v_{r}}{r^{2}} + 2 w_{r} w_{rr} + 2 u_{z} u_{zr} \\ + 2 u_{z} w_{rr} + 2 w_{r} u_{zr} + \frac{4 u_{r} u}{r^{2}} - \frac{4 u^{2}}{r^{3}} + 2 v_{z} v_{zr} + 2 w_{zr} \end{matrix}) (2 u_{r}) \\ + (2 u_{r}^{2} + v_{r}^{2} + \frac{v^{2}}{r^{2}} - \frac{2 v v_{r}}{r} + w_{r}^{2} + u_{z}^{2} + 2 u_{z} w_{r} + 2 \frac{u^{2}}{r^{2}} + v_{z}^{2} + 2 w_{z}) (u_{zz} + w_{rz}) \\ + (\begin{matrix} 4 u_{r} u_{rz} + 2 v_{r} v_{rz} + \frac{2 v_{rz} v}{r^{2}} - 2 \frac{v v_{rz} + v_{r} v_{z}}{r} + 2 w_{r} w_{rz} + 2 u_{z} u_{zz} + 2 u_{z} w_{rz} \\ + 2 w_{r} u_{zz} + \frac{4 u u_{z}}{r^{2}} + 2 v_{z} v_{zz} + 2 w_{zz} \end{matrix}) (u_{z} + w_{r}) \end{matrix}} - \frac{σ {B_{o}}^{2} u}{ρ} \end{matrix}

(2)

\begin{matrix} u v_{r} - \frac{uv}{r} + w v_{z} = ν (1 + M) (\frac{2}{r} (v_{r} - \frac{v}{r}) + v_{rr} \frac{v_{r}}{r} + \frac{v}{r^{2}} + v_{zz}) \\ - \frac{1}{ρ β c^{3}} {\begin{matrix} (2 u_{r}^{2} + v_{r}^{2} + \frac{v^{2}}{r^{2}} - \frac{2 v v_{r}}{r} + w_{r}^{2} + u_{z}^{2} + 2 u_{z} w_{r} + 2 \frac{u^{2}}{r^{2}} + v_{z}^{2} + 2 w_{z}) (\frac{2 v_{r}}{r} - \frac{2 v}{r^{2}}) \\ + (2 u_{r}^{2} + v_{r}^{2} + \frac{v^{2}}{r^{2}} - \frac{2 v v_{r}}{r} + w_{r}^{2} + u_{z}^{2} + 2 u_{z} w_{r} + 2 \frac{u^{2}}{r^{2}} + v_{z}^{2} + 2 w_{z}) (v_{rr} - \frac{v_{r}}{r} + \frac{v}{r^{2}}) \\ + (\begin{matrix} 4 u_{r} u_{rr} + 2 v_{r} v_{rr} + \frac{2 v_{r} v}{r^{2}} - \frac{2 v^{2}}{r^{3}} - 2 \frac{v v_{rr}}{r} - 2 \frac{v_{r}^{2}}{r} + \frac{2 v v_{r}}{r^{2}} + 2 w_{r} w_{rr} + 2 u_{z} u_{zr} \\ + 2 u_{z} w_{rr} + 2 w_{r} u_{zr} + \frac{4 u_{r} u}{r^{2}} - \frac{4 u^{2}}{r^{3}} + 2 v_{z} v_{zr} + 2 w_{zr} \end{matrix}) (v_{r} - \frac{v}{r}) \\ + (2 u_{r}^{2} + v_{r}^{2} + \frac{v^{2}}{r^{2}} - \frac{2 v v_{r}}{r} + w_{r}^{2} + u_{z}^{2} + 2 u_{z} w_{r} + 2 \frac{u^{2}}{r^{2}} + v_{z}^{2} + 2 w_{z}) (v_{zz}) \\ + (\begin{matrix} 4 u_{r} u_{rz} + 2 v_{r} v_{rz} + \frac{2 v_{z} v}{r^{2}} - 2 \frac{v v_{rz} + v_{r} v_{z}}{r} + 2 w_{r} w_{rz} + 2 u_{z} u_{zz} + 2 u_{z} w_{rz} + 2 w_{r} u_{zz} \\ + \frac{4 u u_{z}}{r^{2}} + 2 v_{z} v_{zz} + 2 w_{zz} \end{matrix}) (v_{z}) \end{matrix}} - \frac{σ {B_{o}}^{2} v}{ρ} \end{matrix}

(3)

\begin{matrix} u w_{r} + w w_{z} = ν (1 + M) (\frac{w_{r}}{r} + \frac{u_{z}}{r} + w_{rr} + u_{zr} + 2 w_{zz}) \\ - \frac{1}{ρ β c^{3}} {\begin{matrix} (2 u_{r}^{2} + v_{r}^{2} + \frac{v^{2}}{r^{2}} - \frac{2 v v_{r}}{r} + w_{r}^{2} + u_{z}^{2} + 2 u_{z} w_{r} + 2 \frac{u^{2}}{r^{2}} + v_{z}^{2} + 2 w_{z}) (\frac{w_{r}}{r} + \frac{u_{z}}{r}) \\ + (2 u_{r}^{2} + v_{r}^{2} + \frac{v^{2}}{r^{2}} - \frac{2 v v_{r}}{r} + w_{r}^{2} + u_{z}^{2} + 2 u_{z} w_{r} + 2 \frac{u^{2}}{r^{2}} + v_{z}^{2} + 2 w_{z}) (w_{rr} + u_{zr}) \\ + (\begin{matrix} 4 u_{r} u_{rr} + 2 v_{r} v_{rr} + \frac{2 v_{r} v}{r^{2}} - \frac{2 v^{2}}{r^{3}} - 2 \frac{v v_{rr}}{r} - 2 \frac{v_{r}^{2}}{r} + \frac{2 v v_{r}}{r^{2}} + 2 w_{r} w_{rr} + 2 u_{z} u_{zr} \\ + 2 u_{z} w_{rr} + 2 w_{r} u_{zr} + \frac{4 u_{r} u}{r^{2}} - \frac{4 u^{2}}{r^{3}} + 2 v_{z} v_{zr} + 2 w_{zr} \end{matrix}) (w_{r} + u_{z}) \\ + (2 u_{r}^{2} + v_{r}^{2} + \frac{v^{2}}{r^{2}} - \frac{2 v v_{r}}{r} + w_{r}^{2} + u_{z}^{2} + 2 u_{z} w_{r} + 2 \frac{u^{2}}{r^{2}} + v_{z}^{2} + 2 w_{z}) (2 w_{zz}) \\ + (\begin{matrix} 4 u_{r} u_{rz} + 2 v_{r} v_{rz} + \frac{2 v_{z} v}{r^{2}} - 2 \frac{v v_{rz} + v_{r} v_{z}}{r} + 2 w_{r} w_{rz} + 2 u_{z} u_{zz} + 2 u_{z} w_{rz} + 2 w_{r} u_{zz} \\ + \frac{4 u u_{z}}{r^{2}} + 2 v_{z} v_{zz} + 2 w_{zz} \end{matrix}) (2 w_{z}) \end{matrix}} - \frac{σ {B_{o}}^{2} w}{ρ} \end{matrix}

(4)

u T_{r} + w T_{z} = α (T_{rr} + \frac{T_{r}}{r} + T_{zz}) + \frac{{(ρ c)}_{p}}{{(ρ c)}_{f}} (D_{B} (T_{r} C_{r} + T_{z} C_{z}) + \frac{D_{T}}{T_{\infty}} ({(T_{rr})}^{2} + {(T_{zz})}^{2}))

(5)

u C_{r} + w C_{z} = D_{B} (C_{rr} + \frac{C_{r}}{r} + C_{zz}) + \frac{D_{T}}{T_{\infty}} (T_{rr} + \frac{T_{r}}{r} + T_{zz})

(6)

After applying the boundary layer approximation, equations (1)–(6) can further be rectified as

u u_{r} + \frac{v^{2}}{r} + w u_{z} = ν (1 + M) (u_{zz}) - \frac{1}{ρ β c^{3}} (v_{z}^{2} u_{zz} + 2 u_{z}^{2} u_{zz} + 2 v_{z} u_{z} v_{zz}) - \frac{σ {B_{o}}^{2} u}{ρ}

(7)

u v_{r} + \frac{uv}{r} + w v_{z} = ν (1 + M) (v_{zz}) - \frac{1}{ρ β c^{3}} (u_{z}^{2} v_{zz} + v_{z}^{2} v_{zz} + 2 v_{z} u_{z} u_{zz} + 2 v_{z}^{2} v_{zz}) - \frac{σ B_{o}}{ρ} v

(8)

u T_{r} + w T_{z} = α (T_{zz}) + \frac{{(ρ c)}_{p}}{{(ρ c)}_{f}} (D_{B} (T_{z} C_{z}) + \frac{D_{T}}{T_{\infty}} ({(T_{zz})}^{2}))

(9)

u T_{r} + w T_{z} = α (T_{zz}) + \frac{{(ρ c)}_{p}}{{(ρ c)}_{f}} (D_{B} (T_{z} C_{z}) + \frac{D_{T}}{T_{\infty}} ({(T_{zz})}^{2}))

(10)

The corresponding conditions are

\begin{matrix} T = T_{w}, C = C_{w}, u = L u_{z}, \\ v = r Ω + L v_{z}, w = 0, at z = 0 \\ T \to T_{\infty}, C \to C_{\infty}, u \to 0, v \to 0, as z \to \infty \end{matrix}

(11)

Since we have considered the non-Newtonian fluid model and the destruction toward fluctuation velocity gradients by the action of viscous stresses is assumed to be very small, the viscous dissipation individualities are neglected. Some works in this direction are assessed by Rehman et al.⁴⁰ Furthermore, the involved differential equations (equations (7)–(11)) are highly nonlinear coupled PDEs; therefore, it always remains difficult to solve such a system. So an equivalent system having a less number of independent variables can be attained using the following set of transformations

\begin{matrix} u = r \bar{Ω} \frac{dF (ζ)}{d ζ}, v = r \bar{Ω} G (ζ), w = \sqrt{2 \bar{Ω} ν} F (ζ) \\ T (ζ) = \frac{T - T_{\infty}}{T_{w} - T_{\infty}}, C (ζ) = \frac{C - C_{\infty}}{C_{w} - C_{\infty}}, ζ = \sqrt{\frac{2 \bar{Ω}}{ν} z} \end{matrix}

(12)

By operating equation (12) one can achieve

\begin{matrix} {(\frac{dF (ζ)}{d ζ})}^{2} - {(\frac{dG (ζ)}{d ζ})}^{2} - (2 F (ζ) \frac{d^{2} F (ζ)}{d ζ^{2}}) \\ - (2 (1 + M) \frac{d^{3} F (ζ)}{d^{3} ζ}) + 4 M λ Re {(\frac{dG (ζ)}{d ζ})}^{2} \frac{d^{3} F (ζ)}{d^{3} ζ} \\ + 4 M λ Re (2 {(\frac{d^{2} F (ζ)}{d ζ^{2}})}^{2} (\frac{d^{3} F (ζ)}{d^{3} ζ}) + 2 (\frac{d^{2} G (ζ)}{d ζ^{2}}) (\frac{d^{2} F (ζ)}{d ζ^{2}}) (\frac{dG (ζ)}{d ζ})) \\ + H (\frac{dF (ζ)}{d ζ}) = 0 \end{matrix}

(13)

\begin{matrix} (2 \frac{dF (ζ)}{d ζ}) (G (ζ)) + 2 F (ζ) (\frac{dG (ζ)}{d ζ}) - (2 (1 + M) \frac{d^{2} G (ζ)}{d^{2} ζ}) \\ + 4 M λ Re ({(\frac{d^{2} F (ζ)}{d ζ^{2}})}^{2} \frac{d^{2} G (ζ)}{d ζ^{2}}) \\ + 4 M λ Re (3 {(\frac{dG (ζ)}{d ζ})}^{2} (\frac{d^{2} G (ζ)}{d^{2} ζ}) + 2 (\frac{dG (ζ)}{d ζ}) (\frac{d^{2} F (ζ)}{d ζ^{2}}) (\frac{d^{3} F (ζ)}{d ζ^{3}})) \\ + H (\frac{dF (ζ)}{d ζ}) = 0 \end{matrix}

(14)

\frac{d^{2} T (ζ)}{d ζ^{2}} + 2 \Pr (F (ζ) \frac{dT (ζ)}{d ζ} + Nb \frac{dT (ζ)}{d ζ} \frac{dC (ζ)}{d ζ} + Nt \frac{d^{2} T (ζ)}{d ζ^{2}}) = 0

(15)

\frac{d^{2} C (ζ)}{d ζ^{2}} + ScF (ζ) \frac{dC (ζ)}{d ζ} + \frac{Nt}{Nb} \frac{d^{2} T (ζ)}{d ζ^{2}} = 0

(16)

while the reduced endpoint conditions are as follows

\begin{matrix} F (ζ) = 0, \frac{dF (ζ)}{d ζ} = γ \frac{d^{2} F (ζ)}{d^{2} ζ}, \\ G (ζ) = 1 + γ \frac{dG (ζ)}{d ζ}, T (ζ) = 1, C (ζ) = 1, as ζ \to 0 \\ \frac{dF (ζ)}{d ζ} \to 0, G (ζ) \to 0, T (ζ) \to 0, C (ζ) \to 0, as ζ \to \infty \end{matrix}

(17)

The relations for surface quantities namely, skin-friction coefficient, local Nusselt number, and Sherwood number are defined as

\begin{matrix} C_{f} = {(\frac{τ_{zr}}{ρ {(r Ω)}^{2}})}_{z = 0}, C_{g} = {(\frac{τ_{z θ}}{ρ {(r Ω)}^{2}})}_{z = 0}, \\ N u_{r} = {(\frac{r q_{w}}{k (T_{w} - T_{\infty})})}_{z = 0}, Sh = {(\frac{r q_{m}}{D_{B} (C_{w} - C_{\infty})})}_{z = 0} \end{matrix}

(18)

\begin{matrix} τ_{rz} = {μ + \frac{1}{β c} - \frac{1}{β c^{3}} (2 u_{r}^{2} + v_{r}^{2} + \frac{v^{2}}{r^{2}} - 2 \frac{v v_{r}}{r} + w_{r}^{2} + u_{z}^{2} + 2 u_{z} w_{r} + 2 {(\frac{u}{r})}^{2} + v_{z}^{2} + 2 w_{z})} (u_{z} + w_{r}) \\ τ_{θ z} = {μ + \frac{1}{β c} - \frac{1}{β c^{3}} (2 u_{r}^{2} + v_{r}^{2} + \frac{v^{2}}{r^{2}} - 2 \frac{v v_{r}}{r} + w_{r}^{2} + u_{z}^{2} + 2 u_{z} w_{r} + 2 {(\frac{u}{r})}^{2} + v_{z}^{2} + 2 w_{z})} (v_{z}) \\ q_{w} = - k \frac{\partial T}{\partial z}, q_{m} = - D_{B} \frac{\partial C}{\partial z} \end{matrix}

(19)

The dimensionless form of surface quantities can be written as

\begin{matrix} C_{f} \sqrt{{Re}_{r}} = f ″ (0) {1 + M - ML (3 f' {(0)}^{2} + Re (\frac{1}{4} f ″ {(0)}^{2} + g' {(0)}^{2}))} \\ C_{g} \sqrt{{Re}_{r}} = g' {1 + M - ML (3 f' {(0)}^{2} + Re (\frac{1}{4} f ″ {(0)}^{2} + g' {(0)}^{2}))} \\ \frac{Nu}{\sqrt{{Re}_{r}}} = - \frac{dT (0)}{d ζ}, \frac{Sh}{\sqrt{{Re}_{r}}} = - \frac{dC (0)}{d ζ} \end{matrix}

(20)

One can assess some mathematical analysis on PE fluid model in literature.^11,15,17,32

Computational algorithm

To convert the third- and second-order differential equations to first-order differential equations, we suppose the following assumptions

\begin{matrix} F (ζ) = H_{1}, \frac{dF (ζ)}{d ζ} = H_{2}, \frac{d^{2} F (ζ)}{d^{2} ζ} = H_{3}, \\ G (ζ) = h_{4}, \frac{dG (ζ)}{d ζ} = H_{5} \\ T (ζ) = h_{6}, \frac{dT (ζ)}{d ζ} = h_{7}, \\ C (ζ) = h_{8}, \frac{dC (ζ)}{d ζ} = h_{9} \end{matrix}

(21)

Using equation (21), equations (13)–(16) along with boundary conditions equation (17) has been adapted to a system of nine first-order simultaneous equations

{\begin{matrix} {H'}_{1} = H_{2} \\ {H'}_{2} = H_{3} \\ {H'}_{3} = \frac{(H_{4}^{2} + H_{1} H_{3} - H_{2}^{2} - 8 M λ Re H_{5} H_{3} {H'}_{5} - H H_{2})}{4 M λ Re (H_{5}^{2} + 2 H_{3}^{2} - 2 (1 + M))} \\ {H'}_{4} = H_{5} \\ {H'}_{5} = \frac{(2 H_{2} H_{4} + 2 H_{1} H_{5} + 8 M λ Re H_{5} H_{3} {H'}_{3} - H H_{4})}{- 4 M λ Re (H_{3}^{2} + 3 H_{5}^{2} + 2 (1 + M))} \\ {H'}_{6} = H_{7} \\ {H'}_{7} = - \Pr (H_{1} H_{7} + Nb H_{7} H_{9} + {NtH}_{7}^{2}) \\ {H'}_{8} = H_{9} \\ {H'}_{9} = - Sc H_{1} H_{9} + \frac{Nt}{Nb} {H'}_{7} \end{matrix}}

(22)

\begin{matrix} H_{1} (ζ) = 0, H_{2} (ζ) = γ H_{3} (ζ), \\ H_{4} (ζ) = 1 + γ H_{5} (ζ), H_{6} (ζ) = 1, \\ H_{8} (ζ) = 1, as ζ \to 0 \\ H_{2} (ζ) \to 0, H_{4} (ζ) \to 0, \\ H_{6} (ζ) \to 0, H_{8} (ζ) \to 0, as ζ \to \infty \end{matrix}

(23)

The system given by equations (7)–(11) represents PDEs against PE rotatory fluid flow over a disk. These equations are highly nonlinear; therefore, to seek an exact solution seems impossible. So we reduced the system in terms of ODEs, see equations (13)–(17). To implement shooting scheme along with RK-5 algorithm, we obtained a system of first-order ODEs, see equations (21)–(23). The detail in direction was assessed by Rehman et al.⁴⁰

Graphical results and discussion

The PE fluid flow is mathematically modeled in terms of high-order differential equations. The obtained differential system is further reduced into an ordinary differential system by reducing the number of independent variables using suitable transformation methods. The ultimate flow-narrating equations are solved by shooting methods along with RK-5 algorithm. The involved pertinent flow controlling parameters $((M, λ), γ, H, \Pr, Nt, Nb, Sc)$ are PE, slip parameter, magnetic field parameter, Prandtl number, thermophoresis parameter, Brownian motion parameter, and Schmidt number, respectively. The outcomes subjected to these parameters are shown in Figures 2 –11. In detail, the impact of both PE fluid parameters $(M, λ)$ on PE fluid velocity $F' (ζ)$ is examined, and the observation in this direction is shared through graphical trends, that is, Figures 2 and 3. Particularly, Figure 2 reports the variation in PE fluid velocity $F' (ζ)$ against higher values of the PE parameter $M (= 0.3, 0.5, 0.8, 1.0)$ . It is noticed that PE fluid velocity $F' (ζ)$ shows higher magnitude toward iterative value of M, that is, $M (= 0.3, 0.5, 0.8, 1.0)$ . The relation of PE fluid viscosity is inverse in nature toward M. Higher values of $M (= 0.3, 0.5, 0.8, 1.0)$ cause a reduction in the PE fluid viscosity. This implies that a less resistance will be faced by the PE fluid particles. As a results from Figure 2, one can confirm that the $F' (ζ)$ varies positively over the range $0 \leq ζ \leq 5$ . The impact of PE fluid parameter $(λ)$ on $F' (ζ)$ is noticed and provided in Figure 3. It is seen that the PE fluid velocity $F' (ζ)$ shows a declined nature toward $λ (= 0.1, 1.1, 1.4, 1.7)$ over the range $0 \leq ζ \leq 5$ . The impact of slip parameter on PE fluid viscosity is examined and provided by way of Figure 4. It is noticed that the PE fluid shows decline curves toward higher values of $γ (= 0.5, 0.6, 0.7, 0.8)$ . The effect of applied magnetic field toward PE fluid velocity is examined. Inspection in this direction is organized by iterating the values of magnetic field parameter $H (= 0.5, 0.8, 1.0, 1.2)$ over the range $0 \leq ζ \leq 5$ , and Figure 5 is evident that the PE fluid velocity decreases. In real time, when we iterate $H (= 0.5, 0.8, 1.0, 1.2)$ indirectly, the opposing force named Lorentz force increases. As a result, the PE fluid particles face higher resistance. Such retardation leads to a decrease in an average velocity of PE fluid. The influence of $H, λ, M, and γ$ on the PE fluid $G (ζ)$ . The outputs in this direction are provided by means of Figures 6 and 7. To be more specific, Figure 6 is plotted to examine the effect of magnetic field parameter $(H)$ on $G (ζ)$ over the range $0 \leq ζ \leq 5 .$ It is noted that, when we iterate $H (= 1.2, 1.3, 1.8, 2.2)$ indirectly, Lorentz force increases and the resistance faced by PE fluid increases, resulting in the decrease of fluid velocity $G (ζ)$ . Both Figures 7 and 8 show the impact of PE fluid parameters $λ and M$ on PE fluid velocity $G (ζ)$ . In detail, it is seen that when we iterate $λ (= 0.1, 1.1, 1.4, 1.7)$ the PE fluid velocity $G (ζ)$ reflects decline curves while it shows an inciting nature toward $M (= 0.1, 0.3, 0.5, 0.8)$ . The iteration in $M (= 0.1, 0.3, 0.5, 0.8)$ causes the fluid to be less viscous, due to which the PE fluid particles face less resistance, and as a result, the PE fluid velocity $G (ζ)$ increases. Figure 9 is devoted to highlight the impact of slip parameter $(γ)$ on the PE fluid velocity $G (ζ)$ . It is noticed that when we iterate $γ (= 0.1, 0.3, 0.5, 0.8)$ , the PE fluid velocity $G (ζ)$ reflects decline curves. The influence of both Prandtl number $(\Pr)$ and thermophoresis parameter $(Nt)$ on PE fluid temperature $(T)$ is inspected and offered with the help of graphical trends, that is, Figures 10 and 11, respectively. These results are recorded for the range $0 \leq ζ \leq 5$ . In detail, Figure 10 provided the impact of $(\Pr)$ on PE fluid temperature $(T)$ . It is noted that when we iterate $\Pr (= 0.1, 0.3, 0.7, 1.0)$ , the fluid temperature decreases as expected because Prandtl number admits inverse relation with thermal diffusivity. Furthermore, it is observed from Figures 12 and 13 that both PE fluid temperature and PE fluid concentration show an inciting values toward thermophoresis parameter $(Nt)$ over the range $0 \leq ζ \leq 5$ . Figure 13 provided the effect of Schmidt number $(Sc)$ on PE fluid concentration over the range $0 \leq ζ \leq 5$ . It is noticed that when we iterate $Sc (= 2.0, 2.5, 3.0, 3.5)$ , the PE fluid concentration is found to be a decreasing function of Schmidt number. This is due to inverse relation of Schmidt number with mass diffusivity. Figure 14 is plotted against the parameters $M and H$ . It is noticed that $M and H$ both enhance the value of skin-friction coefficient. Figure 15 shows that the value of Nusselt number decreases versus parameters $Nb and \Pr$ . Figures 16 and 17 show the impact of $Nb, Nt, and Sc$ on Sherwood number. The value of Sherwood number increases by the increasing values of $Nb, Nt, and Sc$ .

Figure 2.

Impact of $M$ on $F' (ζ)$ .

Figure 3.

Impact of $λ$ on $F' (ζ)$ .

Figure 4.

Impact of $γ$ on $F' (ζ)$ .

Figure 5.

Impact of $H$ on $F' (ζ)$ .

Figure 6.

Impact of $H$ on $G (ζ)$ .

Figure 7.

Impact of $λ$ on $G (ζ)$ .

Figure 8.

Impact of $M$ on $G (ζ)$ .

Figure 9.

Impact of $γ$ on $G (ζ)$ .

Figure 10.

Impact of $\Pr$ on $T (ζ)$ .

Figure 11.

Impact of $Nt$ on $T (ζ)$ .

Figure 12.

Impact of $Nb$ on $C (ζ)$ .

Figure 13.

Impact of $Sc$ on $C (ζ)$ .

Figure 14.

Impact of $M$ and $H$ on skin-friction coefficient.

Figure 15.

Impact of $Nb$ and $\Pr$ on Nusselt number.

Figure 16.

Impact of $Nb$ and $Sc$ on Sherwood number.

Figure 17.

Impact of $Nt$ and $Sc$ on Sherwood number.

Conclusion

The current pagination provides analysis on magnetized rotatory flow field subject to non-Newtonian fluid model named PE fluid. The flow-narrating equations are nonlinear; therefore, the exact solution seems impossible. To overcome such difficulty, a computational algorithm is developed, and an approximate solution is presented through graphical trends. The key observations are itemized as follows:

Both axial and tangential velocities $(F' (ζ), G (ζ))$ are normalized by enhancing magnetic field strength.

Tangential velocity of the PE fluid shows decline values toward $(λ, γ)$ but opposite trends are noticed for $(M)$ .

Axial velocity varies positively toward higher values of $(M)$ but inverse is the case for both $(λ, γ)$ .

PE fluid temperature increases significantly toward $(Nt)$ , while it shows a decrease against $\Pr$ as expected.

PE nanoparticle concentration shows higher values for the case of $(Nb)$ and higher values of Schmidt number has opposite behavior.

Footnotes

Appendix 1 Acknowledgements

The authors would like to express their gratitude to King Khalid University, Abha 61413, Saudi Arabia for providing administrative and technical support.

Handling Editor: James Baldwin

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Khalil U Rehman

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