A Framework for the Magnetic Dipole Effect on the Thixotropic Nanofluid Flow Past a Continuous Curved Stretched Surface

The magnetic dipole effect for thixotropic nanofluid with heat and mass transfer, as well as microorganism concentration past a curved stretching surface, is discussed. The flow is in a porous medium, which describes the Darcy–Forchheimer model. Through similarity transformations, the governing equations of the problem are transformed into non-linear ordinary differential equations, which are then processed using an efficient and powerful method known as the homotopy analysis method. All the embedded parameters are considered when analyzing the problem through solution. The dipole and porosity effects reduce the velocity, while the thixotropic nanofluid parameter increases the velocity. Through the dipole and radiation effects, the temperature is enhanced. The nanoparticles concentration increases as the Biot number and curvature, solutal, chemical reaction parameters increase, while it decreases with increasing Schmidt number. The microorganism motile density decreases as the Peclet and Lewis numbers increase. Streamlines demonstrate that the trapping on the curved stretched surface is uniform.


Introduction
Non-Newtonian fluid flows have already captivated the attention of researchers. These materials are used extensively in bioengineering, geophysics, pharmaceuticals, chemical and nuclear industries, polymer solutions, cosmetics, oil storage engineering, paper manufacturing, and other fields. Clearly, no single constitutive relationship can account for all non-Newtonian materials based on behavioral shear stresses. It is distinct from Newtonian and creeping viscous fluids [1]. As a result, several non-Newtonian fluid models have been proposed [2][3][4][5]. One such model is the thixotropic fluid model. The shear thinning fluid differs from the thixotropic fluid in that the shear thinning fluid has study bioconvection through the use of non-linear chemical and thermal radiation in a rotational fluid. Hady et al. [46] studied the unsteady bioconvection thermal boundary layer flow in the presence of gyrotactic microorganisms on a stretching plate and a vertical cone in a porous medium. Recent investigations on bioconvection can be found in the references [47][48][49][50][51][52][53][54][55][56].
The current study discusses the magnetic dipole effect on thixotropic fluid with heat and mass transfer, as well as microorganism concentration passing through a curved stretching surface. The Darcy-Forchheimer model is used to describe the flow in a porous medium. Thermal radiation and viscous dissipation effects are also taken into consideration. Through appropriate similarity transformations, partial differential equations are transformed into ordinary differential equations and solved using a well-known technique, namely homotopy analysis method HAM [57][58][59]. Many researchers [40,47,[60][61][62][63] have used HAM to solve their research problems. The results obtained are used to discuss graphically the effects of all the relevant parameters on all dimensionless profiles.

Methods
Two-dimensional hydrodynamic incompressible ferromagnetic thixotropic nanofluid past a stretched curved sheet under the influence of magnetic dipole is considered. x and y are used for curvilinear coordinates. The stretching surface is curled in a radius circle R . Based on the linear velocity u = Ax (A is constant), the sheet is stretched in the x-direction and y-direction, which is transverse to x-direction. The magnetic field of strength B 0 is perpendicular to the flow direction. The surface is submerged in a non-Darcy porous medium. As the Reynolds number (due to a magnet) is smaller in the present problem, the electrical and induced magnetic fields are ignored. Convective heat and mass transfer conditions are observed. In addition, a chemical reaction of the first order is also considered.
In conjunction with the above assumptions, the boundary layer of the equations involved are governed by the following terms [7,26,27,29,30] ∂{(y + R )v} ∂y ρ v ∂u ∂y with boundary conditions where velocity components are (u,v) in the radial (x-direction) and transverse (y-direction), k m is the mass transfer coefficient, h 1 is the convective heat transfer coefficient, R 1 and R 2 are the material constants, diffusion coefficient is D, constant fluid density is ρ, k T is the thermal conductivity, σ is the electrical conductivity, k o is permeability of porous medium, the effective dynamic viscosity is µ, magnetic permeability is µ o , heat capacitance is (ρc p ), first order chemical reaction parameter is K c , microorganisms diffusion is D m , speed of gyrotactic cell is W c , b is the chemotaxis, C b is the drag coefficient, S 1 is the porosity of porous medium, T is the temperature, C is the concentration, N is the gyrotactic microorganisms concentration, and C ∞ , T ∞ , and N ∞ , respectively, stand for the nanoparticles concentration, temperature, and density of microorganisms far away from the surface. Rosseland and Ozisik approximation allows to write the radiation heat flux q r with σ * Stenfan-Boltzman, and β R mean absorption coefficient [64] as:

Magnetic Dipole
The characteristics of the magnetic field have an effect on the flow of ferrofluid due to the magnetic dipole. Magnetic dipole effects are recognized by the magnetic scalar potential Φ [29] shown in Equation (10) where γ stands for magnetic field strength at the source, c is the distance of the line currents from the leading edge. H x and H y are taken as the components of magnetic field as shown in Equations (11) and (12) The magnetic field H is usually proportional to the components of magnetic field H x and H y , gradient along x and y directions respectively. It is therefore defined in Equation (13) as It is considered that the temperature-dependent variation of magnetization M is linear as shown in Equation (14) where K 1 identifies the coefficient of the ferromagnetic. The physical schematic of the heated ferrofluid can be seen in Figure 1. Considering the following transformations [26], with ν as kinematic viscosity, A is constant: By the application of Equation (15), Equations (2)-(8) provide the following Equations (16), (18)-(25) To eliminate the pressure term, integrating (16) to get p and replacing it, then (17) becomes and the boundary conditions become where A 1 is the ratio of rate constants, α 1 is the curvature parameter, d is the dimensionless distance, Nn 1 and Nn 2 are the non-Newtonian parameters, β is the ferrohydrodynamic interaction parameter, heat dissipation parameter is λ, ε is the curie temperature, Prandtl number is Pr, radiation parameter is Rd, Eckert number is Ec, chemical reaction parameter is δ, the Schmidt number is Sc, local inertia parameter is Li, porosity parameter is P 1 , Lewis number is Pe, Lewis number is Le, thermal Biot number is Bi 1 and concentration Biot number is Bi 2 , which are defined by The quantities of interest, such as coefficient of skin friction, local Nusselt, Sherwood and local density numbers, are determined by where By putting values from Equation (28) in Equation (27), it is obtained that

HAM Solution
The initial guesses and the linear operators are taken as Equation (30) satisfies the properties as given below where E i (i = 1, . . ., 9) indicates the arbitrary constants. The corresponding zeroth order form of the problems are where q ∈ [0, 1] is the embedding parameter while N f , N θ , N φ , and N χ are the nonlinear operators.
The m-th order deformation problems are as follows where The general solutions are given by

Convergence Analysis of the Homotopy Solution
The nonzero auxiliary parameters are involved in the homotopy solution. These parameters are extremely important in controlling and adjusting the convergence acquired by the homotopic series solutions. The h-curves at the 15th order of approximations are sketched to show the acceptable approximate region of convergence. Figure 2 depicts the region as falling within the ranges −1.

Discussion
The velocity behavior with the ferromagnetic hydrodynamic interaction parameter β can be seen in Figure 3. It demonstrates that the velocity decreases as β increases. Ideally, the resistance force known as Lorentz force [65] increases with the β increase, and the velocity field decreases. Figure 4 is used to investigate the effect of curvature parameter α 1 on the velocity profile. It is clearly shown in the figure that the velocity component decreases for larger α 1 . Figures 5 and 6 describe the effects of the thixotropic parameters Nn 1 and Nn 2 on the velocity profile. From these figures, it is observed that Nn 1 and Nn 2 result in an increase in fluid velocity. Ideally, Nn 1 and Nn 2 are associated with the properties of shear thinning, which show a time-dependent changes in viscosity. The higher the fluid under shear stress, the lower the viscosity of nanofluid, which will ultimately lead to an increase in fluid velocity. Figure 7 is used to present the velocity behavior with the porosity parameter P 1 . The presence of porous medium slows down the field of the flow, resulting in an increase in shear stress on the curved surface, and therefore the velocity profile shows a declining trend by increasing the values of P 1 . In contrast to the effect seen with P 1 , change in local inertia parameter Li results in an increase in velocity as shown in Figure 8. Figure 9 is used used to examine the effect of β on temperature. Here, temperature increases with higher values of β. The temperature profile behavior relating to the higher values of thermal Biot number Bi 1 is shown in Figure 10. The parameter Bi 1 significantly promotes the temperature field in a positive manner attributable to the effective convective heat effects. It is also observed that there is no heat transfer at Bi 1 = 0. The effect of the heat dissipation parameter λ on temperature is shown in Figure 11. The temperature is a decreasing function of λ. Physically thermal conductivity of liquid decreases with larger λ, and therefore the temperature decreases. The Eckert number Ec attributes to the temperature profile is shown in Figure 12. For larger Ec, temperature and thermal boundary layer thickness were observed to be effected with the increase in Ec. In this phenomenon, the heat energy stored in the fluid is caused by friction forces that increase the temperature. The Curie temperature parameter ε effect on temperatureprofile is shown in Figure 13. The temperature decreases through larger values of ε. Thermal conductivity of the liquid increases with the larger ε. The effect of Prandtl number Pr on temperature profile is shown in Figure 14. The temperature distribution and thermal boundary layer are reduced by higher values of Pr, due to which thermal diffusion is reduced. In addition, fluids with a smaller values of Pr slowly decay compared to liquids with larger values of Pr. The effect of radiation parameter Rd on temperature profile is discussed in Figure 15. The increase in temperature curves with a larger boundary layer thickness is determined by an increase in Rd. Usually, mean absorption coefficient decays for higher estimation of Rd and diffusion flux occurs as a consequence of the temperature gradient, which therefore increases the temperature.
The effect of the concentration Biot number Bi 2 on the nanoparticles concentration profile is shown in Figure 16. In this case, the concentration is increased in response to increase in the Bi 2 values. Figure 17 shows the effect of the Sc on concentration profile. Since Sc is the ratio of momentum to mass diffusivity, the increase in Sc causes a decay in mass diffusivity, thus leading to a decrease in nanoparticles concentration. Figure 18 shows the effect of the curvature parameter α 1 on the nanoparticles concentration profile. The increase in the curvature parameter results in an increase in the concentration. Figure 19 shows the effect of the chemical reaction parameter δ on the concentration profile. The nanoparticles concentration is observed to increase for the higher estimates of δ. In fact, the consumption of reactive species rapidly declines as δ becomes larger. Figure 20 shows the effect of Peclet number Pe on the microorganisms profile. There is a clear relationship between the reduced density of the microorganisms and the increase in Pe. The higher values of Pe indicate the minimum motile diffusivity. Figure 21 shows the impact of Lewis number Le on microorganisms concentration profile. The decrease in the concentration distribution is shown as the Lewis number increases, since it is inversely proportional to the mass diffusion.
The effect of the dimensionless variable ζ on the streamlines is shown in Figures 22 and 23. It is shown that the number of the trapped boluses increases as the values of ζ increase, and the streamlines have also been identified to be perpendicular to the surface. The increase in the ζ increases the shearing motion, which, in fact, results in a higher precession of the flow to the stretching surface. Table 1 shows a numerical analysis of the skin friction coefficient for β, α 1 , P 1 , Li, Nn 1 , Nn 2 . It is discovered that the skin friction coefficient increases with the increasing values of β, P 1 , Li, Nn 2 , while a reverse trend is observed for α 1 and Nn 1 . Table 2 cross-checks the accurateness of the homotopic solution used in the present investigation. A comparison of skin friction coefficient for the different values of α 1 with the study [66] is shown for Table 3 shows the numerical assessment of the local Nusselt number for various values of β, α 1 , λ, Pr, Rd, ε, Ec, Nn 1 , Nn 2 . It is observed that the local Nusselt number decreases with increasing values of β, α 1 , λ, Nn 1 . Table 4 shows the numerical values of the local Sherwood number for various values of α 1 , Sc, δ. It is observed that the local Sherwood number decreases with the increasing values of parameters. The tables clearly show that the current findings are completely consistent.

Conclusions
The Darcy-Forchheimer hydromagnetic flow of thixotropic nanofluid through a curved stretching sheet with thermal radiation and chemical reaction in the presence of heat and mass transfer, gyrotactic microorganisms, and magnetic dipole is explored. The present study contributes to the findings set out below.

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The velocity decreases with increasing values of ferromagnetic parameter β and a curvature parameter α 1 , while it increases with increasing values of Nn 1 , Nn 2 and P 1 .