Swirling Flow of Chemically Reactive Viscoelastic Oldroyd-B Fluid through Porous Medium with a Convected Boundary Condition Featuring the Thermophoresis Particle Deposition and Soret–Dufour Effects

: In this study, an analysis of the rotating ﬂow of viscoelastic Oldroyd-B ﬂuid along with porous medium featuring the Soret–Dufour effects is explored. The heat transport mechanism is discussed with the involvement of thermal radiation and heat source/sink. Additionally, the thermophoresis of particle deposition and chemical reaction are taken into the concentration equation in order to investigate the mass transportation in the liquid. To formulate the non-linear ordinary differential equations, the von Karman similarity approach is used in the system of partial differential equations and then integrated numerically by the bvp midrich scheme in Maple programming. Results are provided by graphical framework and tabular form. A quick parametric survey is carried out concerning ﬂow ﬁeld, thermal, and solutal distributions through graph representation. The curves show that increasing the values of the retardation time parameter decreases the radial velocity while increasing the angular velocity. Additionally, when the relaxation time parameter becomes powerful, the magnitude of the velocity curves decreases considerably in the radial and axial directions. The presence of a radiation parameter indicates that the ﬂuid will absorb a greater amount of heat, which is equivalent to a higher temperature. Further, an increase in the stretching parameter leads to a reduction in the temperature components.


Introduction
In the field of fluid dynamics, research into rotating disk geometry has attracted a great deal of attention and enthusiasm in recent years due to its many potential technical and industrial applications, including jet engines, hard disks, turbine systems, etc. It is for this reason that the phenomenon of fluid flow by a rotating disk has received a significant amount of attention and has been extensively analyzed by researchers, particularly after von Kerman s seminal work on flow by a rotating disk. The overarching goal of the discussion is to study the convection fluid motion of the Oldroyd-B fluid model [1] due to a porous rotating disk using the novel perspective of thermophoresis particle deposition together with the occurrence of Soret-Dufour impacts and chemical reactions.
Von Kármán [2], in his groundbreaking work, simplified the complete set of equations guiding the solution to the rotating disk problem. After that, Cochran [3] made an attempt to numerically solve the Kármán swirling flow problem. Further, Millsaps and Pohlhausen [4] studied the heat transfer properties to accommodate the supplementary extension. For a viscous fluid, Shevchuk and Buschmann [5] found an exact solution to the heat transfer problem in a rotating disk flow. Awad [6] offered the asymptotic solutions to the heat transport properties for a range of Prandtl numbers. Through the use of a rotating porous disc, Turkyilmazoglu [7] was able to obtain the closed-form solution for an incompressible and viscous fluid. In a separate work, Turkyilmazoglu [8] investigated the impacts of radial electric fields on MHD fluid flow and heat transfer for the rotating disk problem. Recently, the study of viscoelastic fluid flow, along with heat transfer, is discussed by Nuwairan et al. [9]. They demonstrated that heat generation/absorption and thermal radiation contribute to raising the liquid s temperature.
The rheological properties of non-Newtonian fluids are very dissimilar to those of Newtonian fluids. Therefore, many different constitutive equations have been developed to describe these fluids. Of these, a great deal of focus has been placed on rate-type models. As earlier, Oldroyd [10] established a methodical approach to creating models of rate-type viscoelastic fluids. He took great effort to incorporate into his framework the invariance requirements that the model ought to be able to fulfill, but there is no indication that the thermodynamical issue has been taken into account. In 2000, Rajagopal and Srinivasa [11] made a systematic thermodynamic framework within which models of a variety of ratetype viscoelastic fluids can be derived. Notable among these is the Oldroyd-B model, which can adequately describe the behavior of some polymeric liquids. Both theoretical and practical testing of this model is feasible. For this reason, many articles related to these fluids have already been published via Refs. [12][13][14][15].
A survey of the research available shows that, despite the importance of fluid motion over a porous rotating disk to many different types of industries, researchers have paid it relatively little attention. Thus, the main purpose of this research is to examine the flow of Oldroyd-B fluids due to a rotating disk subject to a convection boundary condition, incorporating thermophoresis and Soret-Dufour impacts. In addition, the study elucidates the significance of heat source/sink and thermal radiation, along with the chemical reaction on the heat and mass transport characteristics that occur during fluid motion. The modeled flow problem is solved numerically by a BVP (boundary value problem) midrich scheme in Maple programming. To highlight the impact of active parameters, tabular and graphical trends are obtained and elaborated in detail.

Problem Description
We assume an incompressible magnetized Oldroyd-B fluid flow through porous medium caused by a rotating disk that stretches and rotates at different rates. The surface is considered to be porous, with a mass flux velocity of w 0 (w 0 < 0 for suction and w 0 > 0 for injection). To express the mathematical modelling of the problem, cylindrical coordinates (r, ϕ, z) are used. The stretching and rotating velocities of the disk (positioned at z = 0) are, respectively, a and Ω, as referred in Figure 1. All physical quantities are not depending on ϕ, as the flow is axisymmetric in the z direction. By ignoring the induced electric and magnetic fields, a uniform beam of magnetic field, B 0 , is imposed along the z-axis. The temperature equation is used along with the presence of heat source/sink and radiation to express the heat transportation in the liquid. For the mass transportation, the chemical reaction and thermophoresis particle deposition are both taken into the concentration equation. Additionally, the impact of Soret and Dufour and the convective boundary condition are also considered. From the aforementioned assumptions, the modeled equations [1] are as follows: The radiative flux [13,16] is given by: The thermophoretic velocities are defined by: Figure 1. Flow diagram [13].
From the aforementioned assumptions, the modeled equations [1] are as follows: The radiative flux q rad [13,16] is given by: The thermophoretic velocities are defined by: where the values of k are in the range of 0.2 to 1.2, as expressed by Batchelor and Shen [17], and are defined from the theory of Talbot et al. [18] by: where (C s , C t , C m , C 1 , C 2 , C 3 ) = (1.147, 2.20, 1.146, 1.2, 0.41, 0.88) are, respectively, constants. Additionally, Kn is the Knudsen number and λ p , λ g are the thermal conductivities of the diffusion particles and the fluid. The respective boundary conditions are: Introducing the similarity variables [13] are: Applying Equation (10) into Equations (1)-(5), we have: 1 The transformed boundary conditions (BCs) are: The parameters are expressed as: The physical parameters are defined as: The Nusselt number, Nu r , and Sherwood number, Sh r , are of the form: Their dimensionless forms are: in which, Re = r 2 Ω ν is the local Reynold number.

Results and Discussion
In this section, a discussion is presented in the form of graphs and tables regarding the effect different physical parameters have had on the current investigation. The influence of on the velocity, thermal, and solutal curves on a fixed magnetic parameter and stretching parameter, suction parameter, and porosity parameter are shown in Figure 3a-e. It is obvious that, as the relaxation time parameter, , becomes The elaboration of the curves of the thermal field for a variety of different values of Rd is shown in Figure 4a. The higher rate of Rd cause the thermal field to rise, along with the boundary layer thickness associated with it. As predicted, the presence of a radiation parameter, Rd, indicates that the fluid will absorb a greater amount of heat, which is equivalent to a higher temperature. Figure 4b indicates the peculiarities of the Biot number, Bi, on the fluid temperatures. In a physical point of view, an increase in the Biot num-  The sketches make it clear that the velocity field is radially increasing and azimuthally decreasing. Because R is the stretch to swirl rate ratio, when the rate of the stretching parameter begins to thrive, the stretch rate becomes higher relative to the swirl rate. Therefore, the velocities enhance and reduce in the radial and angular directions, respectively. In addition, the axial velocity component shows a diminishing trend with an increasing rotating parameter, as shown in Figure 2c. Moreover, the evidence shown in Figure 2d indicates that the temperature of the fluid is decreasing as the stretching parameter becomes intensified. A converse trend can be seen for the mass concentration (see Figure 2e).
The influence of β 2 on the velocity, thermal, and solutal curves on a fixed magnetic parameter and stretching parameter, suction parameter, and porosity parameter are shown in Figure 3a-e. It is obvious that, as the relaxation time parameter, β 2 , becomes powerful, the magnitude of the velocity curves decreases considerably in the radial and axial directions. In addition, an increase of β 2 from 0.05 to 0.5 has a positive effect on the angular velocity of the liquid. Moreover, curves are plotted in order to examine the impact that β 2 has on temperature as well as solutal distributions. It can be observed from the curves that the temperature of the liquid rises under the influence of β 2 , while the mass concentration in the liquid reduces, as displayed in Figure 3d Table 1 displays the values of the local Nusselt number on , , Bi, Sr, and Du, respectively. The heat transfer rate increases with the increases in Bi and Sr, while it reduces due to the influence of , , and Du. The changes in mass transfer rate in the liquid due to Nt, , , Sr, and Du can be seen in Table 2. It is noted that the Sherwood number is a monotonically increasing function of Nt, , and Du, while it is a decreasing function of and Sr. The elaboration of the curves of the thermal field for a variety of different values of Rd is shown in Figure 4a. The higher rate of Rd cause the thermal field to rise, along with the boundary layer thickness associated with it. As predicted, the presence of a radiation parameter, Rd, indicates that the fluid will absorb a greater amount of heat, which is equivalent to a higher temperature. Figure 4b indicates the peculiarities of the Biot number, Bi, on the fluid temperatures. In a physical point of view, an increase in the Biot number leads to larger convection at the disk surface, which in turn causes an increase in the fluid's temperature. In addition to this, the higher the Biot number, the more prominent the boundary layer thickness. A similar kind of trend may be seen on the thermal profile, which is caused by an increase in the heat generation parameter (see Figure 4c). Table 1 displays the values of the local Nusselt number on β 1 , β 2 , Bi, Sr, and Du, respectively. The heat transfer rate increases with the increases in Bi and Sr, while it reduces due to the influence of β 1 , β 2 , and Du. The changes in mass transfer rate in the liquid due to Nt, K r , k, Sr, and Du can be seen in Table 2. It is noted that the Sherwood number is a monotonically increasing function of Nt, k, and Du, while it is a decreasing function of K r and Sr. Moreover, we computed F (0), −G (0) and −θ (0), and these results are compared with the available published results of [19,20] in Table 3, and it was found that they are in excellent agreement with each other.  [19,20] with the current problem on fixed Pr = 6.5 and M = 0 = γ = R = s = β = Rd = δ = Du.

Conclusions
The axisymmetric swirling flow of Oldroyd-B fluid through a porous medium featuring the Soret-Dufour impacts is discussed. Further, heat and mass transportations are examined, along with numerous physical features. Numerical solutions are determined with the help of a numerical procedure. Below is a summary of some important findings:

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The magnitude of velocity curves decreases substantially in the radial and axial directions when the relaxation time parameter is changed to dynamic. • A higher rate of radiation parameter causes the thermal field to rise, along with the boundary layer thickness associated with it. • An increasing trend is observed on the thermal profile, which is due to the increase in the heat generation parameter.

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The higher the Biot number, the more pronounced is the thermal boundary layer thickness.

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The heat transfer rate enriches with an increase in the Soret number.

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The Sherwood number is a monotonically increasing function of the Dufour number.